Balakumar Balachandran, Magrab E.B. Vibrations
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EXAMPLE 9.8 Air-damped micro-cantilever beam
Consider a silicon cantilever beam that is undergoing forced harmonic oscil-
lations at a frequency v. In addition, it is assumed that the length L of the
beam can vary between 10 mm L 1 mm and that the width b L/10 and
the height h L/100. Under certain assumptions, it has been shown
38
that, for
these ranges of L, b, and h, the air damping is the dominant form of damping
and it is reasonable to assume that the damping force is proportional to the
transverse velocity of the beam; that is, from Eq. (9.41c)
(a)
where
(b)
and w w(x,t) is the displacement of the beam, r
a
is the density of air
(1.3 kg/m
3
), and m is the viscosity of air (1.8 10
5
Pa s). For 0w 0 L/1000
and for v 100,000 rad/s, the Reynolds number for the airflow at the free end
of the beam is less than 1 and the given damping model is applicable. We shall
determine the form of the forced response in this case when a harmonic forc-
ing is applied at h j. We assume that the initial conditions are zero; that is,
(c)
To determine the form of the response, we make use of Eq. (9.242). For
the given damping coefficient c, the quantity z is evaluated from Eq. (b) and
Eqs. (9.222) to be
(d)
For harmonic forcing at h j, the forcing function g(h,t) is given by
(e)
Substituting Eq. (e) into Eq. (9.242a), using the modes given by Eq. (4b) of
Table 9.3, and making use of Eq. (c) we obtain
(f) g
o
a
q
n1
W
n
1h 2W
n
1j 2
N
n
2
nd
t
0
e
zt¿
sin1
2
nd
t¿ 2sin 3
2
1t t¿ 24dt¿
g
o
a
q
n1
W
n
1h 2
N
n
2
nd
t
0
e
zt¿
sin1
2
nd
t¿ 2sin3
2
1t t¿ 24e
1
0
d1h j 2W
n
1h 2dh fdt¿
w1h,t 2
g1h,t 2 g
o
d1h j 2sin1
2
t2
z
11.5m 0.375pb12r
a
mv2L
3
2c
b
rm
o
w1h,0 2 0
and
w
#
1h,0 2 0
#
c 1.5m 0.375pb 12r
a
mv
f
nc
c
0w
0t
640 CHAPTER 9 Vibrations of Beams
38
H. Hosaka and K. Itao, “Theoretical and Experimental Study on Airflow Damping of Vibrating
Microcantilevers,” J. Vibration Acoustics, Vol. 121, pp. 64–69 (1999).
Making use of Eqs. (5.6) to (5.8) and recognizing that the sinusoidal excita-
tion is present for all time—that is, we can ignore the transient portion of the
solution in Eq. (5.7)—we find that
(g)
where we have used Eq. (9.241) and
EXAMPLE 9.9
Frequency-response functions of a beam
Consider the arrangement shown in Figure 9.37 in which a cantilever beam is ex-
cited by using a shaker at location x x
1
on a beam. We assume that the beam
is initially at rest, that the force imparted to the beam is measured by using a force
transducer, and that the beam response is measured at location x x
2
by using
an accelerometer. We shall construct a frequency-response function based on the
accelerometer measurement and the input force measurement.
The measured force applied at x x
1
is represented by
(a)
and the acceleration measured at x x
2
is represented by
(b)
In terms of the nondimensional variables t t/t
o
. and h x/L, Eqs. (a) and
(b) are written as, respectively,
(c)a1t 2
1
t
2
0
0
2
w1h
2
,t2
0t
2
f
d
1h,t 2 f 1t 2d1h h
1
2
a1t 2
0
2
w1x
2
,t2
0t
2
f
d
1x,t 2 f 1t 2d1x x
1
2
u
n
1 2 tan
1
2z
n
2
/
2
n
1
4
/
4
n
H
n
1 2 ca1
4
4
n
b
2
a2z
n
2
2
n
b
2
d
1/2
w1h,t 2 g
o
a
q
n1
W
n
1h 2W
n
1j 2
N
n
4
n
H
n
1 2sin1
2
t u
n
1 22
9.4 Forced Oscillations 641
x x
2
x x
1
x
0 x L
Force
transducer
Vibration
exciter
Acceleromete
r
FIGURE 9.37
Cantilever beam excited by a shaker
at x x
1
and the resulting accelera-
tion measured at x x
2
.
Let A(s) and F(s) represent the Laplace transforms of a(t) and f(t), re-
spectively. Then, the transfer function G
21
(s) is constructed as
(d)
The required frequency-response function is obtained from Eq. (d) by setting
s j
2
; that is,
(e)
To construct G
21
( j
2
), we start by taking the Laplace transform of the
second of Eqs. (c) to obtain
(f)
We now take the Laplace transform of Eq. (9.228) and obtain
(g)
Upon substituting Eq. (g) into Eq. (f), we arrive at
(h)
To obtain an expression for (s), we use Eq. (9.222) to obtain
(i)
Upon substituting Eq. (i) into Eq. (9.232), we find that
(j)
Upon taking the Laplace transform of Eq. (j), we obtain
(k)
We now substitute Eq. (k) into Eq. (9.237) with the initial conditions set to
zero, and arrive at
G
ˆ
n
1s 2
L
4
N
n
EI
F1s 2W
n
1h
1
2
L
4
N
n
EI
f 1t2W
n
1h
1
2
L
4
f 1t2
N
n
EI
1
0
W
n
1h 2d1h h
1
2dh
G
n
1t 2
1
N
n
1
0
W
n
1h 2g1h, t2dh
g1h,t 2
L
4
EI
f
d
1h,t 2
L
4
EI
f 1t2d1h h
1
2
°
ˆ
A1s 2
s
2
t
2
o
a
q
n1
°
ˆ
n
1s 2W
n
1h
2
2
W
ˆ
1h,s 2
a
q
n1
°
ˆ
n
1s 2W
n
1h 2
A1s 2
s
2
t
2
0
W
ˆ
1h
2
,s2
G
21
1j
2
2
A1j
2
2
F1j
2
2
G
21
1s 2
A1s 2
F1s 2
642 CHAPTER 9 Vibrations of Beams
(l)
Upon substituting Eq. (l) into Eq. (h) and using Eq. (d), we obtain
(m)
Then, the frequency-response function given by Eq. (e) is
(n)
c
2
b
r
2
EI
a
q
n1
c
W
n
1h
1
2W
n
1h
2
2
N
n
dc
4
4
n
4
2jz
n
2
n
2
d
G
21
1j
2
2
A1j
2
2
F1j
2
2
G
21
1s 2
A1s 2
F1s 2
L
4
t
2
o
EI
a
q
n1
c
W
n
1h
1
2W
n
1h
2
2
N
n
dc
s
2
s
2
2z
n
2
n
s
4
n
d
°
ˆ
n
1s 2
L
4
N
n
EI
F1s 2W
n
1h
1
2
s
2
2z
n
2
n
s
4
n
9.4 Forced Oscillations 643
0 5 10
(a) (b)
15 20
10
3
10
2
10
1
10
0
10
1
10
2
10
3
Ω
0510
(c)
15 20
Ω
0510
(d)
15 20
Ω
0 5 10 15 20
Ω
H
21
(Ω)
10
3
10
2
10
1
10
0
10
1
10
2
10
3
H
21
(Ω)
10
3
10
2
10
1
10
0
10
1
10
2
10
4
10
3
H
21
(Ω)
10
3
10
2
10
1
10
0
10
1
10
2
10
4
10
3
H
21
(Ω)
FIGURE 9.38
Magnitude of the transfer function of the acceleration to the driving force at different locations on a cantilever beam: (a) h
1
0.25
and h
2
0.8; (b) h
1
0.25 and h
2
1.0; (c) h
1
0.35 and h
2
0.8; and (d) h
1
0.35 and h
2
1.0.
We see from Eq. (m) that if we were to reverse the drive location, which was
at h h
1
, with the measurement location, which was at h h
2
, the frequency
response function would still be the same. This is an important property of lin-
ear systems and is referred to as the reciprocity property.
The normalized magnitude of Eq. (n) is
This quantity is plotted in Figure 9.38 for h
1
0.25 and 0.35 and for h
2
0.8
and 1.0. We see that one obtains markedly different transfer functions de-
pending on the driving location and the measuring location. However, the lo-
cations of the peaks, which correspond to the
n
, do not change. We have used
20 modes to obtain these results.
EXAMPLE 9.10
Use of a second impact to suppress transient response
We shall extend Example 6.5 to a beam by minimizing the magnitude of the
root-mean-square(rms) response of a point on the beam when the beam is sub-
jected to two impulses, each of magnitude F
o
but applied at different locations
and at different times. Then the force applied to the beam can be expressed as
(a)
where g
o
(F
o
L
4
)/(EI). Assuming that the initial conditions are zero and sub-
stituting Eq. (a) into Eq. (9.242a), we obtain
(b)
where
(c)
and u(t) is the unit step function.
The quantity that is to be minimized is the rms value of the displacement
at a point h, which is given by
(d)w
rms
1h, t
2
2
G
1
t
e
t
2
t
e
t
2
w
2
1h, t2dt
G1h,t,z 2 e
zt
sin1
2
nd
t2W
n
1h 2u1t2
g
o
a
q
n1
W
n
1h 2
N
n
2
nd
3G1h
1
,t,z2 G1h
2
,t t
2
,z24
u1t t
2
2
t
0
e
zt¿
sin 1
2
nd
t¿ 2d1t t¿ t
2
2dt¿
1
0
d1h h
2
2W
n
1h 2dhT
w1h, t2 g
o
a
q
n1
W
n
1h 2
N
n
2
nd
D
t
0
e
zt¿
sin1
2
nd
t¿ 2d1t t¿ 2dt¿
1
0
d1h h
1
2W
n
1h 2dh
g1h,t 2 g
o
d1h h
1
2d1t2 g
o
d1h h
2
2d1t t
2
2
H
21
1 2
EI
c
2
b
r
2
0G
21
1j
2
20
644 CHAPTER 9 Vibrations of Beams
In Eq. (d), t
e
is a time that is selected to provide a sufficient interval over
which one can obtain a meaningful sample of the response after the applica-
tion of the second impulse at h
2
and at time t
2
. The objective, then, is to de-
termine the time t
2
and location h
2
that makes w
rms
a minimum.
We shall illustrate these results for a cantilever beam. From Eq. (4b) of
Table 9.2, we have that the mode shape is
(e)
where
n
are given in Case 4 of Table 9.3. We assume that there is no damp-
ing (i.e., z 0), the first impulse is applied at h
1
0.25, and we use 21 terms
in Eq. (b). We are interested in minimizing the response at h 1. The nu-
merical results
39
are shown in Figure 9.39. It is found that h
2
0.241 at t
2
0.969. It is noted that the nondimensional period of the first natural frequency
is t
p
2p/ 2p/(0.5969p)
2
1.787. Thus, the second impulse is applied
at approximately t
2
/2 at virtually the same location that the first impulse was
applied. For the response shown in Figure 9.39b, w
rms
0.0976 and for re-
sponse shown in Figure 9.39c starting at t
2
0.969, we find that w
rms
0.0429, which is a little more than half of what it was without the second
impulse. It also was found that the peak value of the response shown in Fig-
ure 9.39b is 0.2404, while that in the region t t
2
in Figure 9.39c is 0.108,
attaining a reduction by a factor of approximately 2.4.
2
1
W
n
1h 2
T1
n
2
Q1
n
2
T1
n
h2 S1
n
h2
9.4 Forced Oscillations 645
0 1 2 3 4 5 6 7
0.2
0
0.2
w(1,)
0.2
0
0.2
w(1,)
0.2
0
0.2
w(1,)
(a)
01234567
(b)
01234567
(c)
FIGURE 9.39
Response of the free end of an un-
damped cantilever beam that is sub-
jected to an impulse at h
1
0.25 at
t 0 and another impulse at h
2
0.241 at t
2
0.969: (a) response
to first impulse only; (b) response
to second impulse only; and
(c) response to both impulses.
39
The MATLAB function fminsearch from the Optimization toolbox was used.
EXAMPLE 9.11 Moving load on a beam
A load moving along a beam is often used to determine the response of a
structure (such as a bridge) to a vehicle traveling on it. To illustrate how this
response is obtained, we consider an undamped, uniform beam that is hinged
at both ends as shown in Figure 9.40. The vehicle is represented as a point
force of magnitude F
o
moving with a constant speed V. This point load is as-
sumed to start from the end of the beam at x 0 and travel to the end x L.
Then, the expression for the force on the beam is
where we have used the delta function to locate the position of the load as it
changes with time. In terms of the nondimensional quantities, the forcing
takes the form
where h x/L is the nondimensional spatial location, t t/t
o
is the non-
dimensional time, t
o
is a characteristic time constant given by Eq, (9.67), and
We see that v
o
can be interpreted as the average speed at which a disturbance
propagates along the length of the beam.
If we assume that the beam is initially at rest, then the initial conditions
are zero; that is, 0 and . Then, it follows from Eq. (9.243a)
that the beam response is given by
(a)
For a beam hinged at both ends, the mode shape is given by Eq. (2b) of
Table 9.2; that is
g
1
a
q
n1
W
n
1h 2
N
n
1
2
n
t
0
sin 1
2
n
t¿ 2W
n
1a1t t¿ 22dt¿
0 t 1/a
w1h,
t2
a
q
n1
W
n
1h 2
N
n
1
2
n
t
0
e
1
0
sin1
2
n
t¿ 2g
1
d1h a1t t¿ 22W
n
1h 2dh fdt¿
0 t 1/a
w
#
1h, 0 2w1h, 02
g
1
F
o
L
4
EI
and
a
Vt
o
L
V
v
o
g1h, t2 g
1
d1h at 2
0 t 1/a
f
d
1x, t2 F
o
d1x Vt 2
0 t L/V
646 CHAPTER 9 Vibrations of Beams
F
o
L
V
FIGURE 9.40
Point force traveling with a speed
V along a hinged-hinged beam of
length L.
9.4 Forced Oscillations 647
0
(a)
20
40
60
0
0.2
0.4
0.6
0.8
1
0
0.05
0.1
w
V
(,)
(b)
w
V
(,)
0
2
4
6
0
0.2
0.4
0.6
0.8
1
0
0.05
0.1
0.15
(c)
w
V
(,)
0
0.2
0.4
0.6
0
0.2
0.4
0.6
0.8
1
0
0.05
0.1
0.15
0.2
FIGURE 9.41
Response of a beam to a point force traveling with a speed V: (a) a p/200, (b) a p/20, and (c) a p/2. The thick vertical lines
correspond to the locations of the traveling force.
and, from Case 2 of Table 9.3, the natural frequency coefficients are
n
np,
n 1, 2, . . . . Therefore, from Eq. (9.232) we find that
Then Eq. (a) can be written as
N
n
1
0
sin
2
1
n
h2dh 0.5
W
n
1h 2 sin 1
n
h2
(b)
Upon carrying out the integration in Eq. (b), we obtain
(c)
where
(d)
When a np, the nth term in Eq. (d) has the following limit
(e)
From an examination of the terms in Eq. (d), it is seen that the magnitude
of each contribution to the series given by Eq. (d) is proportional to 1/n
4
.
Thus, only the first few terms of the series are numerically significant. If we
take only the first term of the expansion, then to a close approximation, it is
(f)
Thus, the beam response has the form of the first mode shape with a time-
varying amplitude. When a 1, the function in the brackets varies with
time as the sin(pat). Equation (c) is plotted in Figure 9.41 for a p/200,
a p/20, and a p/2.
9.5
SUMMARY
In this chapter, free and forced oscillations of slender linear elastic beams
were considered. It was illustrated how the governing equations of motion are
obtained from the extended Hamilton’s principle. This approach can also be
used for determining the governing equations for other continua such as
strings and bars. Free oscillations of beams were treated at length for differ-
ent boundary conditions and varying beam properties. The effect of axial
force and elastic foundations were also taken into account. The forced re-
sponse of the beam was determined by using the orthogonality properties of
the modes.
w
V
1h, t2
sin1ph 2
1p a21p a 2
csin1pat2
a
p
sin1p
2
t2d
0 t 1/a and a p
lim
a씮 np
sin 1npat2 1a/np2sin 1n
2
p
2
t2
np a
1
np
sin 1n
2
p
2
t2 npt cos 1n
2
p
2
t2
0 t 1/a
w
V
1h, t2
a
q
n1
sin1nph 2
n
2
1np a21np a 2
csin1npat2
a
np
sin1n
2
p
2
t2d
w1h, t2
2g
1
p
2
w
V
1h, t2
0 t 1/a
w1h,
t2
2g
1
p
2
a
q
n1
sin 1nph2
n
2
t
0
sin 1n
2
p
2
t¿ 2sin 1npa1t t¿ 22dt¿
0 t 1/a
648 CHAPTER 9 Vibrations of Beams
649
Glossary
Absorber—a device that is attached to a system to at-
tenuate its vibration amplitude; see also vibra-
tion absorber
Acceleration—a vector quantity that is the rate of
change of velocity with respect to time
Acceleration, absolute—acceleration with respect to
a fixed point in an inertial reference frame
Accelerance—the ratio of the Laplace transform of a
vibratory system’s acceleration output to the
Laplace transform of the force input; the corre-
sponding frequency-response function is ob-
tained by substituting s jv
Accelerometer—a device whose output is propor-
tional to acceleration
Admittance—the ratio of the Laplace transform of a
vibratory system’s output displacement to the
Laplace transform of the force input; the corre-
sponding frequency-response function is ob-
tained by substituting s jv; used synonymously
with receptance and compliance functions
Amplitude response—the nondimensional
frequency-response function used to relate the
output response of a linear vibratory system to
the input
Angular acceleration—a vector quantity that is the
rate of change of angular velocity with respect to
time; acceleration associated with rotational
motions
Angular momentum—a vector quantity that is the
rotational momentum of a system; moment of
linear momentum
Angular speed—magnitude of the angular velocity
Angular velocity—a vector quantity that is the rate of
rotation around an axis
Aperiodic excitation—an excitation without perio-
dicity; for linear systems, this typically means a
waveform of finite duration or the sum of two or
more sine and/or cosine functions whose indi-
vidual frequencies are incommensurate (i.e., not
related by a rational number)
Band pass filter—a system that allows frequency
components in the input that are within the re-
gion defined by its lower and upper cutoff fre-
quencies to pass relatively unattenuated while
those frequency components outside this region
are attenuated
Bandwidth—the frequency range for a system; for a
band pass filter, this is the difference between the
upper and lower cutoff frequencies; for a low
pass filter it is the upper cutoff frequency; for a
high pass filter it is the lower cutoff frequency
Base excitation—an input applied at the base of a
system
Bernoulli-Euler beam—an elastic beam in which the
bending moment about the cross section of the
beam is linearly proportional to the second