research article propeller excitation of longitudinal vibration characteristics...

Research ArticlePropeller Excitation of Longitudinal Vibration Characteristics ofMarine Propulsion Shafting System

Ganbo Zhang Yao Zhao Tianyun Li and Xiang Zhu

School of Naval Architecture and Ocean Engineering Huazhong University of Science and TechnologyWuhan 430074 China

Correspondence should be addressed to Ganbo Zhang hustzgb126com

Received 29 May 2013 Accepted 27 July 2013 Published 31 March 2014

Academic Editor Valder Steffen

Copyright copy 2014 Ganbo Zhang 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 submarine experiences longitudinal vibration in the propulsion shafting system throughout most of run A transfer matrixmodel of the propulsion shafting system in which the dynamic characteristics of oil film within thrust bearing are considered isestablished to describe the dynamic behavior Using hydrodynamic lubrication theory and small perturbation method the axialstiffness and damping of oil film are deduced in great detail followed by numerical estimation of the foundation stiffness withfinite element method Based upon these values of dynamic parameters the Campbell diagram describing natural frequencies interms of shafting rotating speeds is available and the effect on the 1st natural frequency of considerable variations in thrust bearingstiffness is next investigatedThe results indicate that the amplitude of variation of the 1st natural frequency in range of low rotatingspeeds is great To reduce off-resonance response without drastic changes in propulsion shafting system architecture themeasure ofmoving thrust bearing backward is examined The longitudinal vibration transmission through propulsion shafting system resultsin subsequent axial excitation of hull the thrust load acting on hull is particularly concerned It is observed that the measures ofstructural modification are of little benefit to minimize thrust load transmitted to hull

1 Introduction

Formarine vessels propulsion shafting is an essential changerof engine torque and propeller thrust Modern submarine ismostly equipped with the electric propulsion plant realizingthe mechanical separation between diesel generator andpropulsion motor Thereby the unsteady propeller thrustconstitutes the sole excitation source of longitudinal vibrationin propulsion shafting system It should be mentioned thatthis type of vibration is an inherent vibration attached topropeller propulsion technology Unless the new thrusteris adopted this type of vibration cannot be eliminatedthoroughly The submarine propulsion shafting system con-sists of propeller stern shaft intermediate shaft journalbearing thrust bearing and flexible coupling as shown inFigure 1

Due to existence of nonuniform flow field near the pro-peller caused by asymmetry in hull and protrusions of

control surfaces the axial excitation which occurs at thepropeller is the result of variations in thrust when thepropeller blades rotate through the nonuniform wake Thefrequency of this disturbance is well known as blade passingfrequency which is equal to the shafting rotating speedmultiplied by the number of blades [1] The longitudinalvibration of propulsion shafting is an evidence of fluctuationsin propeller thrust This type of vibration is likely to causeharm to propulsion system including increased wear ofthrust bearing and flexible coupling as a result of increasedrelative motion between fixed and rotating parts Theseequipment failures firstly occurred in surface warships andwere of particular concern to the navy and then theo-retical and experimental researches on longitudinal vibra-tion of geared turbine shafting were subsequently initiated[2]

The transmission path of propeller thrust in propul-sion shafting system can be described as follows propeller

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 413592 19 pageshttpdxdoiorg1011552014413592

2 Shock and Vibration

Stern shaft Intermediate shaft Flexible coupling

Thrust bearing

FoundationJournal bearingJournal bearing

Propeller

Figure 1 Schematic diagram of submarine propulsion shafting system

rarr shafting rarr thrust bearing rarr foundation rarr hullWhile the thrust bearing transmits propeller thrust to hullit also provides the transmission channel of longitudinalvibration from propulsion shafting to different regions ofhull resulting in subsequent reactions of hull Thus thelongitudinal vibration of propulsion shafting is one of thesecondary excitation sources of hull vibration relative to theprimary excitation sources such as propeller diesel genera-tor and auxiliarymachinery [3] It had been reported that theacoustic radiation of hull at low frequencies is associated withlongitudinal vibration transmission through the propulsionshafting system [4 5] Henceforth the minimization oflongitudinal vibration intensity of propulsion shafting andoscillatory thrust load transmitted to hull had received muchresearch attention [6ndash11]

The study of shafting longitudinal vibration ismuchmorecomplex than the torsional vibration The reason is that intorsional vibration the vibratory system is only confined tothe rotating elements [12] while in the case of longitudinalvibration aside from the rotating elements the fixed partsaffect the vibratory behavior and must be taken into accountThe values of thrust bearingmass and axial stiffness propellerfluctuating thrust propeller (includes entrained water) massaxial stiffness and damping of lubricant oil filmwithin thrustbearing are all required to get an appropriate investigation topropulsion shafting longitudinal vibration The fundamentalstudies of longitudinal vibration are evaluation of naturalfrequency and forced response to verify whether the criticalspeed falls within the running range and the vibratoryamplitude is less than the permissible value or not When theresonance happens the vibratory amplitude and thrust loadtransmitted to hull will be multiplied many times and thenthe vibroacoustic response of hull will be increased greatly aswell So the avoidance of resonance is the most basic designcriteria for propulsion shafting

The propulsion shafting and hull are coupled systemCompared with the hull vibration the longitudinal vibrationof propulsion shafting is local vibration Since the mass ofpropulsion shafting system is far less than the hull whilethe shafting nature frequency is larger than the hull thecoupled effect of hull and propulsion shafting is limited

Hence the propulsion shafting can be regarded as an isolatedsystem away from the hull [13 14] One important issue isthe mechanic model of propulsion shafting Formerly thediscrete model involving amounts of mass and spring unitswas used extensively In general the lumped parametermodelis sufficient to estimate the 1st natural frequency as long as thedetermination of parameter values is accurate However it isa bit rough to reveal longitudinal vibration transmission inthe propulsion shafting system Thus the continuous modelof propulsion shafting is necessary to be developed

It should be noted that the problemof propulsion shaftinglongitudinal vibration occurs during the shafting rotatingcourse Once the shafting is still this type of vibration will bestopped immediatelyThat is the propulsion shafting is rotor-bearing system and the study of longitudinal vibration mustbe based on the method of rotor dynamics The propulsionshafting can be deemed as a chained structure with a branchof thrust bearing which is particularly suitable to use thetransfer matrix method (TMM) for analysis Since firstlyproposed by Prohl [15] the TMM had been proved to beone of effective methods of rotor dynamics as well as thefinite element method (FEM) The TMM solves vibrationproblems in frequency domain with a marching procedureNonetheless there exists numerical instability as the increaseof trial frequency resulting in loss of accuracy or evenincorrect solution Except that the Riccati TMM can beutilized to effectively solve this problem the dimensionlessTMM is presented to improve calculation accuracy in thispaper

Thrust bearing is the sole axial support equipment forpropulsion shafting system Virtually all marine vessels bothmerchant and combat use the sliding tilting pad thrustbearing for purpose of transmitting propeller thrust to hull[16] One distinct difference of rotor system from nonrotorsystem is that the rotor system needs to be lubricated Whenthe thrust shaft rotates the lubricant oil adhering to thrustcollar will be continuously drawn in at the entering edgesto build up wedged-shaped films of substantial carryingcapacity Since the pads are pivoted the shape of oil film isself-aligning to correlatewith the propeller thrustTheoil filmnot only serves as thrust transmitter but also eliminates solid

Shock and Vibration 3

friction and wear between thrust collar and pads Unlike thelongitudinal vibration of rod the dynamic characteristics oflubricant oil film between thrust collar and tilting pads mustbe considered in the vibratory model of propulsion shaftingSo an appropriate investigation to dynamic characteristicsof oil film is crucial In early studies a scarcity of researchon this aspect was worked out due to complexity and limitsof calculation tools Schwanecke [17] and Vassilopoulos [1819] deduced the explicit formulas of dynamic characteristicsof oil film based on the simplified infinite length bearingtheory or says the one-dimensional Reynolds equation ofhydrodynamic lubrication Although their works were com-prehensive a cursory derivation was given and the relationassociated with shafting rotating speed was not coveredIordanoff et al [20] applied the small perturbationmethod tocalculate stiffness and damping coefficients of air lubricationthrust bearing which are adapted for small turbines withhigh rotational speeds considering the effect of load andmisalignment but the thermal analysis of air lubricant wasignored It means that the solutions were obtained on thebasis of constant air viscosity at equivalent temperatureIn this work the detailed numerical procedures to solvedynamic coefficients of oil film within marine tilting padthrust bearing considering thermal effect of lubricant oil arepresented

Apart from the thrust bearing itself the axial rigidityof propulsion shafting system is also contributed by themounting foundation of thrust bearing So an estimation offoundation stiffness is significant to predict axial rigidity ofpropulsion shafting system Due to structural complexity offoundation the numerical estimation is made of foundationstiffness using the FEM in this paper In all three setsof models are analyzed to identify effects of surroundingsupport structure Taken as a whole the structural stiffnessof thrust bearing is resultant value of thrust bearing unitstiffness and foundation stiffness The influence of thrustbearing structural stiffness on dynamic characteristics ofpropulsion shafting longitudinal vibration especially the 1stnature frequency and vibratory response has reached anagreement in marine engineering However another factoraffecting the longitudinal vibration level of propulsion shaft-ing which is the location of thrust bearing has drawn littleattention at present In fact the alteration of thrust bearinglocation is an effective measure to raise critical speed andreduce off-resonance response This study is also conductedin the work The ultimate objective of controlling propulsionshafting longitudinal vibration is to minimize thrust loadtransmitted to hull and underwater acoustic radiation Theresults indicate that the measures of structural modificationsuch as reinforcement of thrust bearing structure and alter-nation of thrust bearing location fail to reduce fluctuatingthrust transmission from the propeller-shafting system to thehull

2 Transfer Matrix Model of PropulsionShafting System

With regard to longitudinal vibration the journal bearingsdo not make stiffness and inertia contributions so there

is no necessity to be taken into account Because of thechained characteristic of propulsion shafting system thetransfer matrix model has been established to describedynamic behaviors A modular representation of propulsionshafting system is given in Figure 2 The shaft of propulsionmotor is not considered because of the effective isolation offlexible coupling [21ndash23]Thephysicalmodel has been brokeninto several subsystems in which the characteristic of eachsubsystem is expressed by a transfer matrix

21 Propeller and Flexible Coupling The propeller is fixed atthe end of stern shaft The mass of propeller and entrainedwater around the propeller makes up a large proportion ofthe total mass of propulsion shafting system approximately60 Thus the propeller can be modeled as a lumped massSimilarly the flexible coupling is also interpreted as a lumpedmassThe transfer matrixes of propeller and flexible couplingare given by

Τ119901 = [1 0

minus11987211990112059621

]

Τ119888 = [1 0

minus11987211988812059621

]

(1)

where119872119901 is mass of propeller and entrained water and119872119888 ismass of flexible coupling

22 Drive Shaft The drive shaft is comprised of stern shaftintermediate shaft and thrust shaft but the diameters ofrespective shaft are not consistent at all such as the diameterof flange is much too larger than shaft neck That is thegeometry of drive shaft is stepped Apart from a few tinystructural defects such as chamfer and groove the drive shaftcan be simplified as a series of uniform shafts with differentdiameter and length

The transfer matrix of free-free 119894th uniform shaft under-going longitudinal vibration is expressed as

T119904119894 =[

[

[

cos (119896119871 119894)sin (119896119871 119894)119864119878119894119896

minus119864119878119894119896 sin (119896119871 119894) cos (119896119871 119894)

]

]

]

(2)

where 119878119894 and 119871 119894 represent the cross section area and lengthof uniform shaft respectively 119896 = 120596119888 is the longitudinalwave number and 119888 = radic119864120588119904 is the longitudinal wavespeed 119864 is Youngrsquos modulus and 120588119904 is density When thehysteretic damping of shafting material is considered 119864 canbe replaced by complex modulus 119864(1 + 119895120578119904) where 120578119904 ishysteretic damping ratio

For 119899 numbers of uniform shafts making up of thedrive shaft in series the combined characteristic is given bybackward matrix multiplication of all uniform shafts that is

T119904 = T119904119899 sdot sdot sdotT1199042T1199041 (3)

23 Thrust Bearing and Foundation To some extent thethrust bearing and foundation can be divided into liquidpart of lubricant oil film and solid part of steel structure


Propeller Stern drive shaft Thrust bearing Fore drive shaft Coupling

MpMc

Hull

MbKo

KbCo

Us

FbFs

Us

Fs

Ub

Fp

Up

E 120588 S L E 120588 S L

Figure 2 Modular representation of propulsion shafting system

The thrust bearing is independently mounted on foundationvia bolts rigidly in submarines It means that the thrustbearing and foundation provide axial support for propulsionshafting system together or the thrust bearing and foundationare two sources of axial rigidity in propulsion shafting system

The thrust bearings of submarine are those of Michelltype and Kingsbury type of which the distinct difference isthe supporting form of pivoted padsThe dynamic character-istic of thrust bearing is complicated because of its irregularconfiguration Taking for example the Kingsbury thrustbearing by tracing the load path it is made up of thrustcollar pads buttons upper leveling links lower leveling linksretainer ring and house The structural stiffness of thrustbearing unit can be obtained by computing the total flexi-bility With proper simplification the individual flexibility ofvarious components can be calculated although the geometryof these components is irregular Since these components areconnected in series all flexibilities are summed to get thetotal flexibility This theoretical method was firstly proposedby Vassilopoulos and Hamilton [19] Certainly the moreaccurate value of thrust bearing stiffness can be obtained bystatic experiment test on pedestal

The preceding description is confined to the stiffnessgenerated by thrust bearing unit but it does not incorporatethe stiffness generated by foundation Actually the dominantfactor of deciding whether thrust bearing structure is rigidor flexible is the foundation spring constant The details ofestimating foundation stiffness are described in Section 4

The following expression is made

1

119870119887

=

1

119870119905

+

1

119870119891

(4)

where 119870119905 and 119870119891 are axial structural stiffness of thrustbearing unit and foundation respectively and119870119887 is resultantstiffness

The oil film between thrust collar and tilting pads ismodeled as linear spring stiffness 119870119900 and damping 119862119900 thesteel structure of thrust bearing and foundation is representedas linear spring stiffness 119870119887 and lumped mass119872119887 as shownin Figure 2 To allow for structural damping the complexstiffness119870119887(1+119895120578119887) is alternative where 120578119887 is loss factorThus

Collar

Hull

Ulb Ur

b

Flb Fr

b

Ke

Figure 3 Equivalent mechanic model of thrust bearing and foun-dation

the transfermatrix equation of thrust bearing and foundationcan be expressed as

119880119903

119865119903 =

[

[

1

1

119870119887

0 1

]

]

[

[

1 0

minus11987211988712059621

]

]

[

[

1

1

119870119900 + 119895120596119862119900

0 1

]

]

119880119897

119865119897

(5)

where the superscripts ldquo119903rdquo and ldquo119897rdquo denote the right end andthe left end

The hull is interpreted as the rigid boundary conditionWith introduction of 119880119903 = 0 (5) is manipulated as

119865

119897= minus

(119870119887 minus1198721198871205962) (119870119900 + 119895120596119862119900)

119870119887 minus1198721198871205962+ 119870119900 + 119895120596119862119900

119880

119897= 119870119890119880

119897 (6)

where119870119890 represents the equivalent stiffness of thrust bearingand foundation

Then the thrust bearing and foundation are visualized asa spring attached to thrust collar in Figure 3

The magnitude of thrust collar axial stiffness reaches1011 which amounts to that of a rigid body The relationshipbetween state variables in the left and right sides of thrustcollar has the following form

119880

119903

119887= 119880

119897

119887

119865

119903

119887= 119872119900

119880

119897

119887+ 119870119890119880

119897

119887+ 119865

119897

119887

(7)

where119872119900 is mass of thrust collar


Substitution of expression 119880119897119887= 119880119890119895120596119905 into (7) yields

T119887 = [1 0

minus1198721199001205962+ 119870119890 1

] (8)

where T119887 is the transfer matrix of thrust bearing and founda-tion

24 Dimensionless Transfer Matrix For dimensional TMMthe deviation accumulates as the increase of trial frequencybecause of large numbers in transfer matrix which affectsaccuracy of results seriously The error can be eliminated bytransforming the dimensional transfer matrix to the dimen-sionless form Introducing the following transformation ofstate vector

119880

119865

= [

119871 0

0 119864119878

]

119880

119865

= S119880119865

(9)

where the symbol ldquomdashrdquo denotes dimensionless variables (thesame below) S is transformation matrix

By substitution of (9) into dimensional transfer matrixequation the dimensionless transfer matrix is derived as

T = Sminus1TS (10)

where T is dimensional transfer matrixBy replacingT in (10) by transfermatrix of respective sub-

system of propulsion shafting the following dimensionlesstransfer matrixes are obtained

Τ119901 =[

[

[

[

1 0

minus

1198721199011205962119871

119864119878

1

]

]

]

]

T119904119894 =[

[

[

cos (119896119871 119894)sin (119896119871 119894)119896119871 119894

minus119896119871 119894 sin (119896119871 119894) cos (119896119871 119894)

]

]

]

T119887 =[

[

[

[

1 0

(minus1198721199001205962+ 119870119890) 119871

119864119878

1

]

]

]

]

Τ119888 =[

[

[

1 0

minus

1198721198881205962119871

119864119878

1

]

]

]

(11)

One problem in application of the above dimensionlesstransfer matrixes is that lumped mechanical elements suchas propeller and flexible coupling have no geometricalparameters To overcome this difficulty the cross section areaand length of adjacent uniform shaft are utilized as nominalcross section area and length of those lumped parts

The relationship between state vector of the right and leftends of propulsion shafting can be written as

119880

119903

119865

119903

=[

[

11987911 11987912

11987921 11987922

]

]

119880

119897

119865

119897

(12)

where 11987911 11987912 11987921 and 11987922 are variables of accumulateddimensionless transfer matrix obtained by backward matrixmultiplication of all subsystems

According to the mechanic model of propulsion shaftingshown in Figure 2 the boundary conditions of both endsare free that is 119865119903 = 0 and 119865119897 = 0 So 11987921 is the surplusvariable By sweeping frequency the trial frequencies whichmeet expression 11987921 = 0 are those natural frequencies ofpropulsion shafting longitudinal vibration When the naturalfrequency is confirmed the corresponding mode shape canalso be deduced

25 Expanded Transfer Matrix To calculate vibratory re-sponse of propulsion shafting excited by propeller fluctuatingthrust the transfer matrix of respective subsystem still needsto be expanded The transfer matrix equation of propellerexcited by thrust load is written as

119880119903

119901

119865119903

119901

=[

[

1 0

minus11987211990112059621

]

]

119880119897

119901

119865119897

119901

+

0

119865119901

(13)

where 119865119901 is propeller fluctuating thrustEquation (13) is easily expanded as

119880119903

119901

119865119903

119901

1

=

[

[

[

[

[

1 0 0

minus11987211987512059621 119865119901

0 0 1

]

]

]

]

]

119880119897

119901

119865119897

119901

1

= Te119901

119880119897

119901

119865119897

119901

1

(14)

where Te119901 is the expanded transfer matrix of propellerSimilarly the expanded transfer matrixes of subsequent

subsystems can be deduced with the same procedures whichare represented by Te119904119904 Te119887 Te119904119891 and Te119888 respectively Butthe only difference is that there is no force term in theseexpanded matrixes compared with that of propeller becausethe thrust only acts on the propeller directly

Then the accumulated expanded transfer matrix of thecomplete propulsion shafting system is given by backwardmatrix multiplication of all subsystems

Te = Te119888Te119904119891Te119887Te119904119904Te119901 (15)

Combined with the free-free boundary conditions thesolution of propeller response is

119880119901 = minus11987911989023

11987911989021

(16)

where 11987911989021 and 11987911989023 are the first and third elements in thesecond row of matrix Te respectively

When the propeller reaction is determined the vibratoryresponse of the whole propulsion shafting system can besolved And then the dynamic force of oil film and thrust loadtransmitted to hull are respectively given by

119865119900 = (119870119900 + 119895120596119862119900) (119880119900 minus 119880119887)

119865119887 = 119870119887119880119887

(17)


where 119880119900 and 119880119887 are responses of thrust collar and thrustbearing respectively

As one of the most important indexes of measuringpropeller fluctuating thrust acting on hull through thepropulsion shafting system the force transmissibility is theneasily obtained by 119865119887119865119901

3 Dynamic Characteristics of LubricantOil Film

The values of propeller mass structural stiffness and lumpedmass of thrust bearing stiffness and damping of lubricantoil film within thrust bearing are all necessary to determinecharacteristics of propulsion shafting longitudinal vibrationThese values are easily got fromproducers except the dynamicparameters of oil film The longitudinal vibration of thrustcollar squeezes the oil film resulting in the variation of filmthickness at the equilibrium Generally this variation is sosmall that the small perturbation method can be used tocalculate the stiffness and damping coefficients of oil filmcombined with hydrodynamic lubrication theory

As is shown in Figure 4 thrust bearing is constructedwith two rows of pads one row is designed for driving aheadand the other is for driving astern In Figure 4 119874 and 119874119901 aregeometrical center of pad and supporting center respectively120593 is central angle of pad 120574119901 is tilting angle of pad around itssupporting line connected with points119874 and119874119901 119903119894 and 119903119900 areinner and outer radius of pad respectively 119887 = 119903119900minus119903119894 is widthand 119903119901 and 120579119901 are radial and circumferential coordinatesof the supporting center 119874119901 respectively Because of thecircumference symmetry of tilting pads with respect to thrustcollar the geometrical shape and lubricating property of oilfilm between thrust collar and individual pad is completelythe same That is the cycle is 2120587119885 where 119885 is the numberof pads Thus just the oil film between thrust collar and onepad is taken as study object

31 Solution of Static Characteristics The static characteris-tics of oil film including distribution of thickness viscositypressure and temperature are obtained by solving twoequations of Reynolds and energy as well as two expressionsof film thickness and lubricant oil viscosity temperaturesimultaneously

The steady Reynolds equation in cylindrical coordinate isexpressed in dimensionless form as [24]

120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

(18)

where the dimensionless parameters are as follows119877 is radialcoordinate 120579 is circumferential coordinate 119875 is pressure 119867is film thickness 119880 is viscosity

The energy equation is similarly written as [24]

(6 minus

1198672

1198801198772

120597119875

120597120579

)

120597119879

120597120579

minus

1198672

119880

120597119875

120597119877

120597119879

120597119877

=

1212058301198872120596

119888119900120588119900ℎ21198981199050

1198801198772

1198672+

1198672

12119880

[(

120597119875

119877120597120579

)

2

+ (

120597119875

120597119877

)

2

]

(19)

where 119879 is temperature 120596 is angular speed 119888119900 and 120588119900 arespecific heat capacity and density of lubricant oil respectivelyℎ119898 is minimumfilm thickness and 1205830 is viscosity of lubricantoil corresponding to inlet temperature 1199050

The preceding two equations are derived from the con-ventional form of the equations by means of the followingsubstitutions

119877 =

119903

119887

119867 =

ℎ

ℎ119898

119880 =

120583

1205830

119875 =

119901ℎ2

119898

12058301205961198872 119879 =

119905

1199050

(20)

where 119903 ℎ 120583 119901 and 119905 are dimensional parameters corre-sponding to dimensionless parameters 119877 119867 119880 119875 and 119879respectively

The expression of film thickness can be deduced from thegeometrical shape of oil film and is given by

119867 = 1 +

120574119901119887

ℎ119898

[119877 sin (120579119901 minus 120579) minus 119877119898 sin (120579119901 minus 120579119898)] (21)

where (119877119898 120579119898) is coordinate of the minimum thickness ℎ119898In the end there is necessity to add the expression

of lubricant oil viscosity temperature Here the Waltherequation is used directly [24]

lglg(1205830

120588119900

119880 + 06) = 119860 minus 119861lg (1199050119879 + 27315) (22)

where 119860 and 119861 are constants determined by two sets ofviscosity and temperature values

Note that the two equations of Reynolds and energy arecoupled there is no possibility to obtain analytic solutionbut the numerical solution can be solved With the finitedifference algorithm the oil film attached to the surface ofpad is divided into a mesh pattern as shown in Figure 5 Aslong as themesh scale is not too large the static characteristicsof oil film can be represented by these mesh points

Then the two partial differential equations of Reynoldsand energy are reduced to two sets of algebraic equationsWith introduction of boundary conditions of pressure andtemperature the two algebraic equations are solved by aniterative procedure until the resultsmeet defined convergencecriteria It should be pointed out that not all convergentsolutions are actual static characteristics of oil film exceptthat they meet additional two conditions that is the loadcapacity of oil film must balance with propeller steady thrustand the moment to supporting center is equal to zero Theload capacity and moment of oil film can be solved bythe following formulas By using the Simpson formula innumerical integration it is easy to turn the integrals into thesums

119865119905 = 119885int

120593

0

int

119903119900

119903119894

119901119903d119903 d120579

119872 = int

120593

0

int

119903119900

119903119894

119901119903

2 sin (120579119901 minus 120579) d119903 d120579(23)


Pad

Collar

120574p

ro ri

b

120593

O

OOp(rp 120579p)

Figure 4 Schematic diagram of thrust bearing

OutletInlet

11

2

2

3

3

i

R

j

n +1

m + 1

120579

(i minus 12 j)

(i j + 12)

(i j)(i minus 1 j)

(i j + 1)

(i j minus 1)

(i + 12 j)

(i j minus 12)

(i + 1 j)

Figure 5 Mesh pattern of oil film

The relationship between propeller steady thrust andshafting rotating speed can be established as

119879119901 = 1198701198791205881199081198992119863

4 (24)

where 120588119908 is water density 119899 is rotational speed119863 is propellerdiameter and119870119879 is thrust coefficient obtained by open-watertest of propeller

Due to a scarcity of test data of the thrust coefficient 119870119879at hand a constant is chosen as an initial approximationThus the propeller steady thrust is proportional to square ofrotational speed and can be made

119879119901 = 119879119890(119899

119899119890

)

2

(25)

where 119879119890 is rated propeller steady thrust and 119899119890 is ratedrotational speed

The detailed numerical procedures are presented in theappendix

32 Dynamic Characteristic Calculation By setting1198670 is thedimensionless film thickness at the equilibrium andΔ119867 is thedimensionless perturbation the dynamic film thickness is

119867 = 1198670 + Δ119867 (26)

Assuming that the magnitude of perturbation is small afirst-order expansion of the oil film dynamic pressure can bemade

119875 = 1198750 + (120597119875

120597119867

)

0

Δ119867 + (

120597119875

120597

119867

)

0

Δ

119867 (27)

where the symbol ldquo( )0rdquo denotes the partial derivative at theequilibrium

And then the dimensionless stiffness and damping of oilfilm can be obtained by area integral of dynamic pressure inexpression (27)

119870 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597119867

)

0

119877d119877d120579

119862 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597

119867

)

0

119877d119877d120579

(28)

For dynamic analysis the term of squeezing oil filmmustbe added in the steady Reynolds equation (18) that is

120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

+ 12119877

120597119867

120597119905

(29)


Table 1 Parameters of thrust bearing

Parameter Symbol ValueTilting pad

Inner radius (mm) 119903119894 165Outer radius (mm) 119903119900 325Central angle (rad) 120593 070Number 119885 8

Supporting centerRadial coordinate (mm) 119903119901 245Circumferential coordinate (rad) 120579119901 0405

Lubricant oilDensity (kmm3) 120588119900 890Specific heat capacity (Jkgsdot∘C) 119888119900 19228Inlet temperature (∘C) 1199050 35Viscosity (Pasdots) 1205830 0051

By substitutions of pressure and film thickness by theexpressions (26) and (27) the following two partial differ-ential equations are deduced by keeping only the first-orderterms

120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597119867

)

0

)

= minus

120597

120597119877

(

31198771198672

0

119880

1205971198750

120597119877

) minus

1

119877

120597

120597120579

(

31198672

0

119880

1205971198750

120597120579

)

120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597

119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597

119867

)

0

) = 12119877

(30)

Note that the forms of the above two equations in theleft sides are the same as the steady Reynolds equationin which just 119875 is replaced by (120597119875120597119867)0 and (120597119875120597

119867)0

respectively Thus the two equations can be solved withthe same numerical algorithm as that of steady Reynoldsequation The obtained (120597119875120597119867)0 and (120597119875120597

119867)0 give the

solutions of the dimensionless stiffness and damping of oilfilm in (28) directly and then the actual stiffness and dampingof oil film are solved by

119870119900 =

12058301205961198874

ℎ3119898

119870

119862119900 =

12058301198874

ℎ3119898

119862

(31)

It can be observed that the dynamic characteristics of oilfilm are determined by geometry size of pad (119887) propertyof lubricant oil (1205830) minimum film thickness (ℎ119898) androtational speed (120596)

33 Results The parameters of tilting pad thrust bearing aresummarized in Table 1

A program to solve dynamic characteristics of lubricantoil film has been devised on the platform of MATLAB Since

20 40 60 80 100 120 140 160 180 200Shafting rotating speed (rpm)

Dyn

amic

coeffi

cien

ts

Stiffness (Nm)

1011

1010

109

108

107

Damping (Nlowastsm)

Figure 6 Stiffness and damping of oil film

the propeller steady thrust correlates with shafting rotatingspeed the dynamic characteristics of oil film must be afunction of rotational speed as well The results are given inFigure 6 It clearly shows the nonlinear relationship betweendynamic characteristics of oil film and shafting rotatingspeeds More exactly to say the stiffness and damping of oilfilm keep pace with the increase of shafting rotating speednonlinearly Usually the oil film stiffness is not less than thatof thrust bearing steel structure In terms of the thrust bearingunit the influence of oil film on global stiffness is lessenedas the rotational speed rises Especially the changeability ofglobal stiffness is weak at high rotational speeds Then therecommendation proposed by bearingrsquos producer of a springconstant for thrust bearing unit which is independent ofrotational speed and thrust is feasible

The calculation of stiffness and damping of oil filmis a complicated task in which a large amount of workconcentrates on the solution of static characteristics Forsimplification one approach used previously is to replacethe distribution of film viscosity by an equivalent viscosityat thermal equilibrium temperature [24] In this case thesolution of energy equation can be avoided The formula ofthermal equilibrium temperature is

119905119890 = 1199050 + 120572

119901119900

119888119900120588119900119902119900

(32)

where120572 is correction factor For low-speed bearing the factorcan be set equal to 08 119901119900 and 119902119900 are friction power andlubricant oil flow respectively which can be solved by

119901119900 = int

120593

0

int

119903119900

119903119894

(minus

ℎ

2

120597119901

120597120579

minus

1205831199032120596

ℎ

) 119903120596d119903 d120579

119902119900 = int

119903119900

119903119894

(minus

ℎ3

12120583

120597119901

119903120597120579

+

1

2

ℎ119903120596) d119903 + int120593

0

(minus

ℎ3

12120583

120597119901

120597119903

) 119903d120579

(33)


Table 2 Calculated stiffness and damping of oil film based uponequivalent viscosity and variable viscosity

Shafting rotating speed (rpm) 40 60 80 100Stiffness (times109Nm)

Variable viscosity 059 166 338 606Equivalent viscosity 063 209 451 780

Damping (times108 Nm)Variable viscosity 194 363 543 790Equivalent viscosity 283 920 1614 2232

Here the results at typical shafting rotating speeds basedupon equivalent viscosity and variable viscosity are comparedin Table 2 As can be seen the thermal effect is proneto decrease the stiffness and damping of oil film Onestraightforward explanation is that the distribution of filmtemperature and viscosity is so nonuniform that this simpli-fication is relatively approximateThemore important reasonis that the minimum film thickness obtained by equivalentviscosity is less than the solution corresponding to variableviscosity under the same propeller steady thrust Because ofthe negative correlation between dynamic coefficients of oilfilm andminimumfilm thickness the calculated stiffness anddamping of oil film with equivalent viscosity are certainlygreater The results indicate that the thermal effect should beconsidered for dynamic analysis of oil film

4 Stiffness of Thrust Bearing Foundation

To estimate the longitudinal vibration level in propulsionshafting system it is desirable to know the axial stiffness ofthrust bearing foundation in as-built condition The founda-tion is typical welding assemblymade up of plates Briefly theaxial stiffness of foundation is the result of bend and sheardeflection of plates relative to hull particularly the reactionof shear at the excitation of thrust load Because of thecontinuity of supporting structure the surrounding structurealso contributes to foundation stiffness For such complexstructure analytical method is not deemed to provide suffi-ciently accurate solution It is therefore decided to resort tothe finite element method (FEM) and the commercial finiteelement software ANSYS is used

In view of acoustic design the foundation may be madeas rigid as desired Also to guarantee the normal operationof thrust bearing the stiffness of foundation should be suf-ficient In fact the measure of thrust bearing independentlymounted is to achieve a much more rigid system in theaxial direction From the point of extensive modificationsenvisioned necessary to solve the problem of insufficientfoundation stiffness there is necessity to be aware of effectin any proposed modifications to stiffen the foundation

41 Original Foundation Since the stern hull playsmajor rolein supporting the foundation and reacting to the propellerthrust excitation the structural behavior of foundation canbe analyzed by considering this aft portion of hull Thefinite element model of stern hull including foundation and

supporting structure named Model 1 is shown in Figure 7Themodel is comprised of plates and rib stiffeners which arerepresented by SHELL63 elements and BEAM188 elementsrespectively [25] The rib section on the hull is of T shapewhile on the pedestal of foundation it is of L shape Thethicknesses of face plate web plate and knee plate offoundation are 26mm 20mm and 12mm respectively

In order to identify contribution of various parts tofoundation rigidity additional two models are formulatednamed Model 2 and Model 3 respectively Figure 8 presentsthe twomodels in whichModel 2 is the foundation itself andModel 3 also includes the pedestal of foundation

With Hooker theorem the axial stiffness of foundationcan be obtained by loading in axial direction and measuringthe corresponding deformation Since the thrust bearing ismounted on foundation with bolts the propeller thrust istransmitted to foundation as the form of lumped force at thelocation of each bolt Therefore the static load is distributedequally over eight nodes laying four nodes each on the twosurfaces of the foundation In each case a total axial force of1 times 105N is applied

The assignment of boundary conditions is crucial ForModel 1 all zero translations are located at the nodes in frontof hull For Model 2 only the axial translation is constrainedto zero for the nodes lying on the bottom of plates ForModel 3 all translations are constrained to zero and allrotations are free for the nodes lying on the bottom due toconstraints of hull With these applied boundary conditionsthe axial displacement of nodes at the location of bolts ismeasured By taking the average of these node deformationsthe axial stiffness of three models is evaluated which is8233 times 108Nm 5046 times 109Nm and 8299 times 108Nmrespectively By comparison the rigidity of foundation itselfis sufficient but the surrounding structure connected withfoundation particularly the pedestal is relatively flexibleTheaxial stiffness of marine thrust bearing unit is generally morethan 1times 109Nm itmeans that the thrust bearing unit is stifferthan the mounting foundation which should not occur inships if possible The results of Model 1 and Model 3 indicatethat the hull makes little contribution to foundation stiffnesswhile the pedestal affects the rigidity of foundation greatlyThat is Model 3 is accurate enough to estimate stiffness ofthrust bearing foundation

42 Reinforced Foundation The foundation stiffness shouldnot be much too less than that of thrust bearing unit fromviewpoint of vibration control The foundation and pedestalare vertical structure like a hollow cantilever box Under theaction of thrust the static deformation decreases graduallyfrom the top to the bottom To stiffen the foundation greatlythe measure of thickening plates is not always effective Forinstance when the plate thickness of pedestal increases 8mmthe foundation stiffness is only 1014times 109Nm or an increaseof approximately 222 Obviously this improvement islimited The effort of thickening plates does not achieve theprospective objective of reinforced foundation It seems that itis not technically feasible to increase the foundation stiffness


XZY

(a) Profile

XZ

Y

(b) Front view

(c) Foundation and supporting structure

Figure 7 Illustration of Model 1

(a) Model 2 (b) Model 3

Figure 8 Illustrations of Model 2 and Model 3

largely without drastic changes in structural configuration offoundation

Thus the efforts are directed toward modifying theinternal structural configuration of pedestal One evidentshortcoming of the original internal structural configurationof pedestal is that the stiffeners are almost concentrated onthe top while the bottom seems to be relatively flexible Withreasonable supplement added in this area the total stiffnessof foundation will be improved substantially One proposed

modified internal structural configuration is demonstratedin Figure 9 in which the original is also given The mod-ification includes deletion of local stiffeners and additionof longitudinal and transverse plates These plates are allextended to connect with adjacent platesThe calculated axialstiffness of the modified foundation is found to be 1237 times109Nm or an increase of 4905 compared with the original8299 times 109Nm The result validates the effectiveness of thisintegrated reinforced scheme


Table 3 Parameters of propulsion shafting system

Parameter Symbol ValueDensity (kgm3) 120588119904 7850Youngrsquos modulus (GPa) 119864 20Propeller (includes entrained water) mass (kg) 119872119901 7150Flexible coupling mass (kg) 119872119888 1300Thrust collar mass (kg) 119872119900 445Thrust bearing mass (kg) 119872119887 4435Thrust bearing resultant stiffness (MNm) 119870119887 1100

The reinforced foundation has significant effect toimprove resultant stiffness of thrust bearing Regardingassembly structure the total stiffness depends on individualstiffness of various components as well as the connectionstiffness So the connective quality of thrust bearing andfoundation is also of importance In all the foundation isrequired to be as rigid as possible The reinforced foundationcontributes to reducing the shear deflection of the pedestaland relieving the rotation of stern hull Once the foundationis much too flexible the thrust collar will be misalignedwith respect to the bearing gland face which is seriouslyharmful to lubrication performance of thrust bearing andresults in increased wear and tear of pads Only these factorsare thoroughly implemented the problemof insufficient axialstiffness in propulsion shafting system will not occur at all

5 Longitudinal Vibration Characteristics ofPropulsion Shafting

51 Natural Frequencies of Longitudinal Vibration Theparameters associated with the propulsion shafting systemare listed in Table 3

The 1st and 2nd natural frequencies of propulsion shaftinglongitudinal vibration in terms of shafting rotating speeds areshownby theCampbell diagram inFigure 10 It seems that thenatural frequencies vary with the rotational speeds especiallyin the range of low rotational speeds The finding is contraryto the widely-held understanding that the natural frequenciesof propulsion shafting longitudinal vibration are independentof shafting rotating speed The main reason for this incorrectconclusion is neglect of oil film stiffness Since the oil filmis not considered the previous studies mistook longitudinalvibration of propulsion shafting as longitudinal vibration ofrod If so the rotor characteristic of propulsion shafting wasfactitiously ignored Virtually this finding is consistent withthe result of laboratory measurement on a scaled experimentrig given by Pan et al [21]

However with an increase of rotational speed the 1stand 2nd natural frequencies tend to be constant graduallyThe variation amplitude and range of natural frequencies arerelated to the relativemagnitude of oil film stiffness and thrustbearing stiffness At low speeds the oil film stiffness is lessthan that of thrust bearing and then the effect of rotationalspeeds on natural frequencies is of increased significanceBut the changeability of natural frequencies is slight oncethe oil film stiffness exceeds that of thrust bearing For the

study the stiffness of oil film at shafting rotating speed 60 rpmreaches the thrust bearing stiffness and then the 1st and2nd natural frequencies are almost invariable in the range ofsubsequent rotational speeds Generally it is the case that themore flexible the thrust bearing is the narrower the influencerange of rotational speeds on natural frequencies is available

The excitation frequencies of propeller fluctuating thrustare tonal at the blade passing frequency Obviously thecritical speed corresponding to the 2nd natural frequency isabove the rotational speed Thereby the 1st frequency is ofparticular interest Besides the 1st mode is the most superiormode of propulsion shafting which contributes to vibratoryresponse greatly

52 Effect of Thrust Bearing Stiffness As far as the longitudi-nal vibration is concerned the structural stiffness of thrustbearing is undoubtedly the most important effect factorProvided that the construction of thrust bearing was rein-forced the natural frequency of propulsion shafting wouldrise correspondingly However there are rapid change regionand slow change region with respect to natural frequency ofpropulsion shafting in the variation range of thrust bearingstiffness which is essential to decide whether to modifythrust bearing or shafting rotor when propulsion shaftingencounters severe longitudinal vibration

Most often the submarine sails at low shafting rotatingspeeds Taking the rotational speed 100 rpm as an examplethe effect of thrust bearing stiffness on the 1st naturalfrequency of propulsion shafting longitudinal vibration isgiven in Figure 11 It appears that the slope of curve decreasesgradually with the increase of thrust bearing stiffness Bysetting a stiffness increment of 05 times 109Nm only inthe range of thrust bearing stiffness (1sim25) times 109Nm theincrease rate of the 1st natural frequency is above 1 Takingthe constant 25 times 109Nm as critical value the changeregion is divided into rapid region and slow region asshown in Figure 11Themeasure of altering natural frequencyby reinforcing structural configuration of thrust bearing iseffective if the initial rigidity of propulsion shafting systemis insufficient Nonetheless in case that the spring constantof thrust bearing was higher than the critical value then thescheme of keeping on stiffening thrust bearing to transferthe 1st natural frequency away from the propeller excitationfrequency did not work at all

In the case of as-built submarine the mounted thrustbearing will be operational forever unless there are failuresIt means that the stiffness of thrust bearing unit is establishedthroughout the period of validity while only the foundationstiffness is changeable Thus the variation of thrust bearingstiffness depends on structural modifications of foundationIf the foundation was too flexible initially especially whenthe spring constants of foundation and shafting were of thesame order of magnitude a large improvement in stiffeningof the foundation would lead to an increase of the 1stnatural frequency Figure 12 shows the effect of considerablevariations in foundation stiffness on the 1st natural frequencyThe results validate the conclusion


(a) The original (b) The modified

Figure 9 Illustrations of the original and modified internal structural configuration of pedestal

20 40 60 80 100 120 140 160 180 20010

20

30

40

50

60

70

80

Shafting rotating speed (rpm)

Freq

uenc

y (H

z)

1st2nd

Figure 10 Campbell diagram

Although the 1st natural frequency varies with springconstant of thrust bearing the variation amplitude is reallylimited Even though the thrust bearing stiffness is multiplied7 times the 1st natural frequency only adds 25Hz let alonehow to achieve such large improvement of thrust bearingstiffness in practice However the study is very useful forestimation of the 1st natural frequency in new design periodof propulsion shafting system In the preliminary designstage there is no necessity to determine the thrust bearingstiffness particularly the foundation stiffness with a highdegree of accuracy Furthermore the stiffer the foundationis relative to shafting the less effect an error in estimatingfoundation stiffness will have on the 1st natural frequencyFor example if the estimated stiffness of foundation was

1 15 2 25 3 35 4 45 5 55 6 65 7230

235

240

245

250

255

260

265

Thrust bearing stiffness (Nm)

1st f

requ

ency

(Hz)

Slow region

times109

Rapid region

Figure 11 Effect of thrust bearing stiffness on the 1st natural fre-quency

12 times 109Nm while the actual value was 1 times 109Nm thecalculated 1st natural frequency was 2334Hz compared withthe exact value of 2288Hz Although the error of foundationstiffness reaches up to 20 the error of the 1st naturalfrequency is less than 2

From the viewpoint of vibration control the measureof altering natural frequency is one of the simplest controltechnologies which can be easily achieved by changes instiffness and mass For longitudinal vibration existing inservice the steady response of thrust bearing induced bythrust load at blade passing frequency with respect to variousspring constants of thrust bearing is depicted in Figure 13In calculation the thrust variation which occurs at thepropeller is set as a constant fraction 5 of the steadythrust at any shafting rotating speed The hysteretic damping


08 1 12 14 16 18 2220

225

230

235

240

245

Foundation stiffness (Nm)

1st f

requ

ency

(Hz)

times109

Figure 12 Effect of foundation stiffness on the 1st natural frequency


Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

11 times 109 Nm25 times 109 Nm35 times 109 Nm

Figure 13 Steady response of thrust bearing induced by thrust loadat blade passing frequency

of shafting and structural damping of thrust bearing areincluded with hysteretic damping ratio 002 and loss factor004 respectively On one hand the level of longitudinalvibration is significantly suppressed when the thrust bearingstiffness is largely increased On the other hand the atten-uation amplitude is not the same in different change regionof thrust bearing stiffness Below the critical value 25 times

109Nm the attenuation amplitude is prominent whereas theattenuation amplitude is slight However the effort of achiev-ing reinforced structural configuration of thrust bearing toreduce off-resonance response of longitudinal vibration inpropulsion shafting system is proved to be rewarding

6 Thrust Transmission throughPropulsion Shafting System

The longitudinal vibration transmission through the propul-sion shafting system results in axial excitation of hulland generation of structure-borne noise which constitutesimportant source of low-frequency acoustic signature Thistype of vibration transmission is actually the propagationof stress wave The lubricant oil film between thrust collarand tilting pads within thrust bearing serves as the keytransmitter of propeller steady thrust from propeller to hullBut unfortunately the oscillations in thrust are also passivelytransmitted through the oil film to thrust bearing and hull

Concentrating at the blade passing frequency and con-sidering the existing understanding of square functionalrelationship between propeller steady thrust and shaftingrotating speed the fluctuating thrust is still set as a fixedproportion 5 of steady thrust at different rotational speedsThe propeller fluctuating thrust and dynamic force of oil filmin terms of shafting rotating speeds are shown in Figure 14 aswell as the thrust load transmitted to hull It is interesting tofind that not only the dynamic force of oil film and thrust loadtransmitted to hull do not follow the square relationship withthe shafting rotating speed as the fluctuating thrust does butalso the amplitudes are magnified This special phenomenonis the result of the existence of nonlinear characteristics inpropulsion shafting system which is almost contributed byoil filmwithin thrust bearingDue to the variation of dynamiccharacteristics of oil film with shafting rotating speed thereis no linear relationship between propeller excitation andshafting response The nonlinear characteristics enable thepropulsion shafting to have the force magnification effectThe results well explain why the longitudinal vibration ofpropulsion shafting is so harmful to acoustic stealth ofsubmarine

Further the thrust transmission characteristics in propul-sion shafting system can be described by the force trans-missibility in frequency domain The waterfall plot of forcetransmissibility versus frequency at different shafting rotatingspeeds is presented in Figure 15 The low-frequency range upto 100Hz using a frequency increment of 001Hz has beenconsideredThe peaks are attributed to the propeller-shaftingresonance It is observed that the thrust load transmittedto hull is multiplied several times in low-frequency bandwhether the propulsion shafting is in the resonance stateor not Then the magnification of thrust load at the bladepassing frequency is easy to understand In comparison thelargest peak at high rotational speed is larger The immediatereason is that the dynamic coefficients of oil film are greaterat high rotational speed correspondingly If it was likelyto decrease the dynamic coefficients of oil film activelyby additional control technology then the transmission offluctuating thrust through the oil film could be reduced tosome extent This provides new promise in minimization ofpropeller fluctuating thrust transmitted to hull and subse-quent hull vibration

Figure 16 shows the force transmissibility for the casesof various thrust bearing stiffness at shafting rotating speed100 rpm It appears that the increase of thrust bearing stiffness



Am

plitu

de (N

)

Fluctuating thrustOil-film dynamic forceThrust load transmitted to hull

106

105

104

103

102

101

Figure 14 Dynamic force acting on propeller oil film and hull

02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2

Figure 15 Waterfall plot of force transmissibility

has greater effect to alter the 2nd natural frequency thanthe 1st natural frequency relatively and the span of twonatural frequencies is expanded But most importantly thelargest peak is not attenuated effectively Since the mostsignificant frequency band of propeller excitation is 5sim40Hzthe measure of stiffening thrust bearing to minimize thrustload transmitted to hull is found to be of little benefitalthough it contributes to reduce vibratory response

7 Discussion of Thrust Bearing Location

Although in most of cases stiffening thrust bearing founda-tion is helpful to alter natural frequency and reduce vibratoryresponse on straight course it is not always realistic exceptfor drastic changes in structural configuration Otherwise

0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm

35 times 109 Nm25 times 109 Nm

Figure 16 Effect of variations in thrust bearing stiffness on forcetransmissibility

Table 4 Resultant Stiffness of shafting between thrust bearing andpropeller

Displacement (m) 0 05 10 15 20Stiffness (times108 Nm) 224 237 251 268 286

the stiffness of reinforced thrust bearing steel structure isslightly larger than that of the original Alternatively thechange of thrust bearing location is more technically feasiblein practice

In view of space and arrangement of the stern the thrustbearing is only suitable to move backward to the side ofpropeller and the range of movement is limited Since thestern shaft is placed between pressure hull and nonpressurehull it is not possible to mount thrust bearing there Hencethe range of thrust bearing movement is confined in thelength of the intermediate shaft as shown in Figure 17 Whenthe thrust bearing is moved backward the axial rigidityof shafting between thrust bearing and propeller will beenhanced The axial stiffness of this part of shafting is listedin Table 4

Also the case of shafting rotating speed 100 rpm isexamined The influence of moving thrust bearing backwardby various displacements on the natural frequency and modeshape of propulsion shafting longitudinal vibration are givenin Figures 18 and 19 respectively The prominent effect is todecrease the span of the two natural frequencies while the1st natural frequency is raised largely and the 2nd naturalfrequency is dropped slightly Take the example of seven-bladed skewed propeller once the 1st natural frequencyincreases by 1Hz the critical speed of blade passing frequencywill increase by 86 rpm correspondingly When the thrustbearing was located backward by 1m relative to the presentlocation it appears that the critical speed would increase by


Range of movement

Figure 17 The range of thrust bearing movement

0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)

Backward displacement (m)

68

69

70

2nd

frequ

ency

(Hz)

1st2nd

Figure 18 Natural frequencies versus backward displacement ofthrust bearing

nearly 10 rpm Besides it has an effect to decrease amplitudeof the 1st mode shape To display the axial mode shape clearlyit is plotted with the vertical coordinate in Figure 19 Theamplitude of mode shape in the whole length of shaftingis almost reduced and the degree of amplitude attenuationincreases gradually along the length Compared with thelongitudinal wavelength the shafting is relatively short Thatis why the amplitude of the 1stmode shape varies slightly withthe shafting length

From the standpoint of vibration mode the decreaseof mode shape amplitude means the reduction of vibratoryresponse Under the condition of the same thrust bearingstiffness the calculated steady response of propeller in timedomain is shown in Figure 20 It indicates that the forcedresponse is effectively attenuated when the thrust bearing ismounted backward Besides the more distant the movementof thrust bearing is the better the attenuation effect is Tosome extent it is reasonable to move the thrust bearingbackward as much as possible Viewed from the perspectiveof the same control effect the twomeasures of locating thrust

0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m

Figure 19 1stmode shape of longitudinal vibration versus backwarddisplacement of thrust bearing

bearing backward and reinforcing structural configuration ofthrust bearing are equivalent But the former really does workwith less cost relatively Certainly the attenuation effect willbe maximized with the use of these two measures together

Of particular concern is the effect of minimizing thrustload transmitted to hull with the conjunction of the twomeasures Figure 21 presents the effect of variations in thrustbearing stiffness and thrust bearing location on force trans-missibility It is observed that the effect of altering the 1stnatural frequency is more prominent as a result of the twomeasures but the amplitude of the peaks almost remains thesameThe study proves once again that themeasures of struc-tural modification have no effect to reduce the fluctuatingthrust transmission in the propeller-shafting system Withthe intent to minimize the thrust load transmitted to hullother control technologies must be developed


0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m

Figure 20 Propeller steady response versus various backward dis-placements of thrust bearing at 100 rpm

0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

0m 11 times 109 Nm10m 11 times 109 Nm20m 35 times 109 Nm

Figure 21 Effect of variations in thrust bearing stiffness and thrustbearing location on force transmissibility

8 Conclusions

The study focuses on the problem of longitudinal vibrationin submarine propulsion shafting system and belongs to thescope of rotor dynamics In view of the chain characteristicof propulsion shafting system the modular transfer matrixmodel had been established Firstly the propulsion shaft-ing system was divided into several subsystems of whichdynamic characteristics are all described by transfermatrixesTo improve accuracy the transfer matrix of individualsubsystemwas transformed to the dimensionless formThen

the calculation methods of stiffness and damping of oilfilm and foundation stiffness were proposed Using theseobtained values of dynamic parameters the characteristicsof longitudinal vibration and thrust transmission in thepropulsion shafting system were analyzed In the end theeffect of thrust bearing location was discussed

On the basis of the studies carried out the followingconclusions can be made

(1) As far as longitudinal vibration is concerned the axialdynamic characteristics of lubricant oil film betweenthrust collar and tilting pads within thrust bearingshould be included in mechanic model of propulsionshafting system Combined with the hydrodynamiclubrication theory the stiffness and damping coeffi-cients of oil film can be solved by the small perturba-tionmethodThe thermal effect of lubricant oil affectsthe solutions significantly and should be taken intoaccount for dynamic analysis of oil film The oil filmstiffness and damping keep pace with the increase ofshafting rotating speed nonlinearly

(2) For as-built submarine the foundation is the domi-nant factor of deciding whether the propulsion shaft-ing system is rigid or flexible in axial direction TheFEM is capable of providing sufficient accurate esti-mation of the foundation axial stiffness The pedestalmakes great contribution to foundation stiffnesswhile the effect of hull is slight To achieve reinforcedfoundation the effort should be directed towardmodifying the internal structural configuration ofpedestal One reinforced scheme is proposed but theoptimized design involving the maximum rigidityand the minimum mass addition still needs to beconducted

(3) The natural frequencies of propulsion shafting lon-gitudinal vibration vary with the shafting rotatingspeeds but the variation amplitude decreases gradu-ally The fact that the critical speed of the 1st naturalfrequency falls within the running range needs toattract enough attention Although the 1st naturalfrequency can be altered by changes in thrust bearingstiffness the actual effect of this measure dependson the initial spring constant of thrust bearingGenerally the critical value 25 times 109Nm is proper toserve as reference In any case a large improvementin stiffening of thrust bearing structure contributes toreduce off-resonance response of propulsion shaftinglongitudinal vibration to some extent

(4) The thrust load transmitted to hull is multipliedseveral times in low-frequency band whether thepropulsion shafting is in the resonance state or notThe nonlinear characteristics of oil film enable thepropulsion shafting to have the force magnificationeffect The measure of stiffening thrust bearing is oflittle benefit tominimize the transmission of propellerfluctuating thrust from the propulsion shafting sys-tem to the hull


Mesh oil film number mesh points

Calculate film thickness at mesh points via thickness expression

Calculate film pressure at mesh points via Reynolds equation

Yes

Calculate film viscosity at mesh points via viscosity-temperature expression

Film pressure and temperature meet convergence criterion

Calculate film temperature at mesh points via energy equation

Input initial minimum film thickness and pad tilting angle

Calculate load capacity and moment via numerical integral algorithm

Moment is equal to zero

Load capacity is equal to propeller steady thrust

Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No

Figure 22 Calculation flow chart of hydrodynamic lubrication

(5) The benefits of altering the 1st natural frequencyand reducing vibratory response validate that themeasure of locating thrust bearing backward is effec-tive Viewed from this perspective the measure ofbackward movement of thrust bearing has the sameeffect as reinforcement of thrust bearing structureHowever the twomeasures of structuralmodificationfail to minimize thrust load transmitted to hull andthere is necessity to develop other control technolo-gies

AppendixWith the central finite difference numerical method theequations of Reynolds and energy are reduced to finitedifference forms as follows

119875119894119895 =

119860 119894119895119875119894+1119895 + 119861119894119895119875119894minus1119895 + 119862119894119895119875119894119895+1 + 119863119894119895119875119894119895minus1 minus 119864119894119895

119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =

119886119894119895119879119894minus1119895 minus 119887119894119895119879119894+1119895 minus 119888119894119895119879119894119895+1 + 119889119894119895119879119894119895minus1 + 119890119894119895

119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)


where the coefficients are

119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)

Equation (A1) are two sets of algebraic equations inwhich the pressure at the center point of each mesh isdescribed in terms of pressures viscosities and film thicknessof the surrounding mesh points and then the temperatureat the centers of mesh points are described in terms of thesurrounding pressures viscosities and film thickness Thetwo sets of algebraic equations can be effectively solved by twoiterative procedures To accelerate the convergence speed theoverrelaxation iterative algorithm and low-relaxation itera-tive algorithm are utilized to solve the distribution of pressureand temperature respectively The iterative equations arelisted as follows

119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)

The definitions of convergence criteria are given by

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)

The whole calculation steps can be described with thefollowing flow chart as shown in Figure 22 The programincludes the internal and external circulations which cor-respond to the iterative computation of tilting angle andminimum film thickness respectively

Conflict of Interests

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

Acknowledgments

The authors wish to appreciate the financial support from theAdmiralty under Grant no 401130301

References

[1] S Ishida and T Yokota ldquoMeasurement and calculation of pro-peller vibratory forcerdquo IHI Engineering Review vol 14 no 1 pp1ndash8 1981

[2] C P Rigby ldquoLongitudinal vibrations of marine propeller shaft-ingrdquo Transactions of the Institute of Marine Engineers vol 60pp 67ndash78 1948

[3] G B Zhang and Y Zhao ldquoAnalysis of vibration and acousticradiation of submarine hull induced by longitudinal vibrationof propulsion shaftingrdquoNoise and Vibration Control vol 32 no3 pp 155ndash158 2012

[4] A T Jaques ldquoSubsonic Pressure Variation Produced by Subma-rinesrdquo Report 744 Naval Ordnance Lab 1943

[5] Y S Wei and Y S Wang ldquoUnsteady hydrodynamics of bladeforces and acoustic response of a model scaled submarineexcited by propellerrsquos thrust and side-forcesrdquo Journal of Soundand Vibration vol 332 pp 2038ndash2056 2013

[6] Y Zhao G B Zhang and L W Li ldquoReview of advanceson longitudinal vibration of ship propulsion shafting and itscontrol technologyrdquo Shipbuilding of China vol 52 no 4 pp259ndash269 2011

[7] G B Zhang and Y Zhao ldquoReduced-order modelingmethod forlongitudinal vibration control of propulsion shaftingrdquo in Pro-ceedings of the International Conference on Mechanical Indus-trial and Manufacturing Engineering pp 73ndash80 Singapore2012

[8] A J H Goodwin ldquoThe design of a resonance changer to over-come excessive axial vibration of propeller shaftingrdquo Institute ofMarine Engineers Transactions vol 72 pp 37ndash63 1960

[9] D W Lewis P E Allaire and P W Thomas ldquoActive magneticcontrol of oscillatory axial shaft vibrations in ship shaft trans-mission systems Part 1 system natural frequencies and labo-ratory scale modelrdquo Tribology Transactions vol 32 no 2 pp170ndash178 1989


[10] D W Lewis R R Humphris and P W Thomas ldquoActive mag-netic control of oscillatory axial shaft vibrations in ship shafttransmission systems Part 2 control analysis and response ofexperimental systemrdquo Tribology Transactions vol 32 no 2 pp179ndash188 1989

[11] A Baz J Gilheany and P Steimel ldquoActive vibration control ofpropeller shaftsrdquo Journal of Sound and Vibration vol 136 no 3pp 361ndash372 1990

[12] L Xiang S X Yang and C B Gan ldquoTorsional vibration of ashafting system under electrical disturbancesrdquo Shock andVibra-tion vol 19 pp 1ndash11 2012

[13] P Lu Y Zhao T Y Li and et al ldquoAnalysis of coupled vibration ofpropulsion shafting-hull based on mixed finite element modelrdquoShipbuilding of China vol 53 no 3 pp 151ndash157 2012

[14] L Murawski ldquoAxial vibrations of a propulsion system takinginto account the couplings and the boundary conditionsrdquo Jour-nal of Marine Science and Technology vol 9 no 4 pp 171ndash1812004

[15] M A Prohl ldquoA general method for calculating critical speedsof fiexible rotorsrdquo Journal of Applied Mechanics vol 12 pp 142ndash148 1945

[16] B Sternlicht J C Reid and E B Arwas ldquoReview of propellershaft thrust bearingsrdquo Journal of the American Society for NavalEngineers vol 71 no 2 pp 277ndash290 1959

[17] H Schwanecke ldquoInvestigations on the hydrodynamic stiffnessand damping of thrust bearings in shipsrdquo Transactions of theInstitute of Marine Engineers vol 91 pp 68ndash77 1979

[18] L Vassilopoulos ldquoMethods for computing stiffness and damp-ing properties of main propulsion thrust bearingrdquo InternationalShipbuilding Progress vol 29 no 329 pp 13ndash31 1982

[19] L Vassilopoulos and F M Hamilton ldquoLongitudinal stiffnessanalysis for the propulsion shafting systems of the Polar classicebreakersrdquoNaval Engineers Journal vol 92 no 2 pp 179ndash1951980

[20] I Iordanoff P Stefan R Boudet and D Poirier ldquoDynamicanalysis of a thrust bearing effect of misalignment and loadrdquoProceedings of the Institution of Mechanical Engineers J Journalof Engineering Tribology vol 209 no 3 pp 189ndash194 1995

[21] J Pan N Farag T Lin andR Juniper ldquoPropeller induced struc-tural vibration through the thrust bearingrdquo in Innovation inAcoustics and Vibration pp 390ndash399 Adelaide Australia 2002

[22] P G Dylejko and N J Kessissoglou ldquoMinimization of thevibration transmission through the propeller-shafting system ina submarinerdquo Journal of the Acoustical Society of America vol116 no 4 pp 2569ndash2569 2004

[23] P G Dylejko N J Kessissoglou Y Tso and C J NorwoodldquoOptimisation of a resonance changer tominimise the vibrationtransmission inmarine vesselsrdquo Journal of Sound and Vibrationvol 300 no 1-2 pp 101ndash116 2007

[24] G W Stachowiak and A W Batchelor Engineering TribologyElservier Butterworth-Heinemann Burlington Mass USA3rd edition 2005

[25] F Besnier L Jian L Murawski and M Weryk ldquoEvaluation ofmain engine and propeller excitations of ship hull and super-structure vibrationrdquo International Shipbuilding Progress vol 55no 1-2 pp 3ndash27 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Stern shaft Intermediate shaft Flexible coupling

Thrust bearing

FoundationJournal bearingJournal bearing

Propeller

Figure 1 Schematic diagram of submarine propulsion shafting system

rarr shafting rarr thrust bearing rarr foundation rarr hullWhile the thrust bearing transmits propeller thrust to hullit also provides the transmission channel of longitudinalvibration from propulsion shafting to different regions ofhull resulting in subsequent reactions of hull Thus thelongitudinal vibration of propulsion shafting is one of thesecondary excitation sources of hull vibration relative to theprimary excitation sources such as propeller diesel genera-tor and auxiliarymachinery [3] It had been reported that theacoustic radiation of hull at low frequencies is associated withlongitudinal vibration transmission through the propulsionshafting system [4 5] Henceforth the minimization oflongitudinal vibration intensity of propulsion shafting andoscillatory thrust load transmitted to hull had received muchresearch attention [6ndash11]

The study of shafting longitudinal vibration ismuchmorecomplex than the torsional vibration The reason is that intorsional vibration the vibratory system is only confined tothe rotating elements [12] while in the case of longitudinalvibration aside from the rotating elements the fixed partsaffect the vibratory behavior and must be taken into accountThe values of thrust bearingmass and axial stiffness propellerfluctuating thrust propeller (includes entrained water) massaxial stiffness and damping of lubricant oil filmwithin thrustbearing are all required to get an appropriate investigation topropulsion shafting longitudinal vibration The fundamentalstudies of longitudinal vibration are evaluation of naturalfrequency and forced response to verify whether the criticalspeed falls within the running range and the vibratoryamplitude is less than the permissible value or not When theresonance happens the vibratory amplitude and thrust loadtransmitted to hull will be multiplied many times and thenthe vibroacoustic response of hull will be increased greatly aswell So the avoidance of resonance is the most basic designcriteria for propulsion shafting

The propulsion shafting and hull are coupled systemCompared with the hull vibration the longitudinal vibrationof propulsion shafting is local vibration Since the mass ofpropulsion shafting system is far less than the hull whilethe shafting nature frequency is larger than the hull thecoupled effect of hull and propulsion shafting is limited

Hence the propulsion shafting can be regarded as an isolatedsystem away from the hull [13 14] One important issue isthe mechanic model of propulsion shafting Formerly thediscrete model involving amounts of mass and spring unitswas used extensively In general the lumped parametermodelis sufficient to estimate the 1st natural frequency as long as thedetermination of parameter values is accurate However it isa bit rough to reveal longitudinal vibration transmission inthe propulsion shafting system Thus the continuous modelof propulsion shafting is necessary to be developed

It should be noted that the problemof propulsion shaftinglongitudinal vibration occurs during the shafting rotatingcourse Once the shafting is still this type of vibration will bestopped immediatelyThat is the propulsion shafting is rotor-bearing system and the study of longitudinal vibration mustbe based on the method of rotor dynamics The propulsionshafting can be deemed as a chained structure with a branchof thrust bearing which is particularly suitable to use thetransfer matrix method (TMM) for analysis Since firstlyproposed by Prohl [15] the TMM had been proved to beone of effective methods of rotor dynamics as well as thefinite element method (FEM) The TMM solves vibrationproblems in frequency domain with a marching procedureNonetheless there exists numerical instability as the increaseof trial frequency resulting in loss of accuracy or evenincorrect solution Except that the Riccati TMM can beutilized to effectively solve this problem the dimensionlessTMM is presented to improve calculation accuracy in thispaper

Thrust bearing is the sole axial support equipment forpropulsion shafting system Virtually all marine vessels bothmerchant and combat use the sliding tilting pad thrustbearing for purpose of transmitting propeller thrust to hull[16] One distinct difference of rotor system from nonrotorsystem is that the rotor system needs to be lubricated Whenthe thrust shaft rotates the lubricant oil adhering to thrustcollar will be continuously drawn in at the entering edgesto build up wedged-shaped films of substantial carryingcapacity Since the pads are pivoted the shape of oil film isself-aligning to correlatewith the propeller thrustTheoil filmnot only serves as thrust transmitter but also eliminates solid








Τ119901 = [1 0

minus11987211990112059621

]

Τ119888 = [1 0

minus11987211988812059621

]

(1)




T119904119894 =[

[

[

cos (119896119871 119894)sin (119896119871 119894)119864119878119894119896

minus119864119878119894119896 sin (119896119871 119894) cos (119896119871 119894)

]

]

]

(2)







MpMc

Hull

MbKo

KbCo

Us

FbFs

Us

Fs

Ub

Fp

Up

E 120588 S L E 120588 S L






1

119870119887

=

1

119870119905

+

1

119870119891

(4)



Collar

Hull

Ulb Ur

b

Flb Fr

b

Ke



119880119903

119865119903 =

[

[

1

1

119870119887

0 1

]

]

[

[

1 0

minus11987211988712059621

]

]

[

[

1

1

119870119900 + 119895120596119862119900

0 1

]

]

119880119897

119865119897

(5)



119865

119897= minus

(119870119887 minus1198721198871205962) (119870119900 + 119895120596119862119900)

119870119887 minus1198721198871205962+ 119870119900 + 119895120596119862119900

119880

119897= 119870119890119880

119897 (6)




119880

119903

119887= 119880

119897

119887

119865

119903

119887= 119872119900

119880

119897

119887+ 119870119890119880

119897

119887+ 119865

119897

119887

(7)




T119887 = [1 0

minus1198721199001205962+ 119870119890 1

] (8)



119880

119865

= [

119871 0

0 119864119878

]

119880

119865

= S119880119865

(9)



T = Sminus1TS (10)



Τ119901 =[

[

[

[

1 0

minus

1198721199011205962119871

119864119878

1

]

]

]

]

T119904119894 =[

[

[

cos (119896119871 119894)sin (119896119871 119894)119896119871 119894

minus119896119871 119894 sin (119896119871 119894) cos (119896119871 119894)

]

]

]

T119887 =[

[

[

[

1 0

(minus1198721199001205962+ 119870119890) 119871

119864119878

1

]

]

]

]

Τ119888 =[

[

[

1 0

minus

1198721198881205962119871

119864119878

1

]

]

]

(11)



119880

119903

119865

119903

=[

[

11987911 11987912

11987921 11987922

]

]

119880

119897

119865

119897

(12)




119880119903

119901

119865119903

119901

=[

[

1 0

minus11987211990112059621

]

]

119880119897

119901

119865119897

119901

+

0

119865119901

(13)


119880119903

119901

119865119903

119901

1

=

[

[

[

[

[

1 0 0

minus11987211987512059621 119865119901

0 0 1

]

]

]

]

]

119880119897

119901

119865119897

119901

1

= Te119901

119880119897

119901

119865119897

119901

1

(14)




Te = Te119888Te119904119891Te119887Te119904119904Te119901 (15)


119880119901 = minus11987911989023

11987911989021

(16)



119865119900 = (119870119900 + 119895120596119862119900) (119880119900 minus 119880119887)

119865119887 = 119870119887119880119887

(17)









120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

(18)



(6 minus

1198672

1198801198772

120597119875

120597120579

)

120597119879

120597120579

minus

1198672

119880

120597119875

120597119877

120597119879

120597119877

=

1212058301198872120596

119888119900120588119900ℎ21198981199050

1198801198772

1198672+

1198672

12119880

[(

120597119875

119877120597120579

)

2

+ (

120597119875

120597119877

)

2

]

(19)



119877 =

119903

119887

119867 =

ℎ

ℎ119898

119880 =

120583

1205830

119875 =

119901ℎ2

119898

12058301205961198872 119879 =

119905

1199050

(20)



119867 = 1 +

120574119901119887

ℎ119898




lglg(1205830

120588119900

119880 + 06) = 119860 minus 119861lg (1199050119879 + 27315) (22)




119865119905 = 119885int

120593

0

int

119903119900

119903119894

119901119903d119903 d120579

119872 = int

120593

0

int

119903119900

119903119894

119901119903

2 sin (120579119901 minus 120579) d119903 d120579(23)


Pad

Collar

120574p

ro ri

b

120593

O

OOp(rp 120579p)


OutletInlet

11

2

2

3

3

i

R

j

n +1

m + 1

120579

(i minus 12 j)

(i j + 12)

(i j)(i minus 1 j)

(i j + 1)

(i j minus 1)

(i + 12 j)

(i j minus 12)

(i + 1 j)



119879119901 = 1198701198791205881199081198992119863

4 (24)



119879119901 = 119879119890(119899

119899119890

)

2

(25)




119867 = 1198670 + Δ119867 (26)


119875 = 1198750 + (120597119875

120597119867

)

0

Δ119867 + (

120597119875

120597

119867

)

0

Δ

119867 (27)



119870 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597119867

)

0

119877d119877d120579

119862 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597

119867

)

0

119877d119877d120579

(28)


120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

+ 12119877

120597119867

120597119905

(29)








120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597119867

)

0

)

= minus

120597

120597119877

(

31198771198672

0

119880

1205971198750

120597119877

) minus

1

119877

120597

120597120579

(

31198672

0

119880

1205971198750

120597120579

)

120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597

119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597

119867

)

0

) = 12119877

(30)


119867)0


119867)0 give the


119870119900 =

12058301205961198874

ℎ3119898

119870

119862119900 =

12058301198874

ℎ3119898

119862

(31)





Dyn

amic

coeffi

cien

ts

Stiffness (Nm)

1011

1010

109

108

107

Damping (Nlowastsm)




119905119890 = 1199050 + 120572

119901119900

119888119900120588119900119902119900

(32)


119901119900 = int

120593

0

int

119903119900

119903119894

(minus

ℎ

2

120597119901

120597120579

minus

1205831199032120596

ℎ

) 119903120596d119903 d120579

119902119900 = int

119903119900

119903119894

(minus

ℎ3

12120583

120597119901

119903120597120579

+

1

2

ℎ119903120596) d119903 + int120593

0

(minus

ℎ3

12120583

120597119901

120597119903

) 119903d120579

(33)

















XZY

(a) Profile

XZ

Y

(b) Front view
























20 40 60 80 100 120 140 160 180 20010

20

30

40

50

60

70

80


Freq

uenc

y (H

z)

1st2nd



1 15 2 25 3 35 4 45 5 55 6 65 7230

235

240

245

250

255

260

265


1st f

requ

ency

(Hz)

Slow region

times109

Rapid region





08 1 12 14 16 18 2220

225

230

235

240

245


1st f

requ

ency

(Hz)

times109



Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8












Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















Τ119901 = [1 0

minus11987211990112059621

]

Τ119888 = [1 0

minus11987211988812059621

]

(1)




T119904119894 =[

[

[

cos (119896119871 119894)sin (119896119871 119894)119864119878119894119896

minus119864119878119894119896 sin (119896119871 119894) cos (119896119871 119894)

]

]

]

(2)







MpMc

Hull

MbKo

KbCo

Us

FbFs

Us

Fs

Ub

Fp

Up

E 120588 S L E 120588 S L






1

119870119887

=

1

119870119905

+

1

119870119891

(4)



Collar

Hull

Ulb Ur

b

Flb Fr

b

Ke



119880119903

119865119903 =

[

[

1

1

119870119887

0 1

]

]

[

[

1 0

minus11987211988712059621

]

]

[

[

1

1

119870119900 + 119895120596119862119900

0 1

]

]

119880119897

119865119897

(5)



119865

119897= minus

(119870119887 minus1198721198871205962) (119870119900 + 119895120596119862119900)

119870119887 minus1198721198871205962+ 119870119900 + 119895120596119862119900

119880

119897= 119870119890119880

119897 (6)




119880

119903

119887= 119880

119897

119887

119865

119903

119887= 119872119900

119880

119897

119887+ 119870119890119880

119897

119887+ 119865

119897

119887

(7)




T119887 = [1 0

minus1198721199001205962+ 119870119890 1

] (8)



119880

119865

= [

119871 0

0 119864119878

]

119880

119865

= S119880119865

(9)



T = Sminus1TS (10)



Τ119901 =[

[

[

[

1 0

minus

1198721199011205962119871

119864119878

1

]

]

]

]

T119904119894 =[

[

[

cos (119896119871 119894)sin (119896119871 119894)119896119871 119894

minus119896119871 119894 sin (119896119871 119894) cos (119896119871 119894)

]

]

]

T119887 =[

[

[

[

1 0

(minus1198721199001205962+ 119870119890) 119871

119864119878

1

]

]

]

]

Τ119888 =[

[

[

1 0

minus

1198721198881205962119871

119864119878

1

]

]

]

(11)



119880

119903

119865

119903

=[

[

11987911 11987912

11987921 11987922

]

]

119880

119897

119865

119897

(12)




119880119903

119901

119865119903

119901

=[

[

1 0

minus11987211990112059621

]

]

119880119897

119901

119865119897

119901

+

0

119865119901

(13)


119880119903

119901

119865119903

119901

1

=

[

[

[

[

[

1 0 0

minus11987211987512059621 119865119901

0 0 1

]

]

]

]

]

119880119897

119901

119865119897

119901

1

= Te119901

119880119897

119901

119865119897

119901

1

(14)




Te = Te119888Te119904119891Te119887Te119904119904Te119901 (15)


119880119901 = minus11987911989023

11987911989021

(16)



119865119900 = (119870119900 + 119895120596119862119900) (119880119900 minus 119880119887)

119865119887 = 119870119887119880119887

(17)









120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

(18)



(6 minus

1198672

1198801198772

120597119875

120597120579

)

120597119879

120597120579

minus

1198672

119880

120597119875

120597119877

120597119879

120597119877

=

1212058301198872120596

119888119900120588119900ℎ21198981199050

1198801198772

1198672+

1198672

12119880

[(

120597119875

119877120597120579

)

2

+ (

120597119875

120597119877

)

2

]

(19)



119877 =

119903

119887

119867 =

ℎ

ℎ119898

119880 =

120583

1205830

119875 =

119901ℎ2

119898

12058301205961198872 119879 =

119905

1199050

(20)



119867 = 1 +

120574119901119887

ℎ119898




lglg(1205830

120588119900

119880 + 06) = 119860 minus 119861lg (1199050119879 + 27315) (22)




119865119905 = 119885int

120593

0

int

119903119900

119903119894

119901119903d119903 d120579

119872 = int

120593

0

int

119903119900

119903119894

119901119903

2 sin (120579119901 minus 120579) d119903 d120579(23)


Pad

Collar

120574p

ro ri

b

120593

O

OOp(rp 120579p)


OutletInlet

11

2

2

3

3

i

R

j

n +1

m + 1

120579

(i minus 12 j)

(i j + 12)

(i j)(i minus 1 j)

(i j + 1)

(i j minus 1)

(i + 12 j)

(i j minus 12)

(i + 1 j)



119879119901 = 1198701198791205881199081198992119863

4 (24)



119879119901 = 119879119890(119899

119899119890

)

2

(25)




119867 = 1198670 + Δ119867 (26)


119875 = 1198750 + (120597119875

120597119867

)

0

Δ119867 + (

120597119875

120597

119867

)

0

Δ

119867 (27)



119870 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597119867

)

0

119877d119877d120579

119862 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597

119867

)

0

119877d119877d120579

(28)


120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

+ 12119877

120597119867

120597119905

(29)








120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597119867

)

0

)

= minus

120597

120597119877

(

31198771198672

0

119880

1205971198750

120597119877

) minus

1

119877

120597

120597120579

(

31198672

0

119880

1205971198750

120597120579

)

120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597

119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597

119867

)

0

) = 12119877

(30)


119867)0


119867)0 give the


119870119900 =

12058301205961198874

ℎ3119898

119870

119862119900 =

12058301198874

ℎ3119898

119862

(31)





Dyn

amic

coeffi

cien

ts

Stiffness (Nm)

1011

1010

109

108

107

Damping (Nlowastsm)




119905119890 = 1199050 + 120572

119901119900

119888119900120588119900119902119900

(32)


119901119900 = int

120593

0

int

119903119900

119903119894

(minus

ℎ

2

120597119901

120597120579

minus

1205831199032120596

ℎ

) 119903120596d119903 d120579

119902119900 = int

119903119900

119903119894

(minus

ℎ3

12120583

120597119901

119903120597120579

+

1

2

ℎ119903120596) d119903 + int120593

0

(minus

ℎ3

12120583

120597119901

120597119903

) 119903d120579

(33)

















XZY

(a) Profile

XZ

Y

(b) Front view
























20 40 60 80 100 120 140 160 180 20010

20

30

40

50

60

70

80


Freq

uenc

y (H

z)

1st2nd



1 15 2 25 3 35 4 45 5 55 6 65 7230

235

240

245

250

255

260

265


1st f

requ

ency

(Hz)

Slow region

times109

Rapid region





08 1 12 14 16 18 2220

225

230

235

240

245


1st f

requ

ency

(Hz)

times109



Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8












Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











MpMc

Hull

MbKo

KbCo

Us

FbFs

Us

Fs

Ub

Fp

Up

E 120588 S L E 120588 S L






1

119870119887

=

1

119870119905

+

1

119870119891

(4)



Collar

Hull

Ulb Ur

b

Flb Fr

b

Ke



119880119903

119865119903 =

[

[

1

1

119870119887

0 1

]

]

[

[

1 0

minus11987211988712059621

]

]

[

[

1

1

119870119900 + 119895120596119862119900

0 1

]

]

119880119897

119865119897

(5)



119865

119897= minus

(119870119887 minus1198721198871205962) (119870119900 + 119895120596119862119900)

119870119887 minus1198721198871205962+ 119870119900 + 119895120596119862119900

119880

119897= 119870119890119880

119897 (6)




119880

119903

119887= 119880

119897

119887

119865

119903

119887= 119872119900

119880

119897

119887+ 119870119890119880

119897

119887+ 119865

119897

119887

(7)




T119887 = [1 0

minus1198721199001205962+ 119870119890 1

] (8)



119880

119865

= [

119871 0

0 119864119878

]

119880

119865

= S119880119865

(9)



T = Sminus1TS (10)



Τ119901 =[

[

[

[

1 0

minus

1198721199011205962119871

119864119878

1

]

]

]

]

T119904119894 =[

[

[

cos (119896119871 119894)sin (119896119871 119894)119896119871 119894

minus119896119871 119894 sin (119896119871 119894) cos (119896119871 119894)

]

]

]

T119887 =[

[

[

[

1 0

(minus1198721199001205962+ 119870119890) 119871

119864119878

1

]

]

]

]

Τ119888 =[

[

[

1 0

minus

1198721198881205962119871

119864119878

1

]

]

]

(11)



119880

119903

119865

119903

=[

[

11987911 11987912

11987921 11987922

]

]

119880

119897

119865

119897

(12)




119880119903

119901

119865119903

119901

=[

[

1 0

minus11987211990112059621

]

]

119880119897

119901

119865119897

119901

+

0

119865119901

(13)


119880119903

119901

119865119903

119901

1

=

[

[

[

[

[

1 0 0

minus11987211987512059621 119865119901

0 0 1

]

]

]

]

]

119880119897

119901

119865119897

119901

1

= Te119901

119880119897

119901

119865119897

119901

1

(14)




Te = Te119888Te119904119891Te119887Te119904119904Te119901 (15)


119880119901 = minus11987911989023

11987911989021

(16)



119865119900 = (119870119900 + 119895120596119862119900) (119880119900 minus 119880119887)

119865119887 = 119870119887119880119887

(17)









120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

(18)



(6 minus

1198672

1198801198772

120597119875

120597120579

)

120597119879

120597120579

minus

1198672

119880

120597119875

120597119877

120597119879

120597119877

=

1212058301198872120596

119888119900120588119900ℎ21198981199050

1198801198772

1198672+

1198672

12119880

[(

120597119875

119877120597120579

)

2

+ (

120597119875

120597119877

)

2

]

(19)



119877 =

119903

119887

119867 =

ℎ

ℎ119898

119880 =

120583

1205830

119875 =

119901ℎ2

119898

12058301205961198872 119879 =

119905

1199050

(20)



119867 = 1 +

120574119901119887

ℎ119898




lglg(1205830

120588119900

119880 + 06) = 119860 minus 119861lg (1199050119879 + 27315) (22)




119865119905 = 119885int

120593

0

int

119903119900

119903119894

119901119903d119903 d120579

119872 = int

120593

0

int

119903119900

119903119894

119901119903

2 sin (120579119901 minus 120579) d119903 d120579(23)


Pad

Collar

120574p

ro ri

b

120593

O

OOp(rp 120579p)


OutletInlet

11

2

2

3

3

i

R

j

n +1

m + 1

120579

(i minus 12 j)

(i j + 12)

(i j)(i minus 1 j)

(i j + 1)

(i j minus 1)

(i + 12 j)

(i j minus 12)

(i + 1 j)



119879119901 = 1198701198791205881199081198992119863

4 (24)



119879119901 = 119879119890(119899

119899119890

)

2

(25)




119867 = 1198670 + Δ119867 (26)


119875 = 1198750 + (120597119875

120597119867

)

0

Δ119867 + (

120597119875

120597

119867

)

0

Δ

119867 (27)



119870 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597119867

)

0

119877d119877d120579

119862 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597

119867

)

0

119877d119877d120579

(28)


120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

+ 12119877

120597119867

120597119905

(29)








120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597119867

)

0

)

= minus

120597

120597119877

(

31198771198672

0

119880

1205971198750

120597119877

) minus

1

119877

120597

120597120579

(

31198672

0

119880

1205971198750

120597120579

)

120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597

119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597

119867

)

0

) = 12119877

(30)


119867)0


119867)0 give the


119870119900 =

12058301205961198874

ℎ3119898

119870

119862119900 =

12058301198874

ℎ3119898

119862

(31)





Dyn

amic

coeffi

cien

ts

Stiffness (Nm)

1011

1010

109

108

107

Damping (Nlowastsm)




119905119890 = 1199050 + 120572

119901119900

119888119900120588119900119902119900

(32)


119901119900 = int

120593

0

int

119903119900

119903119894

(minus

ℎ

2

120597119901

120597120579

minus

1205831199032120596

ℎ

) 119903120596d119903 d120579

119902119900 = int

119903119900

119903119894

(minus

ℎ3

12120583

120597119901

119903120597120579

+

1

2

ℎ119903120596) d119903 + int120593

0

(minus

ℎ3

12120583

120597119901

120597119903

) 119903d120579

(33)

















XZY

(a) Profile

XZ

Y

(b) Front view
























20 40 60 80 100 120 140 160 180 20010

20

30

40

50

60

70

80


Freq

uenc

y (H

z)

1st2nd



1 15 2 25 3 35 4 45 5 55 6 65 7230

235

240

245

250

255

260

265


1st f

requ

ency

(Hz)

Slow region

times109

Rapid region





08 1 12 14 16 18 2220

225

230

235

240

245


1st f

requ

ency

(Hz)

times109



Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8












Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











T119887 = [1 0

minus1198721199001205962+ 119870119890 1

] (8)



119880

119865

= [

119871 0

0 119864119878

]

119880

119865

= S119880119865

(9)



T = Sminus1TS (10)



Τ119901 =[

[

[

[

1 0

minus

1198721199011205962119871

119864119878

1

]

]

]

]

T119904119894 =[

[

[

cos (119896119871 119894)sin (119896119871 119894)119896119871 119894

minus119896119871 119894 sin (119896119871 119894) cos (119896119871 119894)

]

]

]

T119887 =[

[

[

[

1 0

(minus1198721199001205962+ 119870119890) 119871

119864119878

1

]

]

]

]

Τ119888 =[

[

[

1 0

minus

1198721198881205962119871

119864119878

1

]

]

]

(11)



119880

119903

119865

119903

=[

[

11987911 11987912

11987921 11987922

]

]

119880

119897

119865

119897

(12)




119880119903

119901

119865119903

119901

=[

[

1 0

minus11987211990112059621

]

]

119880119897

119901

119865119897

119901

+

0

119865119901

(13)


119880119903

119901

119865119903

119901

1

=

[

[

[

[

[

1 0 0

minus11987211987512059621 119865119901

0 0 1

]

]

]

]

]

119880119897

119901

119865119897

119901

1

= Te119901

119880119897

119901

119865119897

119901

1

(14)




Te = Te119888Te119904119891Te119887Te119904119904Te119901 (15)


119880119901 = minus11987911989023

11987911989021

(16)



119865119900 = (119870119900 + 119895120596119862119900) (119880119900 minus 119880119887)

119865119887 = 119870119887119880119887

(17)









120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

(18)



(6 minus

1198672

1198801198772

120597119875

120597120579

)

120597119879

120597120579

minus

1198672

119880

120597119875

120597119877

120597119879

120597119877

=

1212058301198872120596

119888119900120588119900ℎ21198981199050

1198801198772

1198672+

1198672

12119880

[(

120597119875

119877120597120579

)

2

+ (

120597119875

120597119877

)

2

]

(19)



119877 =

119903

119887

119867 =

ℎ

ℎ119898

119880 =

120583

1205830

119875 =

119901ℎ2

119898

12058301205961198872 119879 =

119905

1199050

(20)



119867 = 1 +

120574119901119887

ℎ119898




lglg(1205830

120588119900

119880 + 06) = 119860 minus 119861lg (1199050119879 + 27315) (22)




119865119905 = 119885int

120593

0

int

119903119900

119903119894

119901119903d119903 d120579

119872 = int

120593

0

int

119903119900

119903119894

119901119903

2 sin (120579119901 minus 120579) d119903 d120579(23)


Pad

Collar

120574p

ro ri

b

120593

O

OOp(rp 120579p)


OutletInlet

11

2

2

3

3

i

R

j

n +1

m + 1

120579

(i minus 12 j)

(i j + 12)

(i j)(i minus 1 j)

(i j + 1)

(i j minus 1)

(i + 12 j)

(i j minus 12)

(i + 1 j)



119879119901 = 1198701198791205881199081198992119863

4 (24)



119879119901 = 119879119890(119899

119899119890

)

2

(25)




119867 = 1198670 + Δ119867 (26)


119875 = 1198750 + (120597119875

120597119867

)

0

Δ119867 + (

120597119875

120597

119867

)

0

Δ

119867 (27)



119870 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597119867

)

0

119877d119877d120579

119862 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597

119867

)

0

119877d119877d120579

(28)


120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

+ 12119877

120597119867

120597119905

(29)








120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597119867

)

0

)

= minus

120597

120597119877

(

31198771198672

0

119880

1205971198750

120597119877

) minus

1

119877

120597

120597120579

(

31198672

0

119880

1205971198750

120597120579

)

120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597

119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597

119867

)

0

) = 12119877

(30)


119867)0


119867)0 give the


119870119900 =

12058301205961198874

ℎ3119898

119870

119862119900 =

12058301198874

ℎ3119898

119862

(31)





Dyn

amic

coeffi

cien

ts

Stiffness (Nm)

1011

1010

109

108

107

Damping (Nlowastsm)




119905119890 = 1199050 + 120572

119901119900

119888119900120588119900119902119900

(32)


119901119900 = int

120593

0

int

119903119900

119903119894

(minus

ℎ

2

120597119901

120597120579

minus

1205831199032120596

ℎ

) 119903120596d119903 d120579

119902119900 = int

119903119900

119903119894

(minus

ℎ3

12120583

120597119901

119903120597120579

+

1

2

ℎ119903120596) d119903 + int120593

0

(minus

ℎ3

12120583

120597119901

120597119903

) 119903d120579

(33)

















XZY

(a) Profile

XZ

Y

(b) Front view
























20 40 60 80 100 120 140 160 180 20010

20

30

40

50

60

70

80


Freq

uenc

y (H

z)

1st2nd



1 15 2 25 3 35 4 45 5 55 6 65 7230

235

240

245

250

255

260

265


1st f

requ

ency

(Hz)

Slow region

times109

Rapid region





08 1 12 14 16 18 2220

225

230

235

240

245


1st f

requ

ency

(Hz)

times109



Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8












Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

(18)



(6 minus

1198672

1198801198772

120597119875

120597120579

)

120597119879

120597120579

minus

1198672

119880

120597119875

120597119877

120597119879

120597119877

=

1212058301198872120596

119888119900120588119900ℎ21198981199050

1198801198772

1198672+

1198672

12119880

[(

120597119875

119877120597120579

)

2

+ (

120597119875

120597119877

)

2

]

(19)



119877 =

119903

119887

119867 =

ℎ

ℎ119898

119880 =

120583

1205830

119875 =

119901ℎ2

119898

12058301205961198872 119879 =

119905

1199050

(20)



119867 = 1 +

120574119901119887

ℎ119898




lglg(1205830

120588119900

119880 + 06) = 119860 minus 119861lg (1199050119879 + 27315) (22)




119865119905 = 119885int

120593

0

int

119903119900

119903119894

119901119903d119903 d120579

119872 = int

120593

0

int

119903119900

119903119894

119901119903

2 sin (120579119901 minus 120579) d119903 d120579(23)


Pad

Collar

120574p

ro ri

b

120593

O

OOp(rp 120579p)


OutletInlet

11

2

2

3

3

i

R

j

n +1

m + 1

120579

(i minus 12 j)

(i j + 12)

(i j)(i minus 1 j)

(i j + 1)

(i j minus 1)

(i + 12 j)

(i j minus 12)

(i + 1 j)



119879119901 = 1198701198791205881199081198992119863

4 (24)



119879119901 = 119879119890(119899

119899119890

)

2

(25)




119867 = 1198670 + Δ119867 (26)


119875 = 1198750 + (120597119875

120597119867

)

0

Δ119867 + (

120597119875

120597

119867

)

0

Δ

119867 (27)



119870 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597119867

)

0

119877d119877d120579

119862 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597

119867

)

0

119877d119877d120579

(28)


120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

+ 12119877

120597119867

120597119905

(29)








120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597119867

)

0

)

= minus

120597

120597119877

(

31198771198672

0

119880

1205971198750

120597119877

) minus

1

119877

120597

120597120579

(

31198672

0

119880

1205971198750

120597120579

)

120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597

119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597

119867

)

0

) = 12119877

(30)


119867)0


119867)0 give the


119870119900 =

12058301205961198874

ℎ3119898

119870

119862119900 =

12058301198874

ℎ3119898

119862

(31)





Dyn

amic

coeffi

cien

ts

Stiffness (Nm)

1011

1010

109

108

107

Damping (Nlowastsm)




119905119890 = 1199050 + 120572

119901119900

119888119900120588119900119902119900

(32)


119901119900 = int

120593

0

int

119903119900

119903119894

(minus

ℎ

2

120597119901

120597120579

minus

1205831199032120596

ℎ

) 119903120596d119903 d120579

119902119900 = int

119903119900

119903119894

(minus

ℎ3

12120583

120597119901

119903120597120579

+

1

2

ℎ119903120596) d119903 + int120593

0

(minus

ℎ3

12120583

120597119901

120597119903

) 119903d120579

(33)

















XZY

(a) Profile

XZ

Y

(b) Front view
























20 40 60 80 100 120 140 160 180 20010

20

30

40

50

60

70

80


Freq

uenc

y (H

z)

1st2nd



1 15 2 25 3 35 4 45 5 55 6 65 7230

235

240

245

250

255

260

265


1st f

requ

ency

(Hz)

Slow region

times109

Rapid region





08 1 12 14 16 18 2220

225

230

235

240

245


1st f

requ

ency

(Hz)

times109



Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8












Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Pad

Collar

120574p

ro ri

b

120593

O

OOp(rp 120579p)


OutletInlet

11

2

2

3

3

i

R

j

n +1

m + 1

120579

(i minus 12 j)

(i j + 12)

(i j)(i minus 1 j)

(i j + 1)

(i j minus 1)

(i + 12 j)

(i j minus 12)

(i + 1 j)



119879119901 = 1198701198791205881199081198992119863

4 (24)



119879119901 = 119879119890(119899

119899119890

)

2

(25)




119867 = 1198670 + Δ119867 (26)


119875 = 1198750 + (120597119875

120597119867

)

0

Δ119867 + (

120597119875

120597

119867

)

0

Δ

119867 (27)



119870 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597119867

)

0

119877d119877d120579

119862 = int

120593

0

int

119877119900

119877119894

(

120597119875

120597

119867

)

0

119877d119877d120579

(28)


120597

120597119877

(

1198771198673

119880

120597119875

120597119877

) +

1

119877

120597

120597120579

(

1198673

119880

120597119875

120597120579

) = 6119877

120597119867

120597120579

+ 12119877

120597119867

120597119905

(29)








120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597119867

)

0

)

= minus

120597

120597119877

(

31198771198672

0

119880

1205971198750

120597119877

) minus

1

119877

120597

120597120579

(

31198672

0

119880

1205971198750

120597120579

)

120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597

119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597

119867

)

0

) = 12119877

(30)


119867)0


119867)0 give the


119870119900 =

12058301205961198874

ℎ3119898

119870

119862119900 =

12058301198874

ℎ3119898

119862

(31)





Dyn

amic

coeffi

cien

ts

Stiffness (Nm)

1011

1010

109

108

107

Damping (Nlowastsm)




119905119890 = 1199050 + 120572

119901119900

119888119900120588119900119902119900

(32)


119901119900 = int

120593

0

int

119903119900

119903119894

(minus

ℎ

2

120597119901

120597120579

minus

1205831199032120596

ℎ

) 119903120596d119903 d120579

119902119900 = int

119903119900

119903119894

(minus

ℎ3

12120583

120597119901

119903120597120579

+

1

2

ℎ119903120596) d119903 + int120593

0

(minus

ℎ3

12120583

120597119901

120597119903

) 119903d120579

(33)

















XZY

(a) Profile

XZ

Y

(b) Front view
























20 40 60 80 100 120 140 160 180 20010

20

30

40

50

60

70

80


Freq

uenc

y (H

z)

1st2nd



1 15 2 25 3 35 4 45 5 55 6 65 7230

235

240

245

250

255

260

265


1st f

requ

ency

(Hz)

Slow region

times109

Rapid region





08 1 12 14 16 18 2220

225

230

235

240

245


1st f

requ

ency

(Hz)

times109



Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8












Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597119867

)

0

)

= minus

120597

120597119877

(

31198771198672

0

119880

1205971198750

120597119877

) minus

1

119877

120597

120597120579

(

31198672

0

119880

1205971198750

120597120579

)

120597

120597119877

(

1198771198673

0

119880

120597

120597119877

(

120597119875

120597

119867

)

0

) +

1

119877

120597

120597120579

(

1198673

0

119880

120597

120597120579

(

120597119875

120597

119867

)

0

) = 12119877

(30)


119867)0


119867)0 give the


119870119900 =

12058301205961198874

ℎ3119898

119870

119862119900 =

12058301198874

ℎ3119898

119862

(31)





Dyn

amic

coeffi

cien

ts

Stiffness (Nm)

1011

1010

109

108

107

Damping (Nlowastsm)




119905119890 = 1199050 + 120572

119901119900

119888119900120588119900119902119900

(32)


119901119900 = int

120593

0

int

119903119900

119903119894

(minus

ℎ

2

120597119901

120597120579

minus

1205831199032120596

ℎ

) 119903120596d119903 d120579

119902119900 = int

119903119900

119903119894

(minus

ℎ3

12120583

120597119901

119903120597120579

+

1

2

ℎ119903120596) d119903 + int120593

0

(minus

ℎ3

12120583

120597119901

120597119903

) 119903d120579

(33)

















XZY

(a) Profile

XZ

Y

(b) Front view
























20 40 60 80 100 120 140 160 180 20010

20

30

40

50

60

70

80


Freq

uenc

y (H

z)

1st2nd



1 15 2 25 3 35 4 45 5 55 6 65 7230

235

240

245

250

255

260

265


1st f

requ

ency

(Hz)

Slow region

times109

Rapid region





08 1 12 14 16 18 2220

225

230

235

240

245


1st f

requ

ency

(Hz)

times109



Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8












Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























XZY

(a) Profile

XZ

Y

(b) Front view
























20 40 60 80 100 120 140 160 180 20010

20

30

40

50

60

70

80


Freq

uenc

y (H

z)

1st2nd



1 15 2 25 3 35 4 45 5 55 6 65 7230

235

240

245

250

255

260

265


1st f

requ

ency

(Hz)

Slow region

times109

Rapid region





08 1 12 14 16 18 2220

225

230

235

240

245


1st f

requ

ency

(Hz)

times109



Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8












Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























20 40 60 80 100 120 140 160 180 20010

20

30

40

50

60

70

80


Freq

uenc

y (H

z)

1st2nd



1 15 2 25 3 35 4 45 5 55 6 65 7230

235

240

245

250

255

260

265


1st f

requ

ency

(Hz)

Slow region

times109

Rapid region





08 1 12 14 16 18 2220

225

230

235

240

245


1st f

requ

ency

(Hz)

times109



Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8












Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










08 1 12 14 16 18 2220

225

230

235

240

245


1st f

requ

ency

(Hz)

times109



Am

plitu

de (m

)

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8












Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Am

plitu

de (N

)


106

105

104

103

102

101


02010

403050

7090

6080

100

040 20

8060120100

160140

200180

Frequency (Hz)


Forc

e tra

nsm

issib

ility

102

100

10minus2





0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2

11 times 109 Nm









Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Range of movement


0 025 05 075 1 125 15 175 223

24

25

26

27

1st f

requ

ency

(Hz)


68

69

70

2nd

frequ

ency

(Hz)

1st2nd




0 2 4 6 8 10 12 14088

090

092

094

096

098

100

Length (m)

Am

plitu

de (m

)

0m05m10m

15m20m





0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 001 002 003 004 005 006 007 008

0

05

1

15

Time (s)

Am

plitu

de (m

)

minus15

minus1

minus05

times10minus5

0m05m10m

15m20m


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

Forc

e tra

nsm

issib

ility

102

101

100

10minus1

10minus2



8 Conclusions












Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Yes








Over

Yes

Yes

No

No

Mod

ify p

ad ti

lting

angl

e

Mod

ify m

inim

um fi

lm th

ickn

ess

No




119875119894119895 =


119860 119894119895 + 119861119894119895 + 119862119894119895 + 119863119894119895

119879119894119895 =


119886119894119895 minus 119887119894119895 minus 119888119894119895 + 119889119894119895

(A1)



119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119860 119894119895 = (

1198673

119880

)

119894+(12)119895

119861119894119895 = (1198673

119880

)

119894minus(12)119895

119862119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895+(12)

119863119894119895 = 119877119894119895(Δ120579

Δ119877

)

2

(

1198771198673

119880

)

119894119895minus(12)

119864119894119895 = 6119877

2

119894119895Δ120579 (119867119894+(12)119895 minus 119867119894minus(12)119895)

119886119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119887119894119895 =1

2Δ120579

[(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(6 minus

1198672

1198801198772

120597119875

120597120579

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119888119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

+

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119889119894119895 =1

2Δ119877

[(

1198672

119880

120597119875

120597119877

)

119894119895

minus

1003816100381610038161003816100381610038161003816100381610038161003816

(

1198672

119880

120597119875

120597119877

)

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

]

119890119894119895 =

1212058301198872120596

119888V120588ℎ21198981199050

times (

1198801198772

1198672)

119894119895

+ (

1198672

12119880

)

119894119895

times [(

1

119877119894119895

119875119894+1119895 minus 119875119894minus1119895

2Δ120579

)

2

+(

119875119894119895minus1 minus 119875119894119895+1

2Δ119877

)

2

]

(A2)


119875

119896

119894119895= 119875

119896minus1

119894119895+ 120572 (119875

119896

119894119895minus 119875

119896minus1

119894119895) (1 lt 120572 lt 2)

119879

119896

119894119895= 119879

119896minus1

119894119895+ 120573 (119879

119896

119894119895minus 119879

119896minus1

119894119895) (0 lt 120573 lt 1)

(A3)


sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896

119894119895minus 119875119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119875119896119894119895

10038161003816100381610038161003816

le 10

minus3

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896

119894119895minus 119879119896minus1

119894119895

10038161003816100381610038161003816

sum

119898

119895=2sum

119899

119894=2

10038161003816100381610038161003816119879119896119894119895

10038161003816100381610038161003816

le 10

minus4

(A4)




Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article propeller excitation of longitudinal vibration characteristics...

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