Research ArticleDynamical Analysis on Single Degree-of-Freedom SemiactiveControl System by Using Fractional-Order Derivative

Yongjun Shen1 Minghui Fan1 Xianghong Li2 Shaopu Yang1 and Haijun Xing1

1Department of Mechanical Engineering Shijiazhuang Tiedao University No 17 Bei Erhuan Dong Road Shijiazhuang 050043 China2Department ofMathematics and Physics Shijiazhuang TiedaoUniversity No 17 Bei ErhuanDong Road Shijiazhuang 050043 China

Correspondence should be addressed to Yongjun Shen shenyongjun126com

Received 27 January 2015 Revised 22 March 2015 Accepted 23 March 2015

Academic Editor Roman Lewandowski

Copyright copy 2015 Yongjun Shen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The single degree-of-freedom (SDOF) system under the control of three semiactive methods is analytically studied in this paperwhere a fractional-order derivative is used in the mathematical model The three semiactive control methods are on-off controllimited relative displacement (LRD) control and relative control respectively The averaging method is adopted to provide ananalytical study on the performance of the three different control methods Based on the comparison between the analyticalsolutions with the numerical ones it could be proved that the analytical solutions are accurate enoughThe effects of the fractional-order parameters on the control performance especially the relative and absolute displacement transmissibility are analyzed Theresearch results indicate that the steady-state amplitudes of the three semiactive systems with fractional-order derivative in themodel could be significantly reduced and the control performance can be greatly improved

1 Introduction

Machinery equipment will vibrate if it is excited and moredamage can be caused when it is under the condition ofresonance In order to reduce vibration Frahm invented theoriginal shock absorber based on classical vibration theoryThen Den Hartog propounded the design rule for dampingdynamic shock absorber based on the fixed-points theory [1]Subsequently many improved shock absorbers were pre-sented based on this one At that stage the control methodshould be named as passive control one The damping forceof passive vibration control system could not be adjustedinstantaneously so that it may be limited effective With thedevelopment of science and technology active control andsemiactive controlmethods came into beingThese two kindsof control methods performed noticeably well and they wereadopted in many fields such as vehicle suspension and vibra-tion absorption system [2ndash5]

Vehicle suspension could be simplified to a single degree-of-freedom (SDOF) semiactive control system under someassumptions The study on semiactive control methods orig-inated from Karnopp et al who presented the on-off controlmethod in 1974 [2] After that many scholars carried out

research directing at semiactive suspension system and theresearch efforts were focused on the control methods andordesign of control devices [6ndash8] Some other scholars pre-ferred analytical method than numerical one because theanalytical study may present more information about theadjustment or design of the controllable parameters Shenet al [9] applied averagingmethod to present analytical inves-tigations for different controlmethods Eslaminasab et al [10]adopted averaging method to provide an analytical platformfor analyzing the performance of relative control methodShen et al [11ndash13] studied the approximately analytical solu-tions and parameters optimization of four semiactive on-offdynamic vibration absorbers and researched a single degree-of-freedom semiactive oscillator with time delay

In fact it is rather hard to accurately describe the consti-tutive relations of most real materials by the classical integer-order model especially for some viscoelastic andor rheolog-ical materials Therefore some scholars started to researchfractional-order derivativemodels As far back as 1985 Bagleyand Torvik [14ndash16] introduced fractional-order derivativeinto fluid mechanics and modelled the constitutive relationof the material viscoelasticity in many vibration mitigationsRossikhin and Shitikova [17] proposed a method to analyze

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 272790 13 pageshttpdxdoiorg1011552015272790

2 Mathematical Problems in Engineering

the free damped vibrations of a fractional-order oscillatorTavazoei et al [18] and Pinto and Tenreiro Machado [19]studied the fractional-order van der Pol oscillator and foundmultiple limit cycles existing in the system Shen et al [20ndash23]analytically studied the primary resonance of some nonlinearoscillators with different fractional-order derivatives by theaveraging method and illustrated the effects of fractional-order parameters on dynamical response Wahi and Chat-terjee [24] studied an oscillator with special fractional-orderderivative and time-delay by averaging method Chen andZhu [25]Wang andHe [26] Huang and Jin [27] and Xu et al[28] also investigated different fractional-order systems andpresented some important results by analytical research

In this paper we introduce fractional-order derivativeinto semiactive SDOF systemswhere the semiactivemethodsare on-off control limited relative displacement (LRD) con-trol and relative control respectivelyThe analytical solutionsof the three semiactive systems are obtained by averagingmethodThenwe study the stability of those steady-state solu-tions by Lyapunov stability theory Furthermore the effectsof fractional-order parameters on the steady-state amplitudeand control performance are researched At last the mainconclusions are made

2 Analytical Solutions ofSDOF Semiactive System

Theconsidered SDOF semiactive system is shown in Figure 1If the fractional-order viscoelastic device is used its mathe-matical model should be

119898+ 119896 (119909 minus 1199090) + 119888 ( minus 0) +119870119863119901(119909 minus 1199090) = 0 (1)

where 1199090 = 1198830 cos120596119905 is the external excitation to the con-trolled system with1198830 and 120596 as the excitation amplitude andexcitation frequency119898 119888 and 119896 are the system mass systemdamping coefficient and stiffness respectively119870119863

119901(119909minus1199090) is

the fractional-order derivative of119909minus1199090 to 119905with the fractionalcoefficient119870 (119870 ge 0) and the fractional-order 119901 (0 le 119901 le 1)There are several definitions for fractional-order derivativeand they are equivalent under some conditions for a wideclass of functions In Caputorsquos sense the definition of 119901 orderderivative of 119909(119905) to 119905 is

119863119901[119909 (119905)] =

1Γ (1 minus 119901)

int

119905

0

1199091015840(119906)

(119905 minus 119906)119901119889119906 (2)

where Γ(119911) is Gamma function that satisfies Γ(119911+ 1) = 119911Γ(119911)Defining 119911 = 119909 minus 1199090 (1) becomes

119898 + 119896119911 + 119888 +119870119863119901(119911) = 1198981198830120596

2 cos120596119905 (3)

Using the following transformation of coordinates

12059620 =

119896

119898

120585 =

119888

2119898

ck

x

x0

KDp(x minus x0)

m

Figure 1Themodel of SDOF semiactive systemby using fractional-order derivative

120582 =

119870

119898

119891 = 11988301205962

(4)

equation (3) becomes

+ 12059620119911 + 2120585 + 120582119863

119901(119911) = 119891 cos120596119905 (5)

Supposing the solution of (5) is

119911 = 119886 cos120601

= minus120596119886 sin120601

(6)

where 120601 = 120596119905 + 120579 one could get

= 119886 cos120601minus 119886 (120596+

120579) sin120601

= minus120596 [ 119886 sin120601+ 119886 (120596+

120579) cos120601] (7)

From (5) (6) and (7) one could obtain

119886 cos120601minus 119886

120579 sin120601 = 0

119886 sin120601+ 119886

120579 cos120601 =

119865

120596

(8)

where

119865 = minus [119891 cos120596119905 + 119886 (1205962minus120596

20) cos120601+ 2119886120585120596 sin120601

minus120582119863119901(119886 cos120601)]

(9)

The derivatives of the generalized amplitude 119886 and phase120579 could be solved as

119886 =

119865

120596

sin120601

119886

120579 =

119865

120596

cos120601(10)

Mathematical Problems in Engineering 3

Then one could apply the standard averaging method to (10)in time interval [0 119879]

119886 = lim119879rarrinfin

1119879

int

119879

0

119865

120596

sin120601119889120601

119886

120579 = lim119879rarrinfin

1119879

int

119879

0

119865

120596

cos120601119889120601

(11)

Furthermore (11) could be written as

119886 = 1198861 + 1198862 + 1198863

119886

120579 = 119886

1205791 + 119886

1205792 + 119886

1205793(12)

where

1198861 = lim119879rarrinfin

1119879120596

sdot int

119879

0minus [1198830120596

2 cos (120601 minus 120579) + 119886 (1205962minus120596

20) cos120601]

sdot sin120601119889120601

(13a)

1198862 = lim119879rarrinfin

1119879120596

int

119879

0minus2119886120585120596 sin2120601119889120601 (13b)

1198863 = lim119879rarrinfin

1119879120596

int

119879

0120582119863119901(119886 cos120601) sin120601119889120601 (13c)

119886

1205791 = lim119879rarrinfin

1119879120596

sdot int

119879

0minus [1198830120596

2 cos (120601 minus 120579) + 119886 (1205962minus120596

20) cos120601]

sdot cos120601119889120601

(14a)

119886

1205792 = lim119879rarrinfin

1119879120596

int

119879

0minus119886120585120596 sin 2120601119889120601 (14b)

119886

1205793 = lim119879rarrinfin

1119879120596

int

119879

0120582119863119901(119886 cos120601) cos120601119889120601 (14c)

According to the averaging method one could select thetime terminal 119879 as 119879 = 2120587 if the integrand is periodicfunction or 119879 = infin if the integrand is aperiodic oneAccordingly one could obtain

1198861 = minus

1198830120596 sin 120579

2 (15a)

119886

1205791 =

minus119886 (1205962+ 120596

20) minus 1198830120596

2 cos 1205792120596

(15b)

In order to obtain the definite integral in (13c) and (14c)two important formulae which have been deduced in [20 21]are introduced

1198611 = lim119879rarrinfin

int

119879

0

sin (120596119905)

119905119901

119889119905

= 120596119901minus1

Γ (1minus119901) cos(119901120587

2)

(16a)

1198612 = lim119879rarrinfin

int

119879

0

cos (120596119905)

119905119901

119889119905

= 120596119901minus1

Γ (1minus119901) sin(

119901120587

2)

(16b)

Based on Caputorsquos definition (13c) becomes

1198863 = lim119879rarrinfin

120582

119879120596

int

119879

0119863119901[119886 cos (120596119905 + 120579)] sin (120596119905 + 120579) 119889119905

=

minus120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

sdot int

119879

0[int

119905

0

sin (120596119906 + 120579)

(119905 minus 119906)119901

119889119906] sin (120596119905 + 120579) 119889119905

(17)

Introducing 119904 = 119905 minus 119906 and 119889119904 = minus119889119906 (17) becomes

1198863 =

minus120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

int

119879

0[int

119905

0

sin (120596119905 + 120579 minus 120596119904)

119904119901

119889119904]

sdot sin (120596119905 + 120579) 119889119905 =

minus120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

sdot int

119879

0[int

119905

0

cos (120596119904)

119904119901

119889119904] sin (120596119905 + 120579) sin (120596119905 + 120579) 119889119905

+

120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

int

119879

0[int

119905

0

sin (120596119904)

119904119901

119889119904]

sdot cos (120596119905 + 120579) sin (120596119905 + 120579) 119889119905

(18)

In fact the secondpart in (18)would vanish according to (16a)and (16b) Defining the first part in (18) as1198601 and integratingit by parts one could obtain

1198601

=

minus120582119886

4120596Γ (1 minus 119901)

lim119879rarrinfin

2120596119905 minus sin (2120596119905 + 2120579)119879

[int

119905

0

cos (120596119904)

119904119901

119889119904]

10038161003816100381610038161003816100381610038161003816

119879

0

minus

minus120582119886

4120596Γ (1 minus 119901)

lim119879rarrinfin

1119879

int

119879

0[

[2120596119905 minus sin (2120596119905 + 2120579)] cos (120596119905)

119905119901

] 119889119905

(19)

Based on (16a) and (16b) one could find that the second partin (19) would vanish and the first part in (19) will be

1198601 = minus

minus120582119886120596119901minus1

2sin(

119901120587

2) (20)

That means

1198863 =

minus120582119886120596119901minus1

2sin(

119901120587

2) (21)

After the similar procedure (14c) will be

119886

1205793 =

120582119886120596119901minus1

2cos(

119901120587

2) (22)

4 Mathematical Problems in Engineering

Equations (13b) and (14b) can be rewritten as

1198862 =

12120587

int

2120587

0minus2119886120585sin2120601119889120601 (23a)

119886

1205792 =

12120587

int

2120587

0minus119886120585 sin 2120601119889120601 (23b)

The solutions of (23a) and (23b) depend on the specificsemiactive control methods

Accordingly (12) becomes

119886 = minus

1198830120596 sin 120579

2minus

120582119886120596119901minus1

2sin(

119901120587

2)

+

119886

2120587int

2120587

0minus2120585 sin2120601119889120601

119886

120579 =

minus1198861205962+ 119886120596

20 minus 1198830120596

2 cos 1205792120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

+

119886

2120587int

2120587

0minus120585 sin 2120601119889120601

(24)

which could be rewritten as

119886 = minus

1198830120596 sin 120579

2minus

119886

21198981198621015840

119886

120579 =

minus1198830120596 cos 1205792

minus

120596119886

2+

119886

2120587int

2120587

0minus

119888

2119898sin 2120601119889120601

+

119886

2119898120596

1198701015840

(25)

where

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+int

2120587

0

minus119888 sin2120601119889120601

120587

(26a)

1198701015840= 119870120596

119901 cos(119901120587

2)+ 119896 (26b)

1198621015840 and 119870

1015840 are defined as the equivalent damping coefficientand equivalent stiffness coefficient in semiactive systemsrespectively In the SDOF semiactive system the fractional-order derivative119870119863

119901(119909 minus1199090) serves as damping and stiffness

at the same timeWhen 119901 rarr 0119870119863119901(119909minus1199090)will be changed

into spring force whereas it will be almost the same as damp-ing force if 119901 rarr 1 When 119870 rarr 0 119870119863

119901(119909 minus 1199090) will have

little effect on control performance and the semiactive systemwill degrade into the traditional semiactive one

3 Cases for Three Kinds ofSemiactive Control Systems

31 Analytical Investigation for on-off Control System Thestrategy of semiactive on-off control is

119888 =

119888max ( minus 0) ge 0

119888min else(27)

Originally 119888min is selected as 119888min = 0 in [2] which may beinaccurate in real engineering Here we select it as 119888min =

001119888maxAfter transformation of coordinates (27) becomes

120585 =

120585max ( + 0) ge 0

120585min else(28)

Expanding the critical condition

( + 0) ge 0 (29)

one could get

minus120596119886 sin120601 [minus120596119886 sin120601minus1198830120596 sin (120596119905)]

= 119886119877 sin120601 sin (120601 minus 120573) ge 0(30)

where

120573 = arctan1198830 sin 120579

119886 + 1198830 cos 120579

119877 = 1205962radic(119886 + 1198830 cos 120579)

2+ 119883

20sin2120579

(31)

Equation (28) can be rewritten as

120585 =

120585min 0 lt 120601 lt 120573

120585max 120573 lt 120601 lt 120587

120585min 120587 lt 120601 lt 120587 + 120573

120585max 120587 + 120573 lt 120601 lt 2120587

(32)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) as

1198862 =

minus2119886119860 minus 119886119861

2120587

119886

1205792 =

119886119862 minus 119886119863

2120587

(33)

where

119860 = 120587120585max +120573 (minus120585max + 120585min)

119861 = (120585max minus 120585min) sin 2120573

119862 = 120585max minus 120585min

119863 = (120585max minus 120585min) cos 2120573

(34)

Combining (12) (15a) (15b) (21) (22) and (33) onecould obtain

119886 = minus

1198830120596 sin 120579

2minus

120582119886120596119901minus1

2sin(

119901120587

2)+

minus2119886119860 minus 119886119861

2120587

119886

120579 =

minus1198861205962+ 119886120596

20 minus 1198830120596

2 cos 1205792120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

+

119886119862 minus 119886119863

2120587

(35)

Mathematical Problems in Engineering 5

Obviously the equivalent damping coefficient in this casecould be written as

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+ 119888max +

120573

120587

(minus119888max + 119888min)

+

sin 21205732120587

(119888max minus 119888min)

(36)

From (36) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120573

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain the amplitude and phase

of steady-state response

119886 =

12058711988301205962

radic1198732+ 119871

2

120579 = arctan 119873

119871

(37)

where

119873 = minus 2120596119860minus120596119861minus120587120596119901120582 sin

119901120587

2

119871 = 120596119862minus120596119863+120587120596119901120582 cos

119901120587

2+120587120596

20 minus120587120596

2

(38)

The transmissibility for relative displacement is

1205831 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198732+ 119871

2 (39)

The absolute displacement of on-off control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198831 cos (120596119905 minus 120591)

(40)

where

1198831 = radic

11988320120587

21205964119873

2

(1198732+ 119871

2)2 + (1198830 +

11988301205871205962|119871|

1198732+ 119871

2 )

2

120591 = minus arctan 1205871205962119873

119871 (1205871205962+ (119871

2+ 119873

2) |119871|)

(41)

The displacement transmissibility is

1205891 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988311198830

(42)

Next we study the stability of steady-state solution Let-ting 119886 = 119886 + Δ119886 and 120579 = 120579 + Δ120579 and then substituting theminto (35) one could get

119889Δ119886

119889119905

=

12120587

[119873 sdot Δ119886minus1198830120587120596 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587

[

1205871198830120596 sin 120579

119886

sdot Δ120579 +

1205871198830120596 cos 1205791198862 sdot Δ119886]

(43)

Based on (37) one could eliminate 120579 and obtain the charac-teristic determinant

det[[

[

119873

120596

minus 120572

minus119886119871

120596

119871

119886120596

119873

120596

minus 120572

]

]

]

= 0 (44)

Expanding the determinant one could obtain the character-istic equation

(

119873

120596

minus120572)

2+

1198712

1205962 = 0 (45)

Solving (45) one could obtain characteristic values

12057212 =

119873

120596

plusmn

119871

120596

119894 (46)

Obviously119873 is less than 0 Accordingly all real parts of char-acteristic roots are negative which means the steady-statesolution of this kind of semiactive system is unconditionallystable

32 Analytical Investigation for LRD Control System Thestrategy of LRD control is

119888 =

119888max1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816ge 120575

119888min1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816lt 120575

(47)

where 119888min = 001119888max and 120575 is the space between the massand excitation point Here we select 120575=051198830

After transformation of coordinates (47) becomes

119888 =

119888max |119911| ge 120575

119888min |119911| lt 120575

(48)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt 120576

120585min 120576 lt 120601 lt 120587 minus 120576

120585max 120587 minus 120576 lt 120601 lt 120587 + 120576

120585min 120587 + 120576 lt 120601 lt 2120587 minus 120576

120585max 2120587 minus 120576 lt 120601 lt 2120587

(49)

where 120576 = arccos(051198830119886)Using the averagingmethod one could obtain the simpli-

fied forms of (13b) and (14b)

1198862 =

minus21198861198600 + 211988611986102120587120596

119886

1205792 = 0(50)

where 1198600 = 120596(2120576120585max + 120587120585min minus 2120576120585min) 1198610 = 120596(120585max minus

120585min) sin 2120576

6 Mathematical Problems in Engineering

Combining (12) (15a) (15b) (21) (22) and (50) onecould obtain

119886 =

minus21198861198600 + 21198861198610 minus 11988301205962120587 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579 =

minus119886120587 (1205962minus 120596

20) minus 1198830120596

2120587 cos 120579

2120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(51)

Obviously one could easily obtain the equivalent dampingcoefficient

1198621015840

= 119870120596119901minus1 sin(

119901120587

2)

+

(119888max minus 119888min) sin 2120576 minus (2120576119888max + 119888min120587 minus 2120576119888min)

120587

(52)

From (52) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120576

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain

119886 =

12058711988301205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2

120579 = arctanminus21198600 + 21198610 minus 120582120587120596

119901 sin (1199011205872)minus120587 (120596

2minus 120596

20) + 120582120587120596

119901 cos (1199011205872)

(53)

where 119886 and 120579 are the amplitude and phase of steady-stateresponse respectively

The transmissibility for relative displacement is

1205832 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2 (54)

The absolute displacement of LRD control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198832 cos (120596119905 minus 120594)

(55)

where

119883

22

=

11988320120587

21205964119882

2

[(21198600 minus 21198610)2+ 120587

2(120596

4minus 21205962

12059620 + 120596

40 + 120596

21199011205822) minus 21205872

120596119901(120596

2minus 120596

20) 120582 cos (1199011205872) + 4 (1198600 minus 1198610) 120587120596

119901120582 sin (1199011205872)]

2

+(1198830 +1198830120587120596

2119866

1198662+ 119882

2)

2

120594 = arctan 12058721205962119882

11986612058721205962+ 120587119866

2+ 120587119882

2

(56)

Mathematical Problems in Engineering 7

where

119866 = 120587(minus1205962+120596

20 +120596119901120582 cos

119901120587

2)

119882 = 21198600 minus 21198610 +120587120596119901120582 sin

119901120587

2

(57)

The transmissibility for the absolute displacement of LRDcontrol system is

1205892 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988321198830

(58)

Next we study the stability of the steady-state solutionsLetting 119886 = 119886+Δ119886 and 120579 = 120579+Δ120579 and substituting them into(51) one could get

119889Δ119886

119889119905

=

12120587120596

[minus119882 sdotΔ119886minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862 sdot Δ119886]

(59)

Based on (53) one could eliminate 120579 and obtain the charac-teristic determinant as

det[

[

minus119882 minus 120572 minus119886119866

119866

119886

minus119882 minus 120572

]

]

= 0 (60)

Expanding the determinant one could obtain the character-istic equation as

(minus119882minus120572)2+119866

2= 0 (61)

Solving (61) one could obtain the characteristic values as

12057212 = minus119882plusmn119866119894 (62)

Obviously all real parts of the characteristic roots are neg-ative which means the steady-state solution of this kind ofsemiactive system is also unconditionally stable

33 Analytical Investigation for Relative Control System Thestrategy for semiactive relative control is shown as

119888 =

119888max (119909 minus 1199090) ( minus 0) le 0

119888min else(63)

After transformation of coordinates (63) will become

120585 =

120585max 119911 le 0

120585min else(64)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt

120587

2

120585min120587

2lt 120601 lt 120587

120585max 120587 lt 120601 lt

31205872

120585min31205872

lt 120601 lt 2120587

(65)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) for this semiactive control system

1198862 =

minus120587119886 (120585max + 120585min)

2120587

119886

1205792 =

2119886 (minus120585max + 120585min)

2120587

(66)

Combining (12) (15a) (15b) (21) (22) and (66) one couldobtain

119886

=

minus120587120596119886 (120585max + 120585min) minus 11988301205871205962 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579

=

119886 [120587 (12059620 minus 120596

2) + 2120596 (minus120585max + 120585min)] minus 1198830120587120596

2 cos 1205792120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(67)

Obviously we could easily obtain the equivalent dampingcoefficient

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+

119888max + 119888min2

(68)

From (68) we can find that the equivalent damping coeffi-cient 1198621015840 mainly depends on the fractional-order parameters119870 and 119901 associated with the original damping coefficient

Letting 119886 = 0 and 119886

120579 = 0 one could obtain the amplitudeand phase of the steady-state response

119886 =

12058711988301205962

radic1198802+ 119881

2

120579 = arctan 119880

119881

(69)

where

119880 = minus120582120587120596119901 sin(

119901120587

2)minus120587120596 (120585max + 120585min)

119881 = 120587 (12059620 minus120596

2) + 2120596 (minus120585max + 120585min)

+ 120582120587120596119901 cos(

119901120587

2)

(70)

8 Mathematical Problems in Engineering

The transmissibility for relative displacement in this caseis

1205833 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198802+ 119881

2 (71)

The absolute displacement of relative control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198833 cos (120596119905 minus 120574)

(72)

where

1198833 = radic(1198830 +

11988301205871205962 cos120595

radic119876

)

2

+

11988320120587

21205964sin2120595

119876

120574 = arctan1198830120587120596

2 sin120595

(1198830 + cos120595)radic119876

(73)

where

120595 =

119880

119881

119876 = 1198812+119880

2

(74)

The transmissibility for the absolute displacement is

1205893 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988331198830

(75)

Next we study the stability of steady-state solutionsLetting 119886 = 119886 + Δ119886 120579 = 120579 + Δ120579 and substituting them into(67) one could get

119889Δ119886

119889119905

=

12120587120596

[[minus120587120596 (120585max + 120585min) minus 120582120587120596119901 sin(

119901120587

2)]

sdot Δ119886 minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862

sdot Δ119886]

(76)

Based on (69) one could eliminate 120579 and obtain the charac-teristic determinant

det[

[

119880 minus 120572 minus119886119881

119880

119886

119881 minus 120572

]

]

= 0 (77)

0 20 40 60 80 1000

0005

001

0015

002

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 2 Comparison of analytical solution with numerical oneunder on-off control

Expanding the determinant one could obtain the character-istic equation

(119880minus120572)2+119881

2= 0 (78)

Solving (78) one could obtain the characteristic values

12057212 = 119880plusmn119881119894 (79)

Obviously 119880 lt 0 That is to say all real parts of the char-acteristic values are negative which means the steady-statesolution of this kind of semiactive system is stable uncondi-tionally

4 Numerical Simulation and Analysis

41 Comparison between the Analytical and Numerical Solu-tion In order to verify the precision of the analytical solu-tions we also present the numerical results The numericalscheme [29ndash34] is

119863119901[119909 (119905119897)] asymp ℎ

minus119901119897

sum

119895=0119862119901

119895119909 (119905119897minus119895) (80)

where 119905119897 = 119897ℎ is the time sample points ℎ is the sample stepand119862

119901

119895is the fractional binomial coefficient with the iterative

relationship as

119862119901

0 = 1

119862119901

119895= (1minus

1 + 119901

119895

)119862119901

119895minus1(81)

The selected basic system parameters are 119898 = 240 119896 =

16000 119888 = 10001198830 = 001 119901 = 05119870 = 1000 and 120575=051198830We take 400 seconds as numerical time in the process of

Mathematical Problems in Engineering 9

0 20 40 60 80 1000

0005

001

0015

002

0025

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 3 Comparison of analytical solution with numerical oneunder LRD control

the simulation Considering the maximum value of thenumerical results in the last 100 seconds as the steady-stateamplitude one could get the amplitude-frequency curves forthe numerical results which are denoted by the lines withcircles shown in Figures 2ndash4 Based on (37) (53) and (69)one could obtain the amplitude-frequency curves by theanalytical solutions denoted by the solid lines in Figures 2ndash4From the observation of the three figures one could concludethat the analytical solutions agree very well with the numeri-cal results and could present satisfactory precision Howeverthere are distinct errors between the approximate analyticaland numerical solution in some high-frequency range espe-cially the amplitude These phenomena called chatter insemiactive control system had also been found by Ahmadian[35] numerically and experimentally

42 Effects of 119870 on the Steady-State Amplitudes When thefractional coefficient 119870 is changed the different amplitude-frequency curves are shown in Figures 5ndash7 From the obser-vation of Figures 5ndash7 we could conclude that the largerthe fractional coefficient 119870 is the smaller the steady-stateamplitudes of the relative and absolute displacement of sys-tem mass are At the same time the resonant frequency alsobecomes larger This phenomenon indicates that both theequivalent damping coefficient and equivalent stiffness coef-ficient become larger in this procedure That is to say thefractional-order derivative 119870119863

119901(119909 minus 1199090) serves as damping

and spring simultaneously

43 Effects of 119901 on the Steady-State Amplitudes When thefractional coefficient 119901 is changed the results are shownin Figures 8ndash10 From the observation of Figures 8ndash10 wecould conclude that the larger the fractional coefficient 119901 isthe smaller the steady-state amplitudes of the relative andabsolute displacement of system mass are However it seems

Numerical solutionAnalytical solution

0 20 40 60 80 1000

0005

001

0015

002

0025

003

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 4 Comparison of analytical solution with numerical oneunder relative control

that there is little effect of119901 on the resonant frequency In factby differentiating119870

1015840 and 1198621015840 to 119901 one can get

1198891198621015840

119889119901

= 119896120596119901minus1

radic(119901120596minus1)2+ (

120587

2)

2sin(

119901120587

2+ 1205991)

1198891198701015840

119889119901

= minus 119896120596119901radic(119901120596

minus1)2+ (

120587

2)

2sin(

119901120587

2minus 1205992)

(82)

where

1205991 = arctan 120587120596

2119901

1205992 = arctan2119901120587120596

(83)

Because 0 lt 119901 lt 1 we can conclude that the derivative of1198621015840 with respect to 119901 will be larger than 0 1198621015840 will be mono-

tonically larger if 119901 becomes larger Accordingly the steady-state amplitude will be smaller in the process But the valueof 1015840 is not monotonical in this procedure1198701015840 could increaseor reduce with different 119901 which is affected by the systemparameters

5 Conclusion

In this paper three SDOF semiactive systems are researchedby using averaging method Aiming at these systems wepresent the concepts of equivalent damping coefficient andequivalent stiffness coefficient in order to account for theeffect of the fractional-order derivative on the control perfor-mance

Based on the above study one could find that fractional-order derivative plays an important role in SDOF semiactivesystem Therefore the results in this paper supply valuablereference for engineering practice In addition analyticalmethod in this paper could be adopted in two or multi-degrees-of-freedom system and it also offers a new idea tothe research field on vehicle suspension theory

10 Mathematical Problems in Engineering

0

05

1

15

2

25

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205831

100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 1

100 101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 5 Effect of119870 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205832

101 102

(a) Effect of119870 on relative displacement transmissibility

05

1

15

2

25

3

35

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 2

101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 6 Effect of 119870 on the steady-state amplitudes under LRD control

0

05

1

15

2

25

3

35

4

K = 0

K = 1000

K = 2000

K = 3000

1205833

120596 (rads)100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

2

3

4

5

120577 3

120596 (rads)100 101 102

K = 0

K = 1000

K = 2000

K = 3000

(b) Effect of119870 on absolute displacement transmissibility

Figure 7 Effect of119870 on the steady-state amplitudes under relative control

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the free damped vibrations of a fractional-order oscillatorTavazoei et al [18] and Pinto and Tenreiro Machado [19]studied the fractional-order van der Pol oscillator and foundmultiple limit cycles existing in the system Shen et al [20ndash23]analytically studied the primary resonance of some nonlinearoscillators with different fractional-order derivatives by theaveraging method and illustrated the effects of fractional-order parameters on dynamical response Wahi and Chat-terjee [24] studied an oscillator with special fractional-orderderivative and time-delay by averaging method Chen andZhu [25]Wang andHe [26] Huang and Jin [27] and Xu et al[28] also investigated different fractional-order systems andpresented some important results by analytical research

In this paper we introduce fractional-order derivativeinto semiactive SDOF systemswhere the semiactivemethodsare on-off control limited relative displacement (LRD) con-trol and relative control respectivelyThe analytical solutionsof the three semiactive systems are obtained by averagingmethodThenwe study the stability of those steady-state solu-tions by Lyapunov stability theory Furthermore the effectsof fractional-order parameters on the steady-state amplitudeand control performance are researched At last the mainconclusions are made

2 Analytical Solutions ofSDOF Semiactive System

Theconsidered SDOF semiactive system is shown in Figure 1If the fractional-order viscoelastic device is used its mathe-matical model should be

119898+ 119896 (119909 minus 1199090) + 119888 ( minus 0) +119870119863119901(119909 minus 1199090) = 0 (1)

where 1199090 = 1198830 cos120596119905 is the external excitation to the con-trolled system with1198830 and 120596 as the excitation amplitude andexcitation frequency119898 119888 and 119896 are the system mass systemdamping coefficient and stiffness respectively119870119863

119901(119909minus1199090) is

the fractional-order derivative of119909minus1199090 to 119905with the fractionalcoefficient119870 (119870 ge 0) and the fractional-order 119901 (0 le 119901 le 1)There are several definitions for fractional-order derivativeand they are equivalent under some conditions for a wideclass of functions In Caputorsquos sense the definition of 119901 orderderivative of 119909(119905) to 119905 is

119863119901[119909 (119905)] =

1Γ (1 minus 119901)

int

119905

0

1199091015840(119906)

(119905 minus 119906)119901119889119906 (2)

where Γ(119911) is Gamma function that satisfies Γ(119911+ 1) = 119911Γ(119911)Defining 119911 = 119909 minus 1199090 (1) becomes

119898 + 119896119911 + 119888 +119870119863119901(119911) = 1198981198830120596

2 cos120596119905 (3)

Using the following transformation of coordinates

12059620 =

119896

119898

120585 =

119888

2119898

ck

x

x0

KDp(x minus x0)

m

Figure 1Themodel of SDOF semiactive systemby using fractional-order derivative

120582 =

119870

119898

119891 = 11988301205962

(4)

equation (3) becomes

+ 12059620119911 + 2120585 + 120582119863

119901(119911) = 119891 cos120596119905 (5)

Supposing the solution of (5) is

119911 = 119886 cos120601

= minus120596119886 sin120601

(6)

where 120601 = 120596119905 + 120579 one could get

= 119886 cos120601minus 119886 (120596+

120579) sin120601

= minus120596 [ 119886 sin120601+ 119886 (120596+

120579) cos120601] (7)

From (5) (6) and (7) one could obtain

119886 cos120601minus 119886

120579 sin120601 = 0

119886 sin120601+ 119886

120579 cos120601 =

119865

120596

(8)

where

119865 = minus [119891 cos120596119905 + 119886 (1205962minus120596

20) cos120601+ 2119886120585120596 sin120601

minus120582119863119901(119886 cos120601)]

(9)

The derivatives of the generalized amplitude 119886 and phase120579 could be solved as

119886 =

119865

120596

sin120601

119886

120579 =

119865

120596

cos120601(10)

Mathematical Problems in Engineering 3

Then one could apply the standard averaging method to (10)in time interval [0 119879]

119886 = lim119879rarrinfin

1119879

int

119879

0

119865

120596

sin120601119889120601

119886

120579 = lim119879rarrinfin

1119879

int

119879

0

119865

120596

cos120601119889120601

(11)

Furthermore (11) could be written as

119886 = 1198861 + 1198862 + 1198863

119886

120579 = 119886

1205791 + 119886

1205792 + 119886

1205793(12)

where

1198861 = lim119879rarrinfin

1119879120596

sdot int

119879

0minus [1198830120596

2 cos (120601 minus 120579) + 119886 (1205962minus120596

20) cos120601]

sdot sin120601119889120601

(13a)

1198862 = lim119879rarrinfin

1119879120596

int

119879

0minus2119886120585120596 sin2120601119889120601 (13b)

1198863 = lim119879rarrinfin

1119879120596

int

119879

0120582119863119901(119886 cos120601) sin120601119889120601 (13c)

119886

1205791 = lim119879rarrinfin

1119879120596

sdot int

119879

0minus [1198830120596

2 cos (120601 minus 120579) + 119886 (1205962minus120596

20) cos120601]

sdot cos120601119889120601

(14a)

119886

1205792 = lim119879rarrinfin

1119879120596

int

119879

0minus119886120585120596 sin 2120601119889120601 (14b)

119886

1205793 = lim119879rarrinfin

1119879120596

int

119879

0120582119863119901(119886 cos120601) cos120601119889120601 (14c)

According to the averaging method one could select thetime terminal 119879 as 119879 = 2120587 if the integrand is periodicfunction or 119879 = infin if the integrand is aperiodic oneAccordingly one could obtain

1198861 = minus

1198830120596 sin 120579

2 (15a)

119886

1205791 =

minus119886 (1205962+ 120596

20) minus 1198830120596

2 cos 1205792120596

(15b)

In order to obtain the definite integral in (13c) and (14c)two important formulae which have been deduced in [20 21]are introduced

1198611 = lim119879rarrinfin

int

119879

0

sin (120596119905)

119905119901

119889119905

= 120596119901minus1

Γ (1minus119901) cos(119901120587

2)

(16a)

1198612 = lim119879rarrinfin

int

119879

0

cos (120596119905)

119905119901

119889119905

= 120596119901minus1

Γ (1minus119901) sin(

119901120587

2)

(16b)

Based on Caputorsquos definition (13c) becomes

1198863 = lim119879rarrinfin

120582

119879120596

int

119879

0119863119901[119886 cos (120596119905 + 120579)] sin (120596119905 + 120579) 119889119905

=

minus120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

sdot int

119879

0[int

119905

0

sin (120596119906 + 120579)

(119905 minus 119906)119901

119889119906] sin (120596119905 + 120579) 119889119905

(17)

Introducing 119904 = 119905 minus 119906 and 119889119904 = minus119889119906 (17) becomes

1198863 =

minus120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

int

119879

0[int

119905

0

sin (120596119905 + 120579 minus 120596119904)

119904119901

119889119904]

sdot sin (120596119905 + 120579) 119889119905 =

minus120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

sdot int

119879

0[int

119905

0

cos (120596119904)

119904119901

119889119904] sin (120596119905 + 120579) sin (120596119905 + 120579) 119889119905

+

120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

int

119879

0[int

119905

0

sin (120596119904)

119904119901

119889119904]

sdot cos (120596119905 + 120579) sin (120596119905 + 120579) 119889119905

(18)

In fact the secondpart in (18)would vanish according to (16a)and (16b) Defining the first part in (18) as1198601 and integratingit by parts one could obtain

1198601

=

minus120582119886

4120596Γ (1 minus 119901)

lim119879rarrinfin

2120596119905 minus sin (2120596119905 + 2120579)119879

[int

119905

0

cos (120596119904)

119904119901

119889119904]

10038161003816100381610038161003816100381610038161003816

119879

0

minus

minus120582119886

4120596Γ (1 minus 119901)

lim119879rarrinfin

1119879

int

119879

0[

[2120596119905 minus sin (2120596119905 + 2120579)] cos (120596119905)

119905119901

] 119889119905

(19)

Based on (16a) and (16b) one could find that the second partin (19) would vanish and the first part in (19) will be

1198601 = minus

minus120582119886120596119901minus1

2sin(

119901120587

2) (20)

That means

1198863 =

minus120582119886120596119901minus1

2sin(

119901120587

2) (21)

After the similar procedure (14c) will be

119886

1205793 =

120582119886120596119901minus1

2cos(

119901120587

2) (22)

4 Mathematical Problems in Engineering

Equations (13b) and (14b) can be rewritten as

1198862 =

12120587

int

2120587

0minus2119886120585sin2120601119889120601 (23a)

119886

1205792 =

12120587

int

2120587

0minus119886120585 sin 2120601119889120601 (23b)

The solutions of (23a) and (23b) depend on the specificsemiactive control methods

Accordingly (12) becomes

119886 = minus

1198830120596 sin 120579

2minus

120582119886120596119901minus1

2sin(

119901120587

2)

+

119886

2120587int

2120587

0minus2120585 sin2120601119889120601

119886

120579 =

minus1198861205962+ 119886120596

20 minus 1198830120596

2 cos 1205792120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

+

119886

2120587int

2120587

0minus120585 sin 2120601119889120601

(24)

which could be rewritten as

119886 = minus

1198830120596 sin 120579

2minus

119886

21198981198621015840

119886

120579 =

minus1198830120596 cos 1205792

minus

120596119886

2+

119886

2120587int

2120587

0minus

119888

2119898sin 2120601119889120601

+

119886

2119898120596

1198701015840

(25)

where

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+int

2120587

0

minus119888 sin2120601119889120601

120587

(26a)

1198701015840= 119870120596

119901 cos(119901120587

2)+ 119896 (26b)

1198621015840 and 119870

1015840 are defined as the equivalent damping coefficientand equivalent stiffness coefficient in semiactive systemsrespectively In the SDOF semiactive system the fractional-order derivative119870119863

119901(119909 minus1199090) serves as damping and stiffness

at the same timeWhen 119901 rarr 0119870119863119901(119909minus1199090)will be changed

into spring force whereas it will be almost the same as damp-ing force if 119901 rarr 1 When 119870 rarr 0 119870119863

119901(119909 minus 1199090) will have

little effect on control performance and the semiactive systemwill degrade into the traditional semiactive one

3 Cases for Three Kinds ofSemiactive Control Systems

31 Analytical Investigation for on-off Control System Thestrategy of semiactive on-off control is

119888 =

119888max ( minus 0) ge 0

119888min else(27)

Originally 119888min is selected as 119888min = 0 in [2] which may beinaccurate in real engineering Here we select it as 119888min =

001119888maxAfter transformation of coordinates (27) becomes

120585 =

120585max ( + 0) ge 0

120585min else(28)

Expanding the critical condition

( + 0) ge 0 (29)

one could get

minus120596119886 sin120601 [minus120596119886 sin120601minus1198830120596 sin (120596119905)]

= 119886119877 sin120601 sin (120601 minus 120573) ge 0(30)

where

120573 = arctan1198830 sin 120579

119886 + 1198830 cos 120579

119877 = 1205962radic(119886 + 1198830 cos 120579)

2+ 119883

20sin2120579

(31)

Equation (28) can be rewritten as

120585 =

120585min 0 lt 120601 lt 120573

120585max 120573 lt 120601 lt 120587

120585min 120587 lt 120601 lt 120587 + 120573

120585max 120587 + 120573 lt 120601 lt 2120587

(32)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) as

1198862 =

minus2119886119860 minus 119886119861

2120587

119886

1205792 =

119886119862 minus 119886119863

2120587

(33)

where

119860 = 120587120585max +120573 (minus120585max + 120585min)

119861 = (120585max minus 120585min) sin 2120573

119862 = 120585max minus 120585min

119863 = (120585max minus 120585min) cos 2120573

(34)

Combining (12) (15a) (15b) (21) (22) and (33) onecould obtain

119886 = minus

1198830120596 sin 120579

2minus

120582119886120596119901minus1

2sin(

119901120587

2)+

minus2119886119860 minus 119886119861

2120587

119886

120579 =

minus1198861205962+ 119886120596

20 minus 1198830120596

2 cos 1205792120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

+

119886119862 minus 119886119863

2120587

(35)

Mathematical Problems in Engineering 5

Obviously the equivalent damping coefficient in this casecould be written as

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+ 119888max +

120573

120587

(minus119888max + 119888min)

+

sin 21205732120587

(119888max minus 119888min)

(36)

From (36) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120573

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain the amplitude and phase

of steady-state response

119886 =

12058711988301205962

radic1198732+ 119871

2

120579 = arctan 119873

119871

(37)

where

119873 = minus 2120596119860minus120596119861minus120587120596119901120582 sin

119901120587

2

119871 = 120596119862minus120596119863+120587120596119901120582 cos

119901120587

2+120587120596

20 minus120587120596

2

(38)

The transmissibility for relative displacement is

1205831 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198732+ 119871

2 (39)

The absolute displacement of on-off control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198831 cos (120596119905 minus 120591)

(40)

where

1198831 = radic

11988320120587

21205964119873

2

(1198732+ 119871

2)2 + (1198830 +

11988301205871205962|119871|

1198732+ 119871

2 )

2

120591 = minus arctan 1205871205962119873

119871 (1205871205962+ (119871

2+ 119873

2) |119871|)

(41)

The displacement transmissibility is

1205891 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988311198830

(42)

Next we study the stability of steady-state solution Let-ting 119886 = 119886 + Δ119886 and 120579 = 120579 + Δ120579 and then substituting theminto (35) one could get

119889Δ119886

119889119905

=

12120587

[119873 sdot Δ119886minus1198830120587120596 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587

[

1205871198830120596 sin 120579

119886

sdot Δ120579 +

1205871198830120596 cos 1205791198862 sdot Δ119886]

(43)

Based on (37) one could eliminate 120579 and obtain the charac-teristic determinant

det[[

[

119873

120596

minus 120572

minus119886119871

120596

119871

119886120596

119873

120596

minus 120572

]

]

]

= 0 (44)

Expanding the determinant one could obtain the character-istic equation

(

119873

120596

minus120572)

2+

1198712

1205962 = 0 (45)

Solving (45) one could obtain characteristic values

12057212 =

119873

120596

plusmn

119871

120596

119894 (46)

Obviously119873 is less than 0 Accordingly all real parts of char-acteristic roots are negative which means the steady-statesolution of this kind of semiactive system is unconditionallystable

32 Analytical Investigation for LRD Control System Thestrategy of LRD control is

119888 =

119888max1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816ge 120575

119888min1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816lt 120575

(47)

where 119888min = 001119888max and 120575 is the space between the massand excitation point Here we select 120575=051198830

After transformation of coordinates (47) becomes

119888 =

119888max |119911| ge 120575

119888min |119911| lt 120575

(48)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt 120576

120585min 120576 lt 120601 lt 120587 minus 120576

120585max 120587 minus 120576 lt 120601 lt 120587 + 120576

120585min 120587 + 120576 lt 120601 lt 2120587 minus 120576

120585max 2120587 minus 120576 lt 120601 lt 2120587

(49)

where 120576 = arccos(051198830119886)Using the averagingmethod one could obtain the simpli-

fied forms of (13b) and (14b)

1198862 =

minus21198861198600 + 211988611986102120587120596

119886

1205792 = 0(50)

where 1198600 = 120596(2120576120585max + 120587120585min minus 2120576120585min) 1198610 = 120596(120585max minus

120585min) sin 2120576

6 Mathematical Problems in Engineering

Combining (12) (15a) (15b) (21) (22) and (50) onecould obtain

119886 =

minus21198861198600 + 21198861198610 minus 11988301205962120587 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579 =

minus119886120587 (1205962minus 120596

20) minus 1198830120596

2120587 cos 120579

2120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(51)

Obviously one could easily obtain the equivalent dampingcoefficient

1198621015840

= 119870120596119901minus1 sin(

119901120587

2)

+

(119888max minus 119888min) sin 2120576 minus (2120576119888max + 119888min120587 minus 2120576119888min)

120587

(52)

From (52) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120576

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain

119886 =

12058711988301205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2

120579 = arctanminus21198600 + 21198610 minus 120582120587120596

119901 sin (1199011205872)minus120587 (120596

2minus 120596

20) + 120582120587120596

119901 cos (1199011205872)

(53)

where 119886 and 120579 are the amplitude and phase of steady-stateresponse respectively

The transmissibility for relative displacement is

1205832 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2 (54)

The absolute displacement of LRD control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198832 cos (120596119905 minus 120594)

(55)

where

119883

22

=

11988320120587

21205964119882

2

[(21198600 minus 21198610)2+ 120587

2(120596

4minus 21205962

12059620 + 120596

40 + 120596

21199011205822) minus 21205872

120596119901(120596

2minus 120596

20) 120582 cos (1199011205872) + 4 (1198600 minus 1198610) 120587120596

119901120582 sin (1199011205872)]

2

+(1198830 +1198830120587120596

2119866

1198662+ 119882

2)

2

120594 = arctan 12058721205962119882

11986612058721205962+ 120587119866

2+ 120587119882

2

(56)

Mathematical Problems in Engineering 7

where

119866 = 120587(minus1205962+120596

20 +120596119901120582 cos

119901120587

2)

119882 = 21198600 minus 21198610 +120587120596119901120582 sin

119901120587

2

(57)

The transmissibility for the absolute displacement of LRDcontrol system is

1205892 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988321198830

(58)

Next we study the stability of the steady-state solutionsLetting 119886 = 119886+Δ119886 and 120579 = 120579+Δ120579 and substituting them into(51) one could get

119889Δ119886

119889119905

=

12120587120596

[minus119882 sdotΔ119886minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862 sdot Δ119886]

(59)

Based on (53) one could eliminate 120579 and obtain the charac-teristic determinant as

det[

[

minus119882 minus 120572 minus119886119866

119866

119886

minus119882 minus 120572

]

]

= 0 (60)

Expanding the determinant one could obtain the character-istic equation as

(minus119882minus120572)2+119866

2= 0 (61)

Solving (61) one could obtain the characteristic values as

12057212 = minus119882plusmn119866119894 (62)

Obviously all real parts of the characteristic roots are neg-ative which means the steady-state solution of this kind ofsemiactive system is also unconditionally stable

33 Analytical Investigation for Relative Control System Thestrategy for semiactive relative control is shown as

119888 =

119888max (119909 minus 1199090) ( minus 0) le 0

119888min else(63)

After transformation of coordinates (63) will become

120585 =

120585max 119911 le 0

120585min else(64)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt

120587

2

120585min120587

2lt 120601 lt 120587

120585max 120587 lt 120601 lt

31205872

120585min31205872

lt 120601 lt 2120587

(65)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) for this semiactive control system

1198862 =

minus120587119886 (120585max + 120585min)

2120587

119886

1205792 =

2119886 (minus120585max + 120585min)

2120587

(66)

Combining (12) (15a) (15b) (21) (22) and (66) one couldobtain

119886

=

minus120587120596119886 (120585max + 120585min) minus 11988301205871205962 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579

=

119886 [120587 (12059620 minus 120596

2) + 2120596 (minus120585max + 120585min)] minus 1198830120587120596

2 cos 1205792120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(67)

Obviously we could easily obtain the equivalent dampingcoefficient

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+

119888max + 119888min2

(68)

From (68) we can find that the equivalent damping coeffi-cient 1198621015840 mainly depends on the fractional-order parameters119870 and 119901 associated with the original damping coefficient

Letting 119886 = 0 and 119886

120579 = 0 one could obtain the amplitudeand phase of the steady-state response

119886 =

12058711988301205962

radic1198802+ 119881

2

120579 = arctan 119880

119881

(69)

where

119880 = minus120582120587120596119901 sin(

119901120587

2)minus120587120596 (120585max + 120585min)

119881 = 120587 (12059620 minus120596

2) + 2120596 (minus120585max + 120585min)

+ 120582120587120596119901 cos(

119901120587

2)

(70)

8 Mathematical Problems in Engineering

The transmissibility for relative displacement in this caseis

1205833 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198802+ 119881

2 (71)

The absolute displacement of relative control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198833 cos (120596119905 minus 120574)

(72)

where

1198833 = radic(1198830 +

11988301205871205962 cos120595

radic119876

)

2

+

11988320120587

21205964sin2120595

119876

120574 = arctan1198830120587120596

2 sin120595

(1198830 + cos120595)radic119876

(73)

where

120595 =

119880

119881

119876 = 1198812+119880

2

(74)

The transmissibility for the absolute displacement is

1205893 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988331198830

(75)

Next we study the stability of steady-state solutionsLetting 119886 = 119886 + Δ119886 120579 = 120579 + Δ120579 and substituting them into(67) one could get

119889Δ119886

119889119905

=

12120587120596

[[minus120587120596 (120585max + 120585min) minus 120582120587120596119901 sin(

119901120587

2)]

sdot Δ119886 minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862

sdot Δ119886]

(76)

Based on (69) one could eliminate 120579 and obtain the charac-teristic determinant

det[

[

119880 minus 120572 minus119886119881

119880

119886

119881 minus 120572

]

]

= 0 (77)

0 20 40 60 80 1000

0005

001

0015

002

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 2 Comparison of analytical solution with numerical oneunder on-off control

Expanding the determinant one could obtain the character-istic equation

(119880minus120572)2+119881

2= 0 (78)

Solving (78) one could obtain the characteristic values

12057212 = 119880plusmn119881119894 (79)

Obviously 119880 lt 0 That is to say all real parts of the char-acteristic values are negative which means the steady-statesolution of this kind of semiactive system is stable uncondi-tionally

4 Numerical Simulation and Analysis

41 Comparison between the Analytical and Numerical Solu-tion In order to verify the precision of the analytical solu-tions we also present the numerical results The numericalscheme [29ndash34] is

119863119901[119909 (119905119897)] asymp ℎ

minus119901119897

sum

119895=0119862119901

119895119909 (119905119897minus119895) (80)

where 119905119897 = 119897ℎ is the time sample points ℎ is the sample stepand119862

119901

119895is the fractional binomial coefficient with the iterative

relationship as

119862119901

0 = 1

119862119901

119895= (1minus

1 + 119901

119895

)119862119901

119895minus1(81)

The selected basic system parameters are 119898 = 240 119896 =

16000 119888 = 10001198830 = 001 119901 = 05119870 = 1000 and 120575=051198830We take 400 seconds as numerical time in the process of

Mathematical Problems in Engineering 9

0 20 40 60 80 1000

0005

001

0015

002

0025

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 3 Comparison of analytical solution with numerical oneunder LRD control

the simulation Considering the maximum value of thenumerical results in the last 100 seconds as the steady-stateamplitude one could get the amplitude-frequency curves forthe numerical results which are denoted by the lines withcircles shown in Figures 2ndash4 Based on (37) (53) and (69)one could obtain the amplitude-frequency curves by theanalytical solutions denoted by the solid lines in Figures 2ndash4From the observation of the three figures one could concludethat the analytical solutions agree very well with the numeri-cal results and could present satisfactory precision Howeverthere are distinct errors between the approximate analyticaland numerical solution in some high-frequency range espe-cially the amplitude These phenomena called chatter insemiactive control system had also been found by Ahmadian[35] numerically and experimentally

42 Effects of 119870 on the Steady-State Amplitudes When thefractional coefficient 119870 is changed the different amplitude-frequency curves are shown in Figures 5ndash7 From the obser-vation of Figures 5ndash7 we could conclude that the largerthe fractional coefficient 119870 is the smaller the steady-stateamplitudes of the relative and absolute displacement of sys-tem mass are At the same time the resonant frequency alsobecomes larger This phenomenon indicates that both theequivalent damping coefficient and equivalent stiffness coef-ficient become larger in this procedure That is to say thefractional-order derivative 119870119863

119901(119909 minus 1199090) serves as damping

and spring simultaneously

43 Effects of 119901 on the Steady-State Amplitudes When thefractional coefficient 119901 is changed the results are shownin Figures 8ndash10 From the observation of Figures 8ndash10 wecould conclude that the larger the fractional coefficient 119901 isthe smaller the steady-state amplitudes of the relative andabsolute displacement of system mass are However it seems

Numerical solutionAnalytical solution

0 20 40 60 80 1000

0005

001

0015

002

0025

003

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 4 Comparison of analytical solution with numerical oneunder relative control

that there is little effect of119901 on the resonant frequency In factby differentiating119870

1015840 and 1198621015840 to 119901 one can get

1198891198621015840

119889119901

= 119896120596119901minus1

radic(119901120596minus1)2+ (

120587

2)

2sin(

119901120587

2+ 1205991)

1198891198701015840

119889119901

= minus 119896120596119901radic(119901120596

minus1)2+ (

120587

2)

2sin(

119901120587

2minus 1205992)

(82)

where

1205991 = arctan 120587120596

2119901

1205992 = arctan2119901120587120596

(83)

Because 0 lt 119901 lt 1 we can conclude that the derivative of1198621015840 with respect to 119901 will be larger than 0 1198621015840 will be mono-

tonically larger if 119901 becomes larger Accordingly the steady-state amplitude will be smaller in the process But the valueof 1015840 is not monotonical in this procedure1198701015840 could increaseor reduce with different 119901 which is affected by the systemparameters

5 Conclusion

In this paper three SDOF semiactive systems are researchedby using averaging method Aiming at these systems wepresent the concepts of equivalent damping coefficient andequivalent stiffness coefficient in order to account for theeffect of the fractional-order derivative on the control perfor-mance

Based on the above study one could find that fractional-order derivative plays an important role in SDOF semiactivesystem Therefore the results in this paper supply valuablereference for engineering practice In addition analyticalmethod in this paper could be adopted in two or multi-degrees-of-freedom system and it also offers a new idea tothe research field on vehicle suspension theory

10 Mathematical Problems in Engineering

0

05

1

15

2

25

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205831

100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 1

100 101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 5 Effect of119870 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205832

101 102

(a) Effect of119870 on relative displacement transmissibility

05

1

15

2

25

3

35

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 2

101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 6 Effect of 119870 on the steady-state amplitudes under LRD control

0

05

1

15

2

25

3

35

4

K = 0

K = 1000

K = 2000

K = 3000

1205833

120596 (rads)100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

2

3

4

5

120577 3

120596 (rads)100 101 102

K = 0

K = 1000

K = 2000

K = 3000

(b) Effect of119870 on absolute displacement transmissibility

Figure 7 Effect of119870 on the steady-state amplitudes under relative control

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Then one could apply the standard averaging method to (10)in time interval [0 119879]

119886 = lim119879rarrinfin

1119879

int

119879

0

119865

120596

sin120601119889120601

119886

120579 = lim119879rarrinfin

1119879

int

119879

0

119865

120596

cos120601119889120601

(11)

Furthermore (11) could be written as

119886 = 1198861 + 1198862 + 1198863

119886

120579 = 119886

1205791 + 119886

1205792 + 119886

1205793(12)

where

1198861 = lim119879rarrinfin

1119879120596

sdot int

119879

0minus [1198830120596

2 cos (120601 minus 120579) + 119886 (1205962minus120596

20) cos120601]

sdot sin120601119889120601

(13a)

1198862 = lim119879rarrinfin

1119879120596

int

119879

0minus2119886120585120596 sin2120601119889120601 (13b)

1198863 = lim119879rarrinfin

1119879120596

int

119879

0120582119863119901(119886 cos120601) sin120601119889120601 (13c)

119886

1205791 = lim119879rarrinfin

1119879120596

sdot int

119879

0minus [1198830120596

2 cos (120601 minus 120579) + 119886 (1205962minus120596

20) cos120601]

sdot cos120601119889120601

(14a)

119886

1205792 = lim119879rarrinfin

1119879120596

int

119879

0minus119886120585120596 sin 2120601119889120601 (14b)

119886

1205793 = lim119879rarrinfin

1119879120596

int

119879

0120582119863119901(119886 cos120601) cos120601119889120601 (14c)

According to the averaging method one could select thetime terminal 119879 as 119879 = 2120587 if the integrand is periodicfunction or 119879 = infin if the integrand is aperiodic oneAccordingly one could obtain

1198861 = minus

1198830120596 sin 120579

2 (15a)

119886

1205791 =

minus119886 (1205962+ 120596

20) minus 1198830120596

2 cos 1205792120596

(15b)

In order to obtain the definite integral in (13c) and (14c)two important formulae which have been deduced in [20 21]are introduced

1198611 = lim119879rarrinfin

int

119879

0

sin (120596119905)

119905119901

119889119905

= 120596119901minus1

Γ (1minus119901) cos(119901120587

2)

(16a)

1198612 = lim119879rarrinfin

int

119879

0

cos (120596119905)

119905119901

119889119905

= 120596119901minus1

Γ (1minus119901) sin(

119901120587

2)

(16b)

Based on Caputorsquos definition (13c) becomes

1198863 = lim119879rarrinfin

120582

119879120596

int

119879

0119863119901[119886 cos (120596119905 + 120579)] sin (120596119905 + 120579) 119889119905

=

minus120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

sdot int

119879

0[int

119905

0

sin (120596119906 + 120579)

(119905 minus 119906)119901

119889119906] sin (120596119905 + 120579) 119889119905

(17)

Introducing 119904 = 119905 minus 119906 and 119889119904 = minus119889119906 (17) becomes

1198863 =

minus120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

int

119879

0[int

119905

0

sin (120596119905 + 120579 minus 120596119904)

119904119901

119889119904]

sdot sin (120596119905 + 120579) 119889119905 =

minus120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

sdot int

119879

0[int

119905

0

cos (120596119904)

119904119901

119889119904] sin (120596119905 + 120579) sin (120596119905 + 120579) 119889119905

+

120582119886

Γ (1 minus 119901)

lim119879rarrinfin

1119879

int

119879

0[int

119905

0

sin (120596119904)

119904119901

119889119904]

sdot cos (120596119905 + 120579) sin (120596119905 + 120579) 119889119905

(18)

In fact the secondpart in (18)would vanish according to (16a)and (16b) Defining the first part in (18) as1198601 and integratingit by parts one could obtain

1198601

=

minus120582119886

4120596Γ (1 minus 119901)

lim119879rarrinfin

2120596119905 minus sin (2120596119905 + 2120579)119879

[int

119905

0

cos (120596119904)

119904119901

119889119904]

10038161003816100381610038161003816100381610038161003816

119879

0

minus

minus120582119886

4120596Γ (1 minus 119901)

lim119879rarrinfin

1119879

int

119879

0[

[2120596119905 minus sin (2120596119905 + 2120579)] cos (120596119905)

119905119901

] 119889119905

(19)

Based on (16a) and (16b) one could find that the second partin (19) would vanish and the first part in (19) will be

1198601 = minus

minus120582119886120596119901minus1

2sin(

119901120587

2) (20)

That means

1198863 =

minus120582119886120596119901minus1

2sin(

119901120587

2) (21)

After the similar procedure (14c) will be

119886

1205793 =

120582119886120596119901minus1

2cos(

119901120587

2) (22)

4 Mathematical Problems in Engineering

Equations (13b) and (14b) can be rewritten as

1198862 =

12120587

int

2120587

0minus2119886120585sin2120601119889120601 (23a)

119886

1205792 =

12120587

int

2120587

0minus119886120585 sin 2120601119889120601 (23b)

The solutions of (23a) and (23b) depend on the specificsemiactive control methods

Accordingly (12) becomes

119886 = minus

1198830120596 sin 120579

2minus

120582119886120596119901minus1

2sin(

119901120587

2)

+

119886

2120587int

2120587

0minus2120585 sin2120601119889120601

119886

120579 =

minus1198861205962+ 119886120596

20 minus 1198830120596

2 cos 1205792120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

+

119886

2120587int

2120587

0minus120585 sin 2120601119889120601

(24)

which could be rewritten as

119886 = minus

1198830120596 sin 120579

2minus

119886

21198981198621015840

119886

120579 =

minus1198830120596 cos 1205792

minus

120596119886

2+

119886

2120587int

2120587

0minus

119888

2119898sin 2120601119889120601

+

119886

2119898120596

1198701015840

(25)

where

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+int

2120587

0

minus119888 sin2120601119889120601

120587

(26a)

1198701015840= 119870120596

119901 cos(119901120587

2)+ 119896 (26b)

1198621015840 and 119870

1015840 are defined as the equivalent damping coefficientand equivalent stiffness coefficient in semiactive systemsrespectively In the SDOF semiactive system the fractional-order derivative119870119863

119901(119909 minus1199090) serves as damping and stiffness

at the same timeWhen 119901 rarr 0119870119863119901(119909minus1199090)will be changed

into spring force whereas it will be almost the same as damp-ing force if 119901 rarr 1 When 119870 rarr 0 119870119863

119901(119909 minus 1199090) will have

little effect on control performance and the semiactive systemwill degrade into the traditional semiactive one

3 Cases for Three Kinds ofSemiactive Control Systems

31 Analytical Investigation for on-off Control System Thestrategy of semiactive on-off control is

119888 =

119888max ( minus 0) ge 0

119888min else(27)

Originally 119888min is selected as 119888min = 0 in [2] which may beinaccurate in real engineering Here we select it as 119888min =

001119888maxAfter transformation of coordinates (27) becomes

120585 =

120585max ( + 0) ge 0

120585min else(28)

Expanding the critical condition

( + 0) ge 0 (29)

one could get

minus120596119886 sin120601 [minus120596119886 sin120601minus1198830120596 sin (120596119905)]

= 119886119877 sin120601 sin (120601 minus 120573) ge 0(30)

where

120573 = arctan1198830 sin 120579

119886 + 1198830 cos 120579

119877 = 1205962radic(119886 + 1198830 cos 120579)

2+ 119883

20sin2120579

(31)

Equation (28) can be rewritten as

120585 =

120585min 0 lt 120601 lt 120573

120585max 120573 lt 120601 lt 120587

120585min 120587 lt 120601 lt 120587 + 120573

120585max 120587 + 120573 lt 120601 lt 2120587

(32)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) as

1198862 =

minus2119886119860 minus 119886119861

2120587

119886

1205792 =

119886119862 minus 119886119863

2120587

(33)

where

119860 = 120587120585max +120573 (minus120585max + 120585min)

119861 = (120585max minus 120585min) sin 2120573

119862 = 120585max minus 120585min

119863 = (120585max minus 120585min) cos 2120573

(34)

Combining (12) (15a) (15b) (21) (22) and (33) onecould obtain

119886 = minus

1198830120596 sin 120579

2minus

120582119886120596119901minus1

2sin(

119901120587

2)+

minus2119886119860 minus 119886119861

2120587

119886

120579 =

minus1198861205962+ 119886120596

20 minus 1198830120596

2 cos 1205792120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

+

119886119862 minus 119886119863

2120587

(35)

Mathematical Problems in Engineering 5

Obviously the equivalent damping coefficient in this casecould be written as

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+ 119888max +

120573

120587

(minus119888max + 119888min)

+

sin 21205732120587

(119888max minus 119888min)

(36)

From (36) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120573

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain the amplitude and phase

of steady-state response

119886 =

12058711988301205962

radic1198732+ 119871

2

120579 = arctan 119873

119871

(37)

where

119873 = minus 2120596119860minus120596119861minus120587120596119901120582 sin

119901120587

2

119871 = 120596119862minus120596119863+120587120596119901120582 cos

119901120587

2+120587120596

20 minus120587120596

2

(38)

The transmissibility for relative displacement is

1205831 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198732+ 119871

2 (39)

The absolute displacement of on-off control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198831 cos (120596119905 minus 120591)

(40)

where

1198831 = radic

11988320120587

21205964119873

2

(1198732+ 119871

2)2 + (1198830 +

11988301205871205962|119871|

1198732+ 119871

2 )

2

120591 = minus arctan 1205871205962119873

119871 (1205871205962+ (119871

2+ 119873

2) |119871|)

(41)

The displacement transmissibility is

1205891 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988311198830

(42)

Next we study the stability of steady-state solution Let-ting 119886 = 119886 + Δ119886 and 120579 = 120579 + Δ120579 and then substituting theminto (35) one could get

119889Δ119886

119889119905

=

12120587

[119873 sdot Δ119886minus1198830120587120596 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587

[

1205871198830120596 sin 120579

119886

sdot Δ120579 +

1205871198830120596 cos 1205791198862 sdot Δ119886]

(43)

Based on (37) one could eliminate 120579 and obtain the charac-teristic determinant

det[[

[

119873

120596

minus 120572

minus119886119871

120596

119871

119886120596

119873

120596

minus 120572

]

]

]

= 0 (44)

Expanding the determinant one could obtain the character-istic equation

(

119873

120596

minus120572)

2+

1198712

1205962 = 0 (45)

Solving (45) one could obtain characteristic values

12057212 =

119873

120596

plusmn

119871

120596

119894 (46)

Obviously119873 is less than 0 Accordingly all real parts of char-acteristic roots are negative which means the steady-statesolution of this kind of semiactive system is unconditionallystable

32 Analytical Investigation for LRD Control System Thestrategy of LRD control is

119888 =

119888max1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816ge 120575

119888min1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816lt 120575

(47)

where 119888min = 001119888max and 120575 is the space between the massand excitation point Here we select 120575=051198830

After transformation of coordinates (47) becomes

119888 =

119888max |119911| ge 120575

119888min |119911| lt 120575

(48)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt 120576

120585min 120576 lt 120601 lt 120587 minus 120576

120585max 120587 minus 120576 lt 120601 lt 120587 + 120576

120585min 120587 + 120576 lt 120601 lt 2120587 minus 120576

120585max 2120587 minus 120576 lt 120601 lt 2120587

(49)

where 120576 = arccos(051198830119886)Using the averagingmethod one could obtain the simpli-

fied forms of (13b) and (14b)

1198862 =

minus21198861198600 + 211988611986102120587120596

119886

1205792 = 0(50)

where 1198600 = 120596(2120576120585max + 120587120585min minus 2120576120585min) 1198610 = 120596(120585max minus

120585min) sin 2120576

6 Mathematical Problems in Engineering

Combining (12) (15a) (15b) (21) (22) and (50) onecould obtain

119886 =

minus21198861198600 + 21198861198610 minus 11988301205962120587 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579 =

minus119886120587 (1205962minus 120596

20) minus 1198830120596

2120587 cos 120579

2120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(51)

Obviously one could easily obtain the equivalent dampingcoefficient

1198621015840

= 119870120596119901minus1 sin(

119901120587

2)

+

(119888max minus 119888min) sin 2120576 minus (2120576119888max + 119888min120587 minus 2120576119888min)

120587

(52)

From (52) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120576

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain

119886 =

12058711988301205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2

120579 = arctanminus21198600 + 21198610 minus 120582120587120596

119901 sin (1199011205872)minus120587 (120596

2minus 120596

20) + 120582120587120596

119901 cos (1199011205872)

(53)

where 119886 and 120579 are the amplitude and phase of steady-stateresponse respectively

The transmissibility for relative displacement is

1205832 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2 (54)

The absolute displacement of LRD control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198832 cos (120596119905 minus 120594)

(55)

where

119883

22

=

11988320120587

21205964119882

2

[(21198600 minus 21198610)2+ 120587

2(120596

4minus 21205962

12059620 + 120596

40 + 120596

21199011205822) minus 21205872

120596119901(120596

2minus 120596

20) 120582 cos (1199011205872) + 4 (1198600 minus 1198610) 120587120596

119901120582 sin (1199011205872)]

2

+(1198830 +1198830120587120596

2119866

1198662+ 119882

2)

2

120594 = arctan 12058721205962119882

11986612058721205962+ 120587119866

2+ 120587119882

2

(56)

Mathematical Problems in Engineering 7

where

119866 = 120587(minus1205962+120596

20 +120596119901120582 cos

119901120587

2)

119882 = 21198600 minus 21198610 +120587120596119901120582 sin

119901120587

2

(57)

The transmissibility for the absolute displacement of LRDcontrol system is

1205892 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988321198830

(58)

Next we study the stability of the steady-state solutionsLetting 119886 = 119886+Δ119886 and 120579 = 120579+Δ120579 and substituting them into(51) one could get

119889Δ119886

119889119905

=

12120587120596

[minus119882 sdotΔ119886minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862 sdot Δ119886]

(59)

Based on (53) one could eliminate 120579 and obtain the charac-teristic determinant as

det[

[

minus119882 minus 120572 minus119886119866

119866

119886

minus119882 minus 120572

]

]

= 0 (60)

Expanding the determinant one could obtain the character-istic equation as

(minus119882minus120572)2+119866

2= 0 (61)

Solving (61) one could obtain the characteristic values as

12057212 = minus119882plusmn119866119894 (62)

Obviously all real parts of the characteristic roots are neg-ative which means the steady-state solution of this kind ofsemiactive system is also unconditionally stable

33 Analytical Investigation for Relative Control System Thestrategy for semiactive relative control is shown as

119888 =

119888max (119909 minus 1199090) ( minus 0) le 0

119888min else(63)

After transformation of coordinates (63) will become

120585 =

120585max 119911 le 0

120585min else(64)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt

120587

2

120585min120587

2lt 120601 lt 120587

120585max 120587 lt 120601 lt

31205872

120585min31205872

lt 120601 lt 2120587

(65)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) for this semiactive control system

1198862 =

minus120587119886 (120585max + 120585min)

2120587

119886

1205792 =

2119886 (minus120585max + 120585min)

2120587

(66)

Combining (12) (15a) (15b) (21) (22) and (66) one couldobtain

119886

=

minus120587120596119886 (120585max + 120585min) minus 11988301205871205962 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579

=

119886 [120587 (12059620 minus 120596

2) + 2120596 (minus120585max + 120585min)] minus 1198830120587120596

2 cos 1205792120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(67)

Obviously we could easily obtain the equivalent dampingcoefficient

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+

119888max + 119888min2

(68)

From (68) we can find that the equivalent damping coeffi-cient 1198621015840 mainly depends on the fractional-order parameters119870 and 119901 associated with the original damping coefficient

Letting 119886 = 0 and 119886

120579 = 0 one could obtain the amplitudeand phase of the steady-state response

119886 =

12058711988301205962

radic1198802+ 119881

2

120579 = arctan 119880

119881

(69)

where

119880 = minus120582120587120596119901 sin(

119901120587

2)minus120587120596 (120585max + 120585min)

119881 = 120587 (12059620 minus120596

2) + 2120596 (minus120585max + 120585min)

+ 120582120587120596119901 cos(

119901120587

2)

(70)

8 Mathematical Problems in Engineering

The transmissibility for relative displacement in this caseis

1205833 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198802+ 119881

2 (71)

The absolute displacement of relative control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198833 cos (120596119905 minus 120574)

(72)

where

1198833 = radic(1198830 +

11988301205871205962 cos120595

radic119876

)

2

+

11988320120587

21205964sin2120595

119876

120574 = arctan1198830120587120596

2 sin120595

(1198830 + cos120595)radic119876

(73)

where

120595 =

119880

119881

119876 = 1198812+119880

2

(74)

The transmissibility for the absolute displacement is

1205893 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988331198830

(75)

Next we study the stability of steady-state solutionsLetting 119886 = 119886 + Δ119886 120579 = 120579 + Δ120579 and substituting them into(67) one could get

119889Δ119886

119889119905

=

12120587120596

[[minus120587120596 (120585max + 120585min) minus 120582120587120596119901 sin(

119901120587

2)]

sdot Δ119886 minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862

sdot Δ119886]

(76)

Based on (69) one could eliminate 120579 and obtain the charac-teristic determinant

det[

[

119880 minus 120572 minus119886119881

119880

119886

119881 minus 120572

]

]

= 0 (77)

0 20 40 60 80 1000

0005

001

0015

002

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 2 Comparison of analytical solution with numerical oneunder on-off control

Expanding the determinant one could obtain the character-istic equation

(119880minus120572)2+119881

2= 0 (78)

Solving (78) one could obtain the characteristic values

12057212 = 119880plusmn119881119894 (79)

Obviously 119880 lt 0 That is to say all real parts of the char-acteristic values are negative which means the steady-statesolution of this kind of semiactive system is stable uncondi-tionally

4 Numerical Simulation and Analysis

41 Comparison between the Analytical and Numerical Solu-tion In order to verify the precision of the analytical solu-tions we also present the numerical results The numericalscheme [29ndash34] is

119863119901[119909 (119905119897)] asymp ℎ

minus119901119897

sum

119895=0119862119901

119895119909 (119905119897minus119895) (80)

where 119905119897 = 119897ℎ is the time sample points ℎ is the sample stepand119862

119901

119895is the fractional binomial coefficient with the iterative

relationship as

119862119901

0 = 1

119862119901

119895= (1minus

1 + 119901

119895

)119862119901

119895minus1(81)

The selected basic system parameters are 119898 = 240 119896 =

16000 119888 = 10001198830 = 001 119901 = 05119870 = 1000 and 120575=051198830We take 400 seconds as numerical time in the process of

Mathematical Problems in Engineering 9

0 20 40 60 80 1000

0005

001

0015

002

0025

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 3 Comparison of analytical solution with numerical oneunder LRD control

the simulation Considering the maximum value of thenumerical results in the last 100 seconds as the steady-stateamplitude one could get the amplitude-frequency curves forthe numerical results which are denoted by the lines withcircles shown in Figures 2ndash4 Based on (37) (53) and (69)one could obtain the amplitude-frequency curves by theanalytical solutions denoted by the solid lines in Figures 2ndash4From the observation of the three figures one could concludethat the analytical solutions agree very well with the numeri-cal results and could present satisfactory precision Howeverthere are distinct errors between the approximate analyticaland numerical solution in some high-frequency range espe-cially the amplitude These phenomena called chatter insemiactive control system had also been found by Ahmadian[35] numerically and experimentally

42 Effects of 119870 on the Steady-State Amplitudes When thefractional coefficient 119870 is changed the different amplitude-frequency curves are shown in Figures 5ndash7 From the obser-vation of Figures 5ndash7 we could conclude that the largerthe fractional coefficient 119870 is the smaller the steady-stateamplitudes of the relative and absolute displacement of sys-tem mass are At the same time the resonant frequency alsobecomes larger This phenomenon indicates that both theequivalent damping coefficient and equivalent stiffness coef-ficient become larger in this procedure That is to say thefractional-order derivative 119870119863

119901(119909 minus 1199090) serves as damping

and spring simultaneously

43 Effects of 119901 on the Steady-State Amplitudes When thefractional coefficient 119901 is changed the results are shownin Figures 8ndash10 From the observation of Figures 8ndash10 wecould conclude that the larger the fractional coefficient 119901 isthe smaller the steady-state amplitudes of the relative andabsolute displacement of system mass are However it seems

Numerical solutionAnalytical solution

0 20 40 60 80 1000

0005

001

0015

002

0025

003

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 4 Comparison of analytical solution with numerical oneunder relative control

that there is little effect of119901 on the resonant frequency In factby differentiating119870

1015840 and 1198621015840 to 119901 one can get

1198891198621015840

119889119901

= 119896120596119901minus1

radic(119901120596minus1)2+ (

120587

2)

2sin(

119901120587

2+ 1205991)

1198891198701015840

119889119901

= minus 119896120596119901radic(119901120596

minus1)2+ (

120587

2)

2sin(

119901120587

2minus 1205992)

(82)

where

1205991 = arctan 120587120596

2119901

1205992 = arctan2119901120587120596

(83)

Because 0 lt 119901 lt 1 we can conclude that the derivative of1198621015840 with respect to 119901 will be larger than 0 1198621015840 will be mono-

tonically larger if 119901 becomes larger Accordingly the steady-state amplitude will be smaller in the process But the valueof 1015840 is not monotonical in this procedure1198701015840 could increaseor reduce with different 119901 which is affected by the systemparameters

5 Conclusion

In this paper three SDOF semiactive systems are researchedby using averaging method Aiming at these systems wepresent the concepts of equivalent damping coefficient andequivalent stiffness coefficient in order to account for theeffect of the fractional-order derivative on the control perfor-mance

Based on the above study one could find that fractional-order derivative plays an important role in SDOF semiactivesystem Therefore the results in this paper supply valuablereference for engineering practice In addition analyticalmethod in this paper could be adopted in two or multi-degrees-of-freedom system and it also offers a new idea tothe research field on vehicle suspension theory

10 Mathematical Problems in Engineering

0

05

1

15

2

25

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205831

100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 1

100 101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 5 Effect of119870 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205832

101 102

(a) Effect of119870 on relative displacement transmissibility

05

1

15

2

25

3

35

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 2

101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 6 Effect of 119870 on the steady-state amplitudes under LRD control

0

05

1

15

2

25

3

35

4

K = 0

K = 1000

K = 2000

K = 3000

1205833

120596 (rads)100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

2

3

4

5

120577 3

120596 (rads)100 101 102

K = 0

K = 1000

K = 2000

K = 3000

(b) Effect of119870 on absolute displacement transmissibility

Figure 7 Effect of119870 on the steady-state amplitudes under relative control

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Equations (13b) and (14b) can be rewritten as

1198862 =

12120587

int

2120587

0minus2119886120585sin2120601119889120601 (23a)

119886

1205792 =

12120587

int

2120587

0minus119886120585 sin 2120601119889120601 (23b)

The solutions of (23a) and (23b) depend on the specificsemiactive control methods

Accordingly (12) becomes

119886 = minus

1198830120596 sin 120579

2minus

120582119886120596119901minus1

2sin(

119901120587

2)

+

119886

2120587int

2120587

0minus2120585 sin2120601119889120601

119886

120579 =

minus1198861205962+ 119886120596

20 minus 1198830120596

2 cos 1205792120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

+

119886

2120587int

2120587

0minus120585 sin 2120601119889120601

(24)

which could be rewritten as

119886 = minus

1198830120596 sin 120579

2minus

119886

21198981198621015840

119886

120579 =

minus1198830120596 cos 1205792

minus

120596119886

2+

119886

2120587int

2120587

0minus

119888

2119898sin 2120601119889120601

+

119886

2119898120596

1198701015840

(25)

where

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+int

2120587

0

minus119888 sin2120601119889120601

120587

(26a)

1198701015840= 119870120596

119901 cos(119901120587

2)+ 119896 (26b)

1198621015840 and 119870

1015840 are defined as the equivalent damping coefficientand equivalent stiffness coefficient in semiactive systemsrespectively In the SDOF semiactive system the fractional-order derivative119870119863

119901(119909 minus1199090) serves as damping and stiffness

at the same timeWhen 119901 rarr 0119870119863119901(119909minus1199090)will be changed

into spring force whereas it will be almost the same as damp-ing force if 119901 rarr 1 When 119870 rarr 0 119870119863

119901(119909 minus 1199090) will have

little effect on control performance and the semiactive systemwill degrade into the traditional semiactive one

3 Cases for Three Kinds ofSemiactive Control Systems

31 Analytical Investigation for on-off Control System Thestrategy of semiactive on-off control is

119888 =

119888max ( minus 0) ge 0

119888min else(27)

Originally 119888min is selected as 119888min = 0 in [2] which may beinaccurate in real engineering Here we select it as 119888min =

001119888maxAfter transformation of coordinates (27) becomes

120585 =

120585max ( + 0) ge 0

120585min else(28)

Expanding the critical condition

( + 0) ge 0 (29)

one could get

minus120596119886 sin120601 [minus120596119886 sin120601minus1198830120596 sin (120596119905)]

= 119886119877 sin120601 sin (120601 minus 120573) ge 0(30)

where

120573 = arctan1198830 sin 120579

119886 + 1198830 cos 120579

119877 = 1205962radic(119886 + 1198830 cos 120579)

2+ 119883

20sin2120579

(31)

Equation (28) can be rewritten as

120585 =

120585min 0 lt 120601 lt 120573

120585max 120573 lt 120601 lt 120587

120585min 120587 lt 120601 lt 120587 + 120573

120585max 120587 + 120573 lt 120601 lt 2120587

(32)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) as

1198862 =

minus2119886119860 minus 119886119861

2120587

119886

1205792 =

119886119862 minus 119886119863

2120587

(33)

where

119860 = 120587120585max +120573 (minus120585max + 120585min)

119861 = (120585max minus 120585min) sin 2120573

119862 = 120585max minus 120585min

119863 = (120585max minus 120585min) cos 2120573

(34)

Combining (12) (15a) (15b) (21) (22) and (33) onecould obtain

119886 = minus

1198830120596 sin 120579

2minus

120582119886120596119901minus1

2sin(

119901120587

2)+

minus2119886119860 minus 119886119861

2120587

119886

120579 =

minus1198861205962+ 119886120596

20 minus 1198830120596

2 cos 1205792120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

+

119886119862 minus 119886119863

2120587

(35)

Mathematical Problems in Engineering 5

Obviously the equivalent damping coefficient in this casecould be written as

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+ 119888max +

120573

120587

(minus119888max + 119888min)

+

sin 21205732120587

(119888max minus 119888min)

(36)

From (36) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120573

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain the amplitude and phase

of steady-state response

119886 =

12058711988301205962

radic1198732+ 119871

2

120579 = arctan 119873

119871

(37)

where

119873 = minus 2120596119860minus120596119861minus120587120596119901120582 sin

119901120587

2

119871 = 120596119862minus120596119863+120587120596119901120582 cos

119901120587

2+120587120596

20 minus120587120596

2

(38)

The transmissibility for relative displacement is

1205831 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198732+ 119871

2 (39)

The absolute displacement of on-off control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198831 cos (120596119905 minus 120591)

(40)

where

1198831 = radic

11988320120587

21205964119873

2

(1198732+ 119871

2)2 + (1198830 +

11988301205871205962|119871|

1198732+ 119871

2 )

2

120591 = minus arctan 1205871205962119873

119871 (1205871205962+ (119871

2+ 119873

2) |119871|)

(41)

The displacement transmissibility is

1205891 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988311198830

(42)

Next we study the stability of steady-state solution Let-ting 119886 = 119886 + Δ119886 and 120579 = 120579 + Δ120579 and then substituting theminto (35) one could get

119889Δ119886

119889119905

=

12120587

[119873 sdot Δ119886minus1198830120587120596 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587

[

1205871198830120596 sin 120579

119886

sdot Δ120579 +

1205871198830120596 cos 1205791198862 sdot Δ119886]

(43)

Based on (37) one could eliminate 120579 and obtain the charac-teristic determinant

det[[

[

119873

120596

minus 120572

minus119886119871

120596

119871

119886120596

119873

120596

minus 120572

]

]

]

= 0 (44)

Expanding the determinant one could obtain the character-istic equation

(

119873

120596

minus120572)

2+

1198712

1205962 = 0 (45)

Solving (45) one could obtain characteristic values

12057212 =

119873

120596

plusmn

119871

120596

119894 (46)

Obviously119873 is less than 0 Accordingly all real parts of char-acteristic roots are negative which means the steady-statesolution of this kind of semiactive system is unconditionallystable

32 Analytical Investigation for LRD Control System Thestrategy of LRD control is

119888 =

119888max1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816ge 120575

119888min1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816lt 120575

(47)

where 119888min = 001119888max and 120575 is the space between the massand excitation point Here we select 120575=051198830

After transformation of coordinates (47) becomes

119888 =

119888max |119911| ge 120575

119888min |119911| lt 120575

(48)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt 120576

120585min 120576 lt 120601 lt 120587 minus 120576

120585max 120587 minus 120576 lt 120601 lt 120587 + 120576

120585min 120587 + 120576 lt 120601 lt 2120587 minus 120576

120585max 2120587 minus 120576 lt 120601 lt 2120587

(49)

where 120576 = arccos(051198830119886)Using the averagingmethod one could obtain the simpli-

fied forms of (13b) and (14b)

1198862 =

minus21198861198600 + 211988611986102120587120596

119886

1205792 = 0(50)

where 1198600 = 120596(2120576120585max + 120587120585min minus 2120576120585min) 1198610 = 120596(120585max minus

120585min) sin 2120576

6 Mathematical Problems in Engineering

Combining (12) (15a) (15b) (21) (22) and (50) onecould obtain

119886 =

minus21198861198600 + 21198861198610 minus 11988301205962120587 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579 =

minus119886120587 (1205962minus 120596

20) minus 1198830120596

2120587 cos 120579

2120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(51)

Obviously one could easily obtain the equivalent dampingcoefficient

1198621015840

= 119870120596119901minus1 sin(

119901120587

2)

+

(119888max minus 119888min) sin 2120576 minus (2120576119888max + 119888min120587 minus 2120576119888min)

120587

(52)

From (52) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120576

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain

119886 =

12058711988301205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2

120579 = arctanminus21198600 + 21198610 minus 120582120587120596

119901 sin (1199011205872)minus120587 (120596

2minus 120596

20) + 120582120587120596

119901 cos (1199011205872)

(53)

where 119886 and 120579 are the amplitude and phase of steady-stateresponse respectively

The transmissibility for relative displacement is

1205832 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2 (54)

The absolute displacement of LRD control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198832 cos (120596119905 minus 120594)

(55)

where

119883

22

=

11988320120587

21205964119882

2

[(21198600 minus 21198610)2+ 120587

2(120596

4minus 21205962

12059620 + 120596

40 + 120596

21199011205822) minus 21205872

120596119901(120596

2minus 120596

20) 120582 cos (1199011205872) + 4 (1198600 minus 1198610) 120587120596

119901120582 sin (1199011205872)]

2

+(1198830 +1198830120587120596

2119866

1198662+ 119882

2)

2

120594 = arctan 12058721205962119882

11986612058721205962+ 120587119866

2+ 120587119882

2

(56)

Mathematical Problems in Engineering 7

where

119866 = 120587(minus1205962+120596

20 +120596119901120582 cos

119901120587

2)

119882 = 21198600 minus 21198610 +120587120596119901120582 sin

119901120587

2

(57)

The transmissibility for the absolute displacement of LRDcontrol system is

1205892 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988321198830

(58)

Next we study the stability of the steady-state solutionsLetting 119886 = 119886+Δ119886 and 120579 = 120579+Δ120579 and substituting them into(51) one could get

119889Δ119886

119889119905

=

12120587120596

[minus119882 sdotΔ119886minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862 sdot Δ119886]

(59)

Based on (53) one could eliminate 120579 and obtain the charac-teristic determinant as

det[

[

minus119882 minus 120572 minus119886119866

119866

119886

minus119882 minus 120572

]

]

= 0 (60)

Expanding the determinant one could obtain the character-istic equation as

(minus119882minus120572)2+119866

2= 0 (61)

Solving (61) one could obtain the characteristic values as

12057212 = minus119882plusmn119866119894 (62)

Obviously all real parts of the characteristic roots are neg-ative which means the steady-state solution of this kind ofsemiactive system is also unconditionally stable

33 Analytical Investigation for Relative Control System Thestrategy for semiactive relative control is shown as

119888 =

119888max (119909 minus 1199090) ( minus 0) le 0

119888min else(63)

After transformation of coordinates (63) will become

120585 =

120585max 119911 le 0

120585min else(64)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt

120587

2

120585min120587

2lt 120601 lt 120587

120585max 120587 lt 120601 lt

31205872

120585min31205872

lt 120601 lt 2120587

(65)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) for this semiactive control system

1198862 =

minus120587119886 (120585max + 120585min)

2120587

119886

1205792 =

2119886 (minus120585max + 120585min)

2120587

(66)

Combining (12) (15a) (15b) (21) (22) and (66) one couldobtain

119886

=

minus120587120596119886 (120585max + 120585min) minus 11988301205871205962 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579

=

119886 [120587 (12059620 minus 120596

2) + 2120596 (minus120585max + 120585min)] minus 1198830120587120596

2 cos 1205792120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(67)

Obviously we could easily obtain the equivalent dampingcoefficient

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+

119888max + 119888min2

(68)

From (68) we can find that the equivalent damping coeffi-cient 1198621015840 mainly depends on the fractional-order parameters119870 and 119901 associated with the original damping coefficient

Letting 119886 = 0 and 119886

120579 = 0 one could obtain the amplitudeand phase of the steady-state response

119886 =

12058711988301205962

radic1198802+ 119881

2

120579 = arctan 119880

119881

(69)

where

119880 = minus120582120587120596119901 sin(

119901120587

2)minus120587120596 (120585max + 120585min)

119881 = 120587 (12059620 minus120596

2) + 2120596 (minus120585max + 120585min)

+ 120582120587120596119901 cos(

119901120587

2)

(70)

8 Mathematical Problems in Engineering

The transmissibility for relative displacement in this caseis

1205833 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198802+ 119881

2 (71)

The absolute displacement of relative control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198833 cos (120596119905 minus 120574)

(72)

where

1198833 = radic(1198830 +

11988301205871205962 cos120595

radic119876

)

2

+

11988320120587

21205964sin2120595

119876

120574 = arctan1198830120587120596

2 sin120595

(1198830 + cos120595)radic119876

(73)

where

120595 =

119880

119881

119876 = 1198812+119880

2

(74)

The transmissibility for the absolute displacement is

1205893 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988331198830

(75)

Next we study the stability of steady-state solutionsLetting 119886 = 119886 + Δ119886 120579 = 120579 + Δ120579 and substituting them into(67) one could get

119889Δ119886

119889119905

=

12120587120596

[[minus120587120596 (120585max + 120585min) minus 120582120587120596119901 sin(

119901120587

2)]

sdot Δ119886 minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862

sdot Δ119886]

(76)

Based on (69) one could eliminate 120579 and obtain the charac-teristic determinant

det[

[

119880 minus 120572 minus119886119881

119880

119886

119881 minus 120572

]

]

= 0 (77)

0 20 40 60 80 1000

0005

001

0015

002

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 2 Comparison of analytical solution with numerical oneunder on-off control

Expanding the determinant one could obtain the character-istic equation

(119880minus120572)2+119881

2= 0 (78)

Solving (78) one could obtain the characteristic values

12057212 = 119880plusmn119881119894 (79)

Obviously 119880 lt 0 That is to say all real parts of the char-acteristic values are negative which means the steady-statesolution of this kind of semiactive system is stable uncondi-tionally

4 Numerical Simulation and Analysis

41 Comparison between the Analytical and Numerical Solu-tion In order to verify the precision of the analytical solu-tions we also present the numerical results The numericalscheme [29ndash34] is

119863119901[119909 (119905119897)] asymp ℎ

minus119901119897

sum

119895=0119862119901

119895119909 (119905119897minus119895) (80)

where 119905119897 = 119897ℎ is the time sample points ℎ is the sample stepand119862

119901

119895is the fractional binomial coefficient with the iterative

relationship as

119862119901

0 = 1

119862119901

119895= (1minus

1 + 119901

119895

)119862119901

119895minus1(81)

The selected basic system parameters are 119898 = 240 119896 =

16000 119888 = 10001198830 = 001 119901 = 05119870 = 1000 and 120575=051198830We take 400 seconds as numerical time in the process of

Mathematical Problems in Engineering 9

0 20 40 60 80 1000

0005

001

0015

002

0025

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 3 Comparison of analytical solution with numerical oneunder LRD control

the simulation Considering the maximum value of thenumerical results in the last 100 seconds as the steady-stateamplitude one could get the amplitude-frequency curves forthe numerical results which are denoted by the lines withcircles shown in Figures 2ndash4 Based on (37) (53) and (69)one could obtain the amplitude-frequency curves by theanalytical solutions denoted by the solid lines in Figures 2ndash4From the observation of the three figures one could concludethat the analytical solutions agree very well with the numeri-cal results and could present satisfactory precision Howeverthere are distinct errors between the approximate analyticaland numerical solution in some high-frequency range espe-cially the amplitude These phenomena called chatter insemiactive control system had also been found by Ahmadian[35] numerically and experimentally

42 Effects of 119870 on the Steady-State Amplitudes When thefractional coefficient 119870 is changed the different amplitude-frequency curves are shown in Figures 5ndash7 From the obser-vation of Figures 5ndash7 we could conclude that the largerthe fractional coefficient 119870 is the smaller the steady-stateamplitudes of the relative and absolute displacement of sys-tem mass are At the same time the resonant frequency alsobecomes larger This phenomenon indicates that both theequivalent damping coefficient and equivalent stiffness coef-ficient become larger in this procedure That is to say thefractional-order derivative 119870119863

119901(119909 minus 1199090) serves as damping

and spring simultaneously

43 Effects of 119901 on the Steady-State Amplitudes When thefractional coefficient 119901 is changed the results are shownin Figures 8ndash10 From the observation of Figures 8ndash10 wecould conclude that the larger the fractional coefficient 119901 isthe smaller the steady-state amplitudes of the relative andabsolute displacement of system mass are However it seems

Numerical solutionAnalytical solution

0 20 40 60 80 1000

0005

001

0015

002

0025

003

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 4 Comparison of analytical solution with numerical oneunder relative control

that there is little effect of119901 on the resonant frequency In factby differentiating119870

1015840 and 1198621015840 to 119901 one can get

1198891198621015840

119889119901

= 119896120596119901minus1

radic(119901120596minus1)2+ (

120587

2)

2sin(

119901120587

2+ 1205991)

1198891198701015840

119889119901

= minus 119896120596119901radic(119901120596

minus1)2+ (

120587

2)

2sin(

119901120587

2minus 1205992)

(82)

where

1205991 = arctan 120587120596

2119901

1205992 = arctan2119901120587120596

(83)

Because 0 lt 119901 lt 1 we can conclude that the derivative of1198621015840 with respect to 119901 will be larger than 0 1198621015840 will be mono-

tonically larger if 119901 becomes larger Accordingly the steady-state amplitude will be smaller in the process But the valueof 1015840 is not monotonical in this procedure1198701015840 could increaseor reduce with different 119901 which is affected by the systemparameters

5 Conclusion

In this paper three SDOF semiactive systems are researchedby using averaging method Aiming at these systems wepresent the concepts of equivalent damping coefficient andequivalent stiffness coefficient in order to account for theeffect of the fractional-order derivative on the control perfor-mance

Based on the above study one could find that fractional-order derivative plays an important role in SDOF semiactivesystem Therefore the results in this paper supply valuablereference for engineering practice In addition analyticalmethod in this paper could be adopted in two or multi-degrees-of-freedom system and it also offers a new idea tothe research field on vehicle suspension theory

10 Mathematical Problems in Engineering

0

05

1

15

2

25

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205831

100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 1

100 101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 5 Effect of119870 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205832

101 102

(a) Effect of119870 on relative displacement transmissibility

05

1

15

2

25

3

35

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 2

101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 6 Effect of 119870 on the steady-state amplitudes under LRD control

0

05

1

15

2

25

3

35

4

K = 0

K = 1000

K = 2000

K = 3000

1205833

120596 (rads)100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

2

3

4

5

120577 3

120596 (rads)100 101 102

K = 0

K = 1000

K = 2000

K = 3000

(b) Effect of119870 on absolute displacement transmissibility

Figure 7 Effect of119870 on the steady-state amplitudes under relative control

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Obviously the equivalent damping coefficient in this casecould be written as

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+ 119888max +

120573

120587

(minus119888max + 119888min)

+

sin 21205732120587

(119888max minus 119888min)

(36)

From (36) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120573

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain the amplitude and phase

of steady-state response

119886 =

12058711988301205962

radic1198732+ 119871

2

120579 = arctan 119873

119871

(37)

where

119873 = minus 2120596119860minus120596119861minus120587120596119901120582 sin

119901120587

2

119871 = 120596119862minus120596119863+120587120596119901120582 cos

119901120587

2+120587120596

20 minus120587120596

2

(38)

The transmissibility for relative displacement is

1205831 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198732+ 119871

2 (39)

The absolute displacement of on-off control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198831 cos (120596119905 minus 120591)

(40)

where

1198831 = radic

11988320120587

21205964119873

2

(1198732+ 119871

2)2 + (1198830 +

11988301205871205962|119871|

1198732+ 119871

2 )

2

120591 = minus arctan 1205871205962119873

119871 (1205871205962+ (119871

2+ 119873

2) |119871|)

(41)

The displacement transmissibility is

1205891 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988311198830

(42)

Next we study the stability of steady-state solution Let-ting 119886 = 119886 + Δ119886 and 120579 = 120579 + Δ120579 and then substituting theminto (35) one could get

119889Δ119886

119889119905

=

12120587

[119873 sdot Δ119886minus1198830120587120596 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587

[

1205871198830120596 sin 120579

119886

sdot Δ120579 +

1205871198830120596 cos 1205791198862 sdot Δ119886]

(43)

Based on (37) one could eliminate 120579 and obtain the charac-teristic determinant

det[[

[

119873

120596

minus 120572

minus119886119871

120596

119871

119886120596

119873

120596

minus 120572

]

]

]

= 0 (44)

Expanding the determinant one could obtain the character-istic equation

(

119873

120596

minus120572)

2+

1198712

1205962 = 0 (45)

Solving (45) one could obtain characteristic values

12057212 =

119873

120596

plusmn

119871

120596

119894 (46)

Obviously119873 is less than 0 Accordingly all real parts of char-acteristic roots are negative which means the steady-statesolution of this kind of semiactive system is unconditionallystable

32 Analytical Investigation for LRD Control System Thestrategy of LRD control is

119888 =

119888max1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816ge 120575

119888min1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816lt 120575

(47)

where 119888min = 001119888max and 120575 is the space between the massand excitation point Here we select 120575=051198830

After transformation of coordinates (47) becomes

119888 =

119888max |119911| ge 120575

119888min |119911| lt 120575

(48)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt 120576

120585min 120576 lt 120601 lt 120587 minus 120576

120585max 120587 minus 120576 lt 120601 lt 120587 + 120576

120585min 120587 + 120576 lt 120601 lt 2120587 minus 120576

120585max 2120587 minus 120576 lt 120601 lt 2120587

(49)

where 120576 = arccos(051198830119886)Using the averagingmethod one could obtain the simpli-

fied forms of (13b) and (14b)

1198862 =

minus21198861198600 + 211988611986102120587120596

119886

1205792 = 0(50)

where 1198600 = 120596(2120576120585max + 120587120585min minus 2120576120585min) 1198610 = 120596(120585max minus

120585min) sin 2120576

6 Mathematical Problems in Engineering

Combining (12) (15a) (15b) (21) (22) and (50) onecould obtain

119886 =

minus21198861198600 + 21198861198610 minus 11988301205962120587 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579 =

minus119886120587 (1205962minus 120596

20) minus 1198830120596

2120587 cos 120579

2120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(51)

Obviously one could easily obtain the equivalent dampingcoefficient

1198621015840

= 119870120596119901minus1 sin(

119901120587

2)

+

(119888max minus 119888min) sin 2120576 minus (2120576119888max + 119888min120587 minus 2120576119888min)

120587

(52)

From (52) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120576

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain

119886 =

12058711988301205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2

120579 = arctanminus21198600 + 21198610 minus 120582120587120596

119901 sin (1199011205872)minus120587 (120596

2minus 120596

20) + 120582120587120596

119901 cos (1199011205872)

(53)

where 119886 and 120579 are the amplitude and phase of steady-stateresponse respectively

The transmissibility for relative displacement is

1205832 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2 (54)

The absolute displacement of LRD control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198832 cos (120596119905 minus 120594)

(55)

where

119883

22

=

11988320120587

21205964119882

2

[(21198600 minus 21198610)2+ 120587

2(120596

4minus 21205962

12059620 + 120596

40 + 120596

21199011205822) minus 21205872

120596119901(120596

2minus 120596

20) 120582 cos (1199011205872) + 4 (1198600 minus 1198610) 120587120596

119901120582 sin (1199011205872)]

2

+(1198830 +1198830120587120596

2119866

1198662+ 119882

2)

2

120594 = arctan 12058721205962119882

11986612058721205962+ 120587119866

2+ 120587119882

2

(56)

Mathematical Problems in Engineering 7

where

119866 = 120587(minus1205962+120596

20 +120596119901120582 cos

119901120587

2)

119882 = 21198600 minus 21198610 +120587120596119901120582 sin

119901120587

2

(57)

The transmissibility for the absolute displacement of LRDcontrol system is

1205892 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988321198830

(58)

Next we study the stability of the steady-state solutionsLetting 119886 = 119886+Δ119886 and 120579 = 120579+Δ120579 and substituting them into(51) one could get

119889Δ119886

119889119905

=

12120587120596

[minus119882 sdotΔ119886minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862 sdot Δ119886]

(59)

Based on (53) one could eliminate 120579 and obtain the charac-teristic determinant as

det[

[

minus119882 minus 120572 minus119886119866

119866

119886

minus119882 minus 120572

]

]

= 0 (60)

Expanding the determinant one could obtain the character-istic equation as

(minus119882minus120572)2+119866

2= 0 (61)

Solving (61) one could obtain the characteristic values as

12057212 = minus119882plusmn119866119894 (62)

Obviously all real parts of the characteristic roots are neg-ative which means the steady-state solution of this kind ofsemiactive system is also unconditionally stable

33 Analytical Investigation for Relative Control System Thestrategy for semiactive relative control is shown as

119888 =

119888max (119909 minus 1199090) ( minus 0) le 0

119888min else(63)

After transformation of coordinates (63) will become

120585 =

120585max 119911 le 0

120585min else(64)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt

120587

2

120585min120587

2lt 120601 lt 120587

120585max 120587 lt 120601 lt

31205872

120585min31205872

lt 120601 lt 2120587

(65)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) for this semiactive control system

1198862 =

minus120587119886 (120585max + 120585min)

2120587

119886

1205792 =

2119886 (minus120585max + 120585min)

2120587

(66)

Combining (12) (15a) (15b) (21) (22) and (66) one couldobtain

119886

=

minus120587120596119886 (120585max + 120585min) minus 11988301205871205962 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579

=

119886 [120587 (12059620 minus 120596

2) + 2120596 (minus120585max + 120585min)] minus 1198830120587120596

2 cos 1205792120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(67)

Obviously we could easily obtain the equivalent dampingcoefficient

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+

119888max + 119888min2

(68)

From (68) we can find that the equivalent damping coeffi-cient 1198621015840 mainly depends on the fractional-order parameters119870 and 119901 associated with the original damping coefficient

Letting 119886 = 0 and 119886

120579 = 0 one could obtain the amplitudeand phase of the steady-state response

119886 =

12058711988301205962

radic1198802+ 119881

2

120579 = arctan 119880

119881

(69)

where

119880 = minus120582120587120596119901 sin(

119901120587

2)minus120587120596 (120585max + 120585min)

119881 = 120587 (12059620 minus120596

2) + 2120596 (minus120585max + 120585min)

+ 120582120587120596119901 cos(

119901120587

2)

(70)

8 Mathematical Problems in Engineering

The transmissibility for relative displacement in this caseis

1205833 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198802+ 119881

2 (71)

The absolute displacement of relative control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198833 cos (120596119905 minus 120574)

(72)

where

1198833 = radic(1198830 +

11988301205871205962 cos120595

radic119876

)

2

+

11988320120587

21205964sin2120595

119876

120574 = arctan1198830120587120596

2 sin120595

(1198830 + cos120595)radic119876

(73)

where

120595 =

119880

119881

119876 = 1198812+119880

2

(74)

The transmissibility for the absolute displacement is

1205893 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988331198830

(75)

Next we study the stability of steady-state solutionsLetting 119886 = 119886 + Δ119886 120579 = 120579 + Δ120579 and substituting them into(67) one could get

119889Δ119886

119889119905

=

12120587120596

[[minus120587120596 (120585max + 120585min) minus 120582120587120596119901 sin(

119901120587

2)]

sdot Δ119886 minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862

sdot Δ119886]

(76)

Based on (69) one could eliminate 120579 and obtain the charac-teristic determinant

det[

[

119880 minus 120572 minus119886119881

119880

119886

119881 minus 120572

]

]

= 0 (77)

0 20 40 60 80 1000

0005

001

0015

002

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 2 Comparison of analytical solution with numerical oneunder on-off control

Expanding the determinant one could obtain the character-istic equation

(119880minus120572)2+119881

2= 0 (78)

Solving (78) one could obtain the characteristic values

12057212 = 119880plusmn119881119894 (79)

Obviously 119880 lt 0 That is to say all real parts of the char-acteristic values are negative which means the steady-statesolution of this kind of semiactive system is stable uncondi-tionally

4 Numerical Simulation and Analysis

41 Comparison between the Analytical and Numerical Solu-tion In order to verify the precision of the analytical solu-tions we also present the numerical results The numericalscheme [29ndash34] is

119863119901[119909 (119905119897)] asymp ℎ

minus119901119897

sum

119895=0119862119901

119895119909 (119905119897minus119895) (80)

where 119905119897 = 119897ℎ is the time sample points ℎ is the sample stepand119862

119901

119895is the fractional binomial coefficient with the iterative

relationship as

119862119901

0 = 1

119862119901

119895= (1minus

1 + 119901

119895

)119862119901

119895minus1(81)

The selected basic system parameters are 119898 = 240 119896 =

16000 119888 = 10001198830 = 001 119901 = 05119870 = 1000 and 120575=051198830We take 400 seconds as numerical time in the process of

Mathematical Problems in Engineering 9

0 20 40 60 80 1000

0005

001

0015

002

0025

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 3 Comparison of analytical solution with numerical oneunder LRD control

the simulation Considering the maximum value of thenumerical results in the last 100 seconds as the steady-stateamplitude one could get the amplitude-frequency curves forthe numerical results which are denoted by the lines withcircles shown in Figures 2ndash4 Based on (37) (53) and (69)one could obtain the amplitude-frequency curves by theanalytical solutions denoted by the solid lines in Figures 2ndash4From the observation of the three figures one could concludethat the analytical solutions agree very well with the numeri-cal results and could present satisfactory precision Howeverthere are distinct errors between the approximate analyticaland numerical solution in some high-frequency range espe-cially the amplitude These phenomena called chatter insemiactive control system had also been found by Ahmadian[35] numerically and experimentally

42 Effects of 119870 on the Steady-State Amplitudes When thefractional coefficient 119870 is changed the different amplitude-frequency curves are shown in Figures 5ndash7 From the obser-vation of Figures 5ndash7 we could conclude that the largerthe fractional coefficient 119870 is the smaller the steady-stateamplitudes of the relative and absolute displacement of sys-tem mass are At the same time the resonant frequency alsobecomes larger This phenomenon indicates that both theequivalent damping coefficient and equivalent stiffness coef-ficient become larger in this procedure That is to say thefractional-order derivative 119870119863

119901(119909 minus 1199090) serves as damping

and spring simultaneously

43 Effects of 119901 on the Steady-State Amplitudes When thefractional coefficient 119901 is changed the results are shownin Figures 8ndash10 From the observation of Figures 8ndash10 wecould conclude that the larger the fractional coefficient 119901 isthe smaller the steady-state amplitudes of the relative andabsolute displacement of system mass are However it seems

Numerical solutionAnalytical solution

0 20 40 60 80 1000

0005

001

0015

002

0025

003

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 4 Comparison of analytical solution with numerical oneunder relative control

that there is little effect of119901 on the resonant frequency In factby differentiating119870

1015840 and 1198621015840 to 119901 one can get

1198891198621015840

119889119901

= 119896120596119901minus1

radic(119901120596minus1)2+ (

120587

2)

2sin(

119901120587

2+ 1205991)

1198891198701015840

119889119901

= minus 119896120596119901radic(119901120596

minus1)2+ (

120587

2)

2sin(

119901120587

2minus 1205992)

(82)

where

1205991 = arctan 120587120596

2119901

1205992 = arctan2119901120587120596

(83)

Because 0 lt 119901 lt 1 we can conclude that the derivative of1198621015840 with respect to 119901 will be larger than 0 1198621015840 will be mono-

tonically larger if 119901 becomes larger Accordingly the steady-state amplitude will be smaller in the process But the valueof 1015840 is not monotonical in this procedure1198701015840 could increaseor reduce with different 119901 which is affected by the systemparameters

5 Conclusion

In this paper three SDOF semiactive systems are researchedby using averaging method Aiming at these systems wepresent the concepts of equivalent damping coefficient andequivalent stiffness coefficient in order to account for theeffect of the fractional-order derivative on the control perfor-mance

Based on the above study one could find that fractional-order derivative plays an important role in SDOF semiactivesystem Therefore the results in this paper supply valuablereference for engineering practice In addition analyticalmethod in this paper could be adopted in two or multi-degrees-of-freedom system and it also offers a new idea tothe research field on vehicle suspension theory

10 Mathematical Problems in Engineering

0

05

1

15

2

25

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205831

100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 1

100 101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 5 Effect of119870 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205832

101 102

(a) Effect of119870 on relative displacement transmissibility

05

1

15

2

25

3

35

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 2

101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 6 Effect of 119870 on the steady-state amplitudes under LRD control

0

05

1

15

2

25

3

35

4

K = 0

K = 1000

K = 2000

K = 3000

1205833

120596 (rads)100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

2

3

4

5

120577 3

120596 (rads)100 101 102

K = 0

K = 1000

K = 2000

K = 3000

(b) Effect of119870 on absolute displacement transmissibility

Figure 7 Effect of119870 on the steady-state amplitudes under relative control

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Combining (12) (15a) (15b) (21) (22) and (50) onecould obtain

119886 =

minus21198861198600 + 21198861198610 minus 11988301205962120587 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579 =

minus119886120587 (1205962minus 120596

20) minus 1198830120596

2120587 cos 120579

2120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(51)

Obviously one could easily obtain the equivalent dampingcoefficient

1198621015840

= 119870120596119901minus1 sin(

119901120587

2)

+

(119888max minus 119888min) sin 2120576 minus (2120576119888max + 119888min120587 minus 2120576119888min)

120587

(52)

From (52) it could be found that the equivalent dampingcoefficient1198621015840mainly depends on the fractional-order param-eters 119870 and 119901 associated with the critical angle 120576

Now we study the steady-state solution which is moreimportant and meaningful in vibration engineering Letting119886 = 0 and 119886

120579 = 0 one could obtain

119886 =

12058711988301205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2

120579 = arctanminus21198600 + 21198610 minus 120582120587120596

119901 sin (1199011205872)minus120587 (120596

2minus 120596

20) + 120582120587120596

119901 cos (1199011205872)

(53)

where 119886 and 120579 are the amplitude and phase of steady-stateresponse respectively

The transmissibility for relative displacement is

1205832 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic[minus21198600 + 21198610 minus 120582120587120596119901 sin (1199011205872)]2 + [minus120587 (120596

2minus 1205960

2) + 120582120587120596

119901 cos (1199011205872)]2 (54)

The absolute displacement of LRD control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198832 cos (120596119905 minus 120594)

(55)

where

119883

22

=

11988320120587

21205964119882

2

[(21198600 minus 21198610)2+ 120587

2(120596

4minus 21205962

12059620 + 120596

40 + 120596

21199011205822) minus 21205872

120596119901(120596

2minus 120596

20) 120582 cos (1199011205872) + 4 (1198600 minus 1198610) 120587120596

119901120582 sin (1199011205872)]

2

+(1198830 +1198830120587120596

2119866

1198662+ 119882

2)

2

120594 = arctan 12058721205962119882

11986612058721205962+ 120587119866

2+ 120587119882

2

(56)

Mathematical Problems in Engineering 7

where

119866 = 120587(minus1205962+120596

20 +120596119901120582 cos

119901120587

2)

119882 = 21198600 minus 21198610 +120587120596119901120582 sin

119901120587

2

(57)

The transmissibility for the absolute displacement of LRDcontrol system is

1205892 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988321198830

(58)

Next we study the stability of the steady-state solutionsLetting 119886 = 119886+Δ119886 and 120579 = 120579+Δ120579 and substituting them into(51) one could get

119889Δ119886

119889119905

=

12120587120596

[minus119882 sdotΔ119886minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862 sdot Δ119886]

(59)

Based on (53) one could eliminate 120579 and obtain the charac-teristic determinant as

det[

[

minus119882 minus 120572 minus119886119866

119866

119886

minus119882 minus 120572

]

]

= 0 (60)

Expanding the determinant one could obtain the character-istic equation as

(minus119882minus120572)2+119866

2= 0 (61)

Solving (61) one could obtain the characteristic values as

12057212 = minus119882plusmn119866119894 (62)

Obviously all real parts of the characteristic roots are neg-ative which means the steady-state solution of this kind ofsemiactive system is also unconditionally stable

33 Analytical Investigation for Relative Control System Thestrategy for semiactive relative control is shown as

119888 =

119888max (119909 minus 1199090) ( minus 0) le 0

119888min else(63)

After transformation of coordinates (63) will become

120585 =

120585max 119911 le 0

120585min else(64)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt

120587

2

120585min120587

2lt 120601 lt 120587

120585max 120587 lt 120601 lt

31205872

120585min31205872

lt 120601 lt 2120587

(65)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) for this semiactive control system

1198862 =

minus120587119886 (120585max + 120585min)

2120587

119886

1205792 =

2119886 (minus120585max + 120585min)

2120587

(66)

Combining (12) (15a) (15b) (21) (22) and (66) one couldobtain

119886

=

minus120587120596119886 (120585max + 120585min) minus 11988301205871205962 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579

=

119886 [120587 (12059620 minus 120596

2) + 2120596 (minus120585max + 120585min)] minus 1198830120587120596

2 cos 1205792120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(67)

Obviously we could easily obtain the equivalent dampingcoefficient

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+

119888max + 119888min2

(68)

From (68) we can find that the equivalent damping coeffi-cient 1198621015840 mainly depends on the fractional-order parameters119870 and 119901 associated with the original damping coefficient

Letting 119886 = 0 and 119886

120579 = 0 one could obtain the amplitudeand phase of the steady-state response

119886 =

12058711988301205962

radic1198802+ 119881

2

120579 = arctan 119880

119881

(69)

where

119880 = minus120582120587120596119901 sin(

119901120587

2)minus120587120596 (120585max + 120585min)

119881 = 120587 (12059620 minus120596

2) + 2120596 (minus120585max + 120585min)

+ 120582120587120596119901 cos(

119901120587

2)

(70)

8 Mathematical Problems in Engineering

The transmissibility for relative displacement in this caseis

1205833 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198802+ 119881

2 (71)

The absolute displacement of relative control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198833 cos (120596119905 minus 120574)

(72)

where

1198833 = radic(1198830 +

11988301205871205962 cos120595

radic119876

)

2

+

11988320120587

21205964sin2120595

119876

120574 = arctan1198830120587120596

2 sin120595

(1198830 + cos120595)radic119876

(73)

where

120595 =

119880

119881

119876 = 1198812+119880

2

(74)

The transmissibility for the absolute displacement is

1205893 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988331198830

(75)

Next we study the stability of steady-state solutionsLetting 119886 = 119886 + Δ119886 120579 = 120579 + Δ120579 and substituting them into(67) one could get

119889Δ119886

119889119905

=

12120587120596

[[minus120587120596 (120585max + 120585min) minus 120582120587120596119901 sin(

119901120587

2)]

sdot Δ119886 minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862

sdot Δ119886]

(76)

Based on (69) one could eliminate 120579 and obtain the charac-teristic determinant

det[

[

119880 minus 120572 minus119886119881

119880

119886

119881 minus 120572

]

]

= 0 (77)

0 20 40 60 80 1000

0005

001

0015

002

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 2 Comparison of analytical solution with numerical oneunder on-off control

Expanding the determinant one could obtain the character-istic equation

(119880minus120572)2+119881

2= 0 (78)

Solving (78) one could obtain the characteristic values

12057212 = 119880plusmn119881119894 (79)

Obviously 119880 lt 0 That is to say all real parts of the char-acteristic values are negative which means the steady-statesolution of this kind of semiactive system is stable uncondi-tionally

4 Numerical Simulation and Analysis

41 Comparison between the Analytical and Numerical Solu-tion In order to verify the precision of the analytical solu-tions we also present the numerical results The numericalscheme [29ndash34] is

119863119901[119909 (119905119897)] asymp ℎ

minus119901119897

sum

119895=0119862119901

119895119909 (119905119897minus119895) (80)

where 119905119897 = 119897ℎ is the time sample points ℎ is the sample stepand119862

119901

119895is the fractional binomial coefficient with the iterative

relationship as

119862119901

0 = 1

119862119901

119895= (1minus

1 + 119901

119895

)119862119901

119895minus1(81)

The selected basic system parameters are 119898 = 240 119896 =

16000 119888 = 10001198830 = 001 119901 = 05119870 = 1000 and 120575=051198830We take 400 seconds as numerical time in the process of

Mathematical Problems in Engineering 9

0 20 40 60 80 1000

0005

001

0015

002

0025

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 3 Comparison of analytical solution with numerical oneunder LRD control

the simulation Considering the maximum value of thenumerical results in the last 100 seconds as the steady-stateamplitude one could get the amplitude-frequency curves forthe numerical results which are denoted by the lines withcircles shown in Figures 2ndash4 Based on (37) (53) and (69)one could obtain the amplitude-frequency curves by theanalytical solutions denoted by the solid lines in Figures 2ndash4From the observation of the three figures one could concludethat the analytical solutions agree very well with the numeri-cal results and could present satisfactory precision Howeverthere are distinct errors between the approximate analyticaland numerical solution in some high-frequency range espe-cially the amplitude These phenomena called chatter insemiactive control system had also been found by Ahmadian[35] numerically and experimentally

42 Effects of 119870 on the Steady-State Amplitudes When thefractional coefficient 119870 is changed the different amplitude-frequency curves are shown in Figures 5ndash7 From the obser-vation of Figures 5ndash7 we could conclude that the largerthe fractional coefficient 119870 is the smaller the steady-stateamplitudes of the relative and absolute displacement of sys-tem mass are At the same time the resonant frequency alsobecomes larger This phenomenon indicates that both theequivalent damping coefficient and equivalent stiffness coef-ficient become larger in this procedure That is to say thefractional-order derivative 119870119863

119901(119909 minus 1199090) serves as damping

and spring simultaneously

43 Effects of 119901 on the Steady-State Amplitudes When thefractional coefficient 119901 is changed the results are shownin Figures 8ndash10 From the observation of Figures 8ndash10 wecould conclude that the larger the fractional coefficient 119901 isthe smaller the steady-state amplitudes of the relative andabsolute displacement of system mass are However it seems

Numerical solutionAnalytical solution

0 20 40 60 80 1000

0005

001

0015

002

0025

003

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 4 Comparison of analytical solution with numerical oneunder relative control

that there is little effect of119901 on the resonant frequency In factby differentiating119870

1015840 and 1198621015840 to 119901 one can get

1198891198621015840

119889119901

= 119896120596119901minus1

radic(119901120596minus1)2+ (

120587

2)

2sin(

119901120587

2+ 1205991)

1198891198701015840

119889119901

= minus 119896120596119901radic(119901120596

minus1)2+ (

120587

2)

2sin(

119901120587

2minus 1205992)

(82)

where

1205991 = arctan 120587120596

2119901

1205992 = arctan2119901120587120596

(83)

Because 0 lt 119901 lt 1 we can conclude that the derivative of1198621015840 with respect to 119901 will be larger than 0 1198621015840 will be mono-

tonically larger if 119901 becomes larger Accordingly the steady-state amplitude will be smaller in the process But the valueof 1015840 is not monotonical in this procedure1198701015840 could increaseor reduce with different 119901 which is affected by the systemparameters

5 Conclusion

In this paper three SDOF semiactive systems are researchedby using averaging method Aiming at these systems wepresent the concepts of equivalent damping coefficient andequivalent stiffness coefficient in order to account for theeffect of the fractional-order derivative on the control perfor-mance

Based on the above study one could find that fractional-order derivative plays an important role in SDOF semiactivesystem Therefore the results in this paper supply valuablereference for engineering practice In addition analyticalmethod in this paper could be adopted in two or multi-degrees-of-freedom system and it also offers a new idea tothe research field on vehicle suspension theory

10 Mathematical Problems in Engineering

0

05

1

15

2

25

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205831

100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 1

100 101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 5 Effect of119870 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205832

101 102

(a) Effect of119870 on relative displacement transmissibility

05

1

15

2

25

3

35

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 2

101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 6 Effect of 119870 on the steady-state amplitudes under LRD control

0

05

1

15

2

25

3

35

4

K = 0

K = 1000

K = 2000

K = 3000

1205833

120596 (rads)100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

2

3

4

5

120577 3

120596 (rads)100 101 102

K = 0

K = 1000

K = 2000

K = 3000

(b) Effect of119870 on absolute displacement transmissibility

Figure 7 Effect of119870 on the steady-state amplitudes under relative control

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

where

119866 = 120587(minus1205962+120596

20 +120596119901120582 cos

119901120587

2)

119882 = 21198600 minus 21198610 +120587120596119901120582 sin

119901120587

2

(57)

The transmissibility for the absolute displacement of LRDcontrol system is

1205892 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988321198830

(58)

Next we study the stability of the steady-state solutionsLetting 119886 = 119886+Δ119886 and 120579 = 120579+Δ120579 and substituting them into(51) one could get

119889Δ119886

119889119905

=

12120587120596

[minus119882 sdotΔ119886minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862 sdot Δ119886]

(59)

Based on (53) one could eliminate 120579 and obtain the charac-teristic determinant as

det[

[

minus119882 minus 120572 minus119886119866

119866

119886

minus119882 minus 120572

]

]

= 0 (60)

Expanding the determinant one could obtain the character-istic equation as

(minus119882minus120572)2+119866

2= 0 (61)

Solving (61) one could obtain the characteristic values as

12057212 = minus119882plusmn119866119894 (62)

Obviously all real parts of the characteristic roots are neg-ative which means the steady-state solution of this kind ofsemiactive system is also unconditionally stable

33 Analytical Investigation for Relative Control System Thestrategy for semiactive relative control is shown as

119888 =

119888max (119909 minus 1199090) ( minus 0) le 0

119888min else(63)

After transformation of coordinates (63) will become

120585 =

120585max 119911 le 0

120585min else(64)

and the control strategy can be rewritten as

120585 =

120585max 0 lt 120601 lt

120587

2

120585min120587

2lt 120601 lt 120587

120585max 120587 lt 120601 lt

31205872

120585min31205872

lt 120601 lt 2120587

(65)

Using the averagingmethod one could obtain the simpli-fied forms of (13b) and (14b) for this semiactive control system

1198862 =

minus120587119886 (120585max + 120585min)

2120587

119886

1205792 =

2119886 (minus120585max + 120585min)

2120587

(66)

Combining (12) (15a) (15b) (21) (22) and (66) one couldobtain

119886

=

minus120587120596119886 (120585max + 120585min) minus 11988301205871205962 sin 120579

2120587120596

minus

120582119886120596119901minus1

2sin(

119901120587

2)

119886

120579

=

119886 [120587 (12059620 minus 120596

2) + 2120596 (minus120585max + 120585min)] minus 1198830120587120596

2 cos 1205792120587120596

+

120582119886120596119901minus1

2cos(

119901120587

2)

(67)

Obviously we could easily obtain the equivalent dampingcoefficient

1198621015840= 119870120596

119901minus1 sin(

119901120587

2)+

119888max + 119888min2

(68)

From (68) we can find that the equivalent damping coeffi-cient 1198621015840 mainly depends on the fractional-order parameters119870 and 119901 associated with the original damping coefficient

Letting 119886 = 0 and 119886

120579 = 0 one could obtain the amplitudeand phase of the steady-state response

119886 =

12058711988301205962

radic1198802+ 119881

2

120579 = arctan 119880

119881

(69)

where

119880 = minus120582120587120596119901 sin(

119901120587

2)minus120587120596 (120585max + 120585min)

119881 = 120587 (12059620 minus120596

2) + 2120596 (minus120585max + 120585min)

+ 120582120587120596119901 cos(

119901120587

2)

(70)

8 Mathematical Problems in Engineering

The transmissibility for relative displacement in this caseis

1205833 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198802+ 119881

2 (71)

The absolute displacement of relative control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198833 cos (120596119905 minus 120574)

(72)

where

1198833 = radic(1198830 +

11988301205871205962 cos120595

radic119876

)

2

+

11988320120587

21205964sin2120595

119876

120574 = arctan1198830120587120596

2 sin120595

(1198830 + cos120595)radic119876

(73)

where

120595 =

119880

119881

119876 = 1198812+119880

2

(74)

The transmissibility for the absolute displacement is

1205893 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988331198830

(75)

Next we study the stability of steady-state solutionsLetting 119886 = 119886 + Δ119886 120579 = 120579 + Δ120579 and substituting them into(67) one could get

119889Δ119886

119889119905

=

12120587120596

[[minus120587120596 (120585max + 120585min) minus 120582120587120596119901 sin(

119901120587

2)]

sdot Δ119886 minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862

sdot Δ119886]

(76)

Based on (69) one could eliminate 120579 and obtain the charac-teristic determinant

det[

[

119880 minus 120572 minus119886119881

119880

119886

119881 minus 120572

]

]

= 0 (77)

0 20 40 60 80 1000

0005

001

0015

002

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 2 Comparison of analytical solution with numerical oneunder on-off control

Expanding the determinant one could obtain the character-istic equation

(119880minus120572)2+119881

2= 0 (78)

Solving (78) one could obtain the characteristic values

12057212 = 119880plusmn119881119894 (79)

Obviously 119880 lt 0 That is to say all real parts of the char-acteristic values are negative which means the steady-statesolution of this kind of semiactive system is stable uncondi-tionally

4 Numerical Simulation and Analysis

41 Comparison between the Analytical and Numerical Solu-tion In order to verify the precision of the analytical solu-tions we also present the numerical results The numericalscheme [29ndash34] is

119863119901[119909 (119905119897)] asymp ℎ

minus119901119897

sum

119895=0119862119901

119895119909 (119905119897minus119895) (80)

where 119905119897 = 119897ℎ is the time sample points ℎ is the sample stepand119862

119901

119895is the fractional binomial coefficient with the iterative

relationship as

119862119901

0 = 1

119862119901

119895= (1minus

1 + 119901

119895

)119862119901

119895minus1(81)

The selected basic system parameters are 119898 = 240 119896 =

16000 119888 = 10001198830 = 001 119901 = 05119870 = 1000 and 120575=051198830We take 400 seconds as numerical time in the process of

Mathematical Problems in Engineering 9

0 20 40 60 80 1000

0005

001

0015

002

0025

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 3 Comparison of analytical solution with numerical oneunder LRD control

the simulation Considering the maximum value of thenumerical results in the last 100 seconds as the steady-stateamplitude one could get the amplitude-frequency curves forthe numerical results which are denoted by the lines withcircles shown in Figures 2ndash4 Based on (37) (53) and (69)one could obtain the amplitude-frequency curves by theanalytical solutions denoted by the solid lines in Figures 2ndash4From the observation of the three figures one could concludethat the analytical solutions agree very well with the numeri-cal results and could present satisfactory precision Howeverthere are distinct errors between the approximate analyticaland numerical solution in some high-frequency range espe-cially the amplitude These phenomena called chatter insemiactive control system had also been found by Ahmadian[35] numerically and experimentally

42 Effects of 119870 on the Steady-State Amplitudes When thefractional coefficient 119870 is changed the different amplitude-frequency curves are shown in Figures 5ndash7 From the obser-vation of Figures 5ndash7 we could conclude that the largerthe fractional coefficient 119870 is the smaller the steady-stateamplitudes of the relative and absolute displacement of sys-tem mass are At the same time the resonant frequency alsobecomes larger This phenomenon indicates that both theequivalent damping coefficient and equivalent stiffness coef-ficient become larger in this procedure That is to say thefractional-order derivative 119870119863

119901(119909 minus 1199090) serves as damping

and spring simultaneously

43 Effects of 119901 on the Steady-State Amplitudes When thefractional coefficient 119901 is changed the results are shownin Figures 8ndash10 From the observation of Figures 8ndash10 wecould conclude that the larger the fractional coefficient 119901 isthe smaller the steady-state amplitudes of the relative andabsolute displacement of system mass are However it seems

Numerical solutionAnalytical solution

0 20 40 60 80 1000

0005

001

0015

002

0025

003

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 4 Comparison of analytical solution with numerical oneunder relative control

that there is little effect of119901 on the resonant frequency In factby differentiating119870

1015840 and 1198621015840 to 119901 one can get

1198891198621015840

119889119901

= 119896120596119901minus1

radic(119901120596minus1)2+ (

120587

2)

2sin(

119901120587

2+ 1205991)

1198891198701015840

119889119901

= minus 119896120596119901radic(119901120596

minus1)2+ (

120587

2)

2sin(

119901120587

2minus 1205992)

(82)

where

1205991 = arctan 120587120596

2119901

1205992 = arctan2119901120587120596

(83)

Because 0 lt 119901 lt 1 we can conclude that the derivative of1198621015840 with respect to 119901 will be larger than 0 1198621015840 will be mono-

tonically larger if 119901 becomes larger Accordingly the steady-state amplitude will be smaller in the process But the valueof 1015840 is not monotonical in this procedure1198701015840 could increaseor reduce with different 119901 which is affected by the systemparameters

5 Conclusion

In this paper three SDOF semiactive systems are researchedby using averaging method Aiming at these systems wepresent the concepts of equivalent damping coefficient andequivalent stiffness coefficient in order to account for theeffect of the fractional-order derivative on the control perfor-mance

Based on the above study one could find that fractional-order derivative plays an important role in SDOF semiactivesystem Therefore the results in this paper supply valuablereference for engineering practice In addition analyticalmethod in this paper could be adopted in two or multi-degrees-of-freedom system and it also offers a new idea tothe research field on vehicle suspension theory

10 Mathematical Problems in Engineering

0

05

1

15

2

25

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205831

100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 1

100 101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 5 Effect of119870 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205832

101 102

(a) Effect of119870 on relative displacement transmissibility

05

1

15

2

25

3

35

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 2

101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 6 Effect of 119870 on the steady-state amplitudes under LRD control

0

05

1

15

2

25

3

35

4

K = 0

K = 1000

K = 2000

K = 3000

1205833

120596 (rads)100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

2

3

4

5

120577 3

120596 (rads)100 101 102

K = 0

K = 1000

K = 2000

K = 3000

(b) Effect of119870 on absolute displacement transmissibility

Figure 7 Effect of119870 on the steady-state amplitudes under relative control

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

The transmissibility for relative displacement in this caseis

1205833 =

10038161003816100381610038161003816100381610038161003816

119886

1198830

10038161003816100381610038161003816100381610038161003816

=

1205871205962

radic1198802+ 119881

2 (71)

The absolute displacement of relative control system is

119909 = 1199090 + 119911 = 1198830 cos120596119905 + 119886 cos (120596119905 + 120579)

= 1198833 cos (120596119905 minus 120574)

(72)

where

1198833 = radic(1198830 +

11988301205871205962 cos120595

radic119876

)

2

+

11988320120587

21205964sin2120595

119876

120574 = arctan1198830120587120596

2 sin120595

(1198830 + cos120595)radic119876

(73)

where

120595 =

119880

119881

119876 = 1198812+119880

2

(74)

The transmissibility for the absolute displacement is

1205893 =

10038161003816100381610038161003816100381610038161003816

119909

1198830

10038161003816100381610038161003816100381610038161003816

=

11988331198830

(75)

Next we study the stability of steady-state solutionsLetting 119886 = 119886 + Δ119886 120579 = 120579 + Δ120579 and substituting them into(67) one could get

119889Δ119886

119889119905

=

12120587120596

[[minus120587120596 (120585max + 120585min) minus 120582120587120596119901 sin(

119901120587

2)]

sdot Δ119886 minus11988301205871205962 cos 120579Δ120579]

119889Δ120579

119889119905

=

12120587120596

[

12058711988301205962 sin 120579

119886

sdot Δ120579 +

12058711988301205962 cos 1205791198862

sdot Δ119886]

(76)

Based on (69) one could eliminate 120579 and obtain the charac-teristic determinant

det[

[

119880 minus 120572 minus119886119881

119880

119886

119881 minus 120572

]

]

= 0 (77)

0 20 40 60 80 1000

0005

001

0015

002

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 2 Comparison of analytical solution with numerical oneunder on-off control

Expanding the determinant one could obtain the character-istic equation

(119880minus120572)2+119881

2= 0 (78)

Solving (78) one could obtain the characteristic values

12057212 = 119880plusmn119881119894 (79)

Obviously 119880 lt 0 That is to say all real parts of the char-acteristic values are negative which means the steady-statesolution of this kind of semiactive system is stable uncondi-tionally

4 Numerical Simulation and Analysis

41 Comparison between the Analytical and Numerical Solu-tion In order to verify the precision of the analytical solu-tions we also present the numerical results The numericalscheme [29ndash34] is

119863119901[119909 (119905119897)] asymp ℎ

minus119901119897

sum

119895=0119862119901

119895119909 (119905119897minus119895) (80)

where 119905119897 = 119897ℎ is the time sample points ℎ is the sample stepand119862

119901

119895is the fractional binomial coefficient with the iterative

relationship as

119862119901

0 = 1

119862119901

119895= (1minus

1 + 119901

119895

)119862119901

119895minus1(81)

The selected basic system parameters are 119898 = 240 119896 =

16000 119888 = 10001198830 = 001 119901 = 05119870 = 1000 and 120575=051198830We take 400 seconds as numerical time in the process of

Mathematical Problems in Engineering 9

0 20 40 60 80 1000

0005

001

0015

002

0025

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 3 Comparison of analytical solution with numerical oneunder LRD control

the simulation Considering the maximum value of thenumerical results in the last 100 seconds as the steady-stateamplitude one could get the amplitude-frequency curves forthe numerical results which are denoted by the lines withcircles shown in Figures 2ndash4 Based on (37) (53) and (69)one could obtain the amplitude-frequency curves by theanalytical solutions denoted by the solid lines in Figures 2ndash4From the observation of the three figures one could concludethat the analytical solutions agree very well with the numeri-cal results and could present satisfactory precision Howeverthere are distinct errors between the approximate analyticaland numerical solution in some high-frequency range espe-cially the amplitude These phenomena called chatter insemiactive control system had also been found by Ahmadian[35] numerically and experimentally

42 Effects of 119870 on the Steady-State Amplitudes When thefractional coefficient 119870 is changed the different amplitude-frequency curves are shown in Figures 5ndash7 From the obser-vation of Figures 5ndash7 we could conclude that the largerthe fractional coefficient 119870 is the smaller the steady-stateamplitudes of the relative and absolute displacement of sys-tem mass are At the same time the resonant frequency alsobecomes larger This phenomenon indicates that both theequivalent damping coefficient and equivalent stiffness coef-ficient become larger in this procedure That is to say thefractional-order derivative 119870119863

119901(119909 minus 1199090) serves as damping

and spring simultaneously

43 Effects of 119901 on the Steady-State Amplitudes When thefractional coefficient 119901 is changed the results are shownin Figures 8ndash10 From the observation of Figures 8ndash10 wecould conclude that the larger the fractional coefficient 119901 isthe smaller the steady-state amplitudes of the relative andabsolute displacement of system mass are However it seems

Numerical solutionAnalytical solution

0 20 40 60 80 1000

0005

001

0015

002

0025

003

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 4 Comparison of analytical solution with numerical oneunder relative control

that there is little effect of119901 on the resonant frequency In factby differentiating119870

1015840 and 1198621015840 to 119901 one can get

1198891198621015840

119889119901

= 119896120596119901minus1

radic(119901120596minus1)2+ (

120587

2)

2sin(

119901120587

2+ 1205991)

1198891198701015840

119889119901

= minus 119896120596119901radic(119901120596

minus1)2+ (

120587

2)

2sin(

119901120587

2minus 1205992)

(82)

where

1205991 = arctan 120587120596

2119901

1205992 = arctan2119901120587120596

(83)

Because 0 lt 119901 lt 1 we can conclude that the derivative of1198621015840 with respect to 119901 will be larger than 0 1198621015840 will be mono-

tonically larger if 119901 becomes larger Accordingly the steady-state amplitude will be smaller in the process But the valueof 1015840 is not monotonical in this procedure1198701015840 could increaseor reduce with different 119901 which is affected by the systemparameters

5 Conclusion

In this paper three SDOF semiactive systems are researchedby using averaging method Aiming at these systems wepresent the concepts of equivalent damping coefficient andequivalent stiffness coefficient in order to account for theeffect of the fractional-order derivative on the control perfor-mance

Based on the above study one could find that fractional-order derivative plays an important role in SDOF semiactivesystem Therefore the results in this paper supply valuablereference for engineering practice In addition analyticalmethod in this paper could be adopted in two or multi-degrees-of-freedom system and it also offers a new idea tothe research field on vehicle suspension theory

10 Mathematical Problems in Engineering

0

05

1

15

2

25

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205831

100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 1

100 101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 5 Effect of119870 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205832

101 102

(a) Effect of119870 on relative displacement transmissibility

05

1

15

2

25

3

35

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 2

101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 6 Effect of 119870 on the steady-state amplitudes under LRD control

0

05

1

15

2

25

3

35

4

K = 0

K = 1000

K = 2000

K = 3000

1205833

120596 (rads)100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

2

3

4

5

120577 3

120596 (rads)100 101 102

K = 0

K = 1000

K = 2000

K = 3000

(b) Effect of119870 on absolute displacement transmissibility

Figure 7 Effect of119870 on the steady-state amplitudes under relative control

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 20 40 60 80 1000

0005

001

0015

002

0025

Numerical solutionAnalytical solution

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 3 Comparison of analytical solution with numerical oneunder LRD control

the simulation Considering the maximum value of thenumerical results in the last 100 seconds as the steady-stateamplitude one could get the amplitude-frequency curves forthe numerical results which are denoted by the lines withcircles shown in Figures 2ndash4 Based on (37) (53) and (69)one could obtain the amplitude-frequency curves by theanalytical solutions denoted by the solid lines in Figures 2ndash4From the observation of the three figures one could concludethat the analytical solutions agree very well with the numeri-cal results and could present satisfactory precision Howeverthere are distinct errors between the approximate analyticaland numerical solution in some high-frequency range espe-cially the amplitude These phenomena called chatter insemiactive control system had also been found by Ahmadian[35] numerically and experimentally

42 Effects of 119870 on the Steady-State Amplitudes When thefractional coefficient 119870 is changed the different amplitude-frequency curves are shown in Figures 5ndash7 From the obser-vation of Figures 5ndash7 we could conclude that the largerthe fractional coefficient 119870 is the smaller the steady-stateamplitudes of the relative and absolute displacement of sys-tem mass are At the same time the resonant frequency alsobecomes larger This phenomenon indicates that both theequivalent damping coefficient and equivalent stiffness coef-ficient become larger in this procedure That is to say thefractional-order derivative 119870119863

119901(119909 minus 1199090) serves as damping

and spring simultaneously

43 Effects of 119901 on the Steady-State Amplitudes When thefractional coefficient 119901 is changed the results are shownin Figures 8ndash10 From the observation of Figures 8ndash10 wecould conclude that the larger the fractional coefficient 119901 isthe smaller the steady-state amplitudes of the relative andabsolute displacement of system mass are However it seems

Numerical solutionAnalytical solution

0 20 40 60 80 1000

0005

001

0015

002

0025

003

The s

tead

y-st

ate a

mpl

itude

of

relat

ive d

ispla

cem

ent

120596 (rads)

Figure 4 Comparison of analytical solution with numerical oneunder relative control

that there is little effect of119901 on the resonant frequency In factby differentiating119870

1015840 and 1198621015840 to 119901 one can get

1198891198621015840

119889119901

= 119896120596119901minus1

radic(119901120596minus1)2+ (

120587

2)

2sin(

119901120587

2+ 1205991)

1198891198701015840

119889119901

= minus 119896120596119901radic(119901120596

minus1)2+ (

120587

2)

2sin(

119901120587

2minus 1205992)

(82)

where

1205991 = arctan 120587120596

2119901

1205992 = arctan2119901120587120596

(83)

Because 0 lt 119901 lt 1 we can conclude that the derivative of1198621015840 with respect to 119901 will be larger than 0 1198621015840 will be mono-

tonically larger if 119901 becomes larger Accordingly the steady-state amplitude will be smaller in the process But the valueof 1015840 is not monotonical in this procedure1198701015840 could increaseor reduce with different 119901 which is affected by the systemparameters

5 Conclusion

In this paper three SDOF semiactive systems are researchedby using averaging method Aiming at these systems wepresent the concepts of equivalent damping coefficient andequivalent stiffness coefficient in order to account for theeffect of the fractional-order derivative on the control perfor-mance

Based on the above study one could find that fractional-order derivative plays an important role in SDOF semiactivesystem Therefore the results in this paper supply valuablereference for engineering practice In addition analyticalmethod in this paper could be adopted in two or multi-degrees-of-freedom system and it also offers a new idea tothe research field on vehicle suspension theory

10 Mathematical Problems in Engineering

0

05

1

15

2

25

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205831

100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 1

100 101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 5 Effect of119870 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205832

101 102

(a) Effect of119870 on relative displacement transmissibility

05

1

15

2

25

3

35

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 2

101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 6 Effect of 119870 on the steady-state amplitudes under LRD control

0

05

1

15

2

25

3

35

4

K = 0

K = 1000

K = 2000

K = 3000

1205833

120596 (rads)100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

2

3

4

5

120577 3

120596 (rads)100 101 102

K = 0

K = 1000

K = 2000

K = 3000

(b) Effect of119870 on absolute displacement transmissibility

Figure 7 Effect of119870 on the steady-state amplitudes under relative control

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0

05

1

15

2

25

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205831

100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 1

100 101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 5 Effect of119870 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

1205832

101 102

(a) Effect of119870 on relative displacement transmissibility

05

1

15

2

25

3

35

K = 0

K = 1000

K = 2000

K = 3000

120596 (rads)

120577 2

101 102

(b) Effect of119870 on absolute displacement transmissibility

Figure 6 Effect of 119870 on the steady-state amplitudes under LRD control

0

05

1

15

2

25

3

35

4

K = 0

K = 1000

K = 2000

K = 3000

1205833

120596 (rads)100 101 102

(a) Effect of119870 on relative displacement transmissibility

1

2

3

4

5

120577 3

120596 (rads)100 101 102

K = 0

K = 1000

K = 2000

K = 3000

(b) Effect of119870 on absolute displacement transmissibility

Figure 7 Effect of119870 on the steady-state amplitudes under relative control

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0

05

1

15

2

25

3

35

4p = 01

p = 03

p = 07

p = 09

1205831

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3p = 01

p = 03

p = 07

p = 09

120577 1

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 8 Effect of 119901 on the steady-state amplitudes under on-off control

0

05

1

15

2

25

3p = 01

p = 03

p = 07

p = 091205832

120596 (rads)101 102

(a) Effect of 119901 on relative displacement transmissibility

05

1

15

2

25

3

35p = 01

p = 03

p = 07

p = 09120577 2

120596 (rads)101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 9 Effect of 119901 on the steady-state amplitudes under LRD control

0

1

2

3

4p = 01

p = 03

p = 07

p = 091205833

120596 (rads)100 101 102

(a) Effect of 119901 on relative displacement transmissibility

1

15

2

25

3

35

4

45

p = 01

p = 03

p = 07

p = 09

120577 3

120596 (rads)100 101 102

(b) Effect of 119901 on absolute displacement transmissibility

Figure 10 Effect of 119870 on the steady-state amplitudes under relative control

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the support by National NaturalScience Foundation of China (no 11372198) the program forNewCentury Excellent Talents in university (NCET-11-0936)the cultivation plan for innovation team and leading talent incolleges and universities of Hebei Province (LJRC018) theprogram for advanced talent in the universities of Hebei Pro-vince (GCC2014053) and the program for advanced talent inHebei Province (A201401001)

References

[1] J P Den Hartog Mechanical Vibration McGraw-Hill NewYork NY USA 1947

[2] D Karnopp M J Crosby and R A Harwood ldquoVibrationcontrol using semi-active force generatorsrdquo Journal of Manufac-turing Science and Engineering vol 96 no 2 pp 619ndash626 1974

[3] J KHedlick ldquoRailway vehicle active suspensionrdquoVehicle SystemDynamics vol 10 pp 267ndash272 1981

[4] Y J Shen S P Yang C Z Pan and H J Xing ldquoSemi-active control of hunting motion of locomotive based on mag-netorheological damperrdquo International Journal of InnovativeComputing Information and Control vol 2 no 2 pp 323ndash3292006

[5] D Karnopp ldquoActive and semi-active vibration isolationrdquoASMEJournal of Mechanical Design vol 117 no 6 pp 177ndash185 1995

[6] S M Savaresi V C Poussot C Spelta et al Semi-ActiveSuspension Control Design for Vehicles Elsevier AmsterdamThe Netherlands 2010

[7] G Emanuele S Tudor W S Charles et al Semi-Active Sus-pension Control-Improved Vehicle Ride and Road FriendlinessSpringer London UK 2008

[8] C Fabio M Georges and M Francesco Technology of Semi-Active Devices and Applications in Vibration Mitigation JohnWiley amp Sons West Sussex UK 2008

[9] Y Shen M F Golnaraghi and G R Heppler ldquoSemi-activevibration control schemes for suspension systems using mag-netorheological dampersrdquo Journal of Vibration and Control vol12 no 1 pp 3ndash24 2006

[10] N Eslaminasab O Vahid A and F Golnaraghi ldquoNonlinearanalysis of switched semi-active controlled systemsrdquo VehicleSystem Dynamics vol 49 no 1-2 pp 291ndash309 2011

[11] Y-J Shen L Wang S-P Yang and G-S Gao ldquoNonlineardynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbersrdquo Journal of Vibrationand Control vol 19 no 1 pp 143ndash160 2013

[12] Y J Shen andM Ahmadian ldquoNonlinear dynamical analysis onfour semi-active dynamic vibration absorbers with time delayrdquoShock and Vibration vol 20 no 4 pp 649ndash663 2013

[13] Y-J Shen S-P Yang H-J Xing and C-Z Pan ldquoAnalyticalresearch on a single degree-of-freedom semi-active oscillatorwith time delayrdquo Journal of Vibration and Control vol 19 no12 pp 1895ndash1905 2013

[14] R L Bagley and P J Torvik ldquoFractional calculusmdasha differentapproach to the analysis of viscoe1astically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[15] R L Bagley ldquoPower law and fractional calculus model ofviscoelasticityrdquo AIAA Journal vol 27 no 10 pp 1412ndash1417 1989

[16] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[17] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[18] M S Tavazoei M Haeri M Attari S Bolouki and M SiamildquoMore details on analysis of fractional-order Van der Poloscillatorrdquo Journal of Vibration and Control vol 15 no 6 pp803ndash819 2009

[19] C M A Pinto and J A T Tenreiro Machado ldquoComplex ordervan der Pol oscillatorrdquo Nonlinear Dynamics vol 65 no 3 pp247ndash254 2011

[20] Y J Shen S P Yang H J Xing andG Gao ldquoPrimary resonanceof Duffing oscillator with fractional-order derivativerdquo Commu-nications in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[21] Y J Shen S P Yang H J Xing and HMa ldquoPrimary resonanceof Duffing oscillator with two kinds of fractional-order deriva-tivesrdquo International Journal of Non-LinearMechanics vol 47 no9 pp 975ndash983 2012

[22] Y-J Shen P Wei and S-P Yang ldquoPrimary resonance offractional-order van der Pol oscillatorrdquo Nonlinear Dynamicsvol 77 no 4 pp 1629ndash1642 2014

[23] Y J Shen S P Yang and C Y Sui ldquoAnalysis on limit cycleof fractional-order van der Pol oscillatorrdquo Chaos Solitons ampFractals vol 67 pp 94ndash102 2014

[24] P Wahi and A Chatterjee ldquoAveraging oscillations with smallfractional damping and delayed termsrdquo Nonlinear Dynamicsvol 38 no 1ndash4 pp 3ndash22 2004

[25] L C Chen and W Q Zhu ldquoStochastic jump and bifurcationof Duffing oscillator with fractional derivative damping undercombined harmonic and white noise excitationsrdquo InternationalJournal of Non-Linear Mechanics vol 46 no 10 pp 1324ndash13292011

[26] X Y Wang and Y J He ldquoProjective synchronization of frac-tional order chaotic system based on linear separationrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol372 no 4 pp 435ndash441 2008

[27] Z L Huang and X L Jin ldquoResponse and stability of a SDOFstrongly nonlinear stochastic system with light damping mod-eled by a fractional derivativerdquo Journal of Sound and Vibrationvol 319 no 3ndash5 pp 1121ndash1135 2009

[28] Y Xu Y G Li D Liu W Jia and H Huang ldquoResponses ofDuffing oscillator with fractional damping and random phaserdquoNonlinear Dynamics vol 74 no 3 pp 745ndash753 2013

[29] I Podlubny Fractional Differential Equations Academic PressLondon UK 1998

[30] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Ams-terdam The Netherlands 2006

[31] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[32] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific River Edge NJ USA 2010

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[33] C A Monje Y Q Chen B M Vinagre D Y Xue and VFeliu Fractional-Order Systems andControls Fundamentals andApplications Springer London UK 2010

[34] I Petras Fractional-Order Nonlinear Systems Modeling Anal-ysis and Simulation Higher Education Press Beijing China2011

[35] MAhmadian ldquoOn the isolation properties of semiactive damp-ersrdquo Journal of Vibration and Control vol 5 no 2 pp 217ndash2321999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of