Balakumar Balachandran, Magrab E.B. Vibrations
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where
(k)
Upon combining Eqs. ( j) and (k), and noting that k 34,909 N/m, d
i
0.95d
o
, L 10 m, and E 2.1 10
11
N/m
2
, we obtain
(l)
Thus, the outer diameter of the lamppost is 23.5 cm and its inner diameter
22.3 cm, resulting in a wall thickness of 1.2 cm.
6.3 RESPONSE TO STEP INPUT
The step input to a vibratory system shown in Figure 6.1 is expressed as
(6.23)
where u(t) is the unit step function. After substituting Eq. (6.23) into Eq.
(6.1a) to determine the displacement response, we obtain
f 1t2 F
o
u1t 2
0.235
m
B
4
32 34,909 10
3
3p 2.1 10
11
11 10.952
4
2
d
o
B
4
32kL
3
3Ep11 10.952
4
2
I
p
32
1d
o
4
d
i
4
2
300 CHAPTER 6 Single Degree-of-Freedom Systems Subjected to Transient Excitations
0 2 4 6 8 10
0
2
4
6
8
10
12
f (Hz)
|X( jf
)| (cm)
k
1
20000 N/m
k
2
30000 N/m
k
3
40000 N/m
FIGURE 6.9
Displacement response spectrum of lamppost for three values of stiffness.
6.3 Response to Step Input 301
(6.24)
which is rewritten in terms of the nondimensional time t v
n
t as
(6.25)
where the phase angle w is given by Eq. (6.8) and it has been assumed that the
system is underdamped.
The maximum value x
max
is determined from the response given by
Eq. (6.25) as follows. The extremum values x
max
are obtained by determining
the times t
m
at which the time derivative of x(t) given by Eq. (6.25) is zero;
that is,
(6.26)
where we have made use of Eq. (D.12). Then, setting dx(t)/dt 0, we find
that for t 0, the first extremum occurs at
(6.27)
For the value of t
m
given by Eq. (6.27), Eq. (6.25) evaluates to
8
(6.28)
where we have made use of Eq. (4.15), that is,
sin w 21 z
2
F
o
k
31 e
p/tan w
4
F
o
k
c1
e
zp/21 z
2
21 z
2
sin1p w 2d
x
max
x1t
m
2
F
o
k
c1
e
zt
m
21 z
2
sin1t
m
21 z
2
w2d
t
m
p
21 z
2
F
o
e
zt
k21 z
2
sin1t 21 z
2
2
dx1t 2
dt
F
o
e
zt
k
c
z
21 z
2
sin 1t 21 z
2
w2 cos 1t21 z
2
w2d
x1t 2
F
o
k
c1
e
zt
21 z
2
sin1t 21 z
2
w2du1t2
F
o
k
c1
e
zv
n
t
21 z
2
sin1v
d
t w2du1t2
x1t 2
F
o
e
zv
n
h
mv
d
t
0
e
zv
n
h
sin1v
d
3t h42dh
8
Strictly speaking, to verify that this extremum is indeed a maximum, one needs to verify that the
second derivative is less than zero at t t
m
. We have, instead, determined numerically that the
extremum is a maximum.
302 CHAPTER 6 Single Degree-of-Freedom Systems Subjected to Transient Excitations
The response given by Eq. (6.25) consists of a constant term F
o
/k and an
exponentially decaying sinusoid for 0 z 1. This is plotted in Figure 6.10
for three different values of z. From the responses shown in Figure 6.10, it is
evident that the response of an underdamped system to a step input oscillates
about the final settling position, which is at F
o
/k, before settling down to it.
This settling position, called the steady-state value, can also be determined
from Eq. (6.25) by noting that
(6.29)
To describe the features of the responses seen in Figure 6.10, the follow-
ing definitions are introduced. These descriptors are useful for characterizing
three important aspects of a linear system: (1) speed of response, (2) the mag-
nification of the response by the system, and (3) time taken by the system to
return to its equilibrium state. A graphical illustration of these definitions is
provided in Figure 6.11.
Rise Time
The rise time is the nondimensional time it takes for the response
of a system to a step input to go from 10% of the steady-state value to 90% of
the steady-state value. This quantity is a measure of how fast one can expect
a system to overcome its inertia to a sudden change in the applied force. For
the normalized response considered here—that is,
t
r
v
n
t
r
lim
t씮 q
x1t 2
F
o
k
0 5 10 15 20 25 30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
z
0.06
z
0.25
z
0.7
t
x(t)/(F
o
/k)
FIGURE 6.10
Responses of single degree-of-freedom systems with different damping factors to step inputs.
—the steady-state value is 1. Then
(6.30)
where
In general, the rise time can also be determined for other waveforms. For
example, consider a half-sine wave of frequency v and duration p/v; that is,
. Then the time to reach 90% of the steady-state
value is
Percentage Overshoot
The percentage overshoot is determined from
(6.31)P
o
a
x
max
x
f
x
f
b100%
vt
r
sin
1
1.9 2 sin
1
1.1 2 1.02
x1t 2 sin1vt 2u1t p/v2
x
~
1t
0.1
2 0.1
x
~
1t
0.9
2 0.9
t
r
v
n
t
r
t
0.9
t
0.1
x
~
1t 2
x1t 2
F
o
/k
6.3 Response to Step Input 303
0 5 10 15
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
d/100
t
s
x
max
x(t
m
)
Rise time
t
0.9
t
0.1
0.9
0.1
t
0.1
t
0.9
t
m
t
x(t)/(F
o
/k)
FIGURE 6.11
Definition of rise time, percentage overshoot, and settling time.
where x
f
is the final or steady-state value, which here is F
o
/k, and x
max
is the
maximum value of x(t), which is given by Eq. (6.28). Thus, after substituting
Eq. (6.28) into Eq. (6.31) we obtain
(6.32)
The percentage overshoot is a measure of the magnification in the dis-
placement response to a sudden change in the applied force. This quantity is
important to know in the design of systems where, in a dynamic environment,
interference, wear, undesirable contact between surfaces, and high stresses
must be avoided.
Settling Time
Settling time t
s
is the time it takes for the displacement response of a system
to a step input to decay to within d % of the steady-state (final) value. It is
a measure of a system’s ability to return to an equilibrium state. Thus, from
the envelope of the decay for underdamped oscillations discussed in Section
4.2 and Eq. (6.24), we have
or, equivalently,
(6.33)
Expressions (6.30), (6.32), and (6.33) are evaluated numerically for
underdamped oscillations and the corresponding values are plotted in Fig-
ure 6.12 as a function of z. From these equations and Figure 6.12, one can de-
duce the following design limitations.
v
n
t
s
1
z
ln
d
100
e
zv
n
t
s
d
100
P
o
100e
p/tan w
%
304 CHAPTER 6 Single Degree-of-Freedom Systems Subjected to Transient Excitations
Design Limitations: For a linear single degree-of-freedom system
with a mass m and a natural frequency v
n
, the rise time, bandwidth, and
percentage overshoot are determined by the damping ratio z. Selecting
z to adjust any one characteristic affects the others. Thus, for example,
decreasing z (decreasing c) to decrease rise time increases the percent-
age overshoot and the settling time.
9
The MATLAB function lsqcurvefit was used.
Although an explicit relationship for the rise time could not be obtained,
we can use standard curve fitting procedures
9
to fit the numerical results of
Figure 6.12a to obtain the following expression for the rise time
t
r
1
v
n
10.4377z
2
1.3822z
3
1.0174e
0.8701z
2
6.3 Response to Step Input 305
0 0.2 0.4 0.6 0.8 1
1
1.5
2
2.5
3
3.5
z
v
n
t
r
(a)
0
0.2 0.4 0.6 0.8 1
0
10
20
30
40
50
60
70
80
90
100
z
P
o
%
(b)
0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
z
(c)
v
n
t
s
FIGURE 6.12
Underdamped oscillations: (a) rise time; (b) percentage overshoot; and (c) settling time to within 2%.
EXAMPLE 6.5 Vehicle response to step change in road profile
Consider a car modeled as shown in Figure 6.13 with a mass of 1100 kg, a
suspension stiffness of 400 kN/m, and suspension damping coefficient of
15 kN s/m. The vehicle is traveling at a speed v 64 km/h and the step
change in road elevation a 0.03 m is 160 m away from the vehicle at
t 0. We shall determine the vehicle displacement response.
The governing equation for the vehicle is given by Eq. (3.28), which is
(a)
where x(t) is the displacement response of the vehicle and y(t) is the base mo-
tion. Noting that the vehicle encounters the step change in the road condition
at t
o
(160 m) (3600 s/hr)/(64 10
3
m/hr) 9 s, the base motion char-
acterized by the step change is described by
(b)
To determine the displacement response, we need to find the solution of the
system
(c)
where we have made use of the fact that the derivative of a step function is
the impulse function;
10
that is,
(d)
For the given parameter values, the damping factor is
(e)
Since the system is underdamped, we can make use of Eqs. (4.1), (6.2), and
(6.23) to determine that f
o
2azv
n
m and F
o
ak. Then, the displacement
response is given by
(f)
where the step function u(t t
o
) indicates that the step affects the response
only for t t
o
. For the given parameter values, it is found that
a c1
e
zv1tt
o
2
21 z
2
sin1v
d
1t t
o
2 w2du1t t
o
2
x1t 2
2za
21 z
2
e
zv
n
1tt
o
2
sin1v
d
1t t
o
22u1t t
o
2
z
15 10
3
2 2400 10
3
1100
0.358
du1t t
o
2
dt
d1t t
o
2
d
2
x
dt
2
2zv
n
dx
dt
v
n
2
x 2zv
n
ad1t t
o
2 av
n
2
u1t t
o
2
y1t 2 au1t t
o
2
d
2
x
dt
2
2zv
n
dx
dt
v
n
2
x 2zv
n
dy
dt
v
n
2
y
306 CHAPTER 6 Single Degree-of-Freedom Systems Subjected to Transient Excitations
c
v
k
m
x
y
a 0.03 m
FIGURE 6.13
Car suspension encountering a
step change in road conditions.
10
Papoulis, ibid.
Upon substituting these values into Eq. (f) and noting that a 0.03 m and
t
o
9 s, the graph of the displacement response of the vehicle shown in
Figure 6.14 is obtained. The vehicle response is at the equilibrium position
x 0 before the sudden change due to the step is encountered. This encounter
with the sudden change in road elevation produces transients in the response,
which die out after some time, and the vehicle response settles down to the
new position x a.
w tan
1
21 0.358
2
0.358
1.21 rad
v
d
19.0721 0.358
2
17.81 rad/s
v
n
B
4 10
5
N/m
1100 kg
19.07 rad/s
6.3 Response to Step Input 307
8 8.5 9 9.5 10
0.01
0
0.01
0.02
0.03
0.04
0.05
x(t) (m)
t (s)
FIGURE 6.14
Displacement response of vehicle to step change in roadway.
308 CHAPTER 6 Single Degree-of-Freedom Systems Subjected to Transient Excitations
EXAMPLE 6.6 Response of a foam automotive seat cushion and occupant
In this example, we study the response of a nonlinear system to a step input.
The system is a car seat and its occupant, as shown in Figure 6.15. This
system is subjected to a step input in the form of acceleration. The equation
governing the vibrations of an open-celled polyurethane foam automotive
seat cushion is given by
11
(a)
Linear viscous Quadratic fluid Nonlinear stiffness
damping damping
where the coefficients in Eq. (a) are given by
(b)
The physical quantities associated with the coefficients in Eq. (b) and the
numerical values of the different parameters are given in Table 6.1 for one
type of seat. The second term on the left-hand side of Eq. (a) represents the
linear viscous loss that is caused by the gas in the open cells of the foam be-
ing forced out of the bottom of the cushion. The third term represents the
turbulent flow resistance of air escaping from the cells, and this is the fluid-
damping model discussed in Section 2.4.2. The fourth term represents the
nonlinear elasticity of the walls of the cells and any entrained air that doesn’t
escape. It is an empirically determined function. The quantity a(t) is the input
acceleration to the base of the seat. The input acceleration is assumed to be of
the form a(t) a
o
u(t), where a
o
is the magnitude of the acceleration and u(t)
is the step function.
C
1
mAH
3Kj
,
C
2
rA
3K
b
j
2
,
C
3
E
f
a
1
A,
and
K 0.012d
2
M
d
2
z
dt
2
C
1
dz
dt
C
2
2
dz
dt
2
dz
dt
C
3
1cH d
1
0z 02
p
1
z Ma1t 2
C
1
C
2
M
x
x
y
y
y a(t)
..
z y x
C
3
(cH d
1
⏐
z
⏐
)
p
1
dz
dt
M
(a) (b)
FIGURE 6.15
(a) Automotive seat cushion and (b) its single degree-of-freedom equivalent.
⎫
⎬
⎭
⎫
⎪
⎪
⎬
⎪
⎪
⎭
⎫
⎬
⎭
11
W. N. Patten, S. Sha, and C. Mo, “A Vibration Model of Open Celled Polyurethane Foam
Automotive Seat Cushions,” J. Sound Vibration, Vol. 217, No. 1, pp. 145–161 (1998).
Equation (a) is a nonlinear, ordinary differential equation for which the
solution must be obtained numerically.
12
The numerically obtained solution
of Eq. (a) is used to determine the acceleration of the car seat a
c
(t) and the
scaled acceleration a
cs
(t):
(c)
The two different magnitudes chosen for the acceleration are a
o
0.15 m/s
2
and a
o
0.60 m/s
2
. In Figures 6.16a and 6.16c, the acceleration responses
of the car seat in the time domain for two different values of a
o
and two dif-
ferent values of the system mass M are shown. These time domain accel-
eration responses are then transformed into the frequency domain by using
the discrete Fourier transform
13
to obtain the results shown in Figure 6.16b
and 6.16d.
From the time histories, it is evident that for a given system mass, as the
base acceleration magnitude is increased, the period of oscillation during
the early portion of the motion increases. This characteristic is attributed to
the system nonlinearity. This increase in period of oscillation is reflected as a
decrease in the frequency at which the peak amplitude occurs in the associ-
ated spectra. As discussed in Chapter 4, the period of oscillation of a nonlinear
system can increase or decrease as the amplitude of response grows. Exam-
ining Figures 6.16a and 6.16c, we note that the peak acceleration magnitude
increases with an increase in the base acceleration magnitude for a given
system mass.
a
cs
1t 2
a
c
1t 2
a
o
1
a
o
d
2
z
dt
2
6.3 Response to Step Input 309
TABLE 6.1
Seat Parameters
Quantity Symbol Units Values
Foam cell edge diameter d m 0.0009
Cushion height H m 0.067
Cushion area A m
2
0.095
Mass of load on car seat M kg 36 and 45
Young’s modulus of polymer E
s
N/m
2
6 10
7
Young’s modulus of foam E
f
N/m
2
1.1 10
4
Density of air r kg/m
3
1.22
Viscosity of air m Ns/m
2
1.85 10
5
Volume fraction of open cells j 0.0133
Surface factor K
b
1
Coefficients of shape function a
1
0.0005
b 0.4
c 0.00022
d
1
0.009
p
1
1
Source: Reprinted by permission of Federation of European Biochemical Societies from Journal of Sound
Vibration, 217, W.N.Patten, S.Sha, and C.Ma, FEBS Letters, “A Vibration Model of Open Celled
Polyurethane Foam Automotive Seat Cushions,” pp.145–161, Copyright © 1998, with permission from
Elsevier Science.
12
The MATLAB function ode45 was used.
13
The MATLAB function fft was used.