Ch03 - Harmonically Excited Vibrations_Part1

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336 CHAPTER 3 HARMONICAlLY EXCITED VIBRATION

match the items in the two columns below:

1. Magnification factor of an undamped system

2. Period of beating

3. Magnification factor of a damped system

4. Damped frequency

S. Quality factor

6. Displacement transmissibility

3.6 Match the following equations of motion:

1. m"i + ci + kz = -my2. Mx' + ex + kx = meCli sin wt

3. mx + kx ± pN = F(t)

4. mx + k(1 + i{J)x = Fo sin wt

S. mx + eX + kx = Fosinwt

IPROBLEMS

2 ...a.

00" - £I)

[

1 + (2'r)2 J1/2b. (I _ r2)2 + ( ~ r ) '

Id ' - -

21- r

e . w n ~

a. System with Coulomb damping

b. System with viscous damping

c. System subject to base excitation

d. System with hysteresis damping

e. System with rotating unbalance

Section 3.3 Response of an Undamped System Under Harmonic Force

3.1 A weight of 50 N is suspended from a spring of stiffness 4000 N/m and is subjected to a har·

monic force of amplitude 60 N and frequency 6 Hz. Find (a) the extension of the spring due

to the suspended weight, (b) the static displacement of the spring due to the maximum

applied force, and (c) the amplitude of forced motion of the weight.

3.2 A spring-mass system is subjected to a harmonic force whose frequency is close to the nat

ural frequency of the sys tem.1fthe forcing frequency is 39.8Hz and the natural frequency is

40.0 Hz, determine the period of beating.

3.3 Consider a spring-mass system, with k = 4000N/m and m = 10 kg, subject to a harmonic

force F(t) = 400 cos lOt N. Find and plot the total response of the system under the follow

ing initial conditions:B. Xo = 0.1 m,.:to = 0

b. Xo = 0,;'0 = IOmls

e. Xo = 0.1 m,;,o = IOmls



PROBLEMS 337

3.4 Consider a spring·mass system, with k = 4000 N/m and m = 10 kg, subject to a harmonic

force F(t) = 400 cos 20t N. Find and plot the total response of the system under the fol

lowing initial conditions:

a. Xo = 0.1 In , Xo = 0

b. Xo = 0, Xo = IOmis

c. Xo = O.lm,xo = IOmis

3.S Consider a spring-mass system, with k = 4000 N/m and m = 10 kg, subject to a harmonic

force F(t) = 400 cos 20.1t N. Find and plot the total response of the system under the fol

lowing initial conditions:

a. Xo = 0.1 In , :to = 0


c. Xo = O.lm,xo = IOmis

3.6 Consider a spring-mass system, with k = 4000 N/m and m = 10 kg, subject to a harmonic

force F(t) = 400 cos 30t N. Find and plot the total response of the system under the follow

ing initial conditions:

a. Xo = 0.1 In , Xo = 0


c. Xo = O.lm,:to = IOmis

3.7 A spring-mass system consists of a mass weighing 100 N and a spring with a stiffness of

2000 N/m. The mass is subjected to resonance by a harmonic force F(t) = 25 cos wt N.

Find the amplitude of the forced motion at the end of (a) t cycle, (b) 2 cycles, and (c) cycles.

3.8 A mass m is suspended from a spting of stiffness 4000 N/m and is subjected to a harmonic

force having an amplitude of 100 N and a frequency of 5 Hz. The amplitude of the forced

motion of the mass is observed to be 20 mm. Find the value of In .

3.9 A spring-mass system with m = 10 kg and k = 5000 N/m is subjected to a harmonic force

of amplitude 250 N and frequency w. I f he maximum amplitude of the mass is observed to

be 100 mm, find the value of w.

3.10 In Fig. 3.I(a), a periodic force F(t) = Fo cos wt is applied at a point on the spring that is

located at a distance of 25 percent of its length from the fixed support. Assuming that c = 0,

find the steady-state response of the mass In .

3.11 A spring-mass system, resting on an inclined plane, is subjected to a harmonic force as

shown in Fig. 3.38. Find the response of the system by asswning zero initial conditions.

3.12 The natural frequency of vibration of a person is found to be 5.2 Hz while standing on a hor

izontal floor. Asswning damping to be negligible, dete rmine the following:

a. I f he weight of the person is 70 kg,. determine the equivalent stiffness of his body in the

vertical direction.b. I f he floor is subjected to a vertical harmonic vibration of frequency of 5.3 Hz and ampli

tude of 0.1 m due to an unbalanccdrotating machine operating on the floor, determine the

vertical displacement of the person.




k 4,000 N/m

FIGURE 3.38

3.13 Plot the forced-vibration response of a spring-mass system given by Eq. (3.13) for the following sets of data:

a. Set 1: 88t = 0.1, W = 5,0011 = 6, Xo = 0.1, Xo = 0.5

b. Set 2: 8" ~ 0.1, w ~ 6.1, w. ~ 6, Xo ~ 0.1, Xo ~ 0.5

Co Set 3: 8" ~ 0.1, w ~ 5.9, w. ~ 6, Xo ~ 0.1, Xo ~ 0.5

3.14 A spring-mass system is set to vibrate from zero initial conditions under a harmonic force.

The response is found to exhibit the phenomenon of beats with the period ofbeating equal to

0.5 s and the period of oscillation equal to 0.05 s. Find the natural frequency of the system

and the frequency of the harmonic force.

3.15 A spring-mass system, with m ~ 100 kg and k ~ 400 N/m, is subjected to a harmonic force

f(t} ~ Focos wt withFo ~ ION. Find the response of the system when wis equal to (a) 2

rad/s, (b) 0.2 rad/s, and ( c) 20 radls. Discuss the results.

3.16 An aircraft engine has a rotating unbalanced mass m at radius r. I f he wing can be modeled

as a cantilever beam of uniform cross section a X b, as shown in Fig. 3.39(b), determine the

maximum deflection of the engine at a speed of N rpm. Assume damping and effect of the

wing between the engine and the free end to be negligible.

3.17 A three-bladed wind turbine (Fig. 3.4O(a» has a small unbalanced mass m located at a radius

r in the plane of the blades. The blades are located from the central vertical (y) axis at a dis

tanceR and rotate at an angular velocity of w. I f he supporting truss can bemodeled as a hol

low steel shaft of outer diameter 0.1 m and inner diameter 0.08 m, determine the maxim um

stresses developed at the base of the support (pointA). Themass moment of inertia of the tur

bine system about the vertical (y) axis is 10. AssumeR ~

0.5 m,m ~

0.1 kg,r ~

0.1 m,10 100 kg_m2

, h ~ 8 m, and w ~ 31.416 rad/s.



PROBLEMS 339

I' L- - - - I t LJ!-- - - - - - - - - - . .

f<-#--I------j fE

"", ,, '"-

(b)

FIGURE 3.39

y

r - R ~ y

r-R---jRoto< I I

- Gw%

I--%

, 7 ' ~I /,J,t

"b)

FIGURE 3.40 11uoo-blodod wind twbioe. ~ _ of- . . " '" ' " ' ' ' ' ' ' ' IN", .)




3.18 An electromagnetic fatigue-testing machine is shown in Fig. 3.41 in which an alternating force

is applied to the specimen by passing an alternating current of frequency f through the anna-

1lIre. I f he weight of the anna1llre is 40 lb, the stiffness of the spting (k l ) is 1O,217.0296lblin,.

and the stiffness of the steel specimen is 75 X 104 Ibrm., detemtine the frequency of the alter

nating current that induces a stress in the specimen that is twice the amount generated by

the magnets.

Armature

Specimen

(stiffness, 1<,,)

Spring(stiffness, k ,)

Magnets

FIGURE 3.41 Electromagnetic fatigue-testing machine.

3.19 The spring actuator shown in Fig. 3.42 operates by using the air pressure from a pneumatic

controller (P) as input and providing an output displacement to a valve (x) proportional to the

input air pressure. The diaphragm, made of a fabric-base rubber, has an area A and deflects

under the input air pressure against a spring of stiffness k. Find the response of the valve

under a harmonically fluctuating input air pressure p(t) = Po sin wt for the following data:

Po = 10 psi, w = 8 rad/s, A = J(]() in.2, k = 400 lbrm., weight of spring = 151b, and

weight of valve and valve rod = 20 lb.

3.20 In the cam-follower system shown in Fig. 3.43, the rotation of the cam imparts a vertical

motion to the follower. The pushrod, which acts as a spring, has been compressed by anamount Xo before assembly. Determine the following: (a) equation of motion of the follower,

including the gravitational force; (b) force exerted on the follower by the cam; and (c) condi

tions under which the follower loses contact with the cam.



Input(air under prcssure,p)

C : C ~ + - - - V a l v e rod

Diaphragm(area,A)

8 : ~ - - S p r i n g (stiffness, k)

FIGURE 3.42 A spring actuator.

Ar

FIGURE 3.43

Pushrod (A, E,/),spring constant k AE

I

Follower, mass = m

Output air_ - + ~ (pressurecontroUed

by motion

of valve)

PROBLEMS 341




3.21* Design a solid steel shaft supported in bearings which carries the rotor of a turbine at the

middle. The rotor weighs 500 Ib and delivers a power of 200 hp at 3000 rpm. In order to keep

the stress due to the unbalance in the rotor small, the critical speed of the shaft is to be made

one-fifth of the operating speed of the rotor. The length of the shaft is to be made equal to at

least 30 times its diameter.

3.22 A hollow steel shaft, oflength 100 in., outer diameter 4 in., and inner diameter 3.5 in., carries

the rotor of a turbine, weighing 500 Ib, at the middle aud is supported at the ends in bearings.

The clearance between the rotor and the stator is 0.5 in. The rotor has an eccentricity equiva

lent to a weight of 0.5 Ib at a radius of 2 in. A limit switch is installed to stop the rotor when

ever the rotor touches the stator. If the rotor operates at resonance, how long will it take to

activate the limit switch? Assume the iuitial displacement and velocity of the rotor perpen

dicular to the shaft to be zero.

3.23 A steel cantilever beam, carrying a weight of 0.1 Ib at the free end, is used as a frequency

meter.' The beam has a length of 10 in., width of 0.2 in., and thickness of 0.05 in. The inter

nal friction is equivalent to a damping ratio of 0.01. When the fixed end of the beam is sub

jected to a harmouic displacement Y(I) = 0.05 cos rul, the maximum tip displacement has

been observed to be 2.5 in. Find the forcing frequency.

3.24 Derive the equation of motion aud find !he steady-state response of the system shown in Fig. 3.44

for rotational motion about the binge 0 for the following data: kl = k2 = 5000N/m,

a = 0.25 m, b = 0.5 m, I = I m, M = 50 kg, m = 10 kg, Fa = 500 N, ru = 1000 rpm.

3.25 Derive !he eqoation of motion and find the steady-state solution of he system shown in Fig. 3.45

for rotational motion about the hinge 0 for the following data: k = 5000N/m,

I = 1m , m = 10 kg, Ma = 100 N-m, ru = 1000 rpm.

Uniform rigid bar. mass m

, k ~------+I

I - - - - -b

FIGURE 3.44

F(t) = Fo sin wt

M

*The asterisk denotes a design-type problem or a problem with no unique answer.'The use of cantilever beams as frequency meters is discussed in detail in Section lOA.



PROBLEMS 343

I. 311-----0+----- 4 ------+I.

FIGURE 3.45

Section 3.4 Response of a Damped System under Harmonic Force

3.26 Consider a spring-mass-damper system with k = 4000 N/m, m = 10 kg, and c = 40 N-s/m.

Fiod the steady-state and total responses of the system under the harmonic force

F(t) = 200 cos lOt N and the initial conditions Xo = 0.1 m and:to = O.

3.27 Consider a spring-mass-damper system withk =

4000N/m, m =

10 kg, and c= 40

N-s/m.Find the steady-state and total responses of the system under the harmonic force

F(t) = 200 cos lOt N and the initial conditions Xo = 0 and:to = 10 mls.


Find the steady-state and total responses of the system under the harmonic force

F(t) = 200 cos 20t N and the initial conditions Xo = 0.1 m and:to = o.


Find the steady-state and total responses of the system under the harmonic force

F(t) = 200 cos 20t N and the initial conditions Xo = 0 and:to = 10 mls.

3.30 A four-cylinder automobile engine is to be supported on three shock mounts, as indicated in

Fig. 3.46. The engine-block assembly weighs 500 lb. IT the unbalanced force generated bythe engine is given by 200 sin 100 7rt lb, design the three shock mounts (each of stiffness k

aod viscous-damping constant c) such that the amplitude of vibration is less than 0.1 in.

3.31 The propeller of a ship, of weight lOS N and polar mass moment of inertia 10,000 kg_m2, is

connected to the engine through a hollow stepped steel propeller shaft, as shown in Fig. 3.47.

Assuming that water provides a viscous damping ra tio of 0.1, detennine the torsional vibra

tory response of the propeller when the engine induces a harmonic angular displacement of

0.05 sin 3l4.l6t rad at the base (pointA) of the propeller shaft.

3.32 Find the frequency ratio r = OJ/OJ. at which the amplitude of a single-degree-of-freedom

damped system attains the maximum value. Also find the value of the maximum amplitude.

3.33 Figure 3.48 shows a permanent-magnet moving-coil ammeter. When current (l ) flows throughthe coil wouod on the core, the core rotates by an angle proportional to the magnitude of the

current that is indicated by the pointer on a scale. The core, with the coil, has a mass moment of

inertia 10, the torsional spring constant is k" and the torsional damper has a damping constant




Piston

FIGURE 3.46 Four-cylinder automobile engine.

Engine

block

of Ct. The scaIe of the ammeter is cab'brared such that when a direct currentofmagni1ude 1 ampere

is passed through the coil, the pointer indicates a currentof

1 ampere. The meter has to be recali-brated for measuring the magni1ude of alternating current. Determine the steady-state value of the

current indicatedby the pointerwhen an alternating current of magni1ude 5 amperes and frequency

50 Hz is passed through the coil. Assume 10 = 0.001 N_m2, kt = 62.5 N-mlrad, and

Ct = 0.5 N-m-s/rad.

3.34 A spring-mass-damper system is subjected to a harmonic force. The amplitude is found to be

20 mm at resonance and 10 mm at a frequency 0.75 times the resonant frequency. Find the

damping ratio of the system.

3.35 For the system shown in Fig. 3.49, x and y denote, respectively, the absolute displacements

of the mass m and the end Q of the dashpot C1' (a) Derive the equation of motion of the

mass m, (b) find the steady-state displacement of the mass m, and (c) find the force trans

mitted to the support at P, when the end Q is subjected to the harmonic motiony(t) = Y cos ... .



D(aJ

O.2mPropeller Hollow stepped(10' N) propeller shaft

(b)

FIGURE 3.47 Propellerof a ship .

Permanent magnet

J

Engine

Engine vibratory

disturbance(aocos wt)

Core

orsional damper

B Pivot

FIGURE 3.48 Pennanent-magnet moving-coil ammeter.

PROBLEMS 345




r -x ( t ) yet) = Y cos lOt

r-I-----e.m

Q

/

FIGURE 3.49

3.36 The equation of motion of a spring-mass-damper system subjected to a hannon ic force can

be expressed as

(E.l)

i. Find the steady-state response of the system in the form x,(t) = C, cos wt + C2 sin wt

ii. Find the total response of the system in the form

Assume the initial conditions of the systemasx (t = 0) = xoandx(t = 0) = xo.

3.37 A video camera, of mass 2.0 kg, is mounted on the top of a bank building for surveillance.

The video camera is fixed at one end of a tubular aluminum rod whose other end is fixed to

the building as shown in Fig. 3.50. The wind-induced force acting on the video camera, fit),

is found to be harmonic with [(t) = 25 cos 75.39841 N. Determine the cross-sectional

dimensions of the aluminum tube i f he maximum amplitude of vibration of the video camera

is to be limited to 0.005 m.

Video camera

, F o c o s r u t

r bular aluminum rod/Tu

O.Sm

1Building

FIGURE 3.50



PROBLEMS 347

3.38 A turbine rotoris mounted on a stepped shaft that is fixed at both ends as shown in Fig. 3.51.

The torsional stiffnesses of the two segments of the shaft are given by k,l = 3,000 N-mlrad

and ka = 4,000 N-mlrad The turbine generates aluumonic torque given by M(t) = Mo cos

OJt about the shaft axis with M 0 = 200 N-m and OJ = 500 radls. The mass moment of inertia

of the rotor about the shaft axis is 10 = 0.05 kg_m2. Assuming the equivalent torsional

dsmping constant of the system as c, = 2.5 N-m-s/rad, determine the steady-state response

of the rotor, 6(t).

FIGURE 3.51

~ - - - - - - - - - - - - ~ ~M(I) Mo cos wI

3.39 It is required to design an electromechanical system to achieve a natural frequency of 1000 Hz

and a Q factor of 1200. Determine the dsmping factor and the bandwidth of the system.

3.40 Show that, for small values of dsmping, the dsmping ratio, can be expressed as

fi12 - WI, = ",-,,---,.---::-c

Olz + OJI

where OJI and Olz are the frequencies corresponding to the half-power points.

3.41 A torsional system consists of a disc of mass moment of inertia 10 = 10 kg_m2, a torsional

dsmper of dsmping constant c, = 300 N-m-s/rad, and a steel shaft of diameter 4 em and

length 1 m (fixed at one end and attached to the disc at the other end). A steady angnlar oscil

lation of amplitude 2° is observed when a luumonic torque of magnitude 1000 N-m is

applied to the disc. (a) Find the frequency of the applied torque, an d (b) find the maximum

torque transmitted to the support.

3.42 For a vibratiug system, m = 10 kg, k = 2500 N/m, and c = 45 N-s/m. A luumonic force of

amplitude 180 N and frequency 3.5 Hz acts on the mass. I f the initial displacement and

velocity of the mass are 15 mm and 5 mis, find the complete solution representing the motion

of the mass.

3.43 The peak amplitude of a single-degree-of-freedom system, under a harmonic excitation, isobserved to be 0.2 in. I f he undsmped natural frequency of the system is 5 Hz, and the static

deflection of the mass under the maximum force is 0.1 in., (a) estimate the damping ratio of

the system, and (b) find the frequencies corresponding to the amplitudes at half power.




3.44 The landing gear of an airplane can be idealized as the spring-mass-damper system shown in

Fig. 3.52. I f he runway sutface is described y(I) = Yo cos wI, determine the values of k and

c that limit the amplitude of vibration of the airplane (x) to 0.1 m. Assume m = 2000 kg,

Yo = 0.2 m, and w = 157.08 radls.

3AS A precision grinding machine (Fig. 3.53) is supported on an isolator that has a stiffness of

I MN/m and a viscous damping constant of I kN-s/m. The floor on which the machine is

mounted is subjected to a harmonic disturbance due to the operation of an unbalanced engine

in the vicinity of the grinding machine. Find the maximum acceptable displacement ampli

tude of the floor i f the resulting amplitude of vibration of the grinding wheel is to be

restricted to 10-6m. Assume that the grinding machine and the wheel are a rigid body of

weight 5000 N.

3.46 Derive the equationof motion and find the steady-state response of the system shown in Fig. 3.54

for rotational motion about the hinge 0 for the following data: k = 5000 N/m,

I = I m, C = 1000 N-s/m, m = 10 kg, Mo = 100 N-m, w = 1000 rpm.

Wheel----+

(a)

(b)

FIGURE 3.52 Modeling of landing gear.

Housing withstrut andviscous damping

Mass ofaircraft, m

0- 11M -- TI-I!..:..



Grindffigmachine

x(t)

G r i n d m g ~ - / . ________

wheel

7 7 . ~ ? 7 . ~ ? 7 7 7 - ~ 7 Z ~ ? 7 T / /

y(t) = Ysm",t1 Ysmzoo lTtm

FIGURE 3.53

FIGURE 3.54

Floor

Uniformrigid bar,

PROBLEMS 349

3.47 An air compressor of mass lOO kg is mounted on an elastic foundation. I t has been observed

that, when a harmonic force of amplitude lOO N is applied to the compressor, the maximum

steady-state displacement of 5 mm occurred at a frequency of 300 rpm. Deterntine the eqniv

alent stiffness and dampmg constant of the foundation.

3.48 Find the steady-state response of the system shown in Fig. 3.55 for the following data:

k j = 1000 NIm, k2 = 500 N/m, c = 500 N-s/m, m = 10 kg, r = 5 cm, Jo = 1 kg_m2,

Fo = 50 N, w = 20 radls.

3.49 A uniform slender bar of mass m may be supported in one of two ways as shown in Fig. 3.56.

Deterntine the arrangement that results in a reduced steady-state response of the bar under a

harmonic force, Fo sin wt, applied at the middle of the bar, as shown in the figure.




Pulley, mass moment of inertia 10

FIGURE 3.55

Fa sin wt

e , Ge

Uniform bar,

massm

~ f ,I,(a)

FIGURE 3.56

'-r---r-'tx(t)

c

e ,

Fa sin wt

Ge

Uniform bar,

massm

(b)

Section 3.5 Response of a Damped System Under F(t) = F P

3.50 Derive the expression for the complex frequency response of an undamped torsional system.

3.51 A damped single-degree-of-freedom system, with parameters m = 150 kg, k = 25 kNlm,

and c = 2000 N-s/m, is subjected to the harmonic force fit) = 100 cos 20t N. Find the

amplitude and phase angle of the steady-state response of the system using a graphical

method.

Section 3.6 Response of a System Under the Harmonic Motion of the Base3.52 A single-story building frame is subjected to a harmonic ground acceleration, as shown in

Fig. 3.57. Find the steady-state motion of the floor (mass m).



PROBLEMS 351

m

,----..

3i ,,(t) = A cos ",t

FIGURE 3.57

3.53 Find the horizontal displacement of the floor (mass m) of the building frame shown in Fig. 3.57

when the ground acceleration is given by Xg = 100 sin wt mmlsec: Assume m = 2000 kg,

k = 0.1 MN/m,w = 25rad1s,andxg(t = 0) =Xg (t = 0) = x(t = 0) = i ( t = 0) = O.

3.54 I f the ground in Fig. 3.57, is subjected to a horizontal harmonic displacement with fre

quency w = 200 radls and amplitude Xg = 15 mm, find the amplitude of vibration of the

floor (mass m). Assume the mass of the floor as 2000 kg and the stiffness of the columns as

O.5MN/m.

3.55 An automobile is modeled as a single-degree-of-freedom system vibrating in the vertical

direction. It is driven along a road whose elevation varies sinusoidally. The distance from

peak to trough is 0.2 m and the distance along the road between the peaks is 35 m. I f he nat

ural frequency of the automobile is 2 Hz and the damping ratio of the shock absorbers is

0.15, determine the amplitude of vibration of the automobile at a speed of 60 kmlhour. I f he

speed of the automobile is varied, find the most uofavorable speed for the passengers.

3.56 Derive Eq. (3.74).

3.57 A single-story boilding frame is modeled by a rigid floor of mass m and columns of stiffness

k, as shown in Fig. 3.58. I t s proposed that a damper shown in the figure is attached to absorb

vibrations due to a horizontal ground motion y(t) = Y cos wt. Derive an expression for the

damping constant of the damper that absorbs maximum power.

3.58 A uniform bar of mass m is pivoted at point 0 and supported at the ends by two springs, as

shown in Fig. 3.59. End P of spring PQ is subjected to a sinusoidal displacement,

x(t) = Xo sin wI. Find the steady-state angular displacement of the bar when I = 1 m,

k = 1000 N/m, m = 10 kg, Xo = I em, and w = 10 radls.

3.59 A uniform bar of mass m is pivoted at point 0 and supported at the ends by two springs, as

shown in Fig. 3.60. End P of spring PQ is subjected to a sinusoidal displacement,x(t) = Xo sin wt. Find the steady-state angular displacement of the bar when I = 1m ,

k = 1000 N/m, c = 500 N-slm, m = 10 kg, Xo = I em, and w = 10 radls.




,

m

Ic I

-=r+J.....!_

III ~

y(t) = Y cos Cdt

FIGURE 3.58

Uniform bar,

~ ; - 4 · ~ - · - - - - - ~ - - - - - - ~ · 1FIGURE 3.59

Uniform

k

k

A - - - - - ~ ; + - - - - - ~ - - - - - L - - - - ~ ~o

FIGURE 3.60



PROBLEMS 353

3.60 Find the frequency ratio, T = Tm, at which the displacement transmissibility given by

Eq. (3.68) attains a maximum value.

3.61 An automobile, weighing 1000 Ib empty and 3000 Ib fully loaded, vibrates in a vertical

direction while traveling at 55 mph on a rough road having a sinusoidal waveform with an

amplitude Y ft and a period 12 ft. Assuming that the automobile can be modeled as a

single-degree-of-freedom system with stiffness 30,000 Ib/ft and damping ratio, = 0.2,

determine the amplitude of vibration of the automobile when it is (a) empty and (b) fully

loaded.

3.62 The base of a damped spring-mass system, with m = 25 kg and k = 2500 N/m, is subjected

to a harmonic excitation y(t) = 10 cos wt. The amplitude of the mass is found to be 0.05 m

when the base is excited at the natural frequency of the system with 10 = 0.01 m. Determine

the damping constant of the system.

Section 3.7 Response of a Damped System Under Rotating Unbalance

3.63 A single-cylinder air compressor of mass 100 kg is mounted on rubber mounts, as shown in

Fig. 3.61. The stiffness and damping constants of the rubber mounts are given by 106 N/m

and 2000 N-slm, respectively. I f he unbalance of the compressor is equivalent to a mass0.1 kg located at the end of the crank (point A), determine the response of the compressor at

a crank speed of 3000 rpm. Assume T = 10 cm and I = 4() cm.

I , - t - - - : : : ; ~ ~ P i s t o n

Rubber mounts777 ' 77z777> ' 77777T -777777z7Z

FIGURE 3.61




3.64 One of the tail rotor blades of a helicopter has an unbalanced mass of m ~ 0.5 kg at a dis·

tance of e ~ 0.15 m from the axis of rotation, as shown in Fig. 3.62. The tail section has a

length of 4 m, a mass of 240 kg, a flexural stiffness (El) of 2.5 MN_m2, and a damping ratio

of 0.15. The mass of the tail rotor blades, including their drive system, is 20 kg. Determine

the forced response of the tail section when the blades rotate at 1500 rpm.

Tail rotor blades

FIGURE 3.62

3.65 When an exhaust fan of mass 380 kg is supported on springs with negligible damping, the

resulting static deflection is found to be 45 mm. I f the fan has a rotating unbalance of

0.15 kg-m, find (a) the amplitude of vibration at 1750 rpm, and (b) the force transmitted to

the ground at this speed.

3.66 A fixed-fixed steel beam,oflength

5m,

width 0.5m,

and thickness 0.1m,

carriesan

electricmotor of mass 75 kg and speed 1200 rpm at its mid-span, as shown in Fig. 3.63. A rotating

force of magnitude Fo ~ 5000 N is developed due to the unbalance in the rotor of the motor.

Find the amplitude of steady-state vibrations by disregarding the mass of the beam. What

will be the amplitude i f he mass of the beam is considered?

~ F O~ F = = = = = = = = = ~ = = = = = = = B ~

I ~ . - - - ~ - - ~ . I - . - - - ~ ~FIGURE 3.63

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Ch03 - Harmonically Excited Vibrations_Part1