- ap. 1 Problems 85

simula ted using numerical integrat ion. In mod eling real systems, the nonlinearity isalways present. Whether or not it is imp ortant to include the nonl inear part of themodel in computing the response depends on the initial conditions. If the initial con­ditions are such that the system's nonlinearity comes into play, then these terms shouldbe included . Ot he rwise a linea r response is pe rfectly acceptable. Som ewhat the samecan be said for including damping in a syste m model. Which effects to include andwhich not to include when model ing and anal yzing a vibrat ing system form one of theimpo rtant aspe cts of engineering pr actice .

T his section has int roduced a little about the vibra tions of systems with non lin­ea rities. The imp ort an t po ints are that nonl inear systems potentially ha ve multipleequilibrium positions, ea ch with potentially different sta bility behavior. Nonlinearsyste ms typically do not have closed for m solutions so that th e time history is oftencomputed by num erical int egration. Not addressed here, but nontheless very impor­tant, is that the principle of supe rposition, used in the extensively in Chapters 3 and4, do es not app ly to nonlinear syste ms. Superpo sition is the notion that if an inputX(n result s in a solution Xl (t), and an input of X02 results in a response xz(t), then theresponse to an input of the form a xOi + I3xoz will be a XI (t) + I3xz(t ). The majorityof thi s text focuses on linea r vibr ation problems.The brief int roduction to nonlinearsystems is important to und erscor e that when solving linear problems, init ial condi­tions must be limit ed such that only the linear range is excited. Students are encour­aged to take a course in nonlinear systems and /or nonl inear vibra tions to learn moreabout the ana lysis and behavior of nonlinear vibra tions.

PROBLEMS

T hose proble ms mar ked with an asterisk are int ende d to be solv ed using comp utationalsoftware or the M ATLAB Too lbox .

Section 1.1

1.1. Th e spri ng of Figure lZ is successively loaded with mass and the corresponding (static)displacem ent is recorded as follows. Plot the data and calcu late the spri ng's stiffness.Note that the da ta contain some errors.Al so calculate the standard deviation .

m(kg)

x(m)

10

1.14

11

1.25

12

1.37

13

1.48

14

1.59

15

1.71

16

1.82

1.2. De rive the so lution of mx + k x = 0 and plot the result for at least two per iods for thecase with W Il = 2 ra d/ s. Xo = 1 mm , and va = V5rnrn/ s.

1.3. Solve lIl. t + kx = 0 for k = 4 N/ m, m = 1 kg, X o = 1 mrn, and va = O. Plot the solution.

1.4. The amplit ude of vibra tion of an undamped system is measur ed to be 1 mm.The phaseshift fro m t = 0 is measur ed to be 2 rad and the fre quency is found to be 5 rad/s. Cal ­culate the initia l condition s that caus ed this vibration to occur. Assume the response isof theformx(t ) = Asin (w llt + <1».

86 Introduction to Vibration and the Free Response Chap. 1

1.5. Find the equation of motio n for the hanging spring- mass system of Figure PI.5 and com­pute the natu ral frequency. In pa rt icular. using sta tic eq uilibrium along with New ton'slaw. det ermine wha t effect grav ity has on the equa tion of mo tion and the system'snatur al frequen cy.

TL-_--' x(t)

Figure P1.5

1.6. Find the equation of motion for the system of Figure P1.6 and compute the formula forthe natural frequ ency. In part icular. using static equilibrium along with Newton 's law.determine what effect gravity has on the equation of motion and the system's na turalfrequency. Assume the block slides without friction.

~t)

Figure P1.6

1.7. An und amped system vibra tes with a frequency of 10 Hz and amp litud e 1 mm . Calcu­late the maximum amplitude of the system's velocit y and acceleration .

1.8. Show by calculation that A sin (w,/ + cj» can be represented as B sin w"t + C cos W)

and calculate C and B in term s of A and cj> .

1.9. Using the solution of equa tion (1.2) in the form

x (t ) = Bsin w"t + Ccosw,/

calculate the values of B and C in terms of the initial cond itions .\' 0 and va.

1.10. Using Figure 1.7, verify that equa tion (1.10) satisfies the initi al-velocity condition.

1.11. (a)A 0.5-kg mass is attac hed to a linear spring of stiffness 0.1 N1m.Determine the naturalfrequency of the system in hertz. (b) Repeat this calcula tion for a mass of 50 kg and astiffness of 10 Nl m and compare your result to that of pa rt (a) .

1.12. Derive the solution of the single-degr ee -of-freedom syste m of Figur e 1.5 by writingNewton 's law, ma = - kx , in differential form using a dx = v ltv and integrating twice.

:O J. 1 Problems

1.13. Det ermine the natural frequency of th e two systems illustrated in Figur e P1.13.

87

k j

'r.J¥!'ii 111

(a)

k2

~~~~J~l%

Figure Pl.13

111

(b)

*1.14. Plot the solution given by equa tion (1.10) for the case k = 1000 N/ m and 111 = 10 kg fortwo complete period s for ea ch of th e followi ng se ts of initial condition s: (a) -"0 = 0,Vo = 1 m/ s, (b) -"0 = 0.01 rn, V o = 0, and (c) -"0 = 0.01 m, Vo = 1 m/ s,

*1.15. Make a three-dim ensio nal surface plot of the amplitude A of an und amped oscillatorgiven by equation ( 1.9) versus -"0 and Vo for the range of initial conditions given by

- 0 . 1 :S -"0 :S 0.1 m and -1 :S Vo :S 1 rn / s for a system with natural frequ ency of 10 rad/ s.

1.16. A machine part is modeled as a pendulum connected to a spring as illustra ted inFigure P1.16. Ignore the mass of pendulum 's rod and deri ve the equa tion of motion .Then, following the procedure used in Example 1.1.1, linea rize the equatio n of mot ion,and compute the form ula for the natural frequency. Assume that the ro tation is sma lleno ugh so that the spring only deflect s horizontally.

o

~,

e \,

111

Figure Pl.16

1.17. A pendulum has length of 250 mm. What is the system's natur al frequency in Hertz?

1.18. The pend ulum in Exampl e 1.1.1 is require d to oscillate once every second. Wha t lengthshould it be?

1.19. The approx ima tio n of sin 8 = 8 is reasonable fo r 8 less than 100. If a pendulum oflengt h 0.5 m has an initial posit ion of 8(0) = 0, wha t is the maximum value of the initi alangular veloci ty tha t can be given to the pendulum witho ut viola ting th is small -angleapproximat ion? (Be sure to work in radians.)

88

Section 1.2

Introduction to Vibration and the Free Response Chap. 1

*1.20. Plot the solution of a linear spring-mass system with frequ ency w" = 2 rad /s , Xo = 1 mm ,and va = 2.34 mrn/ s, for at leas t two periods.

*1.21. Co mpu te the natural frequency and plo t the so lution of a spri ng-mass system with massof 1 kg and stiffness of 4 N/m and init ial conditions of Xo = 1 mm and Vo = 0 mm /s. forat least two periods.

1.22. To design a linear spring- mass system it is ofte n a matter of choosing a spring constantsuch that the resulting natural frequency has a specified value. Suppose that the mass ofa syste m is 4 kg and the stiffness is 100 N/m. How mu ch must the spring stiffness becha nged in order to increase the natural frequency by 1O%?

1.23. Refer ring to Figure 1.8, if the maximum peak velocity of a vibra ting system is 200 mm /sat 4 Hz and the maximum allowable peak acceleration is 5000 mrn/s '', wha t will the peakdisplacement be?

1.24. Show that lines of constant displacement and acceleratio n in Figure 1.8 have slopes of+1 and - I, respectively. If rms valu es instead of peak valu es are used , how do es this af­fect the slope?

1.25. A foo t pedal mechan ism for a machin e is cru dely modeled as a pe ndu lum connectedto a spring as illustra ted in Figur e PI .25. T he purpose of th e sp ring is to keep thepedal roughly vertical. Com pute the spring stiffness needed to keep the pendulum at 10

from the horizontal and then compute the correspo nding natural fre quency. Assumethat the angular deflections are sma ll, such th at the spr ing deflection can be approx i­mat ed by the arc length, that the ped al may be treated as a point mass, and th at pendu­lum ro d ha s negligible mass. The values in th e figure are m = 0.5 kg, g = 9.8 rn/s",11 = 0.2 m, and 12 = 0.3 m.

~ I

o 111

k

Figure P1.25

1.26. An automobile is modeled as a 1000-kg mass supported by a spring of stiffnessk = 400.000 N/ m. When it oscillates it does so with a maximum deflection of 10 ern.When loaded with passen gers, the mass increases to as much as 1300 kg. Calcula te thechan ge in frequency, velocity amplitude, and acce leration am plitude if the maximumdefl ection remains 10 em.

1.27. The front suspen sion of some cars contains a torsion ro d as illust rated in Figur e P1.27to improve the ca r 's handling. (a) Co mpute the frequency of vibr ati on of the wheel

r.

r

- 1 Problems

Figure Pl.27

89

assembly given tha t the torsional stiffness is 2000 N m/rad and the wheel assemblyhas a mass of 38 kg. Take the dis tance x = 0.26 m. (b) Som etimes owners put differ ­ent wheels and tires on a car to enhance the appearance or pe rfor ma nce. Suppose athinn er tire is put on with a larger whee l raising the mass to 45 kg. What effe ct doesthis have on the frequency?

1.28. A machine oscillates in simp le harmonic mo tion and appears to be well modeledby an undamped single-degree-of-freedom oscilla tio n. It s acceleration is measuredto have an amplitude of 10.000 mm / s? at 8 Hz. Wh at is the machine 's ma ximumdisplacem en t?

1.29. A simple und amped spring- mass syste m is set into motion from rest by giving it an ini­tial velocity of 100 mm/s, It oscillates with a maximum amplitude of 10 mm. What is itsnatur al frequency?

1.30. A n auto mobile exhib its a vertical oscillating displacemen t of maximum amplitude 5 cmand a meas ured maximum acce lerat ion of 2000 cm/ s2• Assuming that the au tomobilecan be mod eled as a single-deg ree-of-free dom system in the vertic al direction , calculatethe natural frequency of the au tomobile.

Section 1.3

1.31. Solver + 4.r + x = afor Xo = 1 mm. Vo = amrn / s, Sketch your results and det erminewhich root dom inat es.

1.32. Solver + 2.r + 2x = afor Xo = 0 mrn. Vo = 1 mrri/ s and sketc h the response.You maywish to sketch x(t ) = e- t and x(t ) = - e: first.

1.33. Der ive the form of AIand A2 given by equa tio n (1.31) fro m eq ua tion (1.28) and thedefinition of the damping ra tio.

1.34. Use the Euler formulas to derive equ ation (1.36) from eq uation (1.35) and to de terminethe relationships listed in Wind ow 1.5.

1.35. Using equation (1.35) as the form of the solution of the und erdamp ed system, calculatethe values of the constant s G j and G2 in terms of the initia l conditions Xo and vo.

Calculate the constants A and <P in terms of the init ial condition s and thus verify equa­tion (1.38) for the underdamped case.

Ca lculate the con stants (/1 and (/2 in terms of the initial co nditions and thus verify equa ­tion s (1.42) and ( 1.43) for the ove rda mped case.

Ca lculate the con stants (/1 and (/ 2 in terms of the initial con ditions and thus verify equa ­tion (1.46) for the critically da mpe d case.

Using the defini tion of the damping rati o and the undamped natural frequency, deriveequa tion (1.48) from (1.47).

For a damped system, 1Il. C, and k are known to be 1Il = 1 kg, C = 2 kg / s, k = 10 N/ m.Ca lculate the values of ~ and W", Is the system overd amped , underdamped , or criticallydamped ?

Plot X(I) for a damped system of natural frequency W n = 2 rad/ s and init ial condition sXo = I mm , Vo = 0, for the follow ing valu es of the damping ratio: ~ = 0.01, ~ = 0.2.~ = 0.6, ~ = 0.1, ~ = 0.4. and ~ = 0.8.

Plot the respons e X(I) of an underdamped system with WIl = 2 rad /s, ~ = 0.1, and Vo = 0fo r the fo llo wing ini t ia l displacem ents: Xo = 1 mm. Xo = 5 mrn , Xo = 10 mm . andXo = 100 mm .

Solve .r - .\" + x = 0 with Xo = 1 and va = 0 for x(I) and sketch the response.

A spring-mass-damper system has mass of 100 kg, stiffnes s of 3000 N/ m. and dampingcoefficient of 300 kg/soCalcul ate the undamped natural frequency, the damping ratio, andthe damped natural frequency. Doe s the solution oscillate?

A sketch of a valve and rocke r arm system for an intern al co mbustion engine is givenin Figure P1.45 . Mod el the syste m as a pendulum att ach ed to a spring an d a mass andassume the oil provides viscous da mping in the ra nge of ~ = 0.01. Determine the equa­tions of moti on and calculate an exp ression for the natural frequency and the dampedna tural fre que ncy. H ere J is the moment of inertia of the rocke r arm abo ut its pivotpo int, k is the stiffness of the valve spring. and 111 is th e mass of the va lve and ste m.Ignore the mass of the sprin g.

90

1.36.

1.37.

1.38.

1.39.

1.40.

*1.41.

*1.42.

1.43.

1.44.

1.45.

~mlr ~alve

Introduct ion to Vibration and the Free Response

8

Chap. 1

Figure PI.45

Problems 91

1.46. A spring- mass-da mpe r system has mass of 150 kg, stiffness of 1500 N/ m and dampingcoefficient of 200 kg/ so Calculate the undamped natural frequency, the damping ratio, andthe damped natural frequ ency. Is the system overdamped, und erd arnped, or crit icallydam ped ? Does the solution oscillate?

> 1.47. The system of Problem 1.44 is given a zero initial velocity and an init ial displ acement of0.1 m. Ca lculate the form of the response and plot it for as long as it tak es to die out.

"1.48. The syste m of Problem 1.46 is given an initial velocit y of 10 mm/ s and an initi al dis­placement of - 5 mm . Calculate the form of the response and plot it for as long as it takesto die ou t. How long does it take to die out?

"'1.49. Choose the damping coefficient of a spring-mass-damper system with mass of 150 kgand stiffn ess of 2000 N/m such that its response will die out after about 2 s, given a zeroinitial position and an initia l velocity of 10 mrn/ s,

1.50. Derive the equation of motion of the system in Figure PI .50 and discuss the effect of grav­ity on the natural frequency and the damping rati o.

Figure PI .50

1.51. Derive the equation of mot ion of the system in Figure Pl.51 and discuss the effect of grav­ity on the na tural frequency and the dampin g ratio. You may have to make some ap­proximations of the cosine. Assume the bearings provide a viscous damping force onlyin the vertical direct ion . (Fro m A. Diaz-Jimene z, South African Mechanical Engineer,Vol. 26, pp. 65- 69,1 976.)

aA

k

h

I

I,

Figure PI.51

92

Section 1.4

Introduction to Vibration and the Free Res ponse Chap. 1

1.52. Calculate the frequency of the comp ound pendulum of Figure 1.20(b) if a mass lil T isadd ed to the tip, by using the energy method.

1.53. Calculate the total energy in a damped system with frequ ency 2 rad/s and damping ratio~ = 0.01with mass 10kg for the case Xo = 0.1and Vo = O. Plot the total energy versus time.

1.54. Use the energy method to calculate the equation of motio n and natural frequency of anairplane 's steering-gear mechanism for the nose wheel of its landing gear. Th e mecha­nism is mod eled as the single-degree-of-freedom system illustrated in Figure P1.54.

(Stee ring wheel)

x

(Tire-whee lassembly)

Figure P1.54 Single-degree-of-free dom model of a steering mechanism.

T he steering-whee l and tire assembly is mode led as being fixed at ground for thiscalculation.Th e steering-rod gear system is modeled as a linea r spring-and-mass system(Ill, kz) oscillating in the x direction. T he shaft-gear mechanism is modeled as the diskof inertia J and torsion al stiffness k J. T he gea r J turns through the angle esuch that thedisk doe s not slip on the mass. O btain an equati on in the linear mot ion x .

1.55. A cont rol pedal of an aircraft can be modeled as the single-deg ree-o f-freedom system atFigure Pl .55. Conside r the lever as a massless shaft and the pedal as a lumped mass at theend of the shaft. Use the energy method to dete rmine the equatio n of moti on in eand cal­culate the natural frequ ency of the system.Assume the spring to be unstretched at e = 0

~

I !I il l

Io - t--: II I

: II l ize: I

' I

~II

Figure P1.55 Model of a foot pedal us.,to operate an aircraft contro l surface.

1.56. To save space, two large pipes are shipped one stacked inside the other as indicated ­Figure Pl .56. Calculate the natu ral frequ ency of vibration of the smaller pipe (of radiL

Problems 93

R t ) ro lling ba ck and forth insid e the larg er pipe (o f radius R). Use the energy methodand assume that th e insid e pip e roll s with out slipping and has a mass of m .

Large pipe

/

Iruck bed

(a)

e

mg

(b)

Figure Pl.56 (a) Pipes stacked in a truck bed. (b) Vibrat ion mod el of the inside pipe.

1.57. Consider the example of a simp le pendulum given in Example 1.4.2.The pendulum mo­tion is ob ser ved to dec ay with a damping ra tio of ~ = 0.001. Determine a damping co­efficient an d add a viscou s damping term to the pendulum equation.

1.58. Det ermine a damping coefficient for the disk- ro d syste m of Example 1.4.3. Assumingthat the dam pin g is du e to the mat er ial pro perties of the rod, de termine c for the rod ifit is observed to have a damping ratio of ~ = 0.01.

1.59. The rod an d disk of Window 1.1 are in to rsio nal vibration . Calcula te th e dampednatural frequency if J = 1000 m2

• kg. c = 20 N . m s/ rad , and k = 400 N . m /rad.

1.60. Co nside r th e system of Figure P1.60, whic h repres ents a simple model of an aircraftlanding syste m. Assum e. .r = reoWh at is the damped natural fre quency?

Figure Pl.60

94 Introduction to Vibration and the Free Respons e Chap. 1

1.61. Consider Problem 1.60 with k = 400.000 N · m,m = 1500 kg, J = 100 m2• kg, r = 25 em,

and c = 8000 N' m ' s. Calculate the da mping rati o and the damped natural freque ncy.How much effe ct do es the ro tationa l inert ia have on the undamped natural frequenc y?

1.62. Use Lagrange's formulation to calculate th e equation of moti on and the na tural fre ­qu ency of the syste m of Figur e P1.62. Model each of the brack et s as a spring of stiffne ssk, and assum e the inerti al of the pull eys is negligible .

111Figure Pl.62

1.63. Use Lagrange 's formulation to calculate the equa tion of motion and the natural fre­quency of the system of Figure P1.63. Th is figure re pres ents a simplified model of a je tengine mounted to a wing th rough a me chan ism which acts as a spring of stiffness k andmass 111, . Assume the engine has moment of inertia J and ma ss m and that the rot ati onof the engi ne is re lated to the vert ical displa ceme nt of th e engine, x(t), by the "radius"ro (i.e ., x = r08).

Mo unt.x.v»,

\ I /'.01:'.,: \o-- - - --- - --

I

Engine, .I.mFigure P1.63

1.64. Lagrange 's formulation can also be used for non conservative systems by adding the ap­plied nonconser vat ive term to the right side of equatio n (1.64) to ge t

~( aT ) _ aT + au + aR; = adt aq; aq; aq; aq;

Here R;is the Rayleigh dissipation funct ion defined in the case of a viscous damper withone end fixed to ground by

_ I . 2R- - - cel -

I 2 I

Problems 95

Use this extended Lagrange formulation to derive the equation of motion of the damp edautomo bile suspension of Figure P1.64 (here the bod y is treated as gro und and assumex = re) .

Figure Pl.64

1.65. Conside r the disk of Figure P1.90 conn ected to two springs. (a) Use the ene rgy methodto calculate the syste m's natural frequency of oscillatio n for small angles e(t ). (b) UseLagrange's method to derive the equation of motion .

Section 1.5

k k

x(t)

111 = mass

Figure P1.65 Vibra tion model of aro lling disk mounted against twosprings. att ached at point s.

1.66. A helicopter landing gea r consists of a metal frame work rath er than the coil spring­based suspension system used in a fixed-wing aircraft. Th e vibration of the frame inthe vert ical direction can be modeled by a spring made of a slender bar as illustr atedin Figure 1.21. wher e th e hel icoptor is modeled as gro und. Here I = 0.4 m,E = 20 X 1010 N/ m2

• and m = 100 kg. Calculate the cross -sectio nal area that should beused if the natural frequency is to be In= 500 Hz .

1.67. The frequency of oscillation of a pers on on a diving board can be modeled as the trans­verse vibration of a beam as indicated in Figure 1.24. Let m be the mass of the diver(m = 100 kg) and I = 1 m. If the diver wishes to osci llate at 3 Hz. wha t value of EIsho uld the diving -board material have?

1.68. Consider the spring system of Figure 1.29. Let k j = k s = k 2 = 100 N/ m. k -; = 50 N/ m.and kJ, = 1 N/ m. What is the equivalent stiffness?

1.69. Springs are ava ilable in stiffness values of 10.100. and 1000 N/ m. Design a spr ing sys­tem using these values on ly. so that a 100-kg mass is connected to ground with frequencyof about 1.5 rad/s ,

96 Introduction to Vibration and the Free Response Chap. 1

1.70. Calculate the natural frequency of the system in Figure 1.29(a) if k , = k2 = O. Choose mand nonzero values of k 1 , k 4 , and k.; so that the natural frequency is in = 100 Hz.

*1.71. Example 1.4.4 examines the effect of the mass of a spring on the natural frequency of asimple spring-mass system . Use the relationship derived there and plot the natural fre­quency versus the percent that the spring mass is of the oscillating mass. Use your plotto comment on circumstances when it is no longer reasonable to neglect the mass ofthe spr ing.

1.72. Calculate the natural frequency and damping ratio for the system in Figure P1.72 giventhe values m = 10 kg, c = 100 kg/s ,k ] = 4000 N/m, k2 = 200 N/m, and k3 = 1000N/ m.Assume that no friction acts on the rollers. Is the system overdamped, crit ically damped,or underdamped?

c

Figure PI .72

1.73. Repeat Problem 1.72 for the system of Figure P1.73.

Figure Pl.73

1.74. A manufacturer makes a cantilevered leaf spring from a steel (E = 2 X 1011 N/m2) and

sizes the spri ng so that the device has a specific frequency. Later, to save weight.the spr ing is made of aluminum (E = 7.1 x 1010 N/m2

) . Assuming that the mas sof the spring is much smaller than that of the device the spring is attached to, determineif the freq uency increases or decre ases and by how much.

Section 1.6

1.75. Show that the loga rithmic decrement is equal to

1 Xo0= - In-

n XII

where X n is the ampli tude of vibration after n cycles have elaps ed.

1.76. De rive equation (1.70) for the trifi lar suspension system.

1.77. A prototype composite material is formed and hence has unknown modulus. An ex­periment is performed consis ting of forming it into a cantilevered beam of leng th1 m and moment 1 = 10-9 m" with a 6-kg ma ss attached at its end. The system i '

=-ap. 1 Problems 97

given an initi al displacement and found to oscillate with a peri od of 0.5 s. Calculatethe modulus E .

1.78. T he free respon se of a 1000-kg auto mobile with stiffness of k = 400,000 N/m is ob­served to be of the form given in Figure 1.32. Mod eling the automobile as a single-degree­of-free dom oscillation in the ver tical directi on , determine the damping coeffic ient if thedisplacem ent at 'Iis measured to be 2 em and 0.22 em at '2'

1.79. A pendulum decays from 10 em to 1 em over one period . Determine its damping ratio.

1.80. The relation ship be tween the log decrem ent 0 and the damping rati o ~ is ofte n approx­imated as 0 = 21T~ . For what value s of ~ would you conside r th is a good app roximationto eq uat ion (1.74)?

1.81. A damped system is modeled as illustrated in Figur e 1.10. T he mass of the system ismeasured to be 5 kg and its sp ring constant is measured to be 5000 N/ m. It is observedthat during free vibration the amplitude decays to 0.25 of its initial value afte r five cy­cles. Ca lculate the viscous damping coe fficient, c.

Section 1.7 (see also Problems 1.69 and 1.95 through 1.101)

1.82. Choose a dashpot 's viscous dampin g value such th at when placed in parall el with thespring of Ex amp le 1.7.2 it reduces the fre quency of oscillatio n to 9 rad /s.

1.83. For an underdamped system.x., = aand Vo = 10 mrn/s, Deter mine 11l, c, and k such thatthe amplitude is less than 1 mm.

1.84. Rep eat Pro blem 1.83 if the mass is res tr icted to lie between 10 kg < I1l < 15 kg.

1.85. Use the formula for the tor sional stiffness of a shaft from equation (1.66) to design a1-m shaft with tor sional stiffness of 105 N . m/ rad .

1.86. Repeat Example 1.7.2 using aluminum. What difference do you no te?

1.87. Try to design a bar (see Figure 1.21) that has the same stiffness as the spring of Exam­ple 1.7.2. Note th at the bar must remain at least 10 times as long as it is wide in or der tobe mod eled by the formu la of Figure 1.21.

1.88. Repeat Problem 1.87 using plastic (E = 1.40 X 109 N/ m2) and rubber (E= 7 X 106 N/m2

) .

Are any of these feasible?

1.89. Conside r the diving board of Figure P1.89. For divers, a cert ain level of sta tic deflect ion isdesirable, deno ted by ~ . Compute a design formula for the dimensions of the board (b.h.and I) in terms of the static deflection , the aver age diver 's mass,Ill , and the modulus of theboard.

IIlg

~~ _1-------------------------~ - rI U

-~ -I

b

3/ =!!.!!.­

12

'-- -.JI h end view

Figure Pl.89

98 Introduction to Vibration and the Free Response Chap. 1

Section 1.8 (see also Problem 1.43)

1.90. Consider the syste m of Figure P1.90. (a) Wri te the equations of mot ion in term s of theangle, e,the bar makes with the vert ical. Ass ume line ar deflect ions of the springs and lin­ear ize the equa tions of motion. (b) Discuss the stability of the linear system's solutions interms of the physical constants, 111 , k, and I.Assum e the mass of the rod acts at the centeras indicated in the figure .

k k

,mg

! e.g.

Figure Pl.90

1.91. Cons ider the inverted pe nd ulum of Figure 1.37 as disc ussed in E xample 1.8.1. As sum ethat a dashpot (of dampin g rate c) also acts on the pendulum paralle l to the two sprin g'How does th is affect the stabili ty properti es of the pendulum?

1.92. Replace the massles s ro d of the inverted pen dulun of Figure 1.37 with a solid-objec:com po und pe nd ulum of Figure 1.20(b) . Calculate the equations of vibration and discu s­values of the parameter for which the syste m is stable.

1.93. A simple model of a control tab fo r an airplane is sketc he d in Figur e P1.93.The eq u,j'tion of mot ion for the ta b about the hinge point is written in terms of the angle 0 fr orrthe centerli ne to be

18 + (c - I d)6 + kO= 0

Here 1 is the moment of inertia of the tab, k is the rot ational stiffness of the hinge, c :'the ro tational da mping in the hinge, and I de is the negati ve da mping provided by th,ae ro dynamic for ces (indicated by arrows in th e figure ). Discuss the sta bility of the soli.tion in terms of the param eter s c and .f /.

Figure Pl.93 A simple mode l of an air plane con tro l tab.