Mechanical Vibrations in SI Units

Singiresu S. Rao

Chapter 9

Vibration Control - all with Video Answers

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Chapter Questions

Problem 1

A car moving on a rough road, in the form of a sinusoidal surface, is modeled as a springmass system, as shown in Fig. 9.41 . The sinusoidal surface has a wave length of $5 \mathrm{~m}$ and an amplitude of $Y=1 \mathrm{~mm}$. If the mass of the car, including the passengers, is $1500 \mathrm{~kg}$ and the stiffness of the suspension system $(k)$ is $400 \mathrm{kN} / \mathrm{m},$ determine the range of speed $(v)$ of the car in which the passengers perceive the vibration. Suggest possible methods of improving the design for a more comfortable ride of the passengers.

Penny Riley

Penny Riley

Numerade Educator

Problem 2

The root mean square value of a signal $x(t), x_{\mathrm{rms}},$ is defined as
$$
x_{\mathrm{rms}}=\left\{\begin{array}{l}
\left.\lim _{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T} x^{2}(t) d t\right\}^{1 / 2}
\end{array}\right.
$$
Using this definition, find the root mean square values of the displacement $\left(x_{\mathrm{rms}}\right),$ velocity $\left(\dot{x}_{\mathrm{rms}}\right),$ and acceleration $\left(\ddot{x}_{\mathrm{rms}}\right)$ corresponding to $x(t)=X \cos \omega t$.

James Kiss

Numerade Educator

Problem 3

Two identical discs are connected by four bolts of different sizes and mounted on a shaft, as shown in Fig. 9.42 . The masses and locations of three bolts are as follows:
$m_{1}=35 \mathrm{~g}, r_{1}=110 \mathrm{~mm},$ and $\theta_{1}=40^{\circ} ; m_{2}=15 \mathrm{~g}, r_{2}=90 \mathrm{~mm},$ and $\theta_{2}=220^{\circ} ;$ and
$m_{3}=25 \mathrm{~g}, r_{3}=130 \mathrm{~mm}, \theta_{3}=290^{\circ} .$ Find the mass and location of the fourth bolt $\left(m_{c}, r_{c}\right.$
and $\theta_{c}$ ), which results in the static balance of the discs.

Ahmed Kamel

Numerade Educator

Problem 4

Four holes are drilled in a uniform circular disc at a radius of $100 \mathrm{~mm}$ and angles of $0^{\circ}, 60^{\circ},$ $120^{\circ}$, and $180^{\circ} .$ The weight removed at holes 1 and 2 is $100 \mathrm{~g}$ each and the weight removed at holes 3 and 4 is $150 \mathrm{~g}$ each. If the disc is to be balanced statically by drilling a fifth hole at a radius of $125 \mathrm{~mm}$, find the weight to be removed and the angular location of the fifth hole.

James Kiss

Numerade Educator

Problem 5

hree masses, weighing $225 \mathrm{~g}, 320 \mathrm{~g}$, and $550 \mathrm{~g}$, are attached around the rim, of diameter $750 \mathrm{~mm}$, of a flywheel at the angular locations $\theta=10^{\circ}, 100^{\circ}$, and $190^{\circ}$, respectively. Find the weight and the angular location of the fourth mass to be attached on the rim that leads to the dynamic balance of the flywheel.

James Kiss

Numerade Educator

Problem 6

The amplitude and phase angle due to original unbalance in a grinding wheel operating at 1200 rpm are found to be $0.25 \mathrm{~mm}$ and $40^{\circ}$ counterclockwise from the phase mark. When a trial mass $m=170 \mathrm{~g}$ is added at $65^{\circ}$ clockwise from the phase mark and at a radial distance $65 \mathrm{~mm}$ from the center of rotation, the amplitude and phase angle are observed to be $0.5 \mathrm{~mm}$ and $150^{\circ}$ counterclockwise. Find the magnitude and angular position of the balancing weight if it is to be located $65 \mathrm{~mm}$ radially from the center of rotation.

James Kiss

Numerade Educator

Problem 7

An unbalanced flywheel shows an amplitude of $0.165 \mathrm{~mm}$ and a phase angle of $15^{\circ}$ clockwise from the phase mark. When a trial weight of magnitude $50 \mathrm{~g}$ is added at an angular position $45^{\circ}$ counterclockwise from the phase mark, the amplitude and the phase angle become $0.225 \mathrm{~mm}$ and $35^{\circ}$ counterclockwise, respectively. Find the magnitude and angular position of the balancing weight required. Assume that the weights are added at the same radius.

James Kiss

Numerade Educator

Problem 8

In order to determine the unbalance in a grinding wheel, rotating clockwise at $2400 \mathrm{rpm}, \mathrm{a}$ vibration analyzer is used and an amplitude of $0.1 \mathrm{~mm}$ and a phase angle of $45^{\circ}$ are observed with the original unbalance. When a trial mass $m=100 \mathrm{~g}$ is added at $20^{\circ}$ clockwise from the phase mark, the amplitude becomes $0.2 \mathrm{~mm}$ and the phase angle $145^{\circ} .$ If the phase angles are measured counterclockwise from the right-hand horizontal, calculate the magnitude and location of the necessary balancing weight.

James Kiss

Numerade Educator

Problem 9

A turbine rotor is run at the natural frequency of the system. A stroboscope indicates that the maximum displacement of the rotor occurs at an angle $229^{\circ}$ in the direction of rotation. At what angular position must mass be removed from the rotor in order to improve its balancing?

James Kiss

Numerade Educator

Problem 10

A rotor, having three eccentric masses in different planes, is shown in Fig. $9.43 .$ The axial, radial, and angular locations of mass $m_{i}$ are given by $l_{i}, r_{i},$ and $\theta_{i},$ respectively, for $i=1,2,3$. If the rotor is to be dynamically balanced by locating two masses $m_{b 1}$ and $m_{b 2}$ at radii $r_{b 1}$ and $r_{b 2}$ at the angular locations $\theta_{b 1}$ and $\theta_{b 2},$ as shown in Fig. $9.42,$ derive expressions for $m_{b 1} r_{b 1}, m_{b 2} r_{b 2}, \theta_{b 1},$ and $\theta_{b 2}$.

James Kiss

Numerade Educator

Problem 11

The rotor shown in Fig. $9.44($ a) is balanced temporarily in a balancing machine by adding the masses $m_{1}=m_{2}=90 \mathrm{~g}$ in the plane $A$ and $m_{3}=m_{4}=90 \mathrm{~g}$ in the plane $D$ at a radius of $75 \mathrm{~mm}$, as shown in Fig. $9.44(\mathrm{~b})$. If the rotor is permanently balanced by drilling holes at a radius of $100 \mathrm{~mm}$ in planes $B$ and $C,$ determine the position and amount of material to be removed from the rotor. Assume that the adjustable masses $m_{1}$ to $m_{4}$ will be removed from the planes $A$ and $D$.

James Kiss

Numerade Educator

Problem 12

Masses of $1 \mathrm{~kg}, 3 \mathrm{~kg}$, and $2 \mathrm{~kg}$ are located at radii $50 \mathrm{~mm}, 75 \mathrm{~mm}$, and $25 \mathrm{~mm}$ in the planes $C, D,$ and $E,$ respectively, on a shaft supported at the bearings $B$ and $F,$ as shown in Fig. $9.45 .$ Find the masses and angular locations of the two balancing masses to be placed in the end planes $A$ and $G$ so that the dynamic load on the bearings will be zero.

James Kiss

Numerade Educator

Problem 13

The data obtained in a two-plane balancing procedure are given in the table below. Determine the magnitude and angular position of the balancing masses, assuming that all angles are measured from an arbitrary phase mark and all masses are added at the same radius.

James Kiss

Numerade Educator

Problem 14

Figure 9.46 shows a rotating system in which the shaft is supported in bearings at $A$ and $B$. The three masses $m_{1}, m_{2},$ and $m_{3}$ are connected to the shaft as indicated in the figure. (a) Find the bearing reactions at $A$ and $B$ if the speed of the shaft is 1000 rpm. (b) Determine the locations and magnitudes of the balancing masses to be placed at a radius of $0.25 \mathrm{~m}$ in the planes $L$ and $R,$ which can be assumed to pass through the bearings $A$ and $B$.

James Kiss

Numerade Educator

Problem 15

A flywheel, with a mass of $50 \mathrm{~kg}$ and an eccentricity of $12 \mathrm{~mm}$, is mounted at the center of a steel shaft of diameter $25 \mathrm{~mm}$. If the length of the shaft between the bearings is $0.75 \mathrm{~m}$ and the rotational speed of the flywheel is $1200 \mathrm{rpm}$, find (a) the critical frequency in $\mathrm{rad} / \mathrm{s},(\mathrm{b})$ the vibration amplitude of the rotor, and (c) the force transmitted to the bearing supports.

James Kiss

Numerade Educator

Problem 16

Derive the expression for the stress induced in a shaft with an unbalanced concentrated mass located midway between two bearings.

James Kiss

Numerade Educator

Problem 17

A steel shaft of diameter $2.5 \mathrm{~cm}$ and length $1 \mathrm{~m}$ is supported at the two ends in bearings. It carries a turbine disc, of mass $20 \mathrm{~kg}$ and eccentricity $0.005 \mathrm{~m},$ at the middle and operates at $6000 \mathrm{rpm}$. The damping in the system is equivalent to viscous damping with $\zeta=0.01$. Determine the whirl amplitude of the disc at
(a) operating speed,
(b) critical speed, and (c) 1.5 times the critical speed.

James Kiss

Numerade Educator

Problem 18

Find the bearing reactions and the maximum bending stress induced in the shaft at (a) operating speed, (b) critical speed, and (c) 1.5 times the critical speed for the shaft-rotor system described in Problem 9.17 .

James Kiss

Numerade Educator

Problem 18

Find the bearing reactions and the maximum bending stress induced in the shaft at (a) operating speed, (b) critical speed, and (c) 1.5 times the critical speed for the shaft-rotor system described in Problem 9.17 .

James Kiss

Numerade Educator

Problem 19

Solve Problem 9.17 by assuming that the material of the shaft is aluminum rather than steel.

James Kiss

Numerade Educator

Problem 20

Solve Problem 9.18 by assuming that the material of the shaft is aluminum rather than steel.

James Kiss

Numerade Educator

Problem 21

A shaft, having a stiffness of $3.75 \mathrm{MN} / \mathrm{m}$, rotates at $3600 \mathrm{rpm}$. A rotor, having a mass of $60 \mathrm{~kg}$ and an eccentricity of 2000 microns, is mounted on the shaft. Determine (a) the steady-state whirl amplitude of the rotor and (b) the maximum whirl amplitude of the rotor during start-up and stopping conditions. Assume the damping ratio of the system as 0.05 .

James Kiss

Numerade Educator

Problem 22

he cylinders of a four-cylinder in-line engine are placed at intervals of $300 \mathrm{~mm}$ in the axial direction. The cranks have the same length, $100 \mathrm{~mm}$, and their angular positions are given by $0^{\circ}, 180^{\circ}, 180^{\circ},$ and $0^{\circ} .$ If the length of the connecting rod is $250 \mathrm{~mm}$ and the reciprocating mass is $1 \mathrm{~kg}$ for each cylinder, find the unbalanced forces and moments at a speed of 3000 rpm, using the center line through cylinder 1 as the reference plane.

James Kiss

Numerade Educator

Problem 23

The reciprocating mass, crank radius, and connecting-rod length of each of the cylinders in a two-cylinder in-line engine are given by $m, r,$ and $l$, respectively. The crank angles of the two cylinders are separated by $180^{\circ} .$ Find the unbalanced forces and moments in the engine.

James Kiss

Numerade Educator

Problem 24

A four-cylinder in-line engine has a reciprocating weight of $1.5 \mathrm{~kg}$, a stroke of $15 \mathrm{~cm},$ and a connecting-rod length of $25 \mathrm{~cm}$ in each cylinder. The cranks are separated by $10 \mathrm{~cm}$ axially and $90^{\circ}$ radially, as shown in Fig. 9.47 . Find the unbalanced primary and secondary forces and moments with respect to the reference plane shown in Fig. 9.47 at an engine speed of 1500 rpm.

James Kiss

Numerade Educator

Problem 25

The arrangement of cranks in a six-cylinder in-line engine is shown in Fig. 9.48 . The cylinders are separated by a distance $a$ in the axial direction, and the angular positions of the cranks are given by $\alpha_{1}=\alpha_{6}=0^{\circ}, \alpha_{2}=\alpha_{5}=120^{\circ},$ and $\alpha_{3}=\alpha_{4}=240^{\circ} .$ If the crank
length, connecting-rod length, and the reciprocating mass of each cylinder are $r, l,$ and $m,$ respectively, find the primary and secondary unbalanced forces and moments with respect to the reference plane indicated in Fig. 9.48 .

James Kiss

Numerade Educator

Problem 26

A single-cylinder engine has a total mass of $150 \mathrm{~kg}$. Its reciprocating mass is $5 \mathrm{~kg}$, and the rotating mass is $2.5 \mathrm{~kg}$. The stroke $(2 r)$ is $15 \mathrm{~cm}$, and the speed is 600 rpm. (a) If the engine is mounted floating on very weak springs, what is the amplitude of vertical vibration of the engine? (b) If the engine is mounted solidly on a rigid foundation, what is the alternating force amplitude transmitted? Assume the connecting rod to be of infinite length.

James Kiss

Numerade Educator

Problem 27

An electronic instrument is to be isolated from a panel that vibrates at frequencies ranging from $25 \mathrm{~Hz}$ to $35 \mathrm{~Hz}$. It is estimated that at least 80 percent vibration isolation must be achieved to prevent damage to the instrument. If the instrument weights $85 \mathrm{~N},$ find the necessary static deflection of the isolator.

Narayan Hari

Numerade Educator

Problem 28

An exhaust fan, having a small unbalance, weights $800 \mathrm{~N}$ and operates at a speed of $600 \mathrm{rpm} .$ It is desired to limit the response to a transmissibility of 2.5 as the fan passes through resonance during start-up. In addition, an isolation of 90 percent is to be achieved at the operating speed of the fan. Design a suitable isolator for the fan.

James Kiss

Numerade Educator

Problem 29

An air compressor of mass $500 \mathrm{~kg}$ has an eccentricity of $50 \mathrm{~kg}-\mathrm{cm}$ and operates at a speed of 300 rpm. The compressor is to be mounted on one of the following mountings:
(a) an isolator
(b) a shock absorber having a damping consisting of a spring with negligible damping, and ratio of 0.1 and negligible stiffness. Select a suitable mounting and specify the design details by considering the static deflection of the compressor, the transmission ratio, and the amplitude of vibration of the compressor.

James Kiss

Numerade Educator

Problem 30

The armature of a variable-speed electric motor, of mass $200 \mathrm{~kg}$, has an unbalance due to manufacturing errors. The motor is mounted on an isolator having a stiffness of $10 \mathrm{kN} / \mathrm{m}$ and a dashpot having a damping ratio of 0.15 . (a) Find the speed range over which the amplitude of the fluctuating force transmitted to the foundation will be larger than the exciting force. (b) Find the speed range over which the transmitted force amplitude will be less than $10 \%$ of the exciting force amplitude.

James Kiss

Numerade Educator

Problem 31

A dishwashing machine weighing $75 \mathrm{~kg}$ operates at $300 \mathrm{rpm}$. Find the minimum static deflection of an isolator that provides 60 percent isolation. Assume that the damping in the isolator is negligible.

James Kiss

Numerade Educator

Problem 32

A washing machine of mass $50 \mathrm{~kg}$ operates at $1200 \mathrm{rpm}$. Find the maximum stiffness of an isolator that provides 75 percent isolation. Assume that the damping ratio of the isolator is 7 percent.

James Kiss

Numerade Educator

Problem 33

It is found that an exhaust fan, of mass $80 \mathrm{~kg}$ and operating speed $1000 \mathrm{rpm}$, produces a repeating force of $10,000 \mathrm{~N}$ on its rigid base. If the maximum force transmitted to the base is to be limited to $2000 \mathrm{~N}$ using an undamped isolator, determine (a) the maximum permissible stiffness of the isolator that serves the purpose; (b) the steady-state amplitude of the exhaust fan with the isolator that has the maximum permissible stiffness; and (c) the maximum amplitude of the exhaust fan with isolation during start-up.

James Kiss

Numerade Educator

Problem 34

It has been found that a printing press, of mass $300 \mathrm{~kg}$ and operating speed $3000 \mathrm{rpm}$, produces a repeating force of $30,000 \mathrm{~N}$ when attached to a rigid foundation. Find a suitable viscously damped isolator to satisfy the following requirements: (a) the static deflection should be as small as possible; (b) the steady-state amplitude should be less than $2.5 \mathrm{~mm} ;$ (c) the amplitude during start-up conditions should not exceed $20 \mathrm{~mm} ;$ and $($ d) the force transmitted to the foundation should be less than $10,000 \mathrm{~N}$.

James Kiss

Numerade Educator

Problem 35

A compressor of mass $120 \mathrm{~kg}$ has a rotating unbalance of $0.2 \mathrm{~kg}-\mathrm{m} .$ If an isolator of stiffness $0.5 \mathrm{MN} / \mathrm{m}$ and damping ratio 0.06 is used, find the range of operating speeds of the compressor over which the force transmitted to the foundation will be less than $2500 \mathrm{~N}$.

James Kiss

Numerade Educator

Problem 36

An internal combustion engine has a rotating unbalance of $1.0 \mathrm{~kg}-\mathrm{m}$ and operates between 800 rpm and 2000 rpm. When attached directly to the floor, it transmitted a force of $7018 \mathrm{~N}$ at $800 \mathrm{rpm}$ and $43,865 \mathrm{~N}$ at $2000 \mathrm{rpm} .$ Find the stiffness of the isolator that is necessary to reduce the force transmitted to the floor to $6000 \mathrm{~N}$ over the operating-speed range of the engine. Assume that the damping ratio of the isolator is $0.08,$ and the mass of the engine is $200 \mathrm{~kg}$.

James Kiss

Numerade Educator

Problem 37

A small machine tool of mass $100 \mathrm{~kg}$ operates at 600 rpm. Find the static deflection of an undamped isolator that provides 90 percent isolation.

James Kiss

Numerade Educator

Problem 38

A diesel engine of mass $300 \mathrm{~kg}$ and operating speed $1800 \mathrm{rpm}$ is found to have a rotating unbalance of $1 \mathrm{~kg}-\mathrm{m} .$ It is to be installed on the floor of an industrial plant for purposes of emergency power generation. The maximum permissible force that can be transmitted to the floor is $8000 \mathrm{~N}$ and the only type of isolator available has a stiffness of $1 \mathrm{MN} / \mathrm{m}$ and a damping ratio of 5 percent. Investigate possible solutions to the problem.

James Kiss

Numerade Educator

Problem 39

The force transmitted by an internal combustion engine of mass $500 \mathrm{~kg}$, when placed directly on a rigid floor, is given by
$$
F_{t}(t)=(18,000 \cos 300 t+3600 \cos 600 t) \mathrm{N}
$$
Design an undamped isolator so that the maximum magnitude of the force transmitted to the floor does not exceed $12,000 \mathrm{~N}$.

Narayan Hari

Numerade Educator

Problem 40

Design the suspension of a car such that the maximum vertical acceleration felt by the driver is less than $2 g$ at all speeds between $70 \mathrm{~km} / \mathrm{h}$ and $140 \mathrm{~km} / \mathrm{h}$ while traveling on a road whose surface varies sinusoidally as $y(u)=0.5 \sin 2 u \mathrm{~m},$ where $u$ is the horizontal distance in meters. The weight of the car, with the driver, is $700 \mathrm{~kg}$ and the damping ratio of the suspension is to be $0.05 .$ Use a single-degree-of-freedom model for the car.

James Kiss

Numerade Educator

Problem 41

Consider a single-degree-of-freedom system with Coulomb damping (which offers a constant friction force, $F_{c}$ ). Derive an expression for the force transmissibility when the mass is subjected to a harmonic force, $F(t)=F_{0} \sin \omega t$.

James Kiss

Numerade Educator

Problem 42

Consider a single-degree-of-freedom system with Coulomb damping (which offers a constant friction force, $F_{c}$ ). Derive expressions for the absolute and relative displacement transmissibilities when the base is subjected to a harmonic displacement, $y(t)=Y \sin \omega t$.

James Kiss

Numerade Educator

Problem 43

When a washing machine, of mass $200 \mathrm{~kg}$ and an unbalance $0.02 \mathrm{~kg}-\mathrm{m}$, is mounted on an isolator, the isolator deflects by $5 \mathrm{~mm}$ under the static load. Find (a) the amplitude of the washing machine and (b) the force transmitted to the foundation at the operating speed of 1200 rpm.

Narayan Hari

Numerade Educator

Problem 44

An electric motor, of mass $60 \mathrm{~kg}$, rated speed $3000 \mathrm{rpm}$, and an unbalance $0.002 \mathrm{~kg}-\mathrm{m}$, is to be mounted on an isolator to achieve a force transmissibility of less than $0.25 .$ Determine (a) the stiffness of the isolator, (b) the dynamic amplitude of the motor, and (c) the force transmitted to the foundation.

James Kiss

Numerade Educator

Problem 45

An engine is mounted on a rigid foundation through four springs. During operation, the engine produces an excitation force at a frequency of 3000 rpm. If the weight of the engine causes the springs to deflect by $10 \mathrm{~mm}$, determine the reduction in the force transmitted to the foundation.

Narayan Hari

Numerade Educator

Problem 46

A sensitive electronic system, of mass $30 \mathrm{~kg}$, is supported by a spring-damper system on the floor of a building that is subject to a harmonic motion in the frequency range $10 \mathrm{~Hz}$ to $75 \mathrm{~Hz}$. If the damping ratio of the suspension is $0.25,$ determine the stiffness of the suspension if the amplitude of vibration transmitted to the system is to be less than $15 \%$ of the floor vibration over the given frequency range.

James Kiss

Numerade Educator

Problem 47

A machine of mass $1150 \mathrm{~kg}$ is mounted on springs. A piston of mass $m=25 \mathrm{~kg}$ moves up and down in the machine at a speed of 600 rpm with a stroke of $350 \mathrm{~mm}$. Considering the motion to be harmonic, determine the maximum force transmitted to the foundation if (a) $k=1.75 \mathrm{MN} / \mathrm{m},$ and (b) $k=4.5 \mathrm{MN} / \mathrm{m}$

James Kiss

Numerade Educator

Problem 48

A printed circuit board of mass $1 \mathrm{~kg}$ is supported to the base through an undamped isolator. During shipping, the base is subjected to a harmonic disturbance (motion) of amplitude $2 \mathrm{~mm}$ and frequency $2 \mathrm{~Hz} .$ Design the isolator so that the displacement transmitted to the printed circuit board is to be no more than $5 \%$ of the base motion.

Nicholas Mogoi

Numerade Educator

Problem 49

An electronic instrument of mass $10 \mathrm{~kg}$ is mounted on an isolation pad. If the base of the isolation pad is subjected to a shock in the form of a step velocity of $10 \mathrm{~mm} / \mathrm{s},$ find the stiffness of the isolation pad if the maximum permissible values of deflection and acceleration of the instrument are specified as $10 \mathrm{~mm}$ and $20 g$, respectively.

Narayan Hari

Numerade Educator

Problem 50

A water tank of mass $10^{5} \mathrm{~kg}$ is supported on a reinforced cement concrete column, as shown in Fig. $9.49(\mathrm{a}) .$ When a projectile hits the tank, it causes a shock, in the form of a step force, as shown in Fig. $9.49(\mathrm{~b})$. Determine the stiffness of the column if the maximum deflection of the tank is to be limited to $0.5 \mathrm{~m}$. The response spectrum of the shock load is shown in Fig. $9.49(\mathrm{c})$.

James Kiss

Numerade Educator

Problem 51

A viscously damped single-degree-of-freedom system has a body of mass $25 \mathrm{~kg}$ with a spring constant of $70 \mathrm{kN} / \mathrm{m}$. Its base is subjected to harmonic vibration. (a) When the base vibrates with an amplitude of $50 \mathrm{~mm}$ at resonance, the steady-state amplitude of the body is found to be $125 \mathrm{~mm}$. Find the damping ratio of the system. (b) When the base vibrates at a frequency of $10 \mathrm{~Hz}$, the steady-state amplitude of the body is found to be $35 \mathrm{~mm}$. Find the magnitude of the force transmitted to the base.

James Kiss

Numerade Educator

Problem 52

A single-degree-of-freedom system is used to represent an automobile, of mass $m$, damping constant $c$, and stiffness $k$, which travels on a rough road that is in the form of a sinusoidal surface with an amplitude $Y$ and wavelength $l$. If the automobile travels at a velocity $v$, derive an expression for the transmissibility of the vertical motion of the automobile mass $(m)$.

James Kiss

Numerade Educator

Problem 53

A sensitive instrument of mass $100 \mathrm{~kg}$ is installed at a location that is subjected to harmonic motion with frequency $20 \mathrm{~Hz}$ and acceleration $0.5 \mathrm{~m} / \mathrm{s}^{2}$. If the instrument is supported on an isolator having a stiffness $k=25 \times 10^{4} \mathrm{~N} / \mathrm{m}$ and a damping ratio $\zeta=0.05,$ determine the maximum acceleration experienced by the instrument.

James Kiss

Numerade Educator

Problem 54

An electronic instrument of mass $20 \mathrm{~kg}$ is to be isolated from engine vibrations with frequencies ranging from 1000 rpm to 3000 rpm. Find the stiffness of the undamped isolator to be used to achieve a 90 percent isolation.

Narayan Hari

Numerade Educator

Problem 55

A delicate instrument weighing $200 \mathrm{~N}$ is suspended by four identical springs, each with stiffness $50,000 \mathrm{~N} / \mathrm{m}$, in a rigid box as shown in Fig. 9.50 . The box is transported by a truck. If the truck is subjected to a vertical harmonic motion given by $y(t)=0.02 \sin 10 t \mathrm{~m},$ find the maximum displacement, velocity, and acceleration experienced by the instrument.

James Kiss

Numerade Educator

Problem 56

A damped torsional system is composed of a shaft and a rotor (disk). The torsional stiffness and the torsional damping constant of the shaft are given by $k_{t}=6000 \mathrm{~N}-\mathrm{m} / \mathrm{rad}$ and $c_{t}=100 \mathrm{~N}-\mathrm{m}-\mathrm{s} / \mathrm{rad}$. The mass moment of inertia of the rotor is $J_{0}=5 \mathrm{~kg}-\mathrm{m}^{2}$. The rotor is subjected to a harmonically varying torque of magnitude $M_{t}=500 \mathrm{~N}-\mathrm{m},$ which results in a steady-state angular displacement of $5^{\circ} .$ Find the frequency of the harmonically varying torque applied to the rotor and the maximum torque transmitted to the base or support of the system.

James Kiss

Numerade Educator

Problem 57

The force transmissibility of a damped single-degree-of-freedom system with base motion is given by Eq. (9.106):
$$
T_{f}=\frac{F_{t}}{k Y}=r^{2}\left\{\frac{1+(2 \zeta r)^{2}}{\left(1-r^{2}\right)^{2}+(2 \zeta r)^{2}}\right\}^{\frac{1}{2}}
$$
where $F_{t}$ is the magnitude of the force transmitted to the mass. Determine the frequency ratios (r) at which the force transmissibility attains maximum and minimum values. Discuss your results.

James Kiss

Numerade Educator

Problem 58

Derive an expression for the relative displacement transmissibility, $\frac{Z}{Y},$ where $Z=X-Y,$ for a damped single-degree-of-freedom system subjected to the base motion, $y(t)=Y \sin \omega t$.

James Kiss

Numerade Educator

Problem 59

During operation, the compressor unit of a refrigerator, with mass $75 \mathrm{~kg}$ and rotational speed 900 rpm, experiences a dynamic force of $200 \mathrm{~N}$. The compressor unit is supported on four identical springs, each with a stiffness of $k$ and negligible damping. Find the value of $k$ if only 15 percent of the dynamic force is to be transmitted to the support or base. Also, find the clearance space to be provided to the compressor unit.

James Kiss

Numerade Educator

Problem 60

An electronic instrument, of mass $20 \mathrm{~kg}$, is to be isolated to achieve a natural frequency of $15 \mathrm{rad} / \mathrm{s}$ and a damping ratio of 0.95 . The available dashpots can produce a damping constant (c) in the range $10 \mathrm{~N}-\mathrm{s} / \mathrm{m}$ to $80 \mathrm{~N}-\mathrm{s} / \mathrm{m}$. Determine whether the desired damping ratio can be achieved using a passive system. If a passive system cannot be used, design a suitable active control system to achieve the desired damping ratio.

James Kiss

Numerade Educator

Problem 61

A damped single-degree-of-freedom system has a mass ( $m$ ) of $5 \mathrm{~kg}$, stiffness $(k)$ of $20 \mathrm{~N} / \mathrm{m}$, and a damping constant $(c)$ of $5 \mathrm{~N}-\mathrm{s} / \mathrm{m}$. Design an active controller to achieve a settling time less than $15 \mathrm{~s}$ for the closed loop system. Hint: The settling time is defined by Eqs. (4.68) and (4.69).

James Kiss

Numerade Educator

Problem 62

A damped single-degree-of-freedom system has an undamped natural frequency of $20 \mathrm{rad} / \mathrm{s}$ and a damping ratio of $0.20 .$ Design an active control system which achieves an undamped natural frequency of $100 \mathrm{rad} / \mathrm{s}$ and a damping ratio of 0.8 . Assume that the mass, stiffness, and damping constant of the original system remain in place.

James Kiss

Numerade Educator

Problem 63

A printed circuit board $(\mathrm{PCB}),$ made of fiber reinforced plastic composite material, is attached to a chassis that is attached to a motor vibrating at a speed of $3000 \mathrm{rpm}$. The $\mathrm{PCB}$ can be modeled as a fixed-fixed beam, similar to the one shown in Fig. 9.27 , with a length
(l) $20 \mathrm{~cm}$, width $(w) 16 \mathrm{~cm},$ thickness $(t) 0.2 \mathrm{~cm}$, mass (m) $1.0 \mathrm{~kg}$, and Young's modulus $(E)$ $12.5 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}$. Determine the following:
a. Stiffness of the $\mathrm{PCB}$
b. Natural frequency of the $\mathrm{PCB}$
c. Displacement transmissibility of the $\mathrm{PCB}$
Assume the damping to be negligible.

Nicholas Mogoi

Numerade Educator

Problem 64

In the PCB described in Problem $9.63,$ it is desired to reduce the displacement transmissibility to a value of $0.25 .$ If the chassis mass is 50 percent of the mass of the $\mathrm{PCB},$ determine the necessary stiffness $(k)$ and damping constant $(c)$ of the isolator if the damping ratio of the isolator is required to be 0.01

James Kiss

Numerade Educator

Problem 65

A machine with a natural frequency of $4.2 \mathrm{~Hz}$ is subjected to a rotating unbalance force of amplitude $(F)$ of $20 \mathrm{~N}$ at a frequency of $4 \mathrm{~Hz}$. Design a suitable dynamic absorber for the machine assuming that the available clearance for the motion of the absorber mass is $15 \mathrm{~mm}$.

James Kiss

Numerade Educator

Problem 66

Derive an expression for the displacement transmissibility of a damped single-degree-offreedom system whose base is subjected to a general periodic displacement.

James Kiss

Numerade Educator

Problem 67

An air compressor of mass $200 \mathrm{~kg}$, with an unbalance of $0.01 \mathrm{~kg}-\mathrm{m}$, is found to have a large amplitude of vibration while running at $1200 \mathrm{rpm} .$ Determine the mass and spring constant of the absorber to be added if the natural frequencies of the system are to be at least 20 percent from the impressed frequency.

Anand Jangid

Numerade Educator

Problem 68

An electric motor, having an unbalance of $2 \mathrm{~kg}-\mathrm{cm}$, is mounted at the end of a steel cantilever beam, as shown in Fig. 9.51 . The beam is observed to vibrate with large amplitudes at the operating speed of 1500 rpm of the motor. It is proposed to add a vibration absorber to reduce the vibration of the beam. Determine the ratio of the absorber mass to the mass of the motor needed in order to have the lower frequency of the resulting system equal to 75 percent of the operating speed of the motor. If the mass of the motor is $300 \mathrm{~kg}$, determine the stiffness and mass of the absorber. Also find the amplitude of vibration of the absorber mass.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 69

The pipe carrying feedwater to a boiler in a thermal power plant has been found to vibrate violently at a pump speed of 800 rpm. In order to reduce the vibrations, an absorber consisting of a spring of stiffness $k_{2}$ and a trial mass $m_{2}^{\prime}$ of $1 \mathrm{~kg}$ is attached to the pipe. This arrangement is found to give the natural frequencies of the system as $750 \mathrm{rpm}$ and $1000 \mathrm{rpm} .$ It is desired to keep the natural frequencies of the system outside the operating speed range of the pump, which is 700 rpm to 1040 rpm. Determine the values of $k_{2}$ and $m_{2}$ that satisfy this requirement.

James Kiss

Numerade Educator

Problem 70

A reciprocating engine is installed on the first floor of a building, which can be modeled as a rigid rectangular plate resting on four elastic columns. The equivalent mass of the engine and the floor is $1000 \mathrm{~kg}$. At the rated speed of the engine, which is $600 \mathrm{rpm}$, the operators experience large vibration of the floor. It has been decided to reduce these vibrations by suspending a spring-mass system from the bottom surface of the floor. Assume that the spring stiffness is $k_{2}=875 \mathrm{kN} / \mathrm{m}$. (a) Find the mass to be attached to absorb the vibrations. (b) What will be the natural frequencies of the system after the absorber is added?

James Kiss

Numerade Educator

Problem 71

Find the values of $k_{2}$ and $m_{2}$ in Problem 9.54 in order to have the natural frequencies of the system at least 30 percent away from the forcing frequency.

James Kiss

Numerade Educator

Problem 72

A hollow steel shaft of outer diameter $50 \mathrm{~mm}$, inner diameter $38 \mathrm{~mm}$, and length $750 \mathrm{~mm}$ carries a solid disc of diameter $380 \mathrm{~mm}$ and mass $40 \mathrm{~kg}$. Another hollow steel shaft of length $500 \mathrm{~mm},$ carrying a solid disc of diameter $150 \mathrm{~mm}$ and mass $8 \mathrm{~kg}$, is attached to the first disc, as shown in Fig. 9.52 . Find the inner and outer diameters of the shaft such that the attached shaft-disc system acts as an absorber.

James Kiss

Numerade Educator

Problem 73

A rotor, having a mass moment of inertia $J_{1}=15 \mathrm{~kg}-\mathrm{m}^{2},$ is mounted at the end of a steel shaft having a torsional stiffness of $0.6 \mathrm{MN}-\mathrm{m} / \mathrm{rad}$. The rotor is found to vibrate violently when subjected to a harmonic torque of $300 \cos 200 t \mathrm{~N}-\mathrm{m}$. A tuned absorber, consisting of a torsional spring and a mass moment of inertia $\left(k_{t 2}\right.$ and $\left.J_{2}\right),$ is to be attached to the first rotor to absorb the vibrations. Find the values of $k_{t 2}$ and $J_{2}$ such that the natural frequencies of the system are away from the forcing frequency by at least $20 \%$.

James Kiss

Numerade Educator

Problem 74

Plot the graphs of $\left(\Omega_{1} / \omega_{2}\right)$ against $\left(m_{2} / m_{1}\right)$ and $\left(\Omega_{2} / \omega_{2}\right)$ against $\left(m_{2} / m_{1}\right)$ as $\left(m_{2} / m_{1}\right)$ varies from 0 to 1.0 when $\omega_{2} / \omega_{1}=0.1$ and 10.0 .

James Kiss

Numerade Educator

Problem 75

Determine the operating range of the frequency ratio $\omega / \omega_{2}$ for an undamped vibration absorber to limit the value of $\left|X_{1} / \delta_{\mathrm{st}}\right|$ to $0.5 .$ Assume that $\omega_{1}=\omega_{2}$ and $m_{2}=0.1 m_{1}$.

James Kiss

Numerade Educator

Problem 76

When an undamped vibration absorber, having a mass $30 \mathrm{~kg}$ and a stiffness $k,$ is added to a spring-mass system, of mass $40 \mathrm{~kg}$ and stiffness $0.1 \mathrm{MN} / \mathrm{m}$, the main mass ( $40 \mathrm{~kg}$ mass) is found to have zero amplitude during its steady-state operation under a harmonic force of amplitude $300 \mathrm{~N}$. Determine the steady-state amplitude of the absorber mass.

James Kiss

Numerade Educator

Problem 77

An electric motor, of mass $20 \mathrm{~kg}$ and operating speed $1350 \mathrm{rpm}$, is placed on a fixed-fixed steel beam of width $15 \mathrm{~cm}$ and depth $1.2 \mathrm{~cm},$ as shown in Fig. $9.53 .$ The motor has a rotating unbalance of $0.1 \mathrm{~kg}-\mathrm{m} .$ The amplitude of vibration of the beam under steady-state operation of the motor is suppressed by attaching an undamped vibration absorber underneath the motor, as shown in Fig. 9.53. Determine the mass and stiffness of the absorber such that the amplitude of the absorber mass is less than $2 \mathrm{~cm}$.

James Kiss

Numerade Educator

Problem 78

A bridge is found to vibrate violently when a vehicle, producing a harmonic load of magnitude $600 \mathrm{~N},$ crosses it. By modeling the bridge as an undamped spring-mass system with a mass $15,000 \mathrm{~kg}$ and a stiffness $2 \mathrm{MN} / \mathrm{m}$, design a suitable tuned damped vibration absorber. Determine the improvement achieved in the amplitude of the bridge with the absorber.

Narayan Hari

Numerade Educator

Problem 79

A small motor, of mass $50 \mathrm{~kg}$, is found to have a natural frequency of $100 \mathrm{rad} / \mathrm{s}$. It is proposed that an undamped vibration absorber of mass $4 \mathrm{~kg}$ be used to suppress the vibrations when the motor operates at $80 \mathrm{rad} / \mathrm{s}$. Determine the necessary stiffness of the absorber.

Supratim Pal

Numerade Educator

Problem 80

Consider the system shown in Fig. 9.54 in which a harmonic force acts on the mass $m$. Derive the condition under which the steady-state displacement of mass $m$ will be zero.

James Kiss

Numerade Educator

Problem 81

Show the variation of the transmission ratio, $\frac{X_{1}}{\delta_{\mathrm{st}}}$, with $\frac{\omega}{\omega_{1}}$ for an undamped dynamic vibration absorber for $\omega_{2}=\omega_{1}$ and $m_{2}=0.25 m_{1}$.

James Kiss

Numerade Educator

Problem 82

In a dynamic vibration absorber having $\frac{\omega_{2}}{\omega_{1}}=1$ and $\mu=\frac{m_{2}}{m_{1}}=\frac{1}{2},$ determine the frequency range over which the value of the transmission ratio, $\frac{X_{1}}{\delta_{\mathrm{st}}},$ is less than one.

James Kiss

Numerade Educator

Problem 83

Using MATLAB, plot Eq. (9.94) for $\zeta=0,0.25,0.5,0.75$, and 1 over the range $0 \leq r \leq 3$.

James Kiss

Numerade Educator

Problem 84

Using MATLAB, plot Eqs. (9.140) and (9.141) for $f=1, \zeta=0.2,0.3$, and 0.4 , and $\mu=0.2$ and 0.5 over the range $0.6 \leq \omega / \omega_{1}$.

James Kiss

Numerade Educator

Problem 85

Using MATLAB, plot the ratios $\Omega_{1} / \omega_{2}$ and $\Omega_{2} / \omega_{2}$ given by Eq. (9.146) for $\omega_{2} / \omega_{1}=1.5$, 3.0, and 4.5 and $m_{2} / m_{1}=0$ to 1 .

James Kiss

Numerade Educator

Problem 86

Using Program13.m, solve Problem 9.13 .

James Kiss

Numerade Educator

Problem 87

Write a computer program to find the displacement of the main mass and the auxiliary mass of a damped dynamic vibration absorber. Use this program to generate the results of Fig. $9.39 .$

James Kiss

Numerade Educator

Problem 88

Ground vibrations from a crane operation, a forging press, and an air compressor are transmitted to a nearby milling machine and are found to be detrimental to achieving specified accuracies during precision milling operations. The ground vibrations at the locations of the crane, forging press, and air compressor are given by $x_{c}(t)=A_{c} e^{-\omega_{c} \zeta_{c} t} \sin \omega_{c} t, x_{f}(t)=A_{f}$ $\sin \omega_{f} t, \quad$ and respectively, where $A_{c}=20 \mu \mathrm{m}, A_{f}=30 \mu \mathrm{m}$ $A_{a}=25 \mu \mathrm{m}, \omega_{c}=10 \mathrm{~Hz}, \omega_{f}=15 \mathrm{~Hz}, \omega_{a}=20 \mathrm{~Hz},$ and $\zeta_{c}=0.1 .$ The ground vibrations travel at the shear wave velocity of the soil, which is equal to $300 \mathrm{~m} / \mathrm{s}$, and the amplitudes attenuate according to the relation $A_{r}=A_{0} e^{-0.005 r}$, where $A_{0}$ is the amplitude at the source and $A_{r}$ is the amplitude at a distance of $r \mathrm{ft}$ from the source. The crane, forging press, and air compressor are located at a distance of $18 \mathrm{~m}, 24 \mathrm{~m},$ and $12 \mathrm{~m},$ respectively, from the milling machine. The equivalent mass, stiffness, and damping ratio of the machine tool head in vertical vibration (at the location of the cutter) are experimentally determined to be $500 \mathrm{~kg}$, $480 \mathrm{kN} / \mathrm{m},$ and $0.15,$ respectively. The equivalent mass of the machine tool base is $1000 \mathrm{~kg}$. It is proposed that an isolator for the machine tool be used, as shown in Fig. $9.55,$ to improve the cutting accuracies [9.2]. Design a suitable vibration isolator, consisting of a mass, spring, and damper, as shown in Fig. $9.55(\mathrm{~b}),$ for the milling machine such that the maximum vertical displacement of the milling cutter, relative to the horizontal surface being machined, due to ground vibration from all the three sources does not exceed $5 \mu \mathrm{m}$ peak-to-peak.

James Kiss

Numerade Educator