research article one more tool for understanding resonance

Hindawi Publishing CorporationJournal of EngineeringVolume 2013 Article ID 414109 16 pageshttpdxdoiorg1011552013414109

Research ArticleOne More Tool for Understanding Resonance andthe Way for a New Definition

Emanuel Gluskin1 Doron Shmilovitz2 and Yoash Levron2

1 Electrical Engineering Department Faculty of Engineering Kinneret College on the Sea of Galilee Jordan Valley 15132 Israel2 Electrical Engineering Department Faculty of Engineering Tel-Aviv University Israel

Correspondence should be addressed to Emanuel Gluskin gluskineebguacil

Received 20 August 2012 Accepted 2 November 2012

Academic Editor H P S Abdul Khalil

Copyright copy 2013 Emanuel Gluskin et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We propose the application of graphical convolution to the analysis of the resonance phenomenon This time-domain approachencompasses both the finally attained periodic oscillations and the initial transient period It also provides interesting discussionconcerning the analysis of nonsinusoidal waves based not on frequency analysis but on direct consideration of waveforms andthus presenting an introduction to Fourier series Further developing the point of view of graphical convolution we arrive at a newdefinition of resonance in terms of time domain

1 Introduction

11 General The following material fits well into an ldquoIntro-duction to Linear Systemsrdquo or ldquoMechanicsrdquo and is relevantto a wide range of technical and physics courses sincethe resonance phenomenon has long interested physicistsmathematicians chemists engineers and nowadays alsobiologists

The complete resonant response of an initially unexcitedsystem has two different distinguishable parts and there arerespectively two basic definitions of resonance significantlydistanced from each other

In the widely adopted textbook [1] written for physicistsresonance is defined as a linear increase of the amplitudeof oscillations in a lossless oscillatory system obtained whenthe system is pumped with energy by a sinusoidal forceat the correct frequency Figure 1 schematically shows theldquoenveloperdquo of the resonant oscillations being developed

Thus a lossless system under resonant excitation absorbsmore and more energy and a steady state is never reachedIn other words in the lossless system the amplitude of thesteady state and the ldquoquality factorrdquo 119876 (having a somewhatsemantic meaning in such a system) are infinite at resonance

However the slope of the envelope is always finite itdepends on the amplitude of the input function and not on

119876 Though the steady-state response will never be reachedin an ideal lossless system the linear increase in amplitudeby itself has an important sense When a realistic physicalsystem absorbs energy resonantly say in the form of photonsof electromagnetic radiation there indeed is some period(still we can ignore power losses say some back radiation)during which the systemrsquos energy increases linearly in timeThe energy absorption is immediate upon appearance of theinfluence and the rate of the absorption directlymeasures theintensity of the input

One notes that the energy pumping into the system at theinitial stage of the resonance process readily suggests that thesinusoidal waveformof the input function is not necessary forresonance it is obvious (think eg about swinging a swing bykicking it) that the energy pumping can occur for other inputwaveforms as well This is a heuristically important point ofthe definition of [1]

The physical importance of the initial increase in oscilla-tory amplitude is associated not only with the energy pump-ing the informational meaning is also important Assumefor instance that we speak about the start of oscillations ofthe spatial positions of the atoms of a medium caused by anincoming electromagnetic wave Since this start is associatedwith the appearance of the wave it can be also associated with

2 Journal of Engineering

Envelope for theextremal (amplitude)

values

0

119905

Figure 1 The definition of resonance [1] as linear increase of theamplitude (The oscillations fill the angle of the envelope) Theinfinite process of increase of the amplitude is obtained because ofthe assumption of losslessness of the system

the registration of a signal Later on the established steady-state oscillations (that are associated because of the radiationof the atoms with the refraction factor of the medium)influence the velocity of the electromagnetic wave in themedium As [2] stressesmdasheven if this velocity is larger thanthe velocity of light (for refraction factor 119899 lt 1 ie when thefrequency of the incoming wave is slightly higher than that ofthe atoms oscillators)mdashthis does not contradict the theory ofrelativity because there is already no signal Registration ofany signal and its group velocity is associated with a (forced)transient process

A more pragmatic argument for the importance of anal-ysis of the initial transients is that for any application of asteady-state response especially in modern electronics wehave to know how much time is needed for it to be attainedand this relates in particular to the resonant processes Thisis relevant to the frequency range in which the device has tobe operated

Contrary to [1] in textbooks on the theory of electricalcircuits (eg [3ndash5]) and mechanical systems resonance isdefined as the established sinusoidal response with a relativelyhigh amplitude proportional to 119876 Only this definitiondirectly associated with frequency domain analysis is widelyaccepted in the engineering sciences According to thisdefinition the envelope of the resonant oscillations (Figure 2)looks even simpler than in Figure 1 it is given by two horizon-tal lines This would be so for any steady-state oscillationsand the uniqueness is just by the fact that the oscillationamplitude is proportional to 119876

After being attained the steady-state oscillations continueldquoforeverrdquo and the parameters of the ldquofrequency responserdquocan be thus relatively easily measured Nevertheless thesimplicity of Figure 2 is a seeming one because it is notknown when the steady amplitude becomes established andcertainly the ldquofrequency responserdquo is not an immediateresponse to the input signal


values

0

119905

sim119876

Figure 2 The envelope of resonant oscillations according to thedefinition of resonance in [3 4] andmany other technical textbooksWhen this steady state is attained

119891out (119905)

1120574119905

Figure 3 The illustration for 119876 = 10 of the resonant response of asecond-order circuit Note that we show a case when the excitationis precisely at the resonant frequency and the notion of ldquopurelyresonant oscillationsrdquo applies here to thewhole process and not onlyto the final steady-state part

Thus we do not know via the definition of [1] when theslope will finish and we do not know via the definition of [3ndash5] when the steady state is obtained

We shall call the definition of [1] ldquothe ldquoQ-trdquo definitionrdquosince the value of 119876 can be revealed via duration of theinitialtransient process in a real systemThe commonly useddefinition [3ndash5] of resonance in terms of the parameters ofthe sustained response will be called ldquothe ldquoQ-ardquo definitionrdquowhere ldquoardquo is an abbreviation for ldquoamplituderdquo

Figure 3 illustrates the actual development of resonancein a second-order circuit The damping parameter 120574 will bedefined in Section 3

The Q-t and Q-a parts of the resonant oscillations arewell seen For such a not very high 119876 (ie 1120574 not muchlarger than the period of the oscillations) the period of fairinitial linearity of the envelope includes only some half periodof oscillations but for a really high 119876 it can include manyperiods The whole curve shown is the resonant responseThis response can be obtained when the external frequencyis closing the self-frequency of the system from the beats ofthe oscillations (analytically explained by the formulae foundin Section 3) shown in Figure 4

Note that the usual interpretation is somewhat differentIt just says that the linear increase of the envelope shown inFigure 1 can be obtained from the first beat of the periodic

Journal of Engineering 3

119891out (119905)

119905

Figure 4 Possible establishing of the situation shown in Figure 3through beats while adjustment of the frequency We can interpretresonance as ldquofiltrationrdquo of the beats when the resonant frequency isfound

beats observed in a lossless system Contrary to that weobserve the beats in a systemwith losses and after adjustmentof the external frequency obtain the whole resonant responseshown in Figure 3

Our treatment of the topic of resonance for teaching pur-poses is composed of threemain parts shown in Figure 5Thefirst part briefly recalls traditional ldquophasorrdquo material relevantonly to the Q-a part which is necessary for introduction ofthe notations The next part includes some simple thoughusually omitted arguments showing why the phasor analysisis insufficient Finally the third part includes the new toolwhich is complementary to the classical approach of [1]and leads to a nontrivial generalization of the concept ofresonance

Our notations need minor comments As is customary inelectrical engineering the notation for radicminus1 is 119895 The smallitalic Latin ldquovrdquo 119907 is voltage in the time domain (ie a realvalue) means phasor that is a complex number in thefrequency domain 120582 is the dummy variable of integrationin a definite integral of the convolution type It is measuredin seconds and the difference 119905 minus 120582 where 119905 is time oftenappears

2 Some Advice to the Teacher

First we deal here with a lot of pedagogical sciencemdashinprinciple the issues are not new but are often missed in theclassroom as far as we know no such complete scheme of thenecessary arguments for teaching resonance exists Perhapsthis is because some issues indeed require a serious revisitingand time is often limited due to overloaded teaching plans andschedules That the results of this ldquoeconomyrdquo are not bright isseen first of all from the alreadymentioned fact that electricalengineering (EE) students often learn resonance only viaphasors and are not concerned with the time needed for thevery important steady state to be established The resonancephenomenon is so physically important that it is taught to

technical students many times inmechanics in EE in opticsand so forth However all this repeated teaching is actuallyequivalent to the use of phasors that is relates only to theestablished steady state

Furthermore the teachers (almost all of them) missthe very interesting possibility to exhibit the power of theconvolution-integral analysis for studying the development ofa resonant state In our opinion this demonstration makesthe convolution integral a more interesting tool this reallyis one of the best applications of the ldquographical convolutionrdquowhich should not bemissed in any programThe convolutionoutlook well unites the view of resonance as a steady stateby engineers and the view of resonance as energy pumpinginto a system by physicists The arguments of the graphicalconvolution also enable one to easily see (before knowingFourier series) that a nonsinusoidal periodic input wave cancause resonance just as the sinusoidal one does Thus thesearguments can be used also as an explanation of the physicalmeaning of the Fourier expansion Our classroom experienceshows that the average student can understand this materialand finds it interesting

Thus regarding the use of the pedagogical material wewould advise the teacher of the EE students to return to thetopic of resonance (previously taught via phasors) when thestudents start with convolution

Finally the present work includes some new sciencewhich can be also related to teaching but perhaps at graduatelevel depending on the level of the students or the universityWe mean the generalization of the concept of resonanceconsidered in Section 5 It is logical that if the convolutionintegral can show resonance (or resonant conditions) directlynot via Fourier analysis then this ldquoshowingrdquo exposes a generaldefinition of resonance Furthermore since mathematicallythe convolution integral can be seenmdashwith a proper writingof the impulse response in the integrandmdashas a scalar productit is just natural to introduce into the consideration theoutlook of Euclidean space

The latter immediately suggests a geometric interpreta-tion of resonance in functional terms because it is clear whatis the condition (here the resonant one) for optimizationof the scalar product of two normed vectors As a wholewe simply replace the traditional requirement of equality ofsome frequencies to the condition of correlation of two timefunctions which includes the classical sinusoidal (and thesimplest oscillator) case as a particular one

The geometrical consideration leads to a symmetry argu-ment since the impulse response ℎ(119905) is the only givenldquovectorrdquo any optimal input ldquovectorrdquo has to be similarlyoriented there simply is no other selected direction Theassociated writing 119891inp sim ℎ that is often used here just forbrevity precisely means the adjustment of the waveform of119891inp(119905) to that of ℎ(119905) by the following two steps

(1) Set 119891inp(119905) sim minusℎ(119879 minus 119905) in the interval 0 lt 119905 lt 119879(2) Continue this waveform periodically for 119905 gt 119879

It is relevant here that for weak power losses typical for allresonant systems the damping of ℎ(119905) in the first period canbe ignored which should be a simplifying circumstance for


Resonance

The toolgraphical

convolution

Steady stateenergy balance

The toolphasors

The dynamicconsideration

(1) Thetransient partshould not bemissed in the

standarddiscussion (logic

and important timeconstants)

(2) Resonanceas a limiting caseof beats or as a

I II III

Directdefinition ofresonance in

terms ofwaveforms

Establishing of thesteady state

Energy pumping into thesystem during the

transientThe role of periodicity

anda way to Fourier series

ldquofiltrationrdquo of beats

Figure 5 The methodological points regarding the study of resonance in the present work

creation of the periodic 119891inp(119905) The way of the adjustment of119891inp(119905) reflects the fact that the Euclidean space can relate toone period

Both because of the somewhat higher level of the mathe-matical discussion and some connection with the theory ofldquomatched filtersrdquo usually related to special courses (whichcould not be discussed here) it seems that this final materialshould be rather given for graduate students However wealso believe that a teacher will find here some pedagogicalmotivation and will be able to convey more lucid treatmentthan we succeeded to doing Thus the question regardingthe possibility of teaching the generalized resonance toundergraduate students remains open

Some other nontrivial points deserving pedagogicaljudgement or analytical treatment appear already in the useof the convolutionThis means the replacement of the weaklydamping ℎ(119905) of an oscillatory system by the not damping butcut function ℎ

119878(119905) shown in Figure 11 and the problem of

definition of the damping parameter 120574 for the tending to zeroℎ(119905) of a complicated oscillatory circuit A possible way forthe latter can be by observation (this is not yet worked out)of some averages for example how the integral of ℎ2 or of|ℎ| over the fixed-length interval (119905 119905 + Δ) is decreased withincrease in 119905

3 Elementary Approaches

31 The Second-Order Equation The background formulaefor both the Q-t and Q-a parts of the resonant responsecan be given by the Kirchhoff voltage equation for the

electrical current 119894(119905) in a series RLC (resistor-inductor-capacitor) circuit driven from a source of sinusoidal voltagewith amplitude 119907

119898

119871119889119894

119889119905+ 119877119894 +

1

119862int 119894 (119905) 119889119905 = 119907119898 sin120596119905 (1)

Differentiating (1) and dividing by 119871 = 0 we obtain

1198892119894

1198891199052+ 2120574

119889119894

119889119905+ 1205962

119900119894 (119905) =

120596119907119898

119871cos120596119905 (2)

with the damping factor 120574 = 1198772119871 and the resonant frequency120596119900= 1radic119871119862For purely resonant excitation the input sinusoidal func-

tion is at frequency 120596 = 120596119900 or at a very close frequency 120596

119889

as defined below in (6)

32The Time-Domain Argument The full solution of (2) canbe explicitly composed of two terms the first denoted as119894ℎ originates from the homogeneous (ℎ) equation and the

second denoted as 119894119891119904 represents the finally obtained (119891119904)

periodic oscillations that is is the simplest (but not the onlypossible) partial solution of the forced equation

119894 (119905) = 119894ℎ (119905) + 119894119891119904 (119905) (3)

It is important that the zero initial conditions cannot befitted by the second term in (3) 119894

119891119904(119905) continued backward in

time to 119905 = 0 (Indeed no sinusoidal function satisfies boththe conditions 119891(119905) = 0 and 119889119891119889119905 = 0 at any point) Thus


it is obvious that a nonzero term 119894ℎ(119905) is needed in (3) This

term is

119894ℎ (119905) = 119890

minus120574119905(1198701cos120596119889119905 + 1198702sin120596119889119905) (4)

where at least one of the constants1198701 and1198702 nonzero

Furthermore it is obvious from (4) that the time neededfor 119894ℎ(119905) to decay is of the order of 1120574 sim 119876119879119900 (compareto (9)) However according to the two-term structure of (3)the time needed for 119894119891119904(119905) to be established that is for 119894(119905) tobecome 119894

119891119904(119905) is just the time needed for 119894

ℎ(119905) to decay Thus

the established ldquofrequency responserdquo is attained only after thesignificant time of order 119876119879

119900sim 119876

Unfortunately this elementary logic argument followingfrom (3) is missed in [3ndash5] and many other technicaltextbooks that ignore the Q-t part of the resonance anddirectly deal only with the Q-a part

However form (3) is also not optimal here because it isnot explicitly shown that for zero initial conditions not only119894119891119904(119905) but also the decaying 119894

ℎ(119905) are directly proportional to

the amplitude (or scaling factor) 119907119898of the input wave

That is from the general form (3) alone it is not obviousthat when choosing zero initial conditions we make theresponse function as a whole (including the transient) to beproportional to 119907

119898 appearing in (1) that is to be a tool for

studying the input function at least in the scaling senseIt would be better to have one expressionterm from

which this feature of the response is well seen Such a formulaappears in Section 4

33 The Phasor Analysis of the Q-a Part Let us now brieflyrecall the standard phasor (impedance) treatment of the finalQ-a (steady-state) part of a systemrsquos response We can focushere only on the results associated with the amplitude thephase relations follow straightforwardly from the expressionfor the impedance [3 4]

In order to characterize the Q-a part of the response weuse the commonnotations of [3 4] the damping factor of theresponse 120574 equiv 1198772119871 the resonant frequency 120596119900 = 1radic119871119862 thequality factor

119876 =120596119900

2120574=120596119900119871

119877=radic119871119862

119877 (5)

and the frequency at which the system self-oscillates

120596119889equiv radic1205962

119900minus 1205742 asymp 120596

119900minus1205742

2120596119900= 120596119900(1 minus

1

41198762) asymp 120596

119900 (6)

Note that it is assumed that 41198762 ≫ 1 and thus 120596119889and 120596

119900are

practically indistinguishable Thus although we never ignore120574 per se the much smaller value 120574119876 sim 1205742120596119900 can be ignoredWhen speaking about ldquoprecise resonant excitationrdquo we shallmean setting120596with this degree of precision but whenwriting120596 = 120596119900 we shall mean that 120596 minus 120596

119900= 119874(120574) and not 119874(120574119876)

Larger than 119874(120574) deviations of 120596 from 120596119900are irrelevant to

resonance

The impedance of the series circuit is 119885(119895120596) = 119877 + 119895120596119871 +(1119895120596119862) and the phasor approach simply gives the amplitudeof the steady-state solution of (2) as

119894119898 (120596) =

10038161003816100381610038161003816119868 (119895120596)

10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816

119885 (119895120596)

100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119877 + 119895120596119871 + (1119895120596119862)1003816100381610038161003816

=119907119898

radic1198772 + (120596119871 minus (1120596119862))2

(7)

For 120596minus120596119900 ≪ 120596119900 when 120596

2minus1205962

119900asymp 2120596119900(120596 minus 120596119900) asymp 2120596(120596 minus 120596119900)

119894119898 (120596) asymp

119907119898

2119871radic1205742 + (120596 minus 120596119900)2

(8)

From (8) the frequencies at ldquohalf-power levelrdquo for which119894(120596) = (1radic2)(119894

119898)max are defined by the equality (120596 minus 120596119900)

2=

1205742 from which we obtain 1205961 = 120596119900 minus 120574 and 1205962 = 120596119900 + 120574 that

is for the circuitrsquos frequency ldquopass-bandrdquo Δ120596 equiv 1205962 minus 1205961 wehave with the precision taken in the derivation of (8) thatΔ120596 = 2120574

It is remarkable that however small is 120574 it is easywhile working with the steady state to detect differences oforder 120574 between 120596 and 120596119900 using the resonant curveresponsedescribed by (8)

Figure 6 illustrates the resonance curve Though thisfigure is well known it is usually not stressed that since eachpoint of the curve corresponds to some steady state a certaintime is needed for the system to pass on from one point ofthe curve to another one and the sharper the resonance is themore time is needed The physical process is such that for asmall 120574 the establishment of this response takes a (long) timeof the order of

1

120574=120596119900

120574

1

120596119900

= 2119876119879119900

2120587=1

120587119876119879119900sim 119876119879119900 (119879119900 =

2120587

120596119900

) (9)

which is not directly seen from the resonance curveThe relation 1120574 sim 119876119879119900 for the transient period should

be remembered regarding any application of the resonancecurve in any technical device The case of a mistake causedby assuming a quicker performance for measuring inputfrequency by means of passing on from one steady state toanother is mentioned in [2] This mistake is associated withusing only the resonance curve that is thinking only in termsof the frequency response

4 The Use of Graphical Convolution

Wepass on to the constructive point the convolution integralpresenting the resonant response and its graphical treatmentIt is desirable for a good ldquosystem understandingrdquo of the topicthat the concepts of zero input response (ZIR) and zero stateresponse (ZSR) especially the latter one be known to thereader

Briefly ZSR is the partial response of the circuit whichsatisfies the zero initial conditions As 119905 rarr infin (and only


0 1205961 120596120596

2120596119900

(119894119898(120596))max

(119894119898(120596))maxradic2

119894119898(120596) 119894(119905) = 119894119898 cos(120596119905 + 120572)

119894119898 = 119894119898(120596) =119907119898

radic1198772 + (120596119871 minus 1120596119862)2

Figure 6 The resonance curve Δ120596 equiv 1205962minus 1205961= 2120574 119876 = 120596

119900Δ120596

119891inp (119905) ℎ(119905) 119891out (119905)

Figure 7 The input-output map (119891inp rarr 119891out(119905) = ZSR(119905)) givenby ldquoimpulse responserdquo ℎ(119905)

then) it becomes the final steady-steady response that isbecomes the simplest partial response (whose waveform canbe often guessed)

The appendix illustrates the concepts of ZIR and ZSR indetail using a first-order system and stressing the distinctionbetween the forms ZIR + ZSR and (3) of the response

Our system-theory tools are now the impulse (or shock)response ℎ(119905) (or Greenrsquos function) and the integral responseto 119891inp(119905) for zero initial conditions

119891out (119905) = (ℎ lowast 119891inp) (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= (119891inp lowast ℎ) (119905)

(10)

The convolution integral (10) is an example of ZSR andit is the most suitable tool for understanding the resonantexcitation

It is clear (contrary to (3)) that the total response (10) isdirectly proportional to the amplitude of the input function

Figure 7 shows our schematic systemOf course the system-theory outlook does not relate

only to electrical systems this ldquoblock-diagramrdquo can meaninfluence of a mechanical force on the position of a mass ora pressure on a piston or temperature at a point inside a gasand so forth

Note that if the initial conditions are zero they are simplynot mentioned If the input-output map is defined solely byℎ(119905) (eg when one writes in the domain of Laplace variable119865out(119904) = 119867(119904)119865inp(119904)) it is always ZSR

In order to treat the convolution integral it is usefulto briefly recall the simple example [5] of the first-ordercircuit influenced by a single square pulse The involved

A

0 Δ119905

ℎ(119905)119891inp (119905)

1120591

Figure 8The functions for the simplest example of convolution (Afirst-order circuit with an input block pulse)

1205820 119905

Δ

A

ℎ(120582)119891inp (119905 minus 120582)

Figure 9 The functions appearing in the integrand of the convo-lution integral (10) The ldquoblockrdquo 119891inp(119905 minus 120582) is riding (being moved)to the right on the 120582-axes as time passes We multiply the presentcurves in the interval 0 lt 120582 lt 119905 and according to (10) take the areaunder the result in this interval When 119905 lt Δ only the interval (0 119905)is relevant to (10) When 119905 gt Δ only the interval (119905 minus Δ 119905) is actuallyrelevant and because of the decay of ℎ(119905) 119891out(119905) becomes decaying

physical functions are shown in Figure 8 and the associatedldquointegrand situationrdquo of (10) is shown in Figure 9

It is graphically obvious from Figure 9 that the maximalvalue of 119891out(119905) is obtained for 119905 = Δ when the rectangularpulse already fully overlaps with ℎ(120582) but still ldquocatchesrdquo theinitial (highest) part of ℎ(120582) This simple observation showsthe strength of the graphical convolution for a qualitativeanalysis


41 The (Resonant) Case of a Sinusoidal Input Function Actingon the Second-Order System For the second-order systemwith weak losses we use for (10)

ℎ (119905) =1205962

119900

120596119889

119890minus120574119905 sin120596

119889119905

sim 119890minus120574119905 sin120596

119889119905 asymp 119890minus120574119905 sin120596

119900119905 120574 ≪ 120596

119900 (119876 ≫ 1)

(11)

As before we apply

119891inp (119905) = 119891119898 sin120596119889119905 asymp 119891119898 sin120596119900119905 (12)

Figure 10 builds the solution (10) step by step first ourℎ(120582) and119891inp(119905minus120582) (compare to Figure 9) then the product ofthese functions and finally the integral that is 119891out(119905) = 119878(119905)

On the upper graph the ldquotrainrdquo 119891inp(119905 minus 120582) travels to theright starting at 119905 = 0 on the middle graph we have theintegrandof (10)The area119891inp(119905) = 119878(119905)under the integrandrsquoscurve appears as the final result on the third graph

The extreme values of 119878(119905) are 119878(119896(120587120596119889)) obviously For119896 odd these are positive maxima because the overlaps in theupper drawing are then ldquo+rdquo with ldquo+rdquo and ldquondashrdquo with ldquondashrdquo For119896 even these are negative minima because we multiply theopposite polarities in the overlap 119891inp(119905 minus 120582)ℎ(120582) each timeThus 119878(120587120596

119889) gt 0 and 119878(2120587120596

119889) lt 0

In view of the basic role of the overlapping of 119891inp(119905 minus120582) with ℎ(120582) it is worthwhile to look forward a little andcompare Figure 10 to Figures 14 and 15 that relate to the caseof an input square wave For the upper border of integrationin (10) be 119905 = 119896(120587120596

119889) and for very weak damping of ℎ(120582) the

situations being compared are very similar The distinction isthat in order to obtain the extremes of119891inp(119905) we integrate inFigure 15 the absolute value of several sinusoidal pieces (half-waves) while in Figure 10 we integrate the squared sinusoidalpieces Since we integrate in each case 119896 similar pieces (allpositive giving a maximum of 119891out(119905) or all negative givinga minimum) the result of each such integration is directlyproportional to 119896

Thus if 120574 = 0 when ℎ(120582) is strictly periodic from theperiodic nature of also 119891inp(119905) it follows that

119891out (119896120587

120596119889

) sim (minus1)119896+1119896 sim 119896 (13)

for any integer 119896 which is a linear increase in the envelope forthe two very different input waves in the spirit of Figure 1

For a small but finite 120574 0 lt 120574 ≪ 120596119900 the initial linear

increase has high precision only for some first few 119896 when119905 sim 119879119900 sim 1120596119900 ≪ 1120574 that is 120574119905 ≪ 1 or 119890minus120574119905 asymp 1 (Thedamping of ℎ(119905)may be ignored for these 119896)

Observe that the finally obtained periodicity of 119891out(119905)follows only from that of 119891inp(119905) while the linear increaserequires periodicity of both 119891inp(119905) and ℎ(119905)

The above discussion suggests the following simplifi-cation of the impulse response of the circuit useful foranalysis of the resonant systemsThis simplification is a usefulpreparation for the rest of the analysis

42 A Simplified ℎ(119905) and the Associated Envelope of theOscillations Considering that the parameter 1120574 appears inthe above (and in Figure 3) as some symbolic border forthe linearity let us take a constructive step by suggesting ageometrically clearer situation when this border is artificiallymade sharp by introducing an idealizationsimplification ofℎ(119905) which will be denoted as ℎ119878(119905)

In this idealizationmdashthat seems to be no less reasonableand suitable in qualitative analysis than the usual use of thevague expression ldquosomewhere at ldquo119905rdquo of order 1120574rdquo we replaceℎ(119905) by a finite ldquopiecerdquo of nondamping oscillations of totallength 1120574

We thus consider that however weak the damping ofℎ(119905) is for sufficiently large 119905 when 119905 ≫ 1120574 sim 119876119879119900 wehave 119890minus120574119905 ≪ 1 that is the oscillations become stronglydamped with respect to the first oscillation For 119905 gt 1120574 thefurther ldquomovementrdquo of the function 119891inp(119905 minus 120582) to the right(see Figure 10 again) becomes less effective the exponentiallydecreasing tail of the oscillating ℎ(119905) influences (10) via theoverlapmore andmore weakly and as 119905 rarr infin119891out(119905) ceasesto increase and becomes periodic obviously

We simplify this qualitative vision of the process byassuming that up to 119905 = 1120574 there is no damping of ℎ(119905)but starting from 119905 = 1120574 ℎ(119905) completely disappears Thatis we replace the function 119890minus120574119905 sin120596

119889119905 by the function ℎ

119878(119905) =

[119906(119905)minus119906(119905minus1120574)] sin120596119900119905 where 119906(119905) is the unit step function

The factor 119906(119905) minus 119906(119905 minus 1120574) here is a ldquocutting windowrdquofor sin120596

119900119905 This is the formal writing of the ldquopiecerdquo of the

nondamping self-oscillations of the oscillator See Figure 11For ℎ

119878(119905) it is obvious that when the ldquotrainrdquo 119891inp(119905 minus

120582) crosses in Figure 10 the point 119905 = 1120574 the graphicalconstruction of (10) that is 119891out(119905) becomes a periodicprocedure Figuratively speaking we can compare ℎ119878(119905) witha railway station near which the infinite train 119891inp(119905 minus 120582)passes some wagons go away but similar new ones enter andthe total overlapping is repeated periodically

The same is also analytically obvious since when settingfor 119905 gt 1120574 the upper limit of integration in (10) as 1120574 wehave because of the periodicity of 119891inp(sdot) the integral

119891out (119905 gt1

120574) = int

1120574

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582 (14)

as a periodic function of 119905As is illustrated by Figure 12mdashwhich is an approximation

to the envelope shown inFigure 3mdashthe envelope of the outputoscillations becomes completely saturated for 119905 gt 1120574

Figure 12 clearly shows that both the amplitude of thefinally established steady-state oscillations and the timeneeded for establishing these oscillations are proportional to119876 while the initial slope is obviously independent of 119876

It is important that ℎ119878(119905) can be also constructed for more

complicated functions ℎ(119905) (for which it may be for instanceℎ(119905+1198792) = minus ℎ(119905)) and also then the graphical convolution iseasier formulated in terms of ℎ

119878(119905) As an example relevant to

the theoretical investigationsmdashapproximately presenting the


finp(t minus λ) =sinw(t minus λ)

t

t

t

sim eminusγλ sinwdλ

λ

λS

S

π

wd

2πwd

fout(t) =middot finp(t minus λ)dλ

fout( 2πwd

) = minus2 fout( π

wd)

h(λ)

h(λ) middot finp(t minus λ)

γ≪ wd asymp w0

int t0 h(λ)

Figure 10 Graphically obtaining the resonant response for a second-order oscillatory system and a sinusoidal input according to (10) Theenvelope (not shown) has to pass via the maxima and minima of 119891out(119905) appearing in the last graph

λ

1γ

Figure 11 The simplified ℎ(119905) (named ℎ119878(119905)) there is no damping at

0 lt 119905 lt 1120574 but for 119905 gt 1120574 it is identically zero that is we firstignore the damping of the real ℎ(119905) and then cut it completely Thisidealization expresses the undoubted fact that the interval 0 lt 119905 lt1120574 is dominant and makes the treatment simpler A small changein 1120574 which makes the oscillatory part more pleasing by includingin it just the (closest) integer number of the half waves as shownhere may be allowed and when using ℎ

119878(119905) in the following we shall

assume for simplicity that the situation is such

maximal values of the established oscillations obtained for119905119896≫ 1120574

1003816100381610038161003816119891out (119905119896)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816int

infin

0

119891inp (119905119896 minus 120582) ℎ (120582) 11988912058210038161003816100381610038161003816100381610038161003816 119905119896≫ 1120574 (15)

as

10038161003816100381610038161003816119891out (120574

minus1)10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816int

1120574

0

119891inp (120574minus1minus 120582) ℎ

119878 (120582) 119889120582

100381610038161003816100381610038161003816100381610038161003816 (16)

we can easily reduce using periodicity of 119891inp(119905) for anyoscillatory ℎ(119905) (and ℎ

119878(119905)) the analysis of the interval (0 1120574)

to that of a small interval as was for (0 120587120596119889) in Figure 10

1199051120574 sim119876

Figure 12 The envelope of 119891out(119905) obtained for the simplified ℎ(119905)shown in Figure 11

43 Nonsinusoidal Input Waves The advantage of the graph-ical convolution is not so much in the calculation aspect It iseasy for imagination (insight) procedure and it is a flexibletool in the qualitative analysis of the time processes Thegraphical procedure makes it absolutely clear that the reallybasic point for a resonant response is not sinusoidality butperiodicity of the input function Not being derived fromthe spectral (Fourier) approach this observation heuristicallycompletes this approach and may be used (see the following)in an introduction to Fourier analysis


119891inp (119905)

0

1199052120587120596119889

Figure 13 The rectangular wave at the input

ℎ(120582)

119905

119891inp (119905 minus 120582)

1205871199081198892120587119908119889 120582

Figure 14 Convolution with a rectangular wave at the inputCompare to Figures 9 and 10

Thus let us now take 119891inp(119905) as the rectangular waveshown in Figure 13 and follow the way of Figures 9 and 10in the sequential Figures 14 and 15

Here too the envelope of the resonant oscillations can bewell outlined by considering 119891out(119905) at instances 119905119896 = 119896120587120596119889first of all120587120596

119889 2120587120596

119889 and 3120587120596

119889 for whichwe respectively

have the first maximum the first minimum and the secondmaximum of 119891out(119905)

There are absolutely the same qualitative (geometric)reasons for resonance here and Figure 15 explains that if thedamping of ℎ(119905) is weak that is some first sequential half-waves of119891inp(119905minus120582)ℎ(119905) are similar then the respective extremevalues of 119878(119905) = 119891out(119905) form a linear increase in the envelope

Figure 16 shows 119891out(119905) = 119878(119905) at these extreme pointsThough it is not easy to find the precise 119891out(119905) every-

where for the envelope of the oscillations which passesthrough the extreme points the resonant increase in theresponse amplitude is absolutely clear

Figures 10 14 15 and 16 make it clear that many otherwaveforms with the correct period would likewise causeresonance in the circuit Furthermore for the overlapping toremain good we can change not only 119891inp(119905) but also ℎ(119905)Making the form of the impulse response more complicatedmeans making the systemrsquos structure more complicated andthus graphical convolution is also a valuable starting point forstudying resonance in complicated systems in terms of thewaveforms This point of view will be realized in Section 5where we generalize the concept of resonance

ℎ(120582)119891inp (119905 minus 120582

120582

120582

120582

)

ℎ(120582)119891inp (119905 minus 120582)

ℎ(120582)119891inp (119905 minus 120582)

119878119900 120587119908119889

119905 = 120587119908119889 119878 = 119878119900

119905 = 2120587119908119889 119878 = minus2119878119900

2120587119908119889

119905 = 3120587119908119889 119878 = 3119878119900

3120587119908119889

Figure 15 Continuation of the creation of the convolution valueafter Figure 14 The function ℎ(120582)119891inp(119905 minus 120582) is shown at threeintervals 0 lt 120582 lt 119905 = 119896(120587120596

119889) asymp 119896(120587120596

119900) 119896 = 1 2 3 for which

the area under this function of 119905 has local extremes 119878119900= 119878(2120587120596

119889)

denotes the area under a half-wave of ℎ(120582)119891inp(119905 minus 120582) Damping ofℎ(119905) is ignored and we have here the cases of 119878 = 119878

119900 119878 = minus2119878

119900 and

119878 = 3119878119900 which represent the output function at its extremes see

Figure 16

Thus using the algorithm of the graphical convolutionwe make two more methodological steps a pedagogical onein Section 44 and the constructive one in Section 5

44 Let Us Try to ldquoDiscoverrdquo the Fourier Series in Order toUnderstand It Better The conclusion regarding the possi-bility of obtaining resonance using a nonsinusoidal inputreasonably means that when pushing a swing with a childon it it is unnecessary for the father to develop a sinusoidalforce Moreover the nonsinusoidal input even has someobvious advantages While the sinusoidal input wave leadsto resonance only when its frequency has the correct valueexciting resonance by means of a nonsinusoidal wave can bedone at very different frequencies (one need not to kick theswing at every oscillation) which is of course associatedwiththe Fourier expansions of the force

Let us see how using graphical convolution we can revealharmonic structure of a function still not knowing anythingabout Fourier series For that let us continue with the caseof square wave input but take now such a waveform witha period that is 3 times longer than the period of self-oscillations of the oscillator Consider Figure 17

This time the more distant instances 119905 = 3120587120596119889 6120587120596

119889

and 9120587120596119889 are obviously most suitable for understanding

how the envelope of the oscillations looks


3120587119908119889

2120587119908119889

120587119908119889

1199050

119891out (119905)

3119878119900

119878119900

minus2119878119900

Figure 16 Linear increase of the envelope (ideal in the in the losslesssituation) for the square wave input Compare to Figures 1 3 and 12

120582

ℎ(120582)

1205871199080

119905119905 minus 31205871199080

119891inp (119905 minus 120582)

Figure 17 We ldquodiscoverrdquo the Fourier series using graphical convo-lutionThe convolution of ℎ(119905)with the square wave having119879 = 3119879

119900

One sees that also for 119879 = 3119879119900 the same geometric

ldquoresonant mechanismrdquo exists but the transfer from 119879 = 119879119900

to 119879 = 3119879119900makes the excitation significantly less intensive

Indeed see Figure 18 comparing the present extreme case of119905 = 3120587120596

119889to the extreme case of 119905 = 120587120596

119889of Figure 15

We see that each extreme overlap is now only one-thirdas effective as was the respective maximum overlap in theprevious case That is at 119905 = 3120587120596

119889 we now have what we

previously had at 119905 = 120587120596119889 which means a much slower

increase in the amplitude in timeSince 119891out(119905) is now increased at a much slower rate but

1120574 is the same (ie the transient lasts the same time) theamplitude of the final periodic oscillations is respectivelysmaller whichmeansweaker resonance in terms of frequencyresponse

Let us compare the two cases of the square wave thusstudied to the initial case of the sinusoidal function Thecase of the ldquononstretchedrdquo square wave corresponds to theinput sin120596119900119905 while according to the conclusions derived inFigure 18 the case of the ldquostretchedrdquo wave corresponds tothe input (13) sin120596119900119905 We thus simply (and roughly) reducethe change in period of the nonsinusoidal function to theequivalent change in amplitude of the sinusoidal function

Let us now trymdashas a tribute to Joseph Fouriermdashto speaknot about the same circuit influenced by different waves butabout the same wave influencing different circuits Instead

of increasing 119879 we could decrease 119879119900 thus testing the

ability of the same square wave to cause resonance in thedifferent oscillatory circuits For the new circuit the graphicalprocedure remains the same obviously and the ratio 13 ofthe resonant amplitudes in the compared cases of 119879119879

119900= 3

and 119879119879119900= 1 remains

In fact we are thus testing the square wave using two sim-ple oscillatory circuits of different self-frequencies Namelyconnecting in parallel to the source of the square wave voltagetwo simple oscillatory circuits with self-frequencies 120596119900 and3120596119900 we reveal for one of them the action of the square waveas that of sin120596119900119905 and for the other as that of (13) sin 3120596119900119905

This associates the squarewave of height119860 with the series

119891 (119905) sim 119860(sin120596119905 + 13sin 3120596119905 + 1

5sin 5120596119905 sdot sdot sdot) (17)

(which precisely is 119891 = (4119860120587)(sin120596119905 + sdot sdot sdot))Let us check this result by using the arguments in the

inverse order The first sinusoidal term of series (17) roughlycorresponds to the squarewavewith119879 = 119879119900 (ie120596 = 120596119900) andin order to make the second term resonant we have to changethe self-frequency of the circuit to 120596119900 = 3120596 that is make120596 = (13)120596119900 or119879 = 3119879119900 which is our second ldquoexperimentrdquo inwhich the reduced to 13 intensity of the resonant oscillationsis indeed obtained in agreement with (17)

It is possible to similarly graphically analyze a triangularwave at the input or a sequence of periodic pulses of anarbitrary form (more suitable for the father kicking the swing)with a period that is an integer of 119879

119900

One notes that such figures as Figure 18 are relevant to thestandard integral form of Fourier coefficients However onthe way of graphical convolution this similarity arises onlyfor the extremes (119891out(119905))max = |119891out(119905119896)| and this way isindependent and visually very clear

5 A Generalization of the Definition ofResonance in Terms of Mutual Adjustmentof 119891inp(119905) and ℎ(119905)

After working out the examples of the graphical convolutionwe are now in position to formulate a wider 119905-domaindefinition of resonance

In terms of the graphical convolution the analyticalsymmetry of (10)

(ℎ lowast 119891inp) (119905) = (119891inp lowast ℎ) (119905) (18)

means that besides observing the overlapping of 119891inp(119905 minus 120582)and ℎ(120582) we can observe overlapping of ℎ(119905 minus 120582) and 119891inp(120582)In the latter case the graph of ℎ(minus120582) starts tomove to the rightat 119905 = 0 as was in the case with 119891inp(minus120582)

Though equality (18) is a very simple mathematical factsimilar to the equalities 119886119887 = 119887119886 and ( 119886 ) = ( 119886) inthe context of graphical convolution there is a nontrivialityin the motivation given by (18) because the possibility tomove ℎ(minus120582) also suggests changing the form of ℎ(sdot) that isstarting to deal with a complicated system (or structure) to beresonantly excited We thus shall try to define resonance that


Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus

Figure 18 Because of the mutual compensation of two half-waves of ℎ(120582) only each third half-wave of ℎ(119905) contributes to the extreme valueof 119891out(119905) and the maximum overlaps between ℎ(120582) and 119891inp(119905) are now one-third as effective as before The reader is asked (this will soon beneeded) to similarly consider the cases of 119879 = 5119879

119900and so forth

is the optimization of the peaks of 119891out(119905) (or its rms value)in the terms of more arbitrary waveforms of ℎ(119905) while thecase of the sinusoidal ℎ(sdot) that is of simple oscillator appearsas a particular one

51 The Optimization of the Overlapping of 119891(120582) equiv 119891inp(119905minus120582)and ℎ(120582) in a Finite Interval and Creation of the OptimalPeriodic 119891inp(119905) Let us continue to assume that the losses inthe system are small that is that ℎ(119905) is decaying so slowlythat we can speak about at least few oscillatory spikes (labeledby 119896) through which the envelope of the oscillations passesduring its linear increase

Using notation 119878119900 of Figures 15 and 16 we speak about theextreme points of the graph of the resulting function 119891out(119905)that is about the points whose coordinates are (119905119896 119891out(119905119896))or

(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)

In view of the examples studied the extreme points of 119891out(119905)are obtained when 119905

119896are the zero-crossings of ℎ(119905) because

only then the overlapping of119891inp(119905119896minus120582)withℎ(120582) can bemademaximal

Comment Assuming that the parameters of the type 1205741205960of

the different harmonic components of ℎ(119905) are different onesees that for a nonsinusoidal damping ℎ(119905) the distributionof the zero-crossings of ℎ(119905) can be changed with the decay ofthis function and thus for a periodic 119891inp(119905) the condition

sign [119891inp (119905119896 minus 119905)] = sign [ℎ (119905)] (20)

or

sign [119891inp (119905119896 minus 119905)] = minussign [ℎ (119905)] (21)

considered for 119896 ≫ 1 need not be satisfied in the wholeinterval of the integration (0 lt 120582 lt 119905

119896) related to the case of

119905119896≫ 119879 However since both the amplitude-type decays and

the change in the intervals between the zeros are defined bythe same very small damping parameters the resulted effects

of imprecision are of the same smallness Both problems arenot faced when we use the ldquogenerating intervalrdquo and employℎ119878(119905) instead of the precise ℎ(119905) The fact that any use of ℎ

119878(119905)

is anyway associated with error of order 119876minus1 sim 120574119879119900points to

the expected good precision of the generalized definition ofresonance

Thus 119905119896 measured with respect to the time origin that

is with respect to the moment when 119891inp(119905) and ℎ(119905) arise isassumed to be given by the known ℎ(119905) Of course we assumethe system to be an oscillatory one for the parameters 119905

119896and

119878119900 of our graphical constructions to be meaningfulHaving the linearly increasing sequence |119891out(119905119896)| = 119878119900119896

belonging to the envelope of the oscillations and wishing toincrease the finally established oscillations obviously we haveto increase the factor 119878119900

However since 119878119900 and the whole intensity of 119891out(119905) canbe increased not only by the proper wave-form of 119891inp(sdot)but also by an amplitude-type scaling factor for the generaldiscussion some norm for 119891inp(sdot) has to be introduced

For the definitions of the norm and the scalar products ofthe functions appearing during adjustment of 119891inp(119905) to ℎ(119905)it is sufficient to consider a certain (for a fixed not too large 119896)interval (119905

119896 119905119896+1)mdashthe one in which we can calculate 119878

119900 This

interval can be simply (0 1199051) or (0 119879)

The norm over the chosen interval is taken as

10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)

For instance sin120596119905 calculated over interval (0 1198792 =120587120596) or (120587120596 2120587120596) isradic1205872120596 as is easy to find by using theequality sin2120572 = (12)(1 minus cos 2120572)

Respectively the scalar product of two functions is takenas

(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)

With these definitions the set of functions defined for thepurpose of the optimization in the interval (119905

119896 119905119896+1) forms an

(infinite-dimensional) Euclidean space


For the quantities that interest us we have from (23) forthe absolute values

119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)

where (see Figures 10 14 and 15) it is set for simplicity ofwriting

119891 (119905) equiv 119891inp (119905119896+1 minus 119905) 119905119896 lt 119905 lt 119905119896+1 (25)

Not ascribing to ldquo119891(sdot)rdquo index ldquo119896rdquo is justified by the factthat the particular interval (119905

119896 119905119896+1) to be actually used is

finally chosen very naturallyThe basic relation |119891out(119905119896)| = 119878119900119896 means that any local

extremum of 119891out(119905) is a sum of such scalar products as (23)Observe that the physical dimensions of sdot and (sdot sdot) are

[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)

Observe also from (22) and (23) that

(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)

and that if we take 1198912 sim 1198911 that is

1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)

then (27) is generalized to

(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)

Indeed using (27) and then the obvious equalities 119870 =

|119870|sign[119870] and |119870| sdot 119891 = 119870119891 we obtain

(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172

= sign [119870] |119870| sdot 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989111003817100381710038171003817 = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)

The factor sign[119870] means in particular that excitation ofan oscillatory circuit can be equivalently done by either an119891inp(119905) or minus119891inp(119905) (Consider the concept of ldquooverlappingrdquo inthis view)

It follows from (30) that if (28) is provided then1003816100381610038161003816(1198911 1198912)

1003816100381610038161003816 =100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (for (28) provided) (31)

Furthermore we use that the following general inequality1003816100381610038161003816(1198911 1198912)

1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)

takes place In view of (22) and (23) (32) is just the knownCauchy-Bunyakovsky integral inequality

Comparing (32) with (31) we see that condition (28)provides optimization of |(119891

1 1198912)| Applied to 119891 and ℎ that

is to 119878119900= |(119891 ℎ)| this conclusion regarding optimization says

that the condition 119891 sim ℎ optimizes 119878119900 Thus 119891 sim ℎ optimizes

the extremes 119891out(119905119896) sim 119878119900119896 of the systemrsquos response

Thus we finally have the following two points

(a) We find the proper interval (119905119896 119905119896+1) for creating the

optimal periodic 119891inp(119905)(b) The proportionality 119891 sim ℎ in this interval is the

optimal case of the influence of an oscillatory circuitby 119891inp(119905)

Items (a) and (b) are our definition of the generalizedresonance The case of sinusoidal ℎ(119905) is obviously includedsince the proportionality to ℎ(119905) requires 119891(119905) to also besinusoidal of the same period

This mathematical situation is the constructive point butthe discussion of Sections 53 and 55 of the optimization of 119878

119900

from amore physical point of view is useful leading us to verycompact formulation of the extended resonance conditionHowever let us first of all use the simple oscillator checkinghow essential is the direct proportionality of 119891 to ℎ that iswhatmay be the quantitativemiss when the waveform of119891(119905)differs from that of ℎ(119905) in the chosen interval (119905119896 119905119896+1)

52 An Example for a Simple Oscillator Let us compare thecases of the square (Figures 13 14 and 15) and sinusoidal(Figure 10) input waves of the same period for 119878

119900defined in

the interval (0 1198792 = 120587120596) Of course the norms of theinput functions have to be equal for the comparison of therespective responses (Note that in the consideration of theabove figures equality of the norms was not provided andthus the following result cannot be derived from the previousdiscussions)

Let the height of the square wave be 1Then 1198912inp(120587minus120582) =1 everywhere and according to (22) the norm is obtained asradic120587120596 For obtaining the same norm for a sinusoidal inputwe write it as 119870 sin120596119905 and find119870 gt 0 so that

119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)

Because of the symmetry of the sinusoidal and square-wave inputs in both cases 119891inp(120582) = 119891inp(119905 minus 120582) equiv 119891(120582) inthe interval (0 120587120596) For either of the input waveforms thenorm of119891inp(119905) now equalsradic120587120596 and for ℎ(119905) = sin120596119905 of thesimple oscillator (the damping in this interval is ignored) wehave according to (24) and (32)

119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587

120596radic2asymp2221

120596

(35)

as the upper bundThus while for the response to the square wave we have

119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)


only for the response to the input119870 sin120596119905 we have for the119870found

119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587

radic120596 sin120596119905sin1205961199052 = radic120587

120596sin120596119905

(37)

as (35)The ldquorelativemissing the optimalityrdquo in the sence of119891 sim ℎ

in the case of the square wave which we wanted to find is

1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)

53 Analogy with the Usual Vectors In the mathematicalsense the set of functions that can be used for the optimiza-tion of 119878

119900is analogous to the set of usual vectors

For the scalar product ( 119886 ) of two usual vectors 119886 and we have (compare to (32))

10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)

(meaning ldquocos 120579 le 1rdquo) where the equality is obtained onlywhen the vectors are mutually proportional (ldquo120579rdquo = 0) that issimilarly (may be opposite) directed

119886 sim (40)

The latter relation is obvious in particular because it isobvious that while rotation of the usual vector (say a pencil)when directing it in parallel to another vector (anotherpencil) the length of this vector is unchanged This point ismuch more delicate regarding the norm of a function beingadjusted to ℎ(119905) which is the ldquorotationrdquo of the ldquovectorrdquo in thefunction space Since the waveform of the function is beingchanged its norm can be also changed

Thus the usual physical space very simply gives theextreme value of |( 119886 )| as | 119886|||

max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)

Since our ldquovectorsrdquo are the time functions and thefunctional analog of (40) is (for simplicity we sometimeswrite 119905 instead of 119879 minus 119905)

119891 (119905) sim ℎ (119905) (42)

we very simply obtain by the mathematical equivalence of thefunction and the vector spaces condition (31) that is only an119891(119905) that is directly proportional to ℎ(119905) can give an extremevalue for 119878

119900

For the vectors of the same length (eg for unit vectors)|| = | 119886| and the condition of optimality 119886 sim becomes 119886 =

plusmn In the functional space the latter means that if 119891(119905) =ℎ(119905) then in order to have 119878

119900maximalwe should take119891(119905) =

plusmnℎ(119905)

54 Comments One can consider 119891 sim ℎ to be both ageneralization and a direct analogy to the condition 120596 =

120596o of the standard definitions of [1 3ndash5] Then both of theequalities 120596 = 120596o and 119891 = 119870ℎ appear in the associatedtheories as sufficient conditions for obtaining resonance ina linear oscillatory system The norms become important atthe next step namely regarding the theoretical conditionsof systemrsquos linearity which always include some limitationson intensity of the functionprocess in any application Forapplications the real properties of the physical source of119891inp(119905) (eg a voltage source) whose power will here beproportional to1198702 obviously require119870 to be limited

The requirement of preserving the norm 119891(119905) duringrealization of 119891(119905) sim ℎ(119905) also necessarily originates fromthe practically useful formulation of the resonance problemas the optimization problem that requires calculation of theoptimized peaks (or rms value) of 119891inp(119905)

If ℎ(119905+1198792) = minusℎ(119905) then the interval in which the scalarproducts (ie the Euclidean functional space) are defined hasto be taken over the whole period of ℎ(119905) that is as 119878119900 =int119879

0119891inp(119879 minus 120582)ℎ(120582)119889120582 (119879 = 119879119900 is a necessary condition)The interval in which we define 119878119900 can be named the

ldquogeneratingrdquo intervalWe can finally write the optimal119891out(119905) that resulted from

the optimal 119891inp(119905) as

119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)

where the function ℎ(0119879)periodic(119905) is ℎ(119905) in the generating

interval periodically continued for 119905 gt 119879We turn now to an informal ldquophysical abstractionrdquo

suggested by the comparison of the two Euclidean spacesThis abstraction leads us to a very compact formulation ofthe generalized definition of resonance

55 A Symmetry Argument for Formulation of the GeneralizedDefinition of Resonance For the usual vector space we havewell-developed vectorial analysis in which symmetry argu-ments are widely employed The mathematical equivalenceof the two spaces under consideration suggests that suchargumentsmdashas far as they are related to the scalar productsmdashare legitimized also in the functional space

Recall the simple field problem in which the scalar field(eg electrical potential)

120593 ( 119903) = ( 119886 119903) (44)

is given by means of a constant vector 119886 and it is asked inwhat direction to go in order to have the steepest change of120593( 119903)

As the methodological point one need not know howto calculate gradient It is just obvious that only 119886 or aproportional vector can show the direction of the gradientsince there is only one fixed vector given and it is simplyimpossible to ldquoconstructrdquo from the given data any otherconstant vector defining another direction for the gradient


We thus consider the axial symmetry introduced by 119886 inthe physical space that can be seen ad hoc as the ldquospace of theradius-vectorsrdquo and conclude that while catching the steepestincreases of ( 119886 119903) we must go with some 119903 sim 119886

Let us compare this very lucid situation with that of thefunctional space In the problem of making the envelope ofthe convolution 119891out(119905) (for the whole interval 0 lt 120582 lt 119905) toincrease as steep as possible we have in view of the relation|119891out(119905119896)| = 119878119900119896 to optimize the scalar product 119878119900 = (119891 ℎ)This is quite similar to (44) because here ℎ(119905) is the onlyfixed ldquovectorrdquo involved that is no other ldquodirectionsrdquo in thefunctional space are given

Thus by the direct analogy to the fact that the gradientmust be proportional to 119886 the optimal 119891(119905) must be propor-tional to ℎ(119905)

We thus can say that in terms of ZSR that is in termsof the convolution integral response resonance is a use of (orldquoobeyingrdquo) the axial symmetry introduced by ℎ(119905) in the spaceof the input functions convolving with ℎ(119905) or ℎ(119879 minus 119905)

This argument makes the generalized definition compactand easy to remember One just should not forget that weoptimize the factor 119878

119900in a certain interval say the first period

of ℎ(119905)

6 Discussion

The traditional teaching of resonance in technical textbooksin terms of a purely steady-state that is frequency responseand phasors not deepening into the time process that isinto the establishment of the steady-state is seen to beunsatisfactory

The general tendency of engineering teachers to workonly in the frequency domain is explained but is not justifiedby the importance of the fields of communication and signalprocessing A good understanding of the time processesis needed in physics chemistry biology and also powerelectronics We hope that the use of convolution integralsuggested here can to some extent close any such logical gapwhen it appears and can make the topic of resonance moreinteresting to a student The described graphical applicationof convolution is also important for understanding the con-volution integral per se Last but not least we hope that ourgeneralized definition of resonance in terms of optimizationof a scalar product in an interval will be useful

On the way to the generalized definition our hero wasthe father swinging a swing and not the definitions of [13 4] Everything relevant (even the Fourier series) can bedirectly understood from the freedom that the father haswhen enhancing the swingrsquos oscillations

In the historical plane the simplicity of the mathematicaltreatment of the sinusoidal case once defined the generalpoint of view on resonance and the standard classroomtreatment but we see that the convolution integral hasbecome a sufficiently simple and common tool to make thisdefinition wider

The present criticism of the usual teaching resonance wellcorrelates with the ldquooldrdquo pedagogical advice by Guillemin

[6] not to hurry with the frequency-domain analysis and tolet the physical reality first be well understood in the timedomain

Direct study of waveforms (not necessarily using thegraphical convolution) also reveals some specific resonanteffects that are not obtained at all for a sinusoidal input[7 8]Thus for some rectangular-wave periodic input wavesa resonant suppression of the response oscillations of a simpleoscillator can occur at certain periodically repeated timeintervals and only a direct analysis of the waveforms revealsthis suppression [7ndash9] It appears that the singularity of thewaveform and its symmetry [7ndash9] and not Fourier (spectral)representation reveal these ldquopausesrdquo in the oscillatory func-tion Remarkably since singularity and symmetry aspects areapplied also to a nonlinear oscillatory circuit these ldquopausesrdquoin the oscillations can be similarly simply explained [7ndash9] forsuch a nonlinear circuit

The topic of resonance is an important scientific andpedagogical point from which different mathematical andphysical interpretations can be developed and it should berevisited by a teacher

Appendix

A The Representation of the CircuitResponse as ZIR(119905) + ZSR(119905) (Some BasicSystem-Theory Terminology for Physicists)

Besides the standard mathematical representation (3) ofthe solution of a linear equation system theory commonlyuses another representation in which the output function iscomposed of a Zero Input Response (ZIR) and a Zero StateResponse (ZSR)

TheZSR is influenced by the generator inputs and satisfieszero initial conditions (this is the meaning of the wordsldquozero staterdquo) and the ZIR is defined only by nonzero initialconditions that is is not influenced by the generatorrsquos inputswhich is the meaning of the words ldquozero-input responserdquo

Since both the generator-type input functions and the ini-tial conditions can be defined freely they are both legitimizedinputs and altogether form a generalized input

A1 The Superposition with Respect to the Generalized InputThe concept of generalized input (Figure 19) fully explainsthe construction of ZIR and ZSR via the superpositionIndeed in the classical way of (3) 119894ℎ(119905) is found from thehomogeneous equation which is not the given one but isartificially introduced That is the determination of 119894ℎ(119905) isan auxiliary problem in which the generatorrsquos inputs (thatdefine the right-hand side of the given equation) are zeroThe concept of generalized input requires doing the samealso for the initial conditions that is to additionally use thegiven equation with the artificially introduced initial zeroconditionsThus according to the two different groups of theinputs we have two parts of the whole solution obtained fromthe following auxiliary independently solvable problems


Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR

Convolution with ℎ(119905)

Figure 19 The generalized input includes both generator inputsand initial conditions (This becomes trivial when the Laplacetransform is used to transform a system scheme) Considering thesuperposition of the output function of a linear system for such aninput we obtain the associated structure of the solution of the linearsystem in a form that is different from (3) Respectively the outputfunction is written as ZIR(119905) + ZSR(119905)

For ZIR Homogeneous equation (zero generatorinputs) plus the needed initial conditionsFor ZSR Given equation with zero initial conditions

Figure 19 schematically illustrates this presentation of thelinear response

Figure 7 reduces Figure 19 to what we actually needfor the processes with zero initial conditions The logicaladvantage of the presentation ZIR + ZSR over (3) becomesclear in the terms of the superposition

The ZSR includes both the decaying transient neededto satisfy zero initial conditions and the final steady stategiven in its general form by the following integral (A8) Theoscillations shown in Figure 3 are examples of ZSR

The separation of the solution function into ZIR andZSR is advantageous for example when the circuit is usedto analyze the input signal that is when we wish to workonly with the ZSR when nonzero initial conditions are justredundant inputs

The convolution integral (10) is ZSR When speakingabout system with constant parameters having one inputand one output the Laplace transform of ZSR(119905) equals119867(119904)119865inp(119904) where 119867(119904) is the ldquotransfer functionrdquo of thesystem that is the Laplace transform of h(t) Each time whenwe speak about transfer function we speak about ZSR that iszero initial conditions

It is easy to write 119865out(119904) for our problem Using theknown formula for Laplace transform of periodic functionand setting the optimal 119891inp(119879 minus 120582) equiv minusℎ(120582) 120582 isin [0 119879] thatis 119891inp(119905) equiv minusℎ(119879 minus 119905) 119905 isin [0 119879] where 119879 is the period (inthe sense of the generating interval) of ℎ(119905) we have for theperiodically continued 119891inp(119905) the Laplace transform of our119891out(119905) as (see (43))

119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)

(the integration only over the first period and finally ldquo+119904rdquoeverywhere) which is relevant to different oscillatory ℎ(119905)

A2 Example Consider for 119905 gt 0 the following simplestexample of the first-order systemequation

119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)

where 119886 and 119860 are constants Here the solution of type (8)119894ℎ(119905)+ 119894119891119904(119905) is first119870 exp(minus119886119905)+119860119886 and when involving the

initial condition finally

119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)

with the initial conditions and the generator functionldquomixedrdquo in the first term

The ZIR + ZSR representation is obtained by rewritingthis expression as

119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)

The first term depends on the initial condition that is is ZIRand the second term depends on the generator input 119860119906(119905)(119906(119905) is the unit-step function) that is is ZSR

It is easy to check that ZIR can be independently foundfrom the equation 1199101015840 +119886119910 = 0 and the given initial conditionand ZSR can be independently found from the given equation1199101015840+ 119886119910 = 119860 and the zero initial conditionFor 119910(0) = 0 119910(119905) = ZSR(119905) = (119860119886)(1 minus 119890minus119886119905) which can

be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)

that is (as (10)) as

119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)

where ℎ(119905) = 119890minus119886119905119906(119905) is the impulse response of the first-

order circuitConsidering (A4) one sees that the ZSR includes (as 119905 rarr

infin) decaying components of the same type as the ZIR andthat the asymptotic response 119860119886 originates from the ZSR as119905 rarr infin and not at all from the ZIR

A3 119891out(119905) as 119905 rarr infin If (as in the above example) 119891out(infin)exists then it is obtained as

119891out (infin) = lim119905rarrinfin

int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)

but if 119891out(infin) does not exist then as 119905 rarr infin the timefunction of the final state is given by making the upper limitof the integration infinity

119891out (119905) sim119905rarrinfin

int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)


(ie the roles of the argument ldquo119905rdquo in (A6) are different for thedifferent places in which it appears)

The integral in (A8) can be rewritten as

int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)

Dealing with the asymptotic solution (A9) is typical forstochastic problems where contrary to our statement of theresonance problem the initial conditions are not important

When speaking about convolution only in the form

int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)

one misses the effects of the initial conditions which areimportant for our analysis and it is inevitable that only thespectral approach appears be relevant

A4 A Case When ZIR + ZSR Is Directly Obtained When adifferential equation can be directly solved by integration thesolution is directly obtained in the form of ZIR + ZSR Thusfor Newtonrsquos equation written in the usual notations

119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)

which obviously is ZIR + ZSR Superposition with respect tothe force (119905) is realized only by the ZSR

Consider also

1198981198892119903

1198891199052= (119905) (A13)

for 119903(0) and (119889 119903119889119905)(0) givenThe presentation ZIR + ZSR is generally relevant to linear

time-variant (LTV ldquoparametricrdquo) equations that include theequations with constant parameters as a special case Forinstance if the mass in (A11) depends on time the integrandin ZSR in (A12) would be (120582)119898(120582) Generally LTVequations are very difficult but for any linear homogenousequation (eg equation of parametric resonance) for whichZSR need not be found it follows from the linearity that thesolution (which then is just ZIR) has the form

119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)

with all the functions known Since 119910(0) 1199101015840(0) arelegitimized inputs (Figure 19) this is the usual linearsuperposition

References

[1] L D Landau and E M Lifschitz Mechanics Pergamon NewYork Ny USA 1974

[2] L I Mandelstam Lectures on the Theory of Oscillations NaukaMoscow Russia 1972

[3] W H Hayt and J E Kemmerly Engineering Circuit AnalysisMcGraw-Hill New York NY USA 1993

[4] J D Irwin Basic Engineering Circuit Analysis Wiley New YorkNY USA 1998

[5] C A Desoer and E S Kuh Basic Circuit Theory McGraw HillNew York NY USA 1969

[6] E A Guillemin ldquoTeaching of system theory and its impact onother disciplinesrdquo Proceedings of the IRE pp 872ndash878 1961

[7] E Gluskin ldquoThe internal resonance relations in the pause statesof a nonlinear LCR circuitrdquo Physics Letters A vol 175 no 2 pp121ndash132 1993

[8] E Gluskin ldquoThe asymptotic superposition of steady-state elec-trical current responses of a nonlinear oscillatory circuit tocertain input voltage wavesrdquo Physics Letters A vol 159 no 1-2pp 38ndash46 1991

[9] E Gluskin ldquoThe symmetry argument in the analysis of oscilla-tory processesrdquo Physics Letters A vol 144 no 4-5 pp 206ndash2101990

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values

0

119905

Figure 1 The definition of resonance [1] as linear increase of theamplitude (The oscillations fill the angle of the envelope) Theinfinite process of increase of the amplitude is obtained because ofthe assumption of losslessness of the system

the registration of a signal Later on the established steady-state oscillations (that are associated because of the radiationof the atoms with the refraction factor of the medium)influence the velocity of the electromagnetic wave in themedium As [2] stressesmdasheven if this velocity is larger thanthe velocity of light (for refraction factor 119899 lt 1 ie when thefrequency of the incoming wave is slightly higher than that ofthe atoms oscillators)mdashthis does not contradict the theory ofrelativity because there is already no signal Registration ofany signal and its group velocity is associated with a (forced)transient process

A more pragmatic argument for the importance of anal-ysis of the initial transients is that for any application of asteady-state response especially in modern electronics wehave to know how much time is needed for it to be attainedand this relates in particular to the resonant processes Thisis relevant to the frequency range in which the device has tobe operated

Contrary to [1] in textbooks on the theory of electricalcircuits (eg [3ndash5]) and mechanical systems resonance isdefined as the established sinusoidal response with a relativelyhigh amplitude proportional to 119876 Only this definitiondirectly associated with frequency domain analysis is widelyaccepted in the engineering sciences According to thisdefinition the envelope of the resonant oscillations (Figure 2)looks even simpler than in Figure 1 it is given by two horizon-tal lines This would be so for any steady-state oscillationsand the uniqueness is just by the fact that the oscillationamplitude is proportional to 119876

After being attained the steady-state oscillations continueldquoforeverrdquo and the parameters of the ldquofrequency responserdquocan be thus relatively easily measured Nevertheless thesimplicity of Figure 2 is a seeming one because it is notknown when the steady amplitude becomes established andcertainly the ldquofrequency responserdquo is not an immediateresponse to the input signal


values

0

119905

sim119876

Figure 2 The envelope of resonant oscillations according to thedefinition of resonance in [3 4] andmany other technical textbooksWhen this steady state is attained

119891out (119905)

1120574119905

Figure 3 The illustration for 119876 = 10 of the resonant response of asecond-order circuit Note that we show a case when the excitationis precisely at the resonant frequency and the notion of ldquopurelyresonant oscillationsrdquo applies here to thewhole process and not onlyto the final steady-state part

Thus we do not know via the definition of [1] when theslope will finish and we do not know via the definition of [3ndash5] when the steady state is obtained

We shall call the definition of [1] ldquothe ldquoQ-trdquo definitionrdquosince the value of 119876 can be revealed via duration of theinitialtransient process in a real systemThe commonly useddefinition [3ndash5] of resonance in terms of the parameters ofthe sustained response will be called ldquothe ldquoQ-ardquo definitionrdquowhere ldquoardquo is an abbreviation for ldquoamplituderdquo

Figure 3 illustrates the actual development of resonancein a second-order circuit The damping parameter 120574 will bedefined in Section 3

The Q-t and Q-a parts of the resonant oscillations arewell seen For such a not very high 119876 (ie 1120574 not muchlarger than the period of the oscillations) the period of fairinitial linearity of the envelope includes only some half periodof oscillations but for a really high 119876 it can include manyperiods The whole curve shown is the resonant responseThis response can be obtained when the external frequencyis closing the self-frequency of the system from the beats ofthe oscillations (analytically explained by the formulae foundin Section 3) shown in Figure 4

Note that the usual interpretation is somewhat differentIt just says that the linear increase of the envelope shown inFigure 1 can be obtained from the first beat of the periodic


119891out (119905)

119905
















Resonance

The toolgraphical

convolution


The toolphasors






I II III


terms ofwaveforms















119898

119871119889119894

119889119905+ 119877119894 +

1

119862int 119894 (119905) 119889119905 = 119907119898 sin120596119905 (1)


1198892119894

1198891199052+ 2120574

119889119894

119889119905+ 1205962

119900119894 (119905) =

120596119907119898

119871cos120596119905 (2)



119889





119894 (119905) = 119894ℎ (119905) + 119894119891119904 (119905) (3)






term is

119894ℎ (119905) = 119890

minus120574119905(1198701cos120596119889119905 + 1198702sin120596119889119905) (4)






119900sim 119876











119876 =120596119900

2120574=120596119900119871

119877=radic119871119862

119877 (5)


120596119889equiv radic1205962

119900minus 1205742 asymp 120596

119900minus1205742

2120596119900= 120596119900(1 minus

1

41198762) asymp 120596

119900 (6)


119900are


119900= 119874(120574) and not 119874(120574119876)


resonance


119894119898 (120596) =

10038161003816100381610038161003816119868 (119895120596)

10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816

119885 (119895120596)

100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119877 + 119895120596119871 + (1119895120596119862)1003816100381610038161003816

=119907119898

radic1198772 + (120596119871 minus (1120596119862))2

(7)

For 120596minus120596119900 ≪ 120596119900 when 120596

2minus1205962


119894119898 (120596) asymp

119907119898

2119871radic1205742 + (120596 minus 120596119900)2

(8)



2=





1

120574=120596119900

120574

1

120596119900

= 2119876119879119900

2120587=1

120587119876119879119900sim 119876119879119900 (119879119900 =

2120587

120596119900

) (9)







0 1205961 120596120596

2120596119900

(119894119898(120596))max

(119894119898(120596))maxradic2

119894119898(120596) 119894(119905) = 119894119898 cos(120596119905 + 120572)

119894119898 = 119894119898(120596) =119907119898

radic1198772 + (120596119871 minus 1120596119862)2


119900Δ120596

119891inp (119905) ℎ(119905) 119891out (119905)






0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= (119891inp lowast ℎ) (119905)

(10)







A

0 Δ119905

ℎ(119905)119891inp (119905)

1120591


1205820 119905

Δ

A

ℎ(120582)119891inp (119905 minus 120582)






ℎ (119905) =1205962

119900

120596119889

119890minus120574119905 sin120596

119889119905

sim 119890minus120574119905 sin120596

119889119905 asymp 119890minus120574119905 sin120596

119900119905 120574 ≪ 120596

119900 (119876 ≫ 1)

(11)

As before we apply

119891inp (119905) = 119891119898 sin120596119889119905 asymp 119891119898 sin120596119900119905 (12)




119889) gt 0 and 119878(2120587120596

119889) lt 0





119891out (119896120587

120596119889

) sim (minus1)119896+1119896 sim 119896 (13)











119878(119905) =








119891out (119905 gt1

120574) = int

1120574

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582 (14)










t

t

t


λ

λS

S

π

wd

2πwd


fout( 2πwd

) = minus2 fout( π

wd)

h(λ)


γ≪ wd asymp w0

int t0 h(λ)


λ

1γ






1003816100381610038161003816119891out (119905119896)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816int

infin

0

119891inp (119905119896 minus 120582) ℎ (120582) 11988912058210038161003816100381610038161003816100381610038161003816 119905119896≫ 1120574 (15)

as

10038161003816100381610038161003816119891out (120574

minus1)10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816int

1120574

0

119891inp (120574minus1minus 120582) ℎ

119878 (120582) 119889120582

100381610038161003816100381610038161003816100381610038161003816 (16)




1199051120574 sim119876




119891inp (119905)

0

1199052120587120596119889


ℎ(120582)

119905

119891inp (119905 minus 120582)

1205871199081198892120587119908119889 120582




119889 2120587120596

119889 and 3120587120596







ℎ(120582)119891inp (119905 minus 120582

120582

120582

120582

)

ℎ(120582)119891inp (119905 minus 120582)

ℎ(120582)119891inp (119905 minus 120582)

119878119900 120587119908119889

119905 = 120587119908119889 119878 = 119878119900

119905 = 2120587119908119889 119878 = minus2119878119900

2120587119908119889

119905 = 3120587119908119889 119878 = 3119878119900

3120587119908119889


119889) asymp 119896(120587120596

119900) 119896 = 1 2 3 for which


119889)


119900 119878 = minus2119878

119900 and


Figure 16





119889




3120587119908119889

2120587119908119889

120587119908119889

1199050

119891out (119905)

3119878119900

119878119900

minus2119878119900


120582

ℎ(120582)

1205871199080

119905119905 minus 31205871199080

119891inp (119905 minus 120582)


119900






119889of Figure 15










119900= 3

and 119879119879119900= 1 remains



119891 (119905) sim 119860(sin120596119905 + 13sin 3120596119905 + 1





119900









Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus


119900and so forth




(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)







or







119878(119905)





119896and






119900 This



10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)



(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)


119896 119905119896+1) forms an




119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119891out (119905)

119905
















Resonance

The toolgraphical

convolution


The toolphasors






I II III


terms ofwaveforms















119898

119871119889119894

119889119905+ 119877119894 +

1

119862int 119894 (119905) 119889119905 = 119907119898 sin120596119905 (1)


1198892119894

1198891199052+ 2120574

119889119894

119889119905+ 1205962

119900119894 (119905) =

120596119907119898

119871cos120596119905 (2)



119889





119894 (119905) = 119894ℎ (119905) + 119894119891119904 (119905) (3)






term is

119894ℎ (119905) = 119890

minus120574119905(1198701cos120596119889119905 + 1198702sin120596119889119905) (4)






119900sim 119876











119876 =120596119900

2120574=120596119900119871

119877=radic119871119862

119877 (5)


120596119889equiv radic1205962

119900minus 1205742 asymp 120596

119900minus1205742

2120596119900= 120596119900(1 minus

1

41198762) asymp 120596

119900 (6)


119900are


119900= 119874(120574) and not 119874(120574119876)


resonance


119894119898 (120596) =

10038161003816100381610038161003816119868 (119895120596)

10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816

119885 (119895120596)

100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119877 + 119895120596119871 + (1119895120596119862)1003816100381610038161003816

=119907119898

radic1198772 + (120596119871 minus (1120596119862))2

(7)

For 120596minus120596119900 ≪ 120596119900 when 120596

2minus1205962


119894119898 (120596) asymp

119907119898

2119871radic1205742 + (120596 minus 120596119900)2

(8)



2=





1

120574=120596119900

120574

1

120596119900

= 2119876119879119900

2120587=1

120587119876119879119900sim 119876119879119900 (119879119900 =

2120587

120596119900

) (9)







0 1205961 120596120596

2120596119900

(119894119898(120596))max

(119894119898(120596))maxradic2

119894119898(120596) 119894(119905) = 119894119898 cos(120596119905 + 120572)

119894119898 = 119894119898(120596) =119907119898

radic1198772 + (120596119871 minus 1120596119862)2


119900Δ120596

119891inp (119905) ℎ(119905) 119891out (119905)






0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= (119891inp lowast ℎ) (119905)

(10)







A

0 Δ119905

ℎ(119905)119891inp (119905)

1120591


1205820 119905

Δ

A

ℎ(120582)119891inp (119905 minus 120582)






ℎ (119905) =1205962

119900

120596119889

119890minus120574119905 sin120596

119889119905

sim 119890minus120574119905 sin120596

119889119905 asymp 119890minus120574119905 sin120596

119900119905 120574 ≪ 120596

119900 (119876 ≫ 1)

(11)

As before we apply

119891inp (119905) = 119891119898 sin120596119889119905 asymp 119891119898 sin120596119900119905 (12)




119889) gt 0 and 119878(2120587120596

119889) lt 0





119891out (119896120587

120596119889

) sim (minus1)119896+1119896 sim 119896 (13)











119878(119905) =








119891out (119905 gt1

120574) = int

1120574

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582 (14)










t

t

t


λ

λS

S

π

wd

2πwd


fout( 2πwd

) = minus2 fout( π

wd)

h(λ)


γ≪ wd asymp w0

int t0 h(λ)


λ

1γ






1003816100381610038161003816119891out (119905119896)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816int

infin

0

119891inp (119905119896 minus 120582) ℎ (120582) 11988912058210038161003816100381610038161003816100381610038161003816 119905119896≫ 1120574 (15)

as

10038161003816100381610038161003816119891out (120574

minus1)10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816int

1120574

0

119891inp (120574minus1minus 120582) ℎ

119878 (120582) 119889120582

100381610038161003816100381610038161003816100381610038161003816 (16)




1199051120574 sim119876




119891inp (119905)

0

1199052120587120596119889


ℎ(120582)

119905

119891inp (119905 minus 120582)

1205871199081198892120587119908119889 120582




119889 2120587120596

119889 and 3120587120596







ℎ(120582)119891inp (119905 minus 120582

120582

120582

120582

)

ℎ(120582)119891inp (119905 minus 120582)

ℎ(120582)119891inp (119905 minus 120582)

119878119900 120587119908119889

119905 = 120587119908119889 119878 = 119878119900

119905 = 2120587119908119889 119878 = minus2119878119900

2120587119908119889

119905 = 3120587119908119889 119878 = 3119878119900

3120587119908119889


119889) asymp 119896(120587120596

119900) 119896 = 1 2 3 for which


119889)


119900 119878 = minus2119878

119900 and


Figure 16





119889




3120587119908119889

2120587119908119889

120587119908119889

1199050

119891out (119905)

3119878119900

119878119900

minus2119878119900


120582

ℎ(120582)

1205871199080

119905119905 minus 31205871199080

119891inp (119905 minus 120582)


119900






119889of Figure 15










119900= 3

and 119879119879119900= 1 remains



119891 (119905) sim 119860(sin120596119905 + 13sin 3120596119905 + 1





119900









Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus


119900and so forth




(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)







or







119878(119905)





119896and






119900 This



10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)



(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)


119896 119905119896+1) forms an




119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Resonance

The toolgraphical

convolution


The toolphasors






I II III


terms ofwaveforms















119898

119871119889119894

119889119905+ 119877119894 +

1

119862int 119894 (119905) 119889119905 = 119907119898 sin120596119905 (1)


1198892119894

1198891199052+ 2120574

119889119894

119889119905+ 1205962

119900119894 (119905) =

120596119907119898

119871cos120596119905 (2)



119889





119894 (119905) = 119894ℎ (119905) + 119894119891119904 (119905) (3)






term is

119894ℎ (119905) = 119890

minus120574119905(1198701cos120596119889119905 + 1198702sin120596119889119905) (4)






119900sim 119876











119876 =120596119900

2120574=120596119900119871

119877=radic119871119862

119877 (5)


120596119889equiv radic1205962

119900minus 1205742 asymp 120596

119900minus1205742

2120596119900= 120596119900(1 minus

1

41198762) asymp 120596

119900 (6)


119900are


119900= 119874(120574) and not 119874(120574119876)


resonance


119894119898 (120596) =

10038161003816100381610038161003816119868 (119895120596)

10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816

119885 (119895120596)

100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119877 + 119895120596119871 + (1119895120596119862)1003816100381610038161003816

=119907119898

radic1198772 + (120596119871 minus (1120596119862))2

(7)

For 120596minus120596119900 ≪ 120596119900 when 120596

2minus1205962


119894119898 (120596) asymp

119907119898

2119871radic1205742 + (120596 minus 120596119900)2

(8)



2=





1

120574=120596119900

120574

1

120596119900

= 2119876119879119900

2120587=1

120587119876119879119900sim 119876119879119900 (119879119900 =

2120587

120596119900

) (9)







0 1205961 120596120596

2120596119900

(119894119898(120596))max

(119894119898(120596))maxradic2

119894119898(120596) 119894(119905) = 119894119898 cos(120596119905 + 120572)

119894119898 = 119894119898(120596) =119907119898

radic1198772 + (120596119871 minus 1120596119862)2


119900Δ120596

119891inp (119905) ℎ(119905) 119891out (119905)






0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= (119891inp lowast ℎ) (119905)

(10)







A

0 Δ119905

ℎ(119905)119891inp (119905)

1120591


1205820 119905

Δ

A

ℎ(120582)119891inp (119905 minus 120582)






ℎ (119905) =1205962

119900

120596119889

119890minus120574119905 sin120596

119889119905

sim 119890minus120574119905 sin120596

119889119905 asymp 119890minus120574119905 sin120596

119900119905 120574 ≪ 120596

119900 (119876 ≫ 1)

(11)

As before we apply

119891inp (119905) = 119891119898 sin120596119889119905 asymp 119891119898 sin120596119900119905 (12)




119889) gt 0 and 119878(2120587120596

119889) lt 0





119891out (119896120587

120596119889

) sim (minus1)119896+1119896 sim 119896 (13)











119878(119905) =








119891out (119905 gt1

120574) = int

1120574

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582 (14)










t

t

t


λ

λS

S

π

wd

2πwd


fout( 2πwd

) = minus2 fout( π

wd)

h(λ)


γ≪ wd asymp w0

int t0 h(λ)


λ

1γ






1003816100381610038161003816119891out (119905119896)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816int

infin

0

119891inp (119905119896 minus 120582) ℎ (120582) 11988912058210038161003816100381610038161003816100381610038161003816 119905119896≫ 1120574 (15)

as

10038161003816100381610038161003816119891out (120574

minus1)10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816int

1120574

0

119891inp (120574minus1minus 120582) ℎ

119878 (120582) 119889120582

100381610038161003816100381610038161003816100381610038161003816 (16)




1199051120574 sim119876




119891inp (119905)

0

1199052120587120596119889


ℎ(120582)

119905

119891inp (119905 minus 120582)

1205871199081198892120587119908119889 120582




119889 2120587120596

119889 and 3120587120596







ℎ(120582)119891inp (119905 minus 120582

120582

120582

120582

)

ℎ(120582)119891inp (119905 minus 120582)

ℎ(120582)119891inp (119905 minus 120582)

119878119900 120587119908119889

119905 = 120587119908119889 119878 = 119878119900

119905 = 2120587119908119889 119878 = minus2119878119900

2120587119908119889

119905 = 3120587119908119889 119878 = 3119878119900

3120587119908119889


119889) asymp 119896(120587120596

119900) 119896 = 1 2 3 for which


119889)


119900 119878 = minus2119878

119900 and


Figure 16





119889




3120587119908119889

2120587119908119889

120587119908119889

1199050

119891out (119905)

3119878119900

119878119900

minus2119878119900


120582

ℎ(120582)

1205871199080

119905119905 minus 31205871199080

119891inp (119905 minus 120582)


119900






119889of Figure 15










119900= 3

and 119879119879119900= 1 remains



119891 (119905) sim 119860(sin120596119905 + 13sin 3120596119905 + 1





119900









Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus


119900and so forth




(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)







or







119878(119905)





119896and






119900 This



10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)



(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)


119896 119905119896+1) forms an




119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











term is

119894ℎ (119905) = 119890

minus120574119905(1198701cos120596119889119905 + 1198702sin120596119889119905) (4)






119900sim 119876











119876 =120596119900

2120574=120596119900119871

119877=radic119871119862

119877 (5)


120596119889equiv radic1205962

119900minus 1205742 asymp 120596

119900minus1205742

2120596119900= 120596119900(1 minus

1

41198762) asymp 120596

119900 (6)


119900are


119900= 119874(120574) and not 119874(120574119876)


resonance


119894119898 (120596) =

10038161003816100381610038161003816119868 (119895120596)

10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816

119885 (119895120596)

100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119877 + 119895120596119871 + (1119895120596119862)1003816100381610038161003816

=119907119898

radic1198772 + (120596119871 minus (1120596119862))2

(7)

For 120596minus120596119900 ≪ 120596119900 when 120596

2minus1205962


119894119898 (120596) asymp

119907119898

2119871radic1205742 + (120596 minus 120596119900)2

(8)



2=





1

120574=120596119900

120574

1

120596119900

= 2119876119879119900

2120587=1

120587119876119879119900sim 119876119879119900 (119879119900 =

2120587

120596119900

) (9)







0 1205961 120596120596

2120596119900

(119894119898(120596))max

(119894119898(120596))maxradic2

119894119898(120596) 119894(119905) = 119894119898 cos(120596119905 + 120572)

119894119898 = 119894119898(120596) =119907119898

radic1198772 + (120596119871 minus 1120596119862)2


119900Δ120596

119891inp (119905) ℎ(119905) 119891out (119905)






0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= (119891inp lowast ℎ) (119905)

(10)







A

0 Δ119905

ℎ(119905)119891inp (119905)

1120591


1205820 119905

Δ

A

ℎ(120582)119891inp (119905 minus 120582)






ℎ (119905) =1205962

119900

120596119889

119890minus120574119905 sin120596

119889119905

sim 119890minus120574119905 sin120596

119889119905 asymp 119890minus120574119905 sin120596

119900119905 120574 ≪ 120596

119900 (119876 ≫ 1)

(11)

As before we apply

119891inp (119905) = 119891119898 sin120596119889119905 asymp 119891119898 sin120596119900119905 (12)




119889) gt 0 and 119878(2120587120596

119889) lt 0





119891out (119896120587

120596119889

) sim (minus1)119896+1119896 sim 119896 (13)











119878(119905) =








119891out (119905 gt1

120574) = int

1120574

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582 (14)










t

t

t


λ

λS

S

π

wd

2πwd


fout( 2πwd

) = minus2 fout( π

wd)

h(λ)


γ≪ wd asymp w0

int t0 h(λ)


λ

1γ






1003816100381610038161003816119891out (119905119896)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816int

infin

0

119891inp (119905119896 minus 120582) ℎ (120582) 11988912058210038161003816100381610038161003816100381610038161003816 119905119896≫ 1120574 (15)

as

10038161003816100381610038161003816119891out (120574

minus1)10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816int

1120574

0

119891inp (120574minus1minus 120582) ℎ

119878 (120582) 119889120582

100381610038161003816100381610038161003816100381610038161003816 (16)




1199051120574 sim119876




119891inp (119905)

0

1199052120587120596119889


ℎ(120582)

119905

119891inp (119905 minus 120582)

1205871199081198892120587119908119889 120582




119889 2120587120596

119889 and 3120587120596







ℎ(120582)119891inp (119905 minus 120582

120582

120582

120582

)

ℎ(120582)119891inp (119905 minus 120582)

ℎ(120582)119891inp (119905 minus 120582)

119878119900 120587119908119889

119905 = 120587119908119889 119878 = 119878119900

119905 = 2120587119908119889 119878 = minus2119878119900

2120587119908119889

119905 = 3120587119908119889 119878 = 3119878119900

3120587119908119889


119889) asymp 119896(120587120596

119900) 119896 = 1 2 3 for which


119889)


119900 119878 = minus2119878

119900 and


Figure 16





119889




3120587119908119889

2120587119908119889

120587119908119889

1199050

119891out (119905)

3119878119900

119878119900

minus2119878119900


120582

ℎ(120582)

1205871199080

119905119905 minus 31205871199080

119891inp (119905 minus 120582)


119900






119889of Figure 15










119900= 3

and 119879119879119900= 1 remains



119891 (119905) sim 119860(sin120596119905 + 13sin 3120596119905 + 1





119900









Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus


119900and so forth




(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)







or







119878(119905)





119896and






119900 This



10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)



(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)


119896 119905119896+1) forms an




119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1205961 120596120596

2120596119900

(119894119898(120596))max

(119894119898(120596))maxradic2

119894119898(120596) 119894(119905) = 119894119898 cos(120596119905 + 120572)

119894119898 = 119894119898(120596) =119907119898

radic1198772 + (120596119871 minus 1120596119862)2


119900Δ120596

119891inp (119905) ℎ(119905) 119891out (119905)






0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= (119891inp lowast ℎ) (119905)

(10)







A

0 Δ119905

ℎ(119905)119891inp (119905)

1120591


1205820 119905

Δ

A

ℎ(120582)119891inp (119905 minus 120582)






ℎ (119905) =1205962

119900

120596119889

119890minus120574119905 sin120596

119889119905

sim 119890minus120574119905 sin120596

119889119905 asymp 119890minus120574119905 sin120596

119900119905 120574 ≪ 120596

119900 (119876 ≫ 1)

(11)

As before we apply

119891inp (119905) = 119891119898 sin120596119889119905 asymp 119891119898 sin120596119900119905 (12)




119889) gt 0 and 119878(2120587120596

119889) lt 0





119891out (119896120587

120596119889

) sim (minus1)119896+1119896 sim 119896 (13)











119878(119905) =








119891out (119905 gt1

120574) = int

1120574

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582 (14)










t

t

t


λ

λS

S

π

wd

2πwd


fout( 2πwd

) = minus2 fout( π

wd)

h(λ)


γ≪ wd asymp w0

int t0 h(λ)


λ

1γ






1003816100381610038161003816119891out (119905119896)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816int

infin

0

119891inp (119905119896 minus 120582) ℎ (120582) 11988912058210038161003816100381610038161003816100381610038161003816 119905119896≫ 1120574 (15)

as

10038161003816100381610038161003816119891out (120574

minus1)10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816int

1120574

0

119891inp (120574minus1minus 120582) ℎ

119878 (120582) 119889120582

100381610038161003816100381610038161003816100381610038161003816 (16)




1199051120574 sim119876




119891inp (119905)

0

1199052120587120596119889


ℎ(120582)

119905

119891inp (119905 minus 120582)

1205871199081198892120587119908119889 120582




119889 2120587120596

119889 and 3120587120596







ℎ(120582)119891inp (119905 minus 120582

120582

120582

120582

)

ℎ(120582)119891inp (119905 minus 120582)

ℎ(120582)119891inp (119905 minus 120582)

119878119900 120587119908119889

119905 = 120587119908119889 119878 = 119878119900

119905 = 2120587119908119889 119878 = minus2119878119900

2120587119908119889

119905 = 3120587119908119889 119878 = 3119878119900

3120587119908119889


119889) asymp 119896(120587120596

119900) 119896 = 1 2 3 for which


119889)


119900 119878 = minus2119878

119900 and


Figure 16





119889




3120587119908119889

2120587119908119889

120587119908119889

1199050

119891out (119905)

3119878119900

119878119900

minus2119878119900


120582

ℎ(120582)

1205871199080

119905119905 minus 31205871199080

119891inp (119905 minus 120582)


119900






119889of Figure 15










119900= 3

and 119879119879119900= 1 remains



119891 (119905) sim 119860(sin120596119905 + 13sin 3120596119905 + 1





119900









Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus


119900and so forth




(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)







or







119878(119905)





119896and






119900 This



10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)



(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)


119896 119905119896+1) forms an




119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











ℎ (119905) =1205962

119900

120596119889

119890minus120574119905 sin120596

119889119905

sim 119890minus120574119905 sin120596

119889119905 asymp 119890minus120574119905 sin120596

119900119905 120574 ≪ 120596

119900 (119876 ≫ 1)

(11)

As before we apply

119891inp (119905) = 119891119898 sin120596119889119905 asymp 119891119898 sin120596119900119905 (12)




119889) gt 0 and 119878(2120587120596

119889) lt 0





119891out (119896120587

120596119889

) sim (minus1)119896+1119896 sim 119896 (13)











119878(119905) =








119891out (119905 gt1

120574) = int

1120574

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582 (14)










t

t

t


λ

λS

S

π

wd

2πwd


fout( 2πwd

) = minus2 fout( π

wd)

h(λ)


γ≪ wd asymp w0

int t0 h(λ)


λ

1γ






1003816100381610038161003816119891out (119905119896)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816int

infin

0

119891inp (119905119896 minus 120582) ℎ (120582) 11988912058210038161003816100381610038161003816100381610038161003816 119905119896≫ 1120574 (15)

as

10038161003816100381610038161003816119891out (120574

minus1)10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816int

1120574

0

119891inp (120574minus1minus 120582) ℎ

119878 (120582) 119889120582

100381610038161003816100381610038161003816100381610038161003816 (16)




1199051120574 sim119876




119891inp (119905)

0

1199052120587120596119889


ℎ(120582)

119905

119891inp (119905 minus 120582)

1205871199081198892120587119908119889 120582




119889 2120587120596

119889 and 3120587120596







ℎ(120582)119891inp (119905 minus 120582

120582

120582

120582

)

ℎ(120582)119891inp (119905 minus 120582)

ℎ(120582)119891inp (119905 minus 120582)

119878119900 120587119908119889

119905 = 120587119908119889 119878 = 119878119900

119905 = 2120587119908119889 119878 = minus2119878119900

2120587119908119889

119905 = 3120587119908119889 119878 = 3119878119900

3120587119908119889


119889) asymp 119896(120587120596

119900) 119896 = 1 2 3 for which


119889)


119900 119878 = minus2119878

119900 and


Figure 16





119889




3120587119908119889

2120587119908119889

120587119908119889

1199050

119891out (119905)

3119878119900

119878119900

minus2119878119900


120582

ℎ(120582)

1205871199080

119905119905 minus 31205871199080

119891inp (119905 minus 120582)


119900






119889of Figure 15










119900= 3

and 119879119879119900= 1 remains



119891 (119905) sim 119860(sin120596119905 + 13sin 3120596119905 + 1





119900









Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus


119900and so forth




(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)







or







119878(119905)





119896and






119900 This



10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)



(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)


119896 119905119896+1) forms an




119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











t

t

t


λ

λS

S

π

wd

2πwd


fout( 2πwd

) = minus2 fout( π

wd)

h(λ)


γ≪ wd asymp w0

int t0 h(λ)


λ

1γ






1003816100381610038161003816119891out (119905119896)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816int

infin

0

119891inp (119905119896 minus 120582) ℎ (120582) 11988912058210038161003816100381610038161003816100381610038161003816 119905119896≫ 1120574 (15)

as

10038161003816100381610038161003816119891out (120574

minus1)10038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161003816int

1120574

0

119891inp (120574minus1minus 120582) ℎ

119878 (120582) 119889120582

100381610038161003816100381610038161003816100381610038161003816 (16)




1199051120574 sim119876




119891inp (119905)

0

1199052120587120596119889


ℎ(120582)

119905

119891inp (119905 minus 120582)

1205871199081198892120587119908119889 120582




119889 2120587120596

119889 and 3120587120596







ℎ(120582)119891inp (119905 minus 120582

120582

120582

120582

)

ℎ(120582)119891inp (119905 minus 120582)

ℎ(120582)119891inp (119905 minus 120582)

119878119900 120587119908119889

119905 = 120587119908119889 119878 = 119878119900

119905 = 2120587119908119889 119878 = minus2119878119900

2120587119908119889

119905 = 3120587119908119889 119878 = 3119878119900

3120587119908119889


119889) asymp 119896(120587120596

119900) 119896 = 1 2 3 for which


119889)


119900 119878 = minus2119878

119900 and


Figure 16





119889




3120587119908119889

2120587119908119889

120587119908119889

1199050

119891out (119905)

3119878119900

119878119900

minus2119878119900


120582

ℎ(120582)

1205871199080

119905119905 minus 31205871199080

119891inp (119905 minus 120582)


119900






119889of Figure 15










119900= 3

and 119879119879119900= 1 remains



119891 (119905) sim 119860(sin120596119905 + 13sin 3120596119905 + 1





119900









Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus


119900and so forth




(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)







or







119878(119905)





119896and






119900 This



10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)



(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)


119896 119905119896+1) forms an




119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119891inp (119905)

0

1199052120587120596119889


ℎ(120582)

119905

119891inp (119905 minus 120582)

1205871199081198892120587119908119889 120582




119889 2120587120596

119889 and 3120587120596







ℎ(120582)119891inp (119905 minus 120582

120582

120582

120582

)

ℎ(120582)119891inp (119905 minus 120582)

ℎ(120582)119891inp (119905 minus 120582)

119878119900 120587119908119889

119905 = 120587119908119889 119878 = 119878119900

119905 = 2120587119908119889 119878 = minus2119878119900

2120587119908119889

119905 = 3120587119908119889 119878 = 3119878119900

3120587119908119889


119889) asymp 119896(120587120596

119900) 119896 = 1 2 3 for which


119889)


119900 119878 = minus2119878

119900 and


Figure 16





119889




3120587119908119889

2120587119908119889

120587119908119889

1199050

119891out (119905)

3119878119900

119878119900

minus2119878119900


120582

ℎ(120582)

1205871199080

119905119905 minus 31205871199080

119891inp (119905 minus 120582)


119900






119889of Figure 15










119900= 3

and 119879119879119900= 1 remains



119891 (119905) sim 119860(sin120596119905 + 13sin 3120596119905 + 1





119900









Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus


119900and so forth




(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)







or







119878(119905)





119896and






119900 This



10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)



(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)


119896 119905119896+1) forms an




119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










3120587119908119889

2120587119908119889

120587119908119889

1199050

119891out (119905)

3119878119900

119878119900

minus2119878119900


120582

ℎ(120582)

1205871199080

119905119905 minus 31205871199080

119891inp (119905 minus 120582)


119900






119889of Figure 15










119900= 3

and 119879119879119900= 1 remains



119891 (119905) sim 119860(sin120596119905 + 13sin 3120596119905 + 1





119900









Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus


119900and so forth




(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)







or







119878(119905)





119896and






119900 This



10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)



(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)


119896 119905119896+1) forms an




119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Compared to the

previous case

120582120582 1205871199080

31205871199080

++

minus


119900and so forth




(119905119896 (minus1)119896+1119878119900119896) 119896 = 1 2 (19)







or







119878(119905)





119896and






119900 This



10038171003817100381710038171198911003817100381710038171003817 = radicint

119905119896+1

119905119896

1198912 (119905) 119889119905 (22)



(1198911 1198912) = int

119905119896+1

119905119896

1198911 (119905) 1198912 (119905) 119889119905 (23)


119896 119905119896+1) forms an




119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119878119900=1003816100381610038161003816(119891 ℎ)

1003816100381610038161003816 =

100381610038161003816100381610038161003816100381610038161003816int

119905119896+1

119905119896

119891 (119905) ℎ (119905) 119889119905

100381610038161003816100381610038161003816100381610038161003816 (24)







[sdot] = [119891] [11990512] =

[119891]

[12059612]

[(sdot sdot)] = [1198911] [1198912] [119905] =[1198911] [1198912]

[120596]

(26)


(119891 119891) =100381710038171003817100381711989110038171003817100381710038172 (27)


1198912 (119905) = 1198701198911 (119905) 119870 isin R (28)


(1198911 1198912) = sign [119870] 10038171003817100381710038171198911

1003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (29)



(1198911 1198912) = (119891

1 1198701198911) = 119870 (119891

1 1198911) = 119870

1003817100381710038171003817119891110038171003817100381710038172


1003817100381710038171003817 sdot10038171003817100381710038171198701198911

1003817100381710038171003817

= sign [119870] 100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817

(30)





1003816100381610038161003816 le100381710038171003817100381711989111003817100381710038171003817 sdot100381710038171003817100381711989121003817100381710038171003817 (32)













119900



119900defined in



119870 sin120596119905 = radic120587120596 (33)

that is

119870 =radic120587

radic120596 sin120596119905 (34)


119878119900= (119891 ℎ) le (radic

120587

120596) sin120596119905 = radic120587

120596radic120587

2120596=

120587


120596

(35)


119878119900 = int

120587120596

0

1 sdot sin120596119905 119889119905 = 1120596int

120587

0

1 sdot sin119909119889119909 = 2120596

(36)



119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119878119900= int

120587120596

0

(radic120587

radic120596 sin120596119905sin120596119905) sin120596119905 119889119905

=radic120587


120596sin120596119905

(37)



1003816100381610038161003816100381610038161003816

2 minus 2221

2221times 100

1003816100381610038161003816100381610038161003816= |minus995| asymp 10 (38)




10038161003816100381610038161003816( 119886 )

10038161003816100381610038161003816le | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816

(39)


119886 sim (40)



max 10038161003816100381610038161003816( 119886 )10038161003816100381610038161003816= | 119886|

1003816100381610038161003816100381610038161003816100381610038161003816 (41)


119891 (119905) sim ℎ (119905) (42)


119900




plusmnℎ(119905)








119891out (119905) = int119905

0

ℎ (120582) 119891inp (119905 minus 120582) 119889120582

= int

119905

0

ℎ (120582) ℎ(0119879)periodic (120582) 119889120582

(43)






120593 ( 119903) = ( 119886 119903) (44)










of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















of ℎ(119905)

6 Discussion









Appendix







Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Linear system

Initialconditions

Generatorinput (s)

ZIR + ZSR










119865out (119904) = 119867 (119904) 119865inp (119904) = 119867 (119904)int119879

0119891inp (119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879

= 119867 (119904)minus int119879

0ℎ (119879 minus 119905) 119890

minus119904119905119889119905

1 minus 119890minus119904119879= 119867 (119904)

int119879

0ℎ (119905) 119890

119904119905119889119905

1 minus 119890119904119879

(A1)



119889119910

119889119905+ 119886119910 (119905) = 119860 119910 (0) = 0 is given (A2)



119910 (119905) = (119910 (0) minus119860

119886) 119890minus119886119905+119860

119886 119905 gt 0 (A3)



119910 (119905) = 119910 (0) 119890minus119886119905+119860

119886(1 minus 119890

minus119886119905) 119905 gt 0 (A4)



be also written as

119910 (119905) = int

119905

0

119890minus119886(119905minus120582)

119860119889120582 (A5)


119910 (119905) = int

119905

0

ℎ (119905 minus 120582) 119891inp (120582) 119889120582 (A6)






int

119905

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A7)



int

infin

0

ℎ (119905 minus 120582) 119891in (120582) 119889120582 (A8)




int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












int

119905

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A9)



int

infin

minusinfin

ℎ (120582) 119891in (119905 minus 120582) 119889120582 (A10)



119898119889

119889119905= (119905) (A11)

we have

(119905) = (0) +1

119898int

119905

0

(120582) 119889120582 (A12)


Consider also

1198981198892119903

1198891199052= (119905) (A13)



119910 (119905) = ZIR (119905) = 119865 (119910 (0) 1199101015840 (0) 119905)

= 119910 (0) 1198911 (119905) + 1199101015840(0) 1198912 (119905) + sdot sdot sdot

(A14)


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 one more tool for understanding resonance

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