Balakumar Balachandran, Magrab E.B. Vibrations

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where

and, as discussed in Section 2.4.2, the signum function sgn(y) is 1 when

y  0 and is 1 when y  0. Furthermore, we have employed the following

deﬁnitions:

Although it is possible to ﬁnd a solution for this piecewise linear system,

here we obtain a numerical

solution for convenience. We shall determine the

response of this system when it is subjected to an initial (dimensionless) ve-

locity dy(0)/dt  V

/(v

d)  10, the damping factor z  0.15, and the val-

ues of m are 0, 1, and 10. The results are shown in Figure 4.25. We see that the

introduction of the spring-stops decreases the amplitude of the mass. In addi-

tion, it has the effect of decreasing the period of oscillation, which is equiva-

lent to increasing its natural frequency.

t  v



y 

and

2z 

 y  sgn1y2

0y 0 1

h1y 2 0

0y 0 1

170 CHAPTER 4 Single Degree-of-Freedom System

The MATLAB function ode45 was used.

FIGURE 4.24

Single degree-of-freedom system

with additional springs that are not

contacted until the mass displaces

a distance d in either direction.

Source: Reprinted by permission of

Federation of the European Bio-

chemical Societies from Journal of

Sound Vibration, 207, H.Y.Hu., FEBS

Letters, "Primary Resonance of a

Harmonically Forced Oscillator with

a Pair of Symmetric Set-Up Elastic

1997, with permission from Elsevier

Science.

mk mk

FIGURE 4.25

Response of the system shown in Figure 4.24 with prescribed initial velocity V

/(v

d)  10.

0 5 10 15 20 25 30



y(t)



4.5.2 Nonlinear Damping

We compare the free responses of systems with linear viscous damping,

Coulomb damping, and ﬂuid damping, which are obtained from Eqs. (3.23),

(3.24), and (3.25), respectively. They are rewritten here in terms of non-

dimensional time t  v

t as

(4.56)

and the coefﬁcients d

and d

are given by

(4.57)

The ﬁrst of Eqs. (4.56) governs a linear vibratory system with viscous damp-

ing, the second of Eqs. (4.56) governs a vibratory system with nonlinear

damping due to dry friction, and the third of Eqs. (4.56) governs a vibratory

system with nonlinear damping due to a ﬂuid.

We solve Eqs. (4.56) numerically

for the case where the initial velocity

is zero, the initial displacement is 3 units, and z  d

 d

 0.2. The

results are shown in Figure 4.26. The period of damped oscillations about the

system equilibrium position is different in all three cases. For viscous damp-

ing, we have shown in Section 4.2 that the amplitude decays exponentially.

For Coulomb damping, it can be shown

that the amplitude decays linearly

with a slope 2d

/p, where the plus sign is for the envelope of the negative

peaks and the negative sign is for the envelope of positive peaks. The decay

of the amplitude for ﬂuid damping does not have any equivalent expression.

Nonlinear System Response Dependence on Initial Conditions

To illustrate that the long-time response of a nonlinear system depends on the

initial conditions, the system with dry friction governed by the second of



mmg

and



Fluid damping

 d

 x  0

Dry friction

damping

 d

sgn1dx/dt2 x  0

Linear viscous

damping

 2z

 x  0

4.5 Single Degree-of-Freedom Systems With Nonlinear Elements 171

⎫

⎬

⎭

⎫

⎪

⎬

⎪

⎭

⎫

⎪

⎬

⎪

⎭

The MATLAB function ode45 was used.

See, for example, D. J. Inman, Engineering Vibration, 2nd ed., Prentice Hall, Upper Saddle

River, NJ, pp. 65–68 (2001).

Eqs. (4.56) is revisited. During the free oscillations, the system will come to

a stop or reach a rest state when

(4.58)

In other words, this system has multiple equilibrium positions, and the locus

of these positions in the state space is the straight line joining the points

(d

,0) and (d

,0).

The free responses of the system initiated from two sets of initial

conditions x(0)  3 units and (0)  0, and x(0)  5 units and (0)  0,

in the state space are shown in Figure 4.27. Again, these responses can be

determined analytically by noting that the system is linear in the region

(t)  0 and linear in the region (t)  0. However, the responses shown in

Figure 4.27 are determined numerically.

As seen from Figure 4.27, for the

two different initial conditions, the system comes to a rest in ﬁnite time at two

different positions, and the respective rest positions are reached at two differ-

ent times.

 0

and

0x 0 d

172 CHAPTER 4 Single Degree-of-Freedom System

FIGURE 4.26

Comparisons of displacement responses for three different damping models.

The MATLAB function ode45 was used.

0 2 4 6 8 10 12 14 16



4mmg/k



Amplitude

Coulomb

Fluid

Viscous

FIGURE 4.27

(a) Displacement histories and (b) phase portraits for the free response of a system with

dry friction subjected to two different initial displacements: d

 0.86.

0 2 4 6 8 10 12



x(t)



(a)

4 2 0 2 4 6



Displacement

Velocity

(3,0) (5,0)

(b)

174 CHAPTER 4 Single Degree-of-Freedom System

EXERCISES

Section 4.2.1

4.1 A shaft undergoing torsional vibrations is held

ﬁxed at one end, and it has attached at the other end a

disk with a rotary inertia of about the rota-

tional axis. The rotary inertia of the shaft is negligible

compared to that of the disk, and the torsional stiff-

ness of the shaft is /rad. The disk is placed in

an oil housing, and the associated dissipation is mod-

eled by using an equivalent torsional viscous damper

with damping coefﬁcient c

. What should the value

of c

be so that the system is critically damped?

Section 4.2.2

4.2 A load of mass m is suspended by an elastic cable

from the mid-span of a beam, which is held ﬁxed at

one end and is free at the other end, as shown in Fig-

ure E4.2. The beam has a length L

of 4 m and a ﬂex-

ural stiffness EI  60 Nm

. The elastic cable has

a length L

of 8 m, a diameter d of 200 mm, and

it is made from material with a Young’s modulus of

elasticity E

 3  10

N/m

. The load has a mass of

200 kg. Assume that the damping in this vibratory

system is negligible. If the mass is provided an initial

velocity of 0.1 m/s, determine the ensuing motion and

describe it. If the boundary conditions of the beam

were changed so that the beam is now simply sup-

ported at both ends, how does the maximum ampli-

tude of motion change?

8 N

4 kg

4.3 A 25 kg television set is placed on a light table

supported by four cylindrical legs made from a steel

alloy material with a Young’s modulus of elasticity

E  400 MPa. Each of these legs has a length of

0.5 m and a diameter of 10 mm. Consider free mo-

tions of this system in the vertical direction and de-

termine the displacement response when the initial

displacement is zero and the initial velocity is 0.2 m/s.

4.4 Consider two masses shown in Figure E4.4,

which are involved in an impact. The mass is m



2 kg, the spring stiffness is k  500 N/m, and the

damping coefﬁcient is /m. This system is

initially at rest when it is impacted by a 1 kg mass m

which is traveling with a speed of 10 m/s just before

impact. The coefﬁcient of restitution e associated

with these two impacting bodies is 0.8. Determine the

c  15 N

4.6 SUMMARY

In this chapter, the solution for the free response for a linear vibratory system

has been studied. It was shown that the solution is determined by the initial

conditions, and the responses for different types of initial conditions have

been explored and discussed. The notion of logarithmic decrement was intro-

duced and the relationship between the damping factor and the logarithmic

decrement was established. In addition, the notion of stability was brieﬂy ad-

dressed and the problem of machine-tool chatter was examined. The inﬂuence

of nonlinear stiffness and nonlinear damping on the free response of a vibra-

tory system was also studied.

FIGURE E4.2

Beam

Elastic cable

Exercises 175

displacement responses of these bodies after impact

and plot the time histories of their respective re-

sponses. Is there a possibility for another impact?

4.5 Replace the Kelvin-Voigt model used in Example

4.6 with a Maxwell model and determine the govern-

ing equation of motion when the two viscoelastic bod-

ies are in contact with each other.

4.6 Consider the system with Maxwell model treated

in Example 4.7 and treat the case where the spring

with stiffness k is absent. Determine the force trans-

mitted to the ﬁxed base and plot the time histories of

the normalized force as shown in Figure 4.13 for z



0.2 and z

 1.3.

4.7 For the system of Example 4.5, determine design

guidelines that one can use to minimize the relative

acceleration ; that is, the acceleration of the single

degree-of-freedom system inside the container rela-

tive to the container. Present these guidelines in terms

of the coefﬁcient of restitution e, the undamped natu-

ral frequency v

, and the damping of the packaging

material. This type of design guideline is useful for

packaging of electronic components.

4.8 Determine the initial velocity of a damped vibra-

tory system from the time-history data provided in Fig-

ure E4.8 for the displacement of a vibratory system.

Section 4.2.3

4.9 In a certain experiment, assume that you can only

measure the velocity of a vibratory system. Derive an

expression that can be used to determine the logarith-

mic decrement and the damping factor from velocity

data similar to the one derived in Section 4.2.3 for dis-

placement data.

4.10 Consider the translational vibrations of a “small”

rigid lathe along the cutting direction. A model of this

system has a stiffness element k  20  10

N/m, an

inertia element m  22.5 kg, and a damping coefﬁ-

cient /m. Determine the free response

of this system for a 5 mm initial displacement and

zero initial velocity. Plot the response and discuss it.

How does the nature of the response of the system

change when the damping coefﬁcient is increased to

/m?

4.11 Determine the damping factor and natural fre-

quency of a damped vibratory system from the time-

history data provided in Figure E4.11 for the dis-

placement.

4.12 In the system shown in Figure E4.12, mass m

resting on mass m

, which is supported by a spring of

stiffness k. The mass m

is 1 kg, the mass m

is 0.5 kg,

and the stiffness k is 1 kN/m. If the spring is pushed

down 15 mm below the system’s static-equilibrium

position and released, determine if the mass m

will

ever loose contact during the subsequent motion.

4.13 Assume that the hinged door of Figure 4.2 is

0.8 m wide and has a mass of 15 kg. It is found that it

takes a force of 15 N to keep the door opened at an

angle of 90°. When the force is released, the door

takes 2.1 s to reach its closed position. What is the

damping ratio of the system?

43,000 N

c  21205 N

FIGURE E4.4

FIGURE E4.8

0 1 2 3 4 5 6 7 8

0.2

0.4

0.6

0.8

1.2

1.4

t (s)

u (rad)

176 CHAPTER 4 Single Degree-of-Freedom System

the free response of the system initiated from an ini-

tial displacement of X

m and zero initial velocity.

Section 4.2.4

4.16 A microelectromechanical system has a mass of

0.30 g and a stiffness of 0.15 N/m. Assume that the

damping coefﬁcient for the system is negligible and

that the gravity loading is normal to the direction of

motion. Determine the displacement response for this

system, if the system is provided an initial displace-

ment of 2 m and an initial velocity of 5 mm/s.

4.17 An empirical formula used for determining the

natural frequency associated with translational mo-

tions of a steel chimney has the form

rad/s

where d is the diameter of the chimney and h is the

height of the chimney. The damping factor associated

with a chimney of diameter 7 m and height of 100 m

is 0.002. This chimney is located in a region where

wind gusts are capable of producing initial displace-

ments of 0.1 m and initial velocities of 0.4 m/s. De-

termine the maximum displacement experienced by

the chimney and check if it satisﬁes the construction

codes, which require the maximum displacement to

be less than 4% of the diameter. If the present chim-

ney does not satisfy this code, what design changes

would you propose so that the construction code is

satisﬁed?

4.18 A quarter-car model of a heavy vehicle is shown

in Figure E4.18. This vehicle is traveling with a con-

stant speed v on a ﬂat road. It hits a bump, which pro-

duces an initial displacement of 0.2 m and an initial

velocity of 0.1 m/s at the base of the system. If the

mass m of the vehicle is 5000 kg, the stiffness k

is 2800 kN/m, and the damping coefﬁcient c is

18 kN  s

/m, determine the displacement response of

this system and discuss when the system returns to

within 2% of its equilibrium position.

 7100

4.14 Consider the mercury-ﬁlled (r  13.6 

kg/m

) U-tube manometer shown in Figure 2.16.

The total length of the mercury in the manometer is

0.7 m. When the mercury is displaced from its equi-

librium position, damped oscillations are observed.

The oscillations are such that the peak amplitude nine

periods away from the peak amplitude of the ﬁrst cy-

cle has decreased by a factor of eight. What are the

values of the viscous damping factor and the damped

frequency of oscillation?

4.15 A truck tire is characterized with a stiffness of

1.25  10

N/m, a mass of 35 kg, and a damping co-

efﬁcient of 4200 Ns/m. Determine the natural fre-

quency and damping factor for this system, and obtain

FIGURE E4.11

FIGURE E4.12

H. Bachmann et al., Vibration Problems in Structures: Practical

Guidelines, Birkhäuser Verlag, Basel, Germany (1995).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5



Time t (seconds)

Displacement u (radians)

Exercises 177

4.19

Consider the slender tower shown in Figure

E4.19, which vibrates in the transverse direction

shown in the ﬁgure.

It is made from reinforced con-

crete. An estimate for the ﬁrst natural frequency of

this system is 0.15 Hz. The logarithmic decrement

values measured for the tower with uncracked rein-

forced concrete material and cracked reinforced con-

crete material are 0.04 and 0.10, respectively. If a

wind gust induces an initial displacement of 0.5 m and

an initial velocity of 0.2 m/s, determine the peak dis-

placement amplitudes in the cases with uncracked

concrete material and cracked concrete material.

Section 4.3

4.21 Consider the vibratory model of the micro-

electromechanical system treated in Example 3.15. If

the stiffness of the translation spring k, the stiffness of

the torsion spring k

, and the system damping coefﬁ-

cient c are all positive values, would it be possible for

the system to have unstable behavior?

4.22 When there is a ﬂuid moving with a speed V

through a pipe, the transverse vibrations y of a mass m

attached to the pipe can be described by the following

equation

where k

is the equivalent stiffness of the pipe and c is

a constant that depends on the pipe cross-sectional

area, the length of the pipe, and the density of the

ﬂuid. If m is always positive, k

is always positive, c is

positive, and V is positive, determine when the system

can exhibit unstable behavior.

 3k

 cV

4y  0

4.20 Consider the diving platform that is shown in

Figure E4.20. The damping factor associated with this

platform, which is made from reinforced concrete, is

0.012, and the ﬁrst natural frequency of the platform-

slab vibrations is designed to be 12 Hz.

When a

diver jumps off this platform, an initial displacement

of 1 mm and an initial velocity of 0.1 m/s are induced

to the platform. Determine the resulting vibrations.

Bachmann et al., ibid.

FIGURE E4.18

FIGURE E4.19

FIGURE E4.20

Platform slab

Typical design rules require this frequency to be greater than or

equal to 10 Hz.

178 CHAPTER 4 Single Degree-of-Freedom System

4.26

A rigid disk rotates with a constant angular

speed v. A mass m is mounted in a slot in the disk as

shown in Figure E4.26. Assume that gravity acts nor-

mal to the plane of the disk. The mass is restrained by

two identical springs of stiffness k and each spring is

initially stretched by an amount d

. Determine an ex-

pression for the natural frequency of the system and

the effect that d

and v have on the natural frequency

when the displacement of the mass is small. At what

speed does the system become unstable?

4.24 In studies of aircraft wing ﬂutter, the following

simpliﬁed model is used to study the system vibrations

where the system parameters are m, c, and k. The

damping constant a is due to aerodynamic forces,

which change with the angle of attack. Beyond a cer-

tain angle of attack, this constant can assume negative

values. Determine the conditions on m, c, a, and k for

which this system can be unstable.

4.25 Consider the pendulum of length L and mass m

that is rotating about an axis through its hinged point

with an angular speed v. At a given v, the pendulum

swings out an angle a as shown in Figure E4.25.

a) If the pendulum can oscillate an additional angle w

about a, then determine the governing equation of

motion for this system.

b) What are the dynamic equilibrium positions for

this system; that is, when ?

c) Discuss the stability of the system.

 w

 0

 1c  a 2x

 kx  0

4.23 A rigid and uniform bar undergoing rotational

motions in the vertical plane is restrained by a torsional

spring of stiffness k

at the pivot point O, as shown

in Figure E4.23. The bar has a mass m and a length l.

Determine what k

should be, so that small oscillations

about the upright position of the bar (i.e., u  0), are

stable.

FIGURE E4.23



FIGURE E4.25

FIGURE E4.26

4.27 A mass m shown in Figure E4.27 is suspended in

a magnetic ﬁeld by two springs of length L that each

have a tension T

. The magnetic attraction force on the

mass is a function of the magnetic ﬂux 

, and this

force is inversely proportional to the square of the dis-

tance from the magnet; that is,

where b is a constant. For small values of , ﬁnd a

relationship between the magnetic ﬂux and the spring

tension for which the system is stable.

0x 0

mag



b£

1a  x2





Exercises 179

Section 4.5

4.28 Compare the free-oscillation characteristics of

the following systems, when they are set into motion

from the initial displacement of 0.1 m and zero initial

velocity.

Assume that the parameter values are as follows: m 

10 kg, /m, k  10 N/m, and a  25 N/m

Consider another set of free responses, where each

of these three systems is set into motion from an ini-

tial displacement of 0.6 m. Compare these responses

with the previously obtained responses. What can you

conclude?

4.29 Consider the displacement-time history shown

in Figure E4.29 for free oscillations of a vibratory sys-

c  10 N

 cx

 kx  ax

 0

 cx

 kx  ax

 0

 cx

 kx  0

tem. Examine this history and discuss if this system

can be characterized as having viscous damping.

4.30 Consider the system with the piecewise-linear

spring given in Figure 4.24 and choose the damping

factor z to be 0.1. Set the system in motion from the

initial conditions y(0)  0 units and the nondimen-

sional initial velocity .

Obtain the displacement-time histories of the system,

and plot them as shown in Figure 4.25 for the follow-

ing values of the stiffness coefﬁcient m: (a) 0, (b) 1,

and (c) 20. For each case, tabulate the nondimen-

sional period for each cycle of oscillation and discuss

the results.

dy10 2/dt  V

/1v

d2 20

FIGURE E4.27

FIGURE E4.29

North

South

0 1 2 3 4 5 6 7 8



Time t (seconds)

Displacement u (radians)