Balakumar Balachandran, Magrab E.B. Vibrations
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Forced periodic vibrations can occur from rotating systems, such as the gears in the gear reduction unit. The transmission of these
vibrations to their surroundings can be reduced by various vibration isolation and reduction techniques. These techniques are used to
design an air glove to reduce jackhammer vibrations to the hand and soft elastic mounts to reduce rotating machinery vibrations to the
foundation. (Source: Stockxpert.com; Kim Steele / Getty Images.)
181181
5
Single Degree-of-Freedom Systems
Subjected to Periodic Excitations
5.1 INTRODUCTION
5.2 RESPONSE TO HARMONIC EXCITATION
5.2.1 Excitation Applied from t 0
5.2.2 Excitation Present for All Time
5.2.3 Response of Undamped System and Resonance
5.2.4 Magnitude and Phase Information
5.3 FREQUENCY-RESPONSE FUNCTION
5.3.1 Introduction
5.3.2 Curve Fitting and Parameter Estimation
5.3.3 Sensitivity to System Parameters and Filter Characteristics
5.3.4 Relationship of the Frequency-Response Function to the Transfer Function
5.3.5 Alternative Forms of the Frequency-Response Function
5.4 SYSTEMS WITH ROTATING UNBALANCED MASS
5.5 SYSTEMS WITH BASE EXCITATION
5.6 ACCELERATION MEASUREMENT: ACCELEROMETER
5.7 VIBRATION ISOLATION
5.8 ENERGY DISSIPATION AND EQUIVALENT DAMPING
5.9 RESPONSE TO EXCITATION WITH HARMONIC COMPONENTS
5.10 INFLUENCE OF NONLINEAR STIFFNESS ON FORCED RESPONSE
5.11 SUMMARY
EXERCISES
5.1 INTRODUCTION
In Chapter 3, the governing equation of a single degree-of-freedom system
was derived. From this equation, the solution for the system subjected to
forcing and initial conditions was determined in Appendix D. In the absence
of forcing, the responses of a vibratory system subjected to different types of
initial conditions were studied. In this chapter, we address those situations in
which a physical system is subjected to an external force f(t) and the initial
displacement and the initial velocity are zero. In particular, we consider
the responses to harmonic and other periodic excitations. The notion of
frequency-response function is introduced and related to the notion of trans-
fer function, which is used for system design.
In Figure 5.1, a vibratory system subjected to a forcing f(t) is illustrated
along with a conceptual illustration of the system’s input-output relationship.
In this chapter, we determine this relationship in the time domain and in the
transformed domain for periodic excitations. In Chapter 6, we determine this
relationship in the time domain and in transformed domains for arbitrary
excitations.
Responses of vibratory systems with a rotating unbalanced mass are also
studied in this chapter. Physical systems subjected to base motions are ana-
lyzed, and in this context, the device called an accelerometer, which is used
for acceleration measurements, is introduced. Vibration isolation is examined
at length. Energy dissipation for different damping models is discussed and
the notion of equivalent viscous damping is introduced. The influence of stiff-
ness nonlinearities on the forced response is also treated.
Referring to Figure 5.1, we consider the case where f(t) varies periodi-
cally. The governing equation of motion is of the form [recall Eq. (3.22)]
(5.1a)
The initial conditions are taken to be zero; that is,
(5.1b)
Noting that the initial conditions are zero for this system, the general solution
is given by Eq. (D.17) for 0 z 1; that is,
(5.2)
1
mv
d
t
0
e
zv
n
1t h2
sin 1v
d
3t h42f 1h 2dh
x1t 2
1
mv
d
t
0
e
zv
n
h
sin 1v
d
h2f 1t h 2dh
x10 2 0
and
x
#
10 2 0
d
2
x
dt
2
2zv
n
dx
dt
v
2
n
x
f 1t 2
m
182 CHAPTER 5 Single Degree-of-Freedom Systems Subjected to Periodic Excitations
FIGURE 5.1
Vibratory system subjected to forcing and conceptual illustration of system.
f(t)
f(t)
x(t)
x(t)
k
c
m
Input System Output
m, k, c
where h is the variable of integration
1
and
Next, the response of a linear vibratory system subjected to a harmonic
excitation is considered. First, the response is studied when the excitation is
initiated at time t 0, and then the response is studied when the excitation
is present for all time.
In this chapter, we shall show how to:
• Analyze the responses of single degree-of-freedom systems to harmonic
excitations of various durations.
• Determine the frequency response and phase response of a single degree-
of-freedom system.
• Interpret the response of a single degree-of-freedom system for an exci-
tation frequency less than, equal to, and greater than the system’s natural
frequency.
• Determine the system parameters from a measured frequency response.
• Analyze the responses of single degree-of-freedom systems with rotating
unbalance and with base excitation.
• Use accelerometers to measure the responses of single degree-of-freedom
systems.
• Isolate vibrations of single degree-of-freedom systems.
• Analyze the responses of single degree-of-freedom systems to excitations
with multiple harmonic frequency components.
• Determine energy dissipation and define equivalent damping.
5.2 RESPONSE TO HARMONIC EXCITATION
5.2.1 Excitation Applied from t 0
In this section, responses to sine harmonic and cosine harmonic excitation are
considered. It will be shown that although the initial conditions are zero, the
fact that the excitation is suddenly applied at t 0 results in a response that
consists of a transient portion and a steady-state portion. These transients are
typical of situations where a motor is started or where an excitation is inter-
mittently turned on and off. In the absence of damping, the response of a vi-
bratory system cannot be characterized as having a transient portion and a
steady-state portion.
Case 1: Sine Harmonic Excitations
We first consider the periodic forcing function
(5.3)f 1t 2 F
o
sin 1vt 2u1t2
v
d
v
n
21 z
2
5.2 Response to Harmonic Excitation 183
1
In writing Eqs. (5.2), the following property of convolution integrals was used:
t
0
h1h 2f 1t h 2dh
t
0
h1t h2f 1h 2dh
where u(t) is the unit step function
2
(5.4)
When Eq. (5.3) is substituted into Eq. (5.2), the result is
(5.5)
Further, we introduce the dimensionless time t v
n
t and rewrite Eq. (5.5) as
(5.6)
where the nondimensional excitation frequency v/v
n
, the nondimen-
sional time variable of integration j v
n
h, and we have used the definition
of v
d
given by Eq. (4.5). Note that when the excitation frequency is at the
natural frequency—that is, v v
n
or 1.
Solution for forced response After performing the integration,
3
Eq. (5.6)
leads to
(5.7)
where the steady-state portion of the response is given by
(5.8a)
and the transient portion of the response is given by
u12
tan
1
2z
1
2
H12
1
2D12
1
211
2
2
2
12z2
2
D11
2
2
2
12z2
2
x
sss
1t 2
F
o
k
H12sin 1t u1 22
x1t 2 3x
strans
1t 2 x
sss
1t 24u1t2
x 1t2
F
o
e
zt
k21 z
2
t
0
e
zj
sin 121 z
2
3t j42sin 1j2dj
x1t 2
F
o
e
zv
n
t
mv
d
t
0
e
zv
n
h
sin 1v
d
3t h42sin 1vh2dh
u1t 2 1
t 0
u1t 2 0
t 0
184 CHAPTER 5 Single Degree-of-Freedom Systems Subjected to Periodic Excitations
2
The unit step function is used as a compact form that simplifies the way that we can express a
function like Eq. (5.3), which is non-zero only in a specific interval. Without using the unit step
function, Eq. (5.3) would have been written as
3
The MATLAB function int from the Symbolic Toolbox was used.
0
t 0
F1t 2 sin1vt 2
t 0
(5.8b)
After a long period of time, which here means after many cycles of forc-
ing, the response reduces to
(5.9)
In Eqs. (5.8a), the quantity H() is called the amplitude response and the
quantity u() is called the phase response, which provides the phase relative
to the forcing f(t). We see that the steady-state portion varies periodically at
the nondimensional frequency , the frequency of the applied force f (t), and
with an amplitude F
o
H()/k. In addition, the displacement response is de-
layed by an amount u() with respect to the input. The amplitude response
H() and phase response u() are plotted in Figure 5.2 for several values
of z. We shall discuss the significance of these quantities in Section 5.3.
The transient response x
strans
(t) varies periodically with a frequency
v
d
/v
n
and its amplitude decreases exponentially with time as a function of the
damping ratio z. In addition, the response is shifted by an amount u
t
() with
respect to the input.
Duration of transient response For practical purposes, we define the time
duration t
d
beyond which the system can be considered as having reached
steady state; that is,
x(t) x
ss
(t) for t t
d
lim
t씮 q
x1t 2 x
ss
1t 2 x
sss
1t 2
F
o
k
H12sin 1t u122
u
t
1 2 tan
1
2z21 z
2
2z
2
11
2
2
x
strans
1t 2
F
o
k
H12e
zt
21 z
2
sin 1t21 z
2
u
t
1 22
5.2 Response to Harmonic Excitation 185
FIGURE 5.2
Harmonic excitation applied directly to the mass of the system: (a) amplitude response and (b) phase response.
0 0.5 1 1.5 2 2.5
0
1
2
3
4
5
6
Ω
0.02
0.1
0.3
0.5
0.7
0 0.5 1 1.5 2 2.5
0
20
40
60
80
100
120
140
160
180
Ω
(Ω) (deg)
0.02
0.02
0.1
0.3
0.5
0.7
(a)
ζ
ζ
ζ
ζ
ζ
θ
ζ
ζ
ζ
ζ
ζ
ζ
(b)
H(Ω )
To obtain an estimate of this duration, let the envelope of the transient decay
to a value d at a nondimensional time t
d
, which is given by the relation
(5.10a)
If , that is, when the amplitude of the transient portion of the dis-
placement is much less than the amplitude of the steady-state portion of the
displacement, the transient is said to have “died out” and only the steady-state
portion of the response remains. Solving for the nondimensional time t
d
,we
obtain
(5.10b)
We can also express the nondimensional time t
d
, which corresponds to
the dimensional time t
d
, in terms of the number of periods N
d
at the excitation
frequency v that it takes for the normalized displacement x(t)/(F
o
H()/k)
to decay to d. Since the period of the excitation t
v
2p/v, then the period
t
d
N
d
t
v
2pN
d
/v, and
(5.10c)
Therefore, Eqs. (5.10) lead to
(5.11)
Some typical values of N
d
obtained from Eq. (5.11) are provided in Table 5.1.
It is interesting to note from this table that for a given damping factor z, the
number of cycles that the transient lasts increases with the excitation fre-
quency. As expected, for a given excitation frequency, the duration of the tran-
sient portion of the response decreases for an increase in the damping factor.
Representative system responses for sine harmonic excitation The three
sets of graphs shown in Figure 5.3 give the normalized displacement response
x(t)/(F
o
/k) defined by Eq. (5.7) for three values of , and at each value of ,
the response is obtained for three different values of z. The first set shown
in Figure 5.3a corresponds to an excitation frequency that is less than the
natural frequency: 1. The second set, which is shown in Figure 5.3b,
corresponds to an excitation frequency that is equal to the natural frequency:
1. The third set, which is shown in Figure 5.3c, corresponds to an
N
d
2pz
ln c
d21 z
2
d
z 1,
0d 0 1
t
d
v
n
t
d
v
n
2pN
d
v
2pN
d
t
d
1
z
ln c
d21 z
2
d
0d 0
1
e
ztd
21 z
2
d
186 CHAPTER 5 Single Degree-of-Freedom Systems Subjected to Periodic Excitations
TABLE 5.1
Some Values of N
d
for
d 0.02
↓Z/→ 0.01 1 2
0.05 0.51 12.5 29.3
0.3 0.09 02.1 04.9
0.7 0.04 00.97 02.3
2
0
2
H(0.2) 1.04
N
d
1.47, 0.05
→
x
ss
(
τ
)2%
x(
τ
)/(F
o
/k)
2
0
2
H(0.2) 1.04
→
x
ss
(t)2%
x(
τ
)/(F
o
/k)
0 40 80
(a)
120 160 200
2
0
2
H(0.2) 1
→
x
ss
(t)2%
τ
x(
τ
)/(F
o
/k)
20
0
20
H(1) 10
→
x
ss
x(t)/(F
o
/k)
5
0
5
H(1) 2.5
→
x
ss
x(t)/(F
o
/k)
0 20 40 60 80 100
1
0
1
H(1) 0.714
→
x
ss
t
x(t)/(F
o
/k)
(b)
N
d
0.37, 0.2
N
d
0.12, 0.7
N
d
12.5, 0.05
N
d
3.13, 0.2
N
d
0.966, 0.7
(t
)2%
(t
)2%
(t
)2%
FIGURE 5.3
Normalized response of a system to a suddenly applied sine wave forcing function when
the transient envelope parameter d 0.02 and for different values of z: (a) 0.2
and (b) 1.0.
188 CHAPTER 5 Single Degree-of-Freedom Systems Subjected to Periodic Excitations
FIGURE 5.3 (continued )
(c) 3.0.
excitation frequency that is greater than the natural frequency: 1. For each
of these nine combinations of values, the values of H() and N
d
are given.
As t increases, the transient portion dies out, and the amplitude of the
displacement response approaches the magnitude H() (the steady-state
value). The response magnitude is within 2% (d 0.02) or less of this steady-
state value after N
d
periods or, equivalently, when t t
d
. It is noted that
when 1 or 1, the system response decays to within d of its steady-
state value. When 1, the displacement response increases until it reaches
within d of its steady-state value. Furthermore, during the portion of the
response where the transient portion is pronounced, the response is not
periodic. However, when t
d
has elapsed, each of the responses approaches
periodicity with the period determined by the excitation frequency; that is,
t
v
2p/v.
Case 2: Cosine Harmonic Excitations
For completeness, consider the periodic forcing function
f(t) F
o
cos(vt)u(t) (5.12)
After substituting Eq. (5.12) into Eq. (5.2), the result is
0.5
0
0.5
H(3) 0.125
x(t)/(F
o
/k)
0.5
0
0.5
H(3) 0.124
→
x
ss
(t)2%
x(t)/(F
o
/k)
0 12 24 36 48 60
0.5
0
0.5
H(3) 0.111
→
x
ss
t
x(t)/(F
o
/k)
(c)
N
d
47.9, 0.05
N
d
12, 0.2
N
d
3.65, 0.7
(t)2%
(5.13)
where the nondimensional frequency and the variable of integration j are
as defined for Case 1.
Solution for forced response Performing the integration
4
in Eq. (5.13), we
arrive at the displacement response
(5.14)
where the steady-state portion of the response is given by
(5.15a)
and the transient portion of the response is given by
(5.15b)
In Eqs. (5.14) and (5.15), the amplitude response H() and the phase u() are
given by Eqs. (5.8a), and it is noted that the proper quadrant must be consid-
ered for determining u
ct
(). Again, as in the case with the sine harmonic ex-
citation, after a long time (many cycles of forcing) the response settles down
to the steady-state form; that is,
The displacement response given by Eq. (5.14) is plotted in Figure 5.4 for
three values of , and at each value of , the response is determined for three
different values of z. For each of these nine combinations of values, the val-
ues of H() and N
d
are given. As in the case of the sine harmonic excitation,
the transient is initially aperiodic in each of the nine time histories, and then
the response becomes periodic after the system settles down. Although the
shapes of the transient responses are different than those obtained for the sine
wave forcing function, the times it takes for the transients to die out are the
same as in the corresponding cases.
lim
t씮 q
x1t 2 x
ss
1t 2 x
css
1t 2
F
o
k
H12cos 1t u122
u
ct
1 2 tan
1
z11
2
2
1
2
1221 z
2
x
ctrans
1t 2
F
o
k
H12e
zt
21 z
2
cos 1t21 z
2
u
ct
1 22
x
css
1t 2
F
o
k
H12cos1t u122
x1t 2 3x
css
1t 2 x
ctrans
1t 24u1t2
x1t 2
F
o
e
zt
k21 z
2
t
0
e
zj
sin 121 z
2
3t j42cos 1j2dj
5.2 Response to Harmonic Excitation 189
4
The MATLAB function int from the Symbolic Toolbox was used.