Balakumar Balachandran, Magrab E.B. Vibrations
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response of the mass m. Consider the base velocity
as the input, the acceleration response as the out-
put, and determine the frequency-response function of
this system.
5.13 The damped single degree-of-freedom mass-
spring system shown in Figure E5.13 has a mass m
20 kg and a spring stiffness coefficient k 2400 N/m.
Determine the damping coefficient of the system, if it
is given that the mass exhibits a response with an am-
plitude of 0.02 m when the support is harmonically
excited at the natural frequency of the system with an
amplitude Y
o
0.007 m. In addition, determine the
amplitude of the dynamic force transmitted to the
support.
x
$
1t 2
y
#
5.15 Determine the phase shift and amplitude error in
the output of an accelerometer that has a natural fre-
quency of 25 kHz and a damping ratio of 0.1, when it
is measuring vibrations at 1 kHz.
Section 5.7
5.16 A rotating machine runs intermittently. When
it is operating, it is rotating at 4200 rpm. The mass
of the machine is 100 kg, and it is supported by an
elastic structure with an equivalent spring constant of
160 kN/m and a viscous damper of damping coefficient
of 2400 N s/m that is in parallel with the spring. The
engineer would like to keep the maximum transmis-
sion ratio less than 2.3 during the period that it takes the
machine to reach the operating speed. If a spring is in-
serted in series with the damper, then what is the best
value of the stiffness of this spring in order to have a TR
0.08 when the machine is at its operating speed?
5.17 A compressor weighing 1000 kg operates at 1500
rpm. The compressor was originally attached to the
floor of a building, but it produced undesirable vibra-
tions to the building. To reduce these vibrations to the
building, it is proposed that a concrete block be poured
that is separated from the building and that the com-
pressor then be mounted to this block. The location of
the compressor will permit the block to be 1.8 m by
2.2 m. The soil on which the concrete block will rest
has a compression coefficient k
c
20 10
6
N/m
3
.If
the density of the concrete is 23 10
3
N/m
3
, then de-
termine the height of the concrete block so that there is
an 80% reduction in the force transmitted to the soil.
Section 5.8
5.18 Show that the work done per cycle by an har-
monic force acting directly on the mass of a linear
spring-mass-damper system is equal to the energy dis-
sipated by the system per forcing cycle.
5.19 A spring-mass system with m 20 kg and k
8000 N/m vibrates horizontally on a surface with co-
efficient of friction m 0.2. When excited har-
monically at 5 Hz, the steady-state displacement of
the mass is 10 cm. Determine the equivalent viscous
damping.
#
280 CHAPTER 5 Single Degree-of-Freedom Systems Subjected to Periodic Excitations
x(t)
y(t)
c
k
m
FIGURE E5.13
Section 5.6
5.14 An accelerometer is being designed to have a
uniform amplitude frequency response of 1 % up to
f
a
8 kHz. What is the maximum damping ratio al-
lowed if the phase angle is to be less than 2° over this
frequency range?
5.20 The area of the hysteresis loop of a cyclically
loaded system, which is the energy dissipated per
forcing cycle, is measured to be 10 N m, and the
measured maximum response X
o
of the deflection is
2 cm. Calculate the equivalent viscous damping coef-
ficient of this system if the driving force has a fre-
quency of 30 Hz.
Section 5.9
5.21 Torsional oscillations of a vibratory system is
governed by the following equation
where
J
o
20 kg m
2
, k
t
20 N m/rad,
c
t
20 N m/(rad/s)
M
1
10 N m, M
2
20 N m,
v
1
1.0 rad/s, and v
2
2.0 rad/s
Determine the steady-state response of the system.
5.22 Determine an expression for the output of an ac-
celerometer with the damping factor z and natural fre-
quency v
n
, when it is mounted on a system executing
periodic displacement motions of the form
5.23 A single degree-of-freedom system is driven by
the periodic triangular wave excitation as shown in
Case d of Appendix B. If the period T 2 s and the
amplitude f
o
10 N, then find the steady-state re-
sponse of the system by considering the first three
harmonics of the forcing. Assume that the system pa-
rameters are k 10 kN/m, c 10 N s/m and
m 1 kg.
5.24 Consider again the results of Example 5.17,
where the response of a single degree-of-freedom sys-
tem to a periodic pulse train was studied. If we were
to differentiate Eq. (c) of this example with respect to
ƒ and set the result equal to zero, then the value of t
t
max
that satisfies this equation in one period is the
time at which the maximum value x
max
x(t
max
)
#
y A
1
sin v
1
t A
2
sin v
2
t
##
#
##
J
o
w
$
c
t
w
#
k
t
w M
1
cos 1v
1
t2 M
2
cos 1v
2
t2
#
occurs. Use MATLAB (or a similar programming lan-
guage), to determine x
max
as a function of the damping
ratio for
o
0.042426 and 0.4. The results
should look like those shown in Figure E5.24.
5.25 Repeat Exercise 5.24, except for the abscissa use
the nondimensional bandwidth of the system. Explain
your results.
5.26 Consider the base excitation of a single degree-
of-freedom system shown in Figure 3.6 and whose
motion is described by Eq. (5.76). If the displacement
of the base is the periodic sawtooth waveform de-
scribed by Case c in Appendix B, then obtain an ex-
pression for the displacement of the system’s mass.
5.27 Consider a sine wave forcing of magnitude F
o
and frequency v
b
that has a duration of N periods t
p
,
where t
p
2p/v
b
. This sine wave is repeated every
period T, where T Mt
p
2p/v
o
, M N, and N and
M are integers. If this periodic waveform is applied to
the mass of a single degree-of-freedom system, then
(a)
where t
b
Nt
p
and f(t) f(t T). See Figure E5.27a.
0 t T,
0 t
b
T
f1t 2 F
o
sin 1v
b
t23u1t 2 u1t t
b
24
Exercises 281
FIGURE E5.24
0.1 0.2 0.3 0.4 0.5 0.6 0.7
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
z
x
max
a) Expand Eq. (a) in a Fourier series and show that
(b)
These results were obtained by noting that v
b
t
b
2pN, v
o
/v
b
1/M, and v
b
T 2pM.
b) Determine the displacement response of the mass
by using Eq. (b) and Eqs. (5.159) and (5.160).
c) Assume that N 4, M 10,
o
0.0424 or
0.3333, and z 0.1 or 0.7. Using MATLAB, or a
comparable programming language, generate the
equivalent figures for this excitation as those
shown in Figure 5.31. One pair of the results
should look like that shown in Figures E5.27b
and c for
o
0.0424 and z 0.1. Explain
these results.
Section 5.10
5.28 Consider the following nonlinear single degree-
of-freedom system subjected to a harmonic excita-
tion, where f
o
300 N, v v
n
/3, m 100 kg,
c 170 N/m/s, k 2 kN/m, and a 10 kN/m
3
. As-
sume that the initial conditions are x(0) 0.01 m
and v(0) 0.1 m/s. Compute the time response of
the system and associated amplitude spectrum of
steady-state motions and compare it with the corre-
sponding quantities of the linear system; that is,
when a 0.
5.29 Consider the following nonlinear single degree-
of-freedom system subjected to a harmonic excita-
tion, with the same values of parameters as in Exer-
mx
$
cx
#
kx ax
3
f
o
cos vt
b
i
N
M
i M
b
i
1
pM
sin12piN/M2
1 1i/M2
2
i M
a
i
0
i M
a
i
1
pM
1 cos12piN/M 2
1 1i/M2
2
i M
a
0
0
282 CHAPTER 5 Single Degree-of-Freedom Systems Subjected to Periodic Excitations
0
1.5
1
0.5
0
0.5
1
1.5
t
f(t)/F
o
T
Mt
p
2p/v
o
t
b
Nt
p
t
p
2p/v
b
←
→
0 50 100 150 200
1.5
1
0.5
0
0.5
1
1.5
t
Amplitude
f(t)/F
x(t)/(F/k)
(a)
(b)
0 5 10 15 20 25 30
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
H( Ω)
Harmonic number
Amplitude
c
i
(c)
(a
i
2
b
i
2
)
1/2
FIGURE E5.27
Exercises 283
cise 5.28, except that now the excitation frequency is
at v v
n
.
Compute the response of the system and compare it
with that of the corresponding linear system. In ad-
mx
$
cx
#
kx ax
3
f
o
cos vt
dition, for this harmonic excitation, compare the dif-
ferences between the responses of the system of soft-
ening stiffness discussed in Exercise 5.28 and the
system of this exercise with hardening stiffness.
284
Mechanical systems and civil structures must be designed to withstand dynamic environments, which are often non periodic. The
supporting structure of a rock breaker has to endure the intermittent application of non-periodic impulses generated during the
rock breaking process. Structures such as the Seattle Space Needle must be designed to resist intermittent wind-induced forces.
(Source: Stockxpert.com; Richard Nowitz / Getty Images.)
6
Single Degree-of-Freedom Systems
Subjected to Transient Excitations
6.1 INTRODUCTION
6.2 RESPONSE TO IMPULSE EXCITATION
6.3 RESPONSE TO STEP INPUT
6.4 RESPONSE TO RAMP INPUT
6.5 SPECTRAL ENERGY OF THE RESPONSE
6.6 RESPONSE TO RECTANGULAR PULSE EXCITATION
6.7 RESPONSE TO HALF-SINE WAVE PULSE
6.8 IMPACT TESTING
6.9 SUMMARY
EXERCISES
6.1 INTRODUCTION
In Chapter 4, free responses were discussed, and in Chapter 5, responses to
harmonic and other periodic excitations were discussed. As illustrated in the
last two chapters, a “sudden” change in the state of a system brought about by
an initial condition or by a change in the profile of the forcing function results
in transients in the response of the system. Here, the initial conditions are as-
sumed to be zero, and the responses to various types of excitations such as im-
pulse excitations, step inputs, ramp inputs, and pulse excitations are consid-
ered at length. All of these excitations are characterized by sudden changes
285
in their respective profiles of amplitude with time. When transformed to the
frequency domain the responses to such transient excitations can also provide
a basis for determining the characteristics of a system. The resulting dis-
placement response, and several design criteria are established based on this
information. In the systems considered in this chapter, the inertia element or
the base of the system is subjected to a transient forcing.
Solution for Response to Transient Excitation
When the initial displacement X
o
0 and the initial velocity V
o
0, the gov-
erning equation for an underdamped system (0 z 1) with a forcing act-
ing on the mass as shown in Figure 6.1 is given by Eq. (5.2), which is repeated
below.
(6.1a)
In Eq. (6.1a), h is the variable of integration and it is recalled that the damped
natural frequency v
d
is given by
In terms of nondimensional time , Eq. (6.1a) is written as
(6.1b)
where the variable of integration . We use Eqs. (6.1) as a basis for de-
termining the responses to several different forcing functions.
In this chapter, we shall show how to:
• Analyze single degree-of-freedom systems subjected to abruptly chang-
ing forces, such as shocks, pulses, step functions, and ramps that are ap-
plied to the inertia element and to the base of the system.
• Study the significance of the ratio of the duration of a time-varying force
to the period of free oscillation of the system.
• Relate the duration of a shock to its frequency content and study how this
affects the system response.
• Determine the spectral energy of the input disturbance and the response.
j v
n
h
x1t2
1
k21 z
2
t
0
e
z1tj2
sin11t j221 z
2
2f 1j2dj
t v
n
t
v
d
v
n
21 z
2
1
mv
d
t
0
e
zv
n
1th2
sin1v
d
3t h42f 1h2dh
x1t 2
1
mv
d
t
0
e
zv
n
h
sin1v
d
h2f 1t h2dh
286 CHAPTER 6 Single Degree-of-Freedom Systems Subjected to Transient Excitations
f(t)
x(t)
k
c
m
FIGURE 6.1
Spring-mass-damper system
subjected to forcing f(t).
6.2 RESPONSE TO IMPULSE EXCITATION
An impulse excitation is described by
(6.2)
where f
o
is the magnitude of the impulse and has the units
1
of Ns and d(t)
is a generalized function called the delta function, which is defined by the
property
2
(6.3)
where f(t) is assumed to be continuous at t t
o
. After substituting Eq. (6.2)
into Eq. (6.1a) and evaluating the integral, one obtains
(6.4a)
which is rewritten in terms of the non dimensional time t v
n
t as
(6.4b)
In Eqs. (6.4), the unit step function is included to indicate that the response is
limited to those times for which t 0. The displacement response given by
Eq. (6.4b), which has the form of an exponentially decaying sinusoid for
0 z 1, is plotted in Figure 6.2 for z 0.1 and z 0.4.
Similarity to Response to Initial Velocity
The graphs shown in Figure 6.2 are qualitatively similar to those shown in
Section 4.2.2 for a linear vibratory system subjected to no forcing and to an
initial velocity. In fact, the form of the response given by Eq. (6.4a) is identi-
cal to that given by Eq. (4.13), which is given by
(6.5)x 1t 2
V
o
e
zv
n
t
v
d
sin1v
d
t2
x1t2
f
o
mv
n
e
zt
21 z
2
sin121 z
2
t2u1t 2
f
o
mv
d
e
zv
n
t
sin1v
d
t2u1t 2
x 1t 2
f
o
mv
d
t
0
e
zv
n
h
sin1v
d
h2d1t h 2dh
q
q
d1t t
o
2f 1t2dt f 1t
o
2
f 1t 2 f
o
d1t 2
6.2 Response to Impulse Excitation 287
1
Based on Eqs. (6.2) and (6.3), the magnitude f
o
of the impulse acting on the system is given by
hence, the units Ns.
2
See, for example, A. Papoulis, The Fourier Integral and Its Applications, McGraw-Hill, NY,
p. 270 ff (1962).
f
o
t
0
f 1t2dt
Comparing Eqs. (6.4a) and (6.5), we see that
which should be expected since the change in linear momentum of a system
is equal to the impulse applied to a system; that is, from Eq. (1.11), we have
Assuming that prior to applying the impulse the system is at rest, then V
o
is
the velocity of the mass after the impulse has acted on the system. In other
words, the response of a system to an impulse is equivalent to the response of
a system with the corresponding initial velocity.
Extremum of Response to an Impulse
In view of the identical nature of the response of the system to an impulse and
that due to an initial velocity given by Eqs. (6.4a) and (6.5), respectively, the
maximum displacement x
max
of the impulse response can be obtained directly
from Eq. (4.18). Thus, since V
o
f
o
/m, the first extremum is
(6.6)
which, from Eq. (4.17), occurs at
(6.7)t
m
w
21 z
2
x
max
x1t
m
2
f
o
mv
n
e
w/tan w
Change in momentum
t
0
f 1t 2dt
V
o
f
o
m
288 CHAPTER 6 Single Degree-of-Freedom Systems Subjected to Transient Excitations
0 5 10 15 20 25 30
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
x(t)/[
f
o
/(mv
n
)]
t
z
0.1
z
0.4
FIGURE 6.2
Response of a system to an impulse.
where
(6.8)
Force Transmitted to Boundary
The force transmitted to the fixed boundary of Figure 6.1 is given by
Eq. (3.10). Thus,
(6.9)
After using Eq. (6.4a) and Eq. (D.12), we obtain for t 0 that
(6.10)
where the phase angle c is given by
The maximum force transmitted to the fixed boundary occurs at the time
t
m
at which dF
b
/dt 0. Thus, Eq. (6.10) leads to
(6.11)
where w is given by Eq. (6.8). Thus, the maximum normalized force
3
trans-
mitted to the fixed boundary is given by
(6.12)
This result is plotted in Figure 6.3, where it is seen that is a
minimum at where It is noted that as the
damping increases beyond the magnitude of the transmitted force
increases. From Eqs. (6.10) and (6.12) and Figure 6.3, one can deduce the fol-
lowing design guideline.
z 0.25,
F
b,max
/1f
o
v
n
2 0.81.z 0.25,
F
b,max
/1f
o
v
n
2
F
b,max
f
o
v
n
e
1wc2/tan w
v
d
t
m
w c
c tan
1
2z21 z
2
1 2z
2
F
b
f
o
v
n
21 z
2
e
zv
n
t
sin1v
d
t c2
F
b
kx1t 2 c
dx1t 2
dt
k cx1t2
2z
v
n
dx1t 2
dt
d
w tan
1
21 z
2
z
6.2 Response to Impulse Excitation 289
Design Guideline: To obtain the maximum decrease in the amount of
force transmitted to the base of a linear single degree-of-freedom sys-
tem due to an impulse (shock) loading, one can choose the damping ra-
tio to be between 0.2 and 0.35. For this range of damping values, the
maximum attenuation of the force transmitted to the fixed boundary
will be around 18%. For a given damping ratio and magnitude of the
impulse force, decreasing the natural frequency decreases the magni-
tude of the force transmitted to the fixed boundary.
3
Strictly speaking, the force transmitted has an extremum at t t
m
. In order to determine that this
force is a maximum here, we use numerical means.