Chapter 1 - Inmann Mechanical Vibration

Chap.1

77

Problems

9Xi

8 ;

x1, -0.1

-0.2 Figure 1.44 The response of both the linear (dashed line) and nonlinear system (solid line) of Example 1.10.4with the initial velocitics set to zero.

model in computingthe responsedependson the initial conditions.If the initial conditions are suchthat the system'snonlinearitycomesinto play,then thesetermsshould Somewhatthe same be included.Otherwisea linearresponseis perfectlyacceptable. to include and effects Which model. can be said for includingdampingin a system one of the form vibrating system a whichnot to includewhenmodelingand analyzing important aspectsof engineeringpractice.

PROBLEMS Those problems marked with an asterisk are intended to be solved using computational software or the Meu-,A.e Toolbox. Section 7.1 1.1. The spring of Figure 1.2 is successivelyloaded with mass and the corresponding (static) displacement is recorded as follows. Plot the data and calculate the spring's stiffness. Note that the data contain some error. Also calculate the standard deviation.

n(ke)

10

t1

12

13

14

15

16

x(m)

l.l4

1.25

r.37

1.48

1.59

r.7l

r.82

1.2. Derive the solutionof mi * kx : 0 and plot the resultfor at leasttwo periodsfor the casewith @n: 2rad/s, xs : 1 mm, and os : 16 m-/s. 1 . 3 . S o l v em i ' r k x : 0 f o r k = 4 N l m , m : 1 k g , x o : l m m , a n d o o= 0 . P l o t t h e s o l u t i o n ' 1.4. The amplitudeof vibrationof an undampedsystemis measuredto be 1 mm.The phase shift from t : 0 is measuredtobe? rad and the frequencyis found to be 5 rad/s. Calculate the initial conditionsthat causedthis vibration to occur.

Introduction to Vibration andtheFreeResponse

78

Chap.1

1.5. An undamped system vibrates with a frequency of 10 Hz and amplitude L mm. Calculate the maximum amplitude of the system'svelocity and acceleration. 1.6. Show by calculation that,4sin(or,,t + S) can be representedas Bsinon/ * Ccosrrr,/ and calculate C and B in terms of ,4 and $. 1.7. Using the solution of equation (1.2) in the forni x(t):

B s i n o r n /* C c o s r , . r n /

calculate the values of B and C in terms of the initial conditions x6 and o0. 1.8. Using Figure 1.6,verify that equation (1.10) satisfiesthe initial-velocity condition. L.9. (a) A 0.5-kg mass is attached to a linear spring of stiffness0.1 N/m. Determine the natural frequency of the system in hertz. (b) Repeat this calculation for a mass of 50 kg and a stiffness of 10 N/m and compare your result to that of part (a). 1.10. Derive the solution of the single-degree-of-freedomsystemof Figure 1.4 by writing Newton's law, ma : -kx, in differential form using a dx : u da and integrating twice. 1.L1. Determine the natural frequency of the two systemsillustrated in Figure P1.11.

(b)

(a)

FigureP1.11 * 1 . 1 2 .P l o t t h e s o l u t i o n g i v e n b y e q u a t i o n ( 1 . 1 0 ) f o r t h:e c1a0s0e0kN / ma n d m : l 0 k g f o r two completeperiodsfor eachof the following setsof initial conditions:(a) xu : 0, t r u: 1 m / s , ( b )r u : 0 . 0 1m , u o: O , a n d ( c ) r o: 0 . 0 1m , o . : 1 m / s . *1.13. Make a three-dimensional surfaceplot of the amplitudeA of an undampedoscillator given by equation(1.9)versusxs and tl6for the rangeofinitial conditionsgiven by - 1 < x u < 0 . 1m a n d - 1 < r r n< 1 m / s f o r a s y s t e m w i t h n a t u r a l f r e q u e n c1y0orfa d / s . 1.14. Use a free-bodydiagramof the pendulumof Window 1.1and derive the equationof motion givenin the windowby usingthe approximationsin(0) : 0 for small0. 1.15. A pendulumhaslengthof 250mm.What is the system'snaturalfrequencyinHertz? 1.16. The pendulumin Window 1".1is requiredto oscillateonceeverysecond.What length shouldit be? 1.17. The approximationof sin 0 : 0 is reasonable for 0 lessthan10'.If a pendulumof length 0.5m hasan initial positionof 0(0) : 0, what is the maximumvalueof the initial angular velocity that can be given to the pendulumwithout violatingthis small-angleapproximation?(Be sureto work in radians.) Section 1.2 *1,.18.Plotthesolutionofalinearspring-masssystemwithfrequencycon :2rad/s,xs: and ao : 2.34mm/s,for at leasttwo periods.

1mm,

Chao.1

Problems

79

*1.19. Compute the natural frequency and plot the solution of a spring-mass system with mass of 1 kg and stiffness of 4 N/m and initial conditions of xs : 1 mm and t'6 : 0 mm/s, for at least two periods. 'lb design a linear spring-mass system it is often a matter of choosing a spring constanl 1.20. such that the resulting natural frequency has a specified value. Suppose that the mass of a system is 4 kg and the stiffness is 100 N/m. Horv much must the spring stiffness be changed in order to increasc the natural frequency lty I0%,? 1.21. Referring to Figure 1.7,if the maximum peak velocity of a vibrating system is 200 mrn/s at 4Hz and the maximum allowable peak acceleration is 5000 mm/s2, what wilt the peak displacernent be? 1.22. Show that lines of constant displacement and acceleration in Figure 1.7 have slopes of *1 and -1, respectively.If rms values instead of peak values are used, how does this affect the slope? 1.23. An automobile is modeled as a 1000-kg mass supported by a spring of stiffness k : 400,000 N/m. When it oscillates it does so with a maximum deflcction of 10 cm. When loaded with passengers,themassincreascsto as much as 1300kg. Calculatc the change in frequency,velocity amplitude, zrndacccleration amplitude if the ntaxin"rumdeflectii-rnremains L0 cm. 1.24. A machine oscillatesin simple harmonic motion and appears to be well rnodeled by an undamped single-degree-of-freedomoscillittion. Its accoleration is measured to have an amplitude of 10,000 mm/sz at 8 Hz. What is the machine's maximurn clisplacement? 1.25. A sirnple undamped spring-mass system is set into motion from rest by giving it an initial velocity of 100 mm/s. It oscillateswith a maximum amplitudc: of 10 mrn. What is its natural frequency? 1.26. An automobile exhibits a vertical oscillating displacement of maximum arnplitude 5 cnr and a measurcd maximum accelerationof 2000 cm/s2.Assuming that the automobilc systcrnin the vertical direction,calculate can be modeled as a single-degrec-of-frecdom automobile. of the the natural frequency Section 7.3 I mm.o,,:0mm/s.Sketchyourrcsultsanddeternrine 1 . 2 7 . S o l v ei + 1 i * - r : 0 f o r x 6 : which root dominates. lmrn/sandsketchtheresponse.Youmay 1 . 2 8 . S o l v ei + 2 * - l 2 x : 0 f o r x 6 : 0 m m , o o : wish to sketch x(t) : e-' and x(/) = -g-rfi1s1. 1.29. Derive the form of tr1,and \2 given by equation (1.31) fiom equation (1.28) and the definition of the damping ratio. 1.30. Use the Euler formulas to derive equation (1.36) from equation (1.35) and to determine the relationships listed in Window 1.4. 1.31. Using equation (1.35) as the form of the solution of the underdamped system,calculatc 'u6. the values of the constants a, and a2 in terms of the initial conditions x6 and 1.32. Calculate the constants A and $ in terms of the initial conditions and thus verify equation (1.38) for the underdamped case' 1.33. Calculate the constants a1 and a2in terms of the initial conditions and thus verify equations (1.42) and (1.43) for the overdamped case.

Introductionto Vibration and the Free Response

80

Chap. 1

1.34. Calculate the constants a1and a2in terms of the initial conditions anci thus verify equation (1.46)for the critically damped case. 1.35. Using the definition of the damping ratio and the undamped natural frequency, derive equation (1.48)from (1.47). 1 . 3 6 . F o r a d a m p e d s y s t e mm , ,c,andkareknowntobenr = 1kg,c =2kg/s,k = 10N/m. Calculate the values of ( and o,,.Is the systemoverdamped,underdamped,or critically damped? *1.37. Plot;r(r) for a damped system of natural frequency @n: 2 rad/s and initial conditions xo: I mm, o0 : 0, for the fiiilowing values of the damping ratio: ( : 0.01, E: 0.2' ( : 0 . 6 , ( - 0 . 1 ,t : 0 . 4 , a n d ( : 9 . 3 . 8L.38. Plot the responsex(t) of an underdampedsystemwith a, : 2radf s,( : 0'1, and oo : 0 for the following initial displacements:xe : 1 mm, -rn : 5 mm,"to : 10 mm, and xo : 100 mrn. landtre:0forx(r) andsketchtheresponse. i + x:0withxo: 1.40. A spring-mass-damper system has mass of 100 kg, stiffnessof 3000 N/m, and damping coefficient of 300 kg/s. Calculate the undamped natural frequency,the damping ratio, and the damped natural frequency.I)oes the solution oscillate? 1.41. A spring mass-dampersystcm has massof 150 kg, stiffnessof 1500N/m and damping coefficient of 200 kg/s. Calculate the undamped natural frequency, the damping ratio and the damped natural freqtrency.Is the systemoverdamped,underdamped,or critically damped? Does the solution oscillate? *1.42. The system of Problem 1.40is given a zero initial velocity and an initial displacement of 0.1 rn. Calculate the form clf the responseand plot it for as long as it takes to die 1 . 3 9 . S o l v ei

or.lt. *1.43. 'fhe system of Problern l.4l is givcn an initial velocity of 10 mm/s and an initial displacement of -5 mm. Clalculatethe form of the responseand plot it for as long as it takes to dic out. How long docs it take to die oul'l *1.44. Choose the damping coefficientof a spring-mass*dampersystemwith mass of 150 kg and stiffness ol 2000 N/m such that its responscwill die out after about 2 s, given a zero initial position and an initial velocity of 10 mn.r/s. 1.45. Derive the equation ol motion of the systcm in Figure P L45 and discussthe effect of gravity on the natural frequencyand the damping ratio.

t, FigureP1.45 the effectof grav1.46. Derivethe equationof motionof the systemin FigureP1.46anddiscuss to make someapmay have ratio. You and the damping the natural frequency ity on proximationsof the cosine.Assumethe bearingsprovidea viscousdampingforce only

Chap.1

8l

Problems

' SouthAfrican MechanicalEngineer, in the verticaldirection.(From A' Diaz-Jimenez Vol. 26, pP.65-69,1916')

FigureP1.46

Section 1.4 pendulum of Figure 1'19(b) if a mass nrt ts 1.47. Calculate the lrequency of the compound added to the tip, by using the energy method' with frequency 2 rad/s and damping ratto 1.4g. Calculate the total en"rgy in a damped system : 0.1 and 'uo: 0. Plot the total energy versustinte' ( : 0.01with mass10kffor the cur" tn of motion and natural frequency of an 1.4g. Use the energy method to calculate the equation airplane,ssteering-gearmechanismfortlrenosewheeloiitslandinggear.Themechasystemillustratedi' Figure P1'49' nism is modeled asihe single-degree-of-freeclom

( S t e e r i n gw h e e l )

(Tire-wheel assemblY)

modelof a steeringmechanism' FigureP1.49 Single-degree-of-freedorn

Thesteering.wheelancltireassemblyaremodeledasbeingfixedatgroundforthis as a linear spring-and-masssystem calculation. The steering-rod gear system is modeled nechanism is modeled as the disk shaft-gear The diiection' f"ting in th"ex i-, frl "*if through the angle 0 such that the of inertia -/ and torsional stiffnessk1.The gear,I turns diskdoesnotsliponthemass.obtainanequationinthelinearmotionx.

82

Introduction to Vibration andtheFreeResponse

Chap.1

1.50. A control pedal of an aircraft can be modeled as the single-degree-of-freedom system of F'igure P1.50.Consider the lever as a masslessshaft and the pedal as a lumped mass at the end of the shaft. Use the energy method to determine the equation of motion in 0 and calculate the natural frequency of the system.Assume the spring to be unstretched at0 = 0.

Figure P1.50 Model of a foot pedal used to operate an aircraft control surface.

1.51.

'lb

savespace,two largepipesare shippedone stackedinsidethe other as indicatedin FigureP1.51.Calculatethe naturalfrequencyof vibrationof the smallerpipe (of radius R1) rolling back and forth insidethe largerpipe (of radiusR). Use the energymethod and assumethat the insidepipe rolls without slippingand hasa massof rn.

Large pipe

oo

r,"----*;too

Tiuck bed (a) Figure P1.51 prpe.

(b) (a) Pipes stacked in a truck bed. (b) Vibration model of the inside

1.52. Considerthe exampleof a simplependulumgivenin ExampleI.4.2.Thependulummotion is observedto decaywith a dampingratio of ( : 0.001.Determinea dampingcoefficientand add a viscousdampingterm to the pendulumequation.

84

Introduction to Vibrationand the FreeResponse

Chap.1

1.64. RepeatProblen.r1.63for the svstemof FigureP1.{r,1.

FigureP1.64

1.65. A manufacturer nrakes a leaf spring from a steel (e - Z x 1011N/m2) and sizesthe spring so that the device has a specificfrequency.Later, to save weight, the spring is made of aluminurn(C - 7.1 x 10'0N/m2). Assumingthat the massof the springis much smaller then that of the device the spring is attachedto, determine if the frequency incrcasesor decreasesand hv how much.

Section 1.6 1.66. Show that the logarithmic decrementis equal tcr a =

ltnll

where ,r, is the an'rplitude of vibratio n aftern rJ". nn". elapsed. 1.67. Derive equation (1.68) for the trililar suspensionsystem. 1.68. A prototype compositcmalerial is fornred and hence has unknown modulus.An experiment is perfornrcd consistingo[ f ornring it into a cantileveredbeam of length 1 nt ancl m o m e n t 1 . = 1 0 ' " m a w i t h a 6 - k g m a s sa t t a c h c da t i t s c n d . T h e s y s t e mi s g i v e n a n i n i t i a l displacementar"rdfound to oscillatcwith a period of 0.5 s.Calculatethe modulus E. 'I'he 1.69. free responseof a 1000-kgautomobile with stiffnessof k - 400,000N/m is observed to be of the form given in Figure 1.31.Modeling the autornobile as a single-degrce-of-freedon.r oscillationin the vertical direction,determine the damping coefficient if the displacenlentat I, is rnelrsuredto be 2 crn nnd 0.22cm at tr. 1.70. A pendulum decaysfrom l0 crn to I cnr over one pcriod. Dctermine its damping ratio. 1.7L. The relationshipbetween the log decrement6 and the damping ratio ( is often approximated as 6 : 2n(. For what valuesof ( would you considerthis a good approximation to equation (1.72)'! 1.72. A damped systemis nrodeledas illustratedin Figure 1.9.The massof the systemis measured to be -5kg and its spring constant is measured to be 5000 N/m. It is observed that during free vibration the amplitude decays to 0.25 of its initial value after five cycles. Calculate the viscousdamping coefficient,c. Section 1.7 (seealso Problems 1.60 and 1.85 through 1.90) 1.73. Choose a dashpot'sviscousdamping value such that when placed in parallel with the spring of Example 1.7.2reduces the frequency of oscillation to 9 rad/s.

Chap.1

Problems

85

1.74. For an underdampedsystem,'xo= 0 andoo : 10mrn/s'Determinem' c'andk suchthat the amplitudeis lessthan 1 mm. 1.75. RepeatProblem1.74if the massis restrictedto lie between10 kg < m < 15 kg. 1.76. Use the formula for the torsionalstiffnessof a shaft frotn equation(1.6a)to designa of 105N'm/rad. 1-mshaftwith torsionalstiffness L.77. RepeatExample1.7.2usingaluminum.Whatdifferencedo you note? 1.78. Try to designa bar (seeFigure1.20)that hasthe samestiffnessas the springof Examplel.7.2.Note that the bar mustremainat least10 timesaslong asit is wide in order to be modeledby the formulaof Figure1.20. =l x 106N/m2)' 1.79. RepeatProblem1.78usingplastic(E: 1.40x 10eN/m2) andrubber(E feasible? of these any Are Section 1.8 (seealso Problem 1.j9) in Example1.8'1.Assume 1.80. Considerthe invertedpendulumof Figure1.36as discussed parallel to the two springs. pendulum on the also acts rate c) (of damping that a dashpot How doesthis affectthe stabilitypropertiesof the pendulum? rod of the invertedpenclulunof Figure 1.36with a solid-object 1.81. Replacethe massless compoundpendulumof Figure1.19(b).Calculatethe equationsof vibrationand discttss valuesof the parameterfor whichthe systemis stable:. 1.82. Considerthe disk of FigureP1.82connectedto two springs.Use the energymethodto calculatethe system'snaturalfrequencyof oscillationfor smallangles0(r).

Figure P1.82 Vibration model of a rolling disk mounted against two sprlngs, attached at point s.

Section1.9 *1.83. ReproduceFigure l'37 for the varioustime stepsindicated' +L.84. Use numerical integrationto solve the systemof Example 1.7.3with m = 136I kg, : I k : 2.688 x 10sN/m, c : 3.81x 103kgls subjectto the initial conditionsx(0) just plotting to ,r.'(0) integration = numerical using your result 0.01m/s. Compare and the analyticalsolution(usingthe appropriateformulafrom Section1.3)by plottingboth on the samegraph. *1.85. Consideragainthe dampedsystemof Problem1.84and designa dampersuchthat the There are at leasttwo waysto do this.Here it is inoscillationdiesout afte;2 seconds. tendedto solvefor the responsenumerically,following ExamplesI.9.2,1.9.3,or!.9-4' usingdifferentvaluesof the dampingparameterc until the desiredresponseis achieved. *1.86. Consideragainthe dampedsystemof Example1.9.2anddesigna dampersuchthat the oscillation dies out aftei 25 seconds.There are at least two ways to do this. Here it is

lntroduction toVibration andtheFreeResponse Chap.1

86

1.9.4, followingExamplesI.9.2,1..9.3,or intendedto solvefor the responsenumerically, is achieved.Is usingdifferentvaluesof the dampingparameterc until the desiredresponse your resultoverdamped, underdamped, or criticallydamped? *1.87. RepeatProblem1.85for the initialconditions-r(0): 0.1m ando(0) = 0.01mm/s. *1.88. A springand damperare attachedto a massof 100kg in the arrangementgivenin Figr(0) : 0.Lm andc'(0) : 1 mm/s.Deure 1.9.Thesystemis giventheinitialconditions signthe springand damper(i.e.,choosek and c) suchthat the systemwill cometc rest in 2 secondsand not oscillatemore than two completecycles.Tiyto keep c as small as possible.Also compute(. *1.89. RepeatExample1.7.L by usingthe numericalapproachof the previousfive problems. *1.90. RepeatExample1.7.Lfor the initial conditionsx(0) : 0.01m and o(0) : L mm/s.

Section1.10 to a springof stiffnessl0r N/m hasa dry-slidingfriction force 1.91. A 2-kg massconnectecl 20 cm in 15 cycles.How long its amplitudedecreases (itr) of 3 N.As the massoscillates, doesthis take? 9 x l0rN/mwithafriction m:Skgandk: 1 . 9 2 . C o n s i d e r t h e s y s t e mFoifg u r e l . 4 0 w i t h forcc of magnitude6 N. If the initial amplitudeis 4 cm, determinethe amplitudeone cyclelater aswell asthe dampedfrequency. *1.93. Compute and plot the responseof the systemof Figure P1.93for tire casewhere r o : 0 . 1 r r , o o : 0 . 1 m / s , p = 0 . 0 5 , n : 2 5 0 k g , o = 2 0 " , a n d f=t 3 0 0 0 N / m . H o w l o n g doesit take for the vibrationto die out?

FigureP1.93

*1.94. Computeand plot the responseof a systemwith Coulombdampingof equation(1.88) for the casewhere xo : 0.5 rl,oo : 0 m/s, p : 0.1.,m : 100kg' and ft : 1500N/m. How long doesit take for the vibrationto die out'? *1.95. A massmovesin a fluid againstslidingfriction asillustratedin FigureP1.95.Model the dampingforce asa slowfluid (i.e.,linearviscousdamping)plus Coulombfriction because of the sliding,with the followingparameters:m : 250kg, p = 0.01,c : 25 kg/s, and k : 3000N/m. (a) Computeand plot the responseto the initial conditionsxs = 0.1m, oo : 0.1 m/s (b) Computeand plot the responseto the initial conditionsro : 0.1 m, oo : 1 m/s. How long doesit take for the vibrationto die out in eachcase?

87 Chap.1

Problems

FigureP1'95 new dampingccefficient'c' that *1.g6. considerthe systemof Problem1.95(a) andcomputea oscillation' will causethe vibrationto die out after one of i + ,jf,x + Bxz : 0'How many are there? 1.97. Computethe equilibriumpositions : are there? i * .ilznx- |'xt 1 1'rs 0' How many L.98. Computethe equilibriumpositionsof of 1 m and initial conditionsof *1.gg. consider the pendulumo1E*ample1.10.3;ith a length 0o:rr./10..0"'ao,:o.Comparethe'differencebetweentheresponseofthelinear versionofthependulumequation[i.e.,withsin(0):0]andttieresponseofthenonlinear versionofthependulumequationbvplottingth".".pon'"ofbothforfourperiods. is 8o : n /2 rad' *1.100. RepeatProblem1'99if the initial displacement

l . . l 0 l . . I f t h e p e n d u l u m o f E x a m p l e l . l 0 . 3 i s g i v e n a n i n i t i a lthis c o nequilibrium? ditionneartheequilibriumpo around : sition of 0o : T tuOunAdo 0 doesit oscillate * l . l 0 2 . C a l c u l a t e t h e r e s p o n s e o f t h e s y s t e m o f P r o b l e m l . g 8 f o r t :h e100'r i n i t i a: l c0'o n d i t i o n s of.3rad/s aTl for B xo : 0.01-,oo :'6 -7t'and a naturalfrequency *l.l03.RepeatProblemt.l02andplottheresponseofthelinearversionofthesystem(B:0) onthesameplottocomparethedifferencebetweenthelinearandnonlinearverslon of this equationof motion'

TOOLBOX Mo,Lo". ENGINEERING VIBRATION

a Mnrr-ee Toolbox has Dr. JosephC. Slater of Wright State University "{!g:9 Tittlbox (EVT) is organizedby chapkeyed to this text.The EngiTeerfngVllratiln Problemsfound at the end of eachchapter and may be usedto soivethe ioolbox thosehomeworkproblemssuggested ter. In addition,the EvT may be usedto solve forcomputerusageinSectionsl.gandl.l0'rathertlranusingMerLeedirectly.M,+ intendedto assisiin analysis'parametric Lae and the EVT ur"lni"ru.tive and are s t u d i e s a n d d e s i g n , a s w e l l a s i n s o l v i n g h o m e w o r kuse' p r oFor b l eprofessional m s . T h e E n use' gineerin for educational bration Toolbox is tlcenseafree of charge Toolbox author directly' usersshouldcontacttn" E"gi"""ring Vibration TheEVTisupdatedandimprovedregularlyandcanbedownloadedforfree. or current revision'go to the on To downloaa,upout",o' oUiulniniormatiori "'ug" pageat EngineeringVibration Toolboxhome http ://www.cs.wright'edu/-vtoolbo.x on earlier versionsof M.lrt-ns, aswell This siteincludeslinks to editionsthat ,irn a s t h e m o s t r e c e n t v e r s i o n . A n e - m a i l l i s t o f i n s t r u c t updates. o r s w h oThe u s eEVT t h e Eis V deTismai ,"""iu" e-mailnotificationof the latest tained Sousers andVMS) "u,, supportedby Merlen (including Macintosh signedto run on any;i;tf-of Merversion current the *ittt and is regularlyupdatedto mainiaincompatibility LAB'Abriefintrodu"u"'.Merl-,ceandUNIXisavailableonthehomepageas well.PleasereadthefileReadme.txttogetstartedandtypehelpvtoo.lbox

88

Introduction to Vibrationand the FreeResponse

Chap.1

installation to obtain au overview. Once installed, typing vtbud will display the current revision status of your installation and attempt to download the current revision status from the anonymous FTP site. Updates can then be downloaded incrementally as desired. Please seeAppendix G for further information.

TOOLBOX PROBLEMS T81.1. Fix [your choiceor usethe valuesfrom Example1.3.1with x(0) : 1 mm] the values of m, c, k, and x(0) and plot the responses x(t) for a rangeof valuesof the initial velocity i(0) to seehow the responsedependson the initial velocity.Rememberto use numberswith consislent units. TBl.2. Usingthe valuesfrom ProblemTBl.l andi(0) : 0, plot the response x(r) for a range of valuesof x(0) to seehow the response depends on the initialdisplacement. T81.3. ReproduceFigures1.10,1.11,and 1.12. T81.4. consider solvingProblem1.32andcomparethe time for eachresponseto reachand staybelow0.01mm. TBl.5' SolveProblems1.88,1.89,and 1.90usingthe Engineering vibrationToolbox. T81.6. SolveProblems1.93,1.94,and 1.95usingthe EngineeringVibrationToolbox.