Balakumar Balachandran, Magrab E.B. Vibrations
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650 GLOSSARY
derivative of the transverse displacement of the
beam
Boundary conditions—the state of a spatially dis-
tributed system at its boundaries
Characteristic equation—an equation whose roots
provide the eigenvalues or natural frequencies of
a system
Critically damped—damping factor is equal to one
Coulomb damping—a damping model for dry fric-
tion in which the damping force has a constant
magnitude and a direction opposite to that of the
motion
Cutoff frequency—for a vibratory system, the fre-
quency at which the magnitude of the displace-
ment response has decreased to 0.7071 (1/ )
of its maximum value; that is, the system has the
half the power it has at the maximum value
Damping-dominated response—the excitation fre-
quency range in which the system response is
heavily influenced by the viscous damping of the
system; in this range, the response is inversely
proportional to the damping coefficient
Damping factor—a nondimensional quantity relat-
ing the amount of viscous dissipation in a system
to the stiffness and mass of the system
Degrees of freedom—the minimum number of inde-
pendent coordinates needed to describe the mo-
tion of a system
Dissipation element—a device that provides resist-
ance to motion in the form of an irrecoverable
loss of energy
Displacement—a vector quantity that is used to spec-
ify the position of a system
Energy density spectrum—the square of the magni-
tude of the frequency-response function
Euler-Bernoulli beam—see Bernoulli-Euler beam
Free response—motion of a system in the absence of
an externally applied force; that is, a system sub-
ject only to initial conditions
Frequency-response function—the ratio of the out-
put of a system to the input of the system as a
function of frequency; can be obtained from the
transfer function by substituting s jv
Generalized coordinates—the minimum number of
independent coordinates needed to describe a
system
12
Half-power points—excitation frequencies for
which the displacement response has half the
power that it has at its maximum value; the fre-
quencies corresponding to the half-power points
are used to define the cutoff frequencies
Harmonic—any frequency that is an integer multiple
of a basic frequency
Harmonic excitation—in mechanical vibrations it
refers to either a sine function of given amplitude
and frequency, a cosine function of given ampli-
tude and frequency, or the combination of a sine
function of given amplitude and cosine function
of given amplitude each with the same frequency
High pass filter—a system that allows frequency
components in the input that are greater than its
cutoff frequency to pass relatively unattenuated
while those frequency components below it are
attenuated
Impact—a forceful contact that occurs in a relatively
short amount of time
Impulse—a mathematical representation of a very
short-duration force; it is usually represented in
the physical world by a pulse whose duration is
very short, typically around one-hundredth of a
system’s natural period
Impulse response—time-varying response of a sys-
tem to an impulse
Impulse-response function—displacement response
of a system to an impulse of unit magnitude
Inertia-dominated response—the excitation fre-
quency range in which the system response is
heavily influenced by the inertia of the system; in
this range, the response is inversely proportional
to the system mass
Inertia force—the resistance force experienced by a
mass due to a change in absolute velocity; this
force has a magnitude equal to the product of the
mass and the magnitude of absolute acceleration
and is in a direction opposite to that of the motion;
similar notion applies for rotational motions
Inertial reference frame—a fixed frame that does
not share any of the motions of the system under
consideration; usually taken as “ground” or fixed
in space
Initial conditions—the state of a system at time equal
to zero
Kinetic energy—energy associated with system
motion; a scalar quantity determined by using
the translational and rotational velocities of the
system
Lagrange’s equations—equations of motions de-
rived from a formulation based on the system
kinetic energy, potential energy, and work ex-
pressed in terms of generalized coordinates
Linear momentum—a vector quantity that is the
translational momentum of a system; the prod-
uct of the body’s mass and velocity
Logarithmic decrement—the natural logarithm of
the ratio of any two successive amplitudes of the
response of a linear system that occur one pe-
riod apart
Low pass filter—a system that allows frequency
components in the input that are less than its
cutoff frequency to pass relatively unattenuated
while those frequency components above it are
attenuated
Mass—a measure of the amount of material a body
contains; its weight is the mass times the grav-
ity force in a gravitational field; for translational
motions, the ratio of force to acceleration
Mass moment of inertia—a measure of the resist-
ance of a body to angular acceleration about a
given axis; the ratio of moment to angular ac-
celeration
Mode shape—provides a spatial description of mo-
tion during free oscillations of an undamped
system; any of various stationary vibration pat-
terns in which an oscillatory undamped system
or an undamped elastic body oscillates at one of
the natural frequencies of the system
N degree-of-freedom system—a vibratory system
whose motions are described by a set of N gen-
eralized coordinates
Natural frequency—the frequency at which an un-
damped system will vibrate in the absence of
external forces
Node point—a point that remains stationary when a
multi-degree-of-freedom or spatially continu-
ous system is vibrating at one of its natural
frequencies
Normal-mode solution method—a mathematical
procedure that uses the orthogonal properties of
Glossary 651
mode shapes to decouple the equations of mo-
tion and obtain the solution for the response
Orthogonality of mode shapes—the special matrix
or integral relations that mode shapes satisfy to
ensure that each mode is independent of another
mode
Overdamped system—damping factor is greater
than one
Percentage overshoot—a measure of the magnifica-
tion of the displacement response of a system to
a step input
Period of oscillation—the reciprocal of a system’s
oscillation frequency given in Hertz
Periodic excitation—a time-varying excitation
whose waveform repeats itself every constant
time interval called the period; the simplest type
of periodic excitation is a waveform that varies
with time as a sine or cosine function
Phase response—the phase lead or lag of the ampli-
tude response with respect to the input as a func-
tion of frequency
Potential energy—a scalar quantity that represents
the energy that is associated with elastic forces
and gravitational forces.
Proportional damping—damping in an N degree-of-
freedom system that is proportional in a specific
manner to the mass and/or stiffness properties of
the system
Resonance—the condition at which the excitation
frequency is equal to one of the natural frequen-
cies of a system; also refers to the response of an
undamped or lightly damped linear system at
frequencies close to or at one of the system’s
natural frequencies; for a nonlinear system, res-
onances can occur at rational multiples of a nat-
ural frequency
Rise time—the time it takes for the displacement re-
sponse of a system to a step input force to go
from 10% of its steady-state value to 90% of its
steady-state value
Settling time—the time it takes for the displacement
response of a system to a step input to decay to
within a specified percentage of its steady-state
value
Spring—a device that recovers its original shape
when released after being distorted
Stable system—a system whose response is finite
for all time when all initial conditions and
externally applied forces and moments have fi-
nite magnitudes
Static-equilibrium position—the position of a sys-
tem at rest
Step input—an input force or displacement that goes
from zero to its maximum magnitude virtually
instantaneously and remains at this maximum
magnitude thereafter
Stiffness-dominated response—the excitation fre-
quency range in which the system response is
heavily influenced by the stiffness of the system;
in this range, the response is inversely propor-
tional to the system stiffness
Stiffness element—a device that stores and releases
potential energy
Transfer function—the ratio of the output of a sys-
tem to the input of the system in the Laplace
domain expressed as a function of the complex
variable s; the frequency-response function is
obtained from the transfer function by setting
s jv
Transient response—system response to initial
conditions and/or to suddenly applied external
forces or moments
652 GLOSSARY
Transmissibility ratio—a measure of the amount of
the applied force to the mass that is transmitted
to the ground or the amount of displacement ap-
plied to the base that is transmitted to the mass
Undamped system—a system without damping
Underdamped system—damping factor is greater
than zero and less than one
Unstable system—a system whose response to inputs
of finite magnitude grows in an unbounded man-
ner with time
Velocity—a vector quantity that is the rate of change
of position with respect to time
Velocity, absolute—velocity with respect to a point
fixed in an inertial reference frame
Vibration absorber—a secondary vibratory system
added to a vibrating primary system to cancel or
attenuate the motion of the primary system at a
specific frequency or frequency region; see also
absorber
Vibration isolation—means used to reduce a sys-
tem’s transmissibility ratio
Viscous damping—a damping model in which the
damping force has a magnitude linearly propor-
tional to the system speed and a direction oppo-
site to that of the motion
653
Appendix A
Laplace Transform Pairs
DEFINITION OF LAPLACE TRANSFORM
The Laplace transform is defined as
(A.1)
where the variable s is a complex variable represented as where
In writing this integral transform definition, it is assumed that the
time-dependent function g(t) is defined for all values of time t 0 and that
this function is such that this integral exists; that is,
(A.2)
where a is a positive real number. This restriction means that a function g(t)
that satisfies Eq. (A.2) does not increase with time more rapidly than the
exponential function e
at
. In addition, the function g(t) is required to be piece-
wise continuous. For the functions g(t) considered in this book, these condi-
tions are satisfied.
If the time-dependent function is a displacement response x(t), which has
units of a meter (m), then the corresponding Laplace transform X(s) has units
of meter-seconds (m s) when t has the units of time. In this case, the variable s
has units of 1/time so that the product st is nondimensional.
EVALUATION OF LAPLACE TRANSFORMS
We shall illustrate the evaluation of the Laplace transform with two examples.
First we consider the function
(A.3)g1t 2 cos 1t2
#
q
0
0g1t 20e
at
dt q
j 11
.
s s jv,
L3g1t 24 G1s 2
q
0
e
st
g1t 2dt
654 APPENDIX A Laplace Transform Pairs
which appears in Section 5.2. By using the definition (A.1), the Laplace trans-
form of Eq. (A.3) is
(A.4)
which is tabulated as entry 18 of Table A. This integration can also be carried
out symbolically by using MATLAB, MAPLE, or Mathematica.
Next, we consider the Laplace transform of the first and second time de-
rivatives of the function h(t), respectively; that is,
Then, from Eq. (A.1),
(A.5)
where integration by parts was used. For the second derivative of h(t), we have
(A.6) s
2
H1s 2 sh10 2 h
#
10 2
dh
dt
`
t 0
s3sH1s 2 h10 24
e
st
dh
dt
`
q
0
s
q
0
dh
dt
e
st
dt
L c
d
2
h
dt
2
d
q
0
d
2
h
dt
2
e
st
dt
h10 2 sH1s 2
e
st
h1t 2`
q
0
sH1s 2
e
st
h1t 2`
q
0
s
q
0
h1t 2e
st
L c
dh
dt
d
q
0
dh
dt
e
st
dt
dh
dt
and
d
2
h
dt
2
s
s
2
2
e
st
s
2
2
3s cos1t 2sin1t24`
q
0
L3g1t 24 G1s 2
q
0
cos1t 2e
st
dt
(continued)
g(t)
tt
o
1
TABLE A
Laplace Transform Pairs
g(t)
t
t
o
2t
o
1
G(s) g(t) Description
1 G(s/a) ag(at) Scaling of variable
2 nth-order derivative,
n 1,2,...
3 g(t t
o
) u(t t
o
) Time shifting
4 G(s)H(s) Convolution
5 e
st
o
d(t t
o
) Delta function
6 u(t t
o
) Unit step function
7 e
at
Exponential
8 u(t) u(t t
o
) Rectangular pulse
of duration t
o
9 u(t) 2u(t t
o
) u(t 2t
o
)
10 sin(v
o
t)[u (t) u(t p/v
o
)] Half sine wave of
frequency v
o
11
g(t) (t/t
o
)[u(t) u (t t
o
)]
(1 e
st
1
)
[u(t t
o
) u(t t
1
)]
(t/t
o
1 t
1
/t
o
)
[u(t t
1
) u(t t
1
t
o
)]
12 g(t) (t/t
o
)[u(t) u(t t
o
)]
(t/t
o
2)[u(t t
o
)
u(t 2t
o
)]
13 (1 t/t
o
)[u(t) u(t t
o
)]
e
t
o
s
t
o
s 1
t
o
s
2
G1s 2
1
t
o
s
2
11 e
st
o
2
2
G1s 2
1
t
o
s
2
11 e
st
o
2
v
o
11 e
ps/v
o
2
s
2
v
2
o
11 e
t
o
s
2
2
s
1 e
st
o
s
1
s a
e
st
o
s
t
0
g1h 2h1t h 2dh
t
0
g1t h2h1h 2dh
e
t
0
s
G1s 2
g
n
1t 2
d
n
g
dt
n
s
n
G1s 2
a
n
k1
s
nk
g
k1
102
g(t)
t
/
o
1
g(t)
t
o
t
1
t
1
t
o
1
g(t)
t
o
t
2t
o
1
t
o
t
g(t)
1
TABLE A
(continued)
G(s) g(t) Description
14 Impulse response of
single degree-of-
freedom system
15 g(t) is step response
of single degree-
of-freedom system
16
17 Response of single
degree-of-freedom
system to initial
displacement
18 cos(vt) Special case of 16
19 sin(vt) Special case of 14
20 cosh(vt)
21 sinh(vt)
22
"
n 1,2, . . .
23 sinh(at) sin(at) Term in solution to
Euler beam
equation
24 cosh(at) cos(at) Term in solution to
Euler beam
equation
25 sinh(at) sin(at) Term in solution to
Euler beam
equation
26 cosh(at) cos(at) Term in solution to
Euler beam
equation
27 Generalization of 14
28 Generalization of 16
29 Linearity theorem
30 Initial value theorem
assuming that
exists
lim
s씮 q
3sG1s 24
lim
t씮 0
3g1t 24lim
s씮 q
3sG1s 24
a
1
g
1
1t 2 a
2
g
2
1t 2a
1
G
1
1s 2 a
2
G
2
1s 2
1
1b a 2
3be
bt
ae
at
4
s
1s a 21s b2
1
1b a 2
3e
at
e
bt
4
1
1s a 21s b2
2s
3
s
4
a
4
2as
2
s
4
a
4
2a
2
s
s
4
a
4
2a
3
s
4
a
4
t
n1
1n 1 2!
1
s
n
v
s
2
v
2
s
s
2
v
2
v
s
2
v
2
s
s
2
v
2
v
n
v
d
e
zv
n
t
sin1v
d
t w2
s 2zv
n
s
2
2zv
n
s v
2
n
v
n
v
d
e
zv
n
t
sin1v
d
t w2
s
s
2
2zv
n
s v
2
n
w cos
1
z
z 1
1
v
n
v
d
e
zv
n
t
sin1v
d
t w2
v
2
n
s1s
2
2zv
n
s v
2
n
2
v
d
v
n
11 z
2
1
v
d
e
zv
n
t
sin1v
d
t2
1
s
2
2zv
n
s v
2
n
Use of Partial Fractions 657
TABLE A
(continued)
G(s) g(t) Description
31 Final value theorem
assuming that
exists
32 s-domain shiftingg1t 2e
at
G1s a2
lim
t씮 q
3g1t 24
lim
t씮 q
3g1t 24lim
s씮 0
3sG1s 24
where integration by parts and Eq. (A.5) were used. The results given by
Eqs. (A.5) and (A.6) are summarized along with higher-order derivatives in
entry 2 of Table A. The evaluation of the inverse Laplace theorem is not
straightforward, and for details of this, the reader is referred to Widder.
1
USE OF PARTIAL FRACTIONS
The method of partial fractions is frequently used to reduce higher order poly-
nomials to lower order polynomials whose form corresponds to one of those
appearing in Table A. To illustrate this method, consider the following
equation that describes the motion of a thin elastic beam on an elastic founda-
tion that is subjected to an axial load and a harmonically excited external
force:
2
(A.7)
Using Eq. (A.6) and entry 2 of Table A, the Laplace transform of
Eq. (A.7) is
(A.8)
where the prime denotes the derivative with respect to x,
(A.9)
and we assume that b
2
4
k 0. Note that 2b P
2
d
2
.
Consider a ratio of two polynomials N(s)/D(s,n), where n is the order of
the polynomial. The first step in the method of partial fractions is to factor D
into the product of lower order polynomials and
then find an equivalent sum of polynomial ratios, such that
(A.10)
N
D
a
m
k1
A
k
s
n
k
1
D1s, n
k2
D1s,n
1
2D1s,n
2
2 . . . D1s,n
m
2
P
2
b 2b
2
4
k
d
2
b 2b
2
4
k
D1s 2 s
4
2bs
2
k
4
1s
2
d
2
21s
2
P
2
2
Y1s 2
1
D1s 2
31s
3
2bs2y10 2 1s
2
2b2y¿102 sy–10 2 y‡102 F1s 24
d
4
y
dx
4
2b
d
2
y
dx
2
1k
4
2y f 1x 2
1
David Widder, The Laplace Transform, Princeton University Press, Princeton, NJ (1941).
2
See Section 9.3.6.
where A
k
is determined by equating the coefficients of the like powers of s on
both sides of Eq. (A.10). We illustrate this procedure for the first term on the
right-hand side of Eq. (A.8), where we see that m 2 and n
1
n
2
2.
Thus,
(A.11)
Upon equating the coefficients of the s
3
term and the s term in Eq. (A.11), we
find that
Solving for A
1
and A
2
, we obtain
(A.12)
On substituting Eqs. (A.12) in Eq. (A.11), we arrive at
(A.13)
Using entries 18 and 20 from Table A, the inverse Laplace transform of
Eq. (A.13) is
(A.14)
When b k 0, P d , and Eq. (A.13) simplifies to
(A.15)
The inverse Laplace transform of Eq. (A.15) is obtained using entries 18 and
20 from Table A to arrive at
1
2
c
s
1s
2
2
2
s
1s
2
2
2
d
s
3
s
4
4
s
3
1s
2
2
21s
2
2
2
1
P
2
d
2
3d
2
cos P x P
2
cosh d x 4
L
1
c
s
3
2bs
1s
2
d
2
21s
2
P
2
2
d
1
P
2
d
2
L
1
c
sP
2
1s
2
d
2
2
sd
2
1s
2
P
2
2
d
s
3
2bs
1s
2
d
2
21s
2
P
2
2
1
P
2
d
2
c
sP
2
1s
2
d
2
2
sd
2
1s
2
P
2
2
d
A
2
d
2
P
2
d
2
A
1
P
2
P
2
d
2
A
1
P
2
A
2
d
2
2b
1A
1
A
2
2 1
1A
1
A
2
2s
3
s1A
1
P
2
A
2
d
2
2
1s
2
d
2
21s
2
P
2
2
A
1
s1s
2
P
2
2 A
2
s1s
2
d
2
2
1s
2
d
2
21s
2
P
2
2
s
3
2bs
1s
2
d
2
21s
2
P
2
2
c
A
1
s
1s
2
d
2
2
A
2
s
1s
2
P
2
2
d
658 APPENDIX A Laplace Transform Pairs
Thus, we have verified entry 26 of Table A. This result could have also been
obtained directly from Eq. (A.14).
In a similar manner, the inverse Laplace transform of the remaining terms
on the right-hand side Eq. (A.8) are obtained. The result is
(A.16)
where we have used entry 4 in Table A and
(A.17)
The method of partial fractions
3
is quite often needed for the design of con-
trol systems.
There is a one-to-one transformation from g(t) to G(s), which enables us
to establish a catalog of the different functions one is likely to use in Vibra-
tions and Control. One such catalog is Table A. A large compendium of
Laplace transforms and their inverse transforms has been collected by
Roberts and Kaufman.
4
T
o
1x 2
1
d
2
P
2
c
1
P
sin1Px 2
1
d
sinh1dx 2d
S
o
1x 2
1
d
2
P
2
3cos1Px2 cosh1dx 24
R
o
1x 2
1
d
2
P
2
c
d
2
P
sin1Px 2
P
2
d
sinh1dx 2d
U
o
1x 2
1
d
2
P
2
3d
2
cos1Px 2 P
2
cosh1dx 24
y–102S
o
1x 2 y‡10 2T
o
1x 2
x
0
f 1h2T
o
1x m2dh
y1x 2 y10 2U
o
1x 2 y¿10 2R
o
1x 2
1
2
3cos x cosh x4
L
1
c
s
3
1s
2
2
21s
2
2
2
d
1
2
L
1
c
s
1s
2
2
2
s
1s
2
2
2
d
Use of Partial Fractions 659
3
See, for example, R. V. Churchill, Operational Mathematics, McGraw-Hill, NY (1958).
4
G. E. Roberts and H. Kaufman, Table of Laplace Transforms, W. B. Saunders Co, Philadelphia
(1966).