اسم المؤلف
غير معروف
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1 أكتوبر 2022
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263
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محاضرة بعنوان
Free Vibration of 1-DOF System
2.0 Outline
 Free Response of Undamped System
 Free Response of Damped System
 Natural Frequency, Damping Ratio
Ch. 2: Free Vibration of 1-DOF System
2.1 Free Response of Undamped System
Free vibration is the vibration of a system in response
to initial excitations, consisting of initial displacements/
velocities. To obtain the free response, we must solve
system of homogeneous ODEs, i.e. ones with zero
applied forces. The standard form of MBK EOM is
2.1 Free Response of Undamped System
mx cx kx x x t   + + = = 0, ( )Ch. 2: Free Vibration of 1-DOF System
If the system is undamped, c = 0. The EOM becomes
2.1 Free Response of Undamped System
subject to the initial conditions 0 , 0
The solutions of homogeneous ODE are in the form
x t Ae A s ( ) = st , is the amplitude and is constant
Subs. the solution into ODE, we get
0 0 ** characteristic equation **
** characteristic roots, eigenvalues **
general solution: by superposition n n
= +Ch. 2: Free Vibration of 1-DOF System
We can now apply the given i.c. to solve for A1 and A2.
However we will use some facts to arrange the solutions
into a more appealing form.
2.1 Free Response of Undamped System
Because is real, .
Let . Therefore .
The given i.c. are then used to solve for the amplitude C
and the phase angle Φ . Note ω n, known as natural
frequency, is the system parameter. The system is
called harmonic oscillator because of its response to i.c.
is the oscillation at harmonic frequency forever.Ch. 2: Free Vibration of 1-DOF System
2.1 Free Response of Undamped System
If the initial conditions are 0 and 0
cos and sin and tan
cos sin as the function of i.c. and system parameter
Ch. 2: Free Vibration of 1-DOF System
2.2 Free Response of Damped System
We normalize the standard MBK EOM by mass m:
2.2 Free Response of Damped System
/ natural frequency
/ 2 viscous damping factor
subject to the initial conditions 0 and 0
The solutions of homogeneous ODE are in the form
x t Ae A s ( ) = st , is the amplitude and is constant
Subs. the solution into ODE, we getCh. 2: Free Vibration of 1-DOF System
2.2 Free Response of Damped System
( )
( ) 1 2
2 2 2 2
2
12
1 2
2 0 2 0 CHE
1 ** characteristic roots, eigenvalue **
general solution: by superposition
= +Ch. 2: Free Vibration of 1-DOF System
2.2 Free Response of Damped System
12
Response of 0, i.c. 0 and 0
i) 0 :
cos
harmonic oscillation with frequency
entially decaying amplitude oscillation
with damped frequency and envelope ωd Ce−ζωntCh. 2: Free Vibration of 1-DOF System
2.2 Free Response of Damped SystemCh. 2: Free Vibration of 1-DOF System
2.2 Free Response of Damped System
Response of 0, i.c. 0 and 0
iii) 1: 1
aperiodic decay with peak more suppressed and
decay further slow down as inc
aperiodic decay with highest peak and fastest decay
= +Ch. 2: Free Vibration of 1-DOF System
2.2 Free Response of Damped SystemCh. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
2.3 Natural Frequency, Damping Ratio
The response will partly be dictated by the roots s12,
which depend on Root locus diagram gives
a complete picture of the manner in which s12 change
with The focus will be the left half s-plane
where the system response is stable.
and .
ζ ωn
and .
ζ ωnCh. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System
constant, vary
i) 0 undamped , on the Im-axis
far from origin by . The motion is
with natural frequency .
ii) 0 1 underdamped , 1
harmonic oscillation
ir of symmetric points moving on a semicircle
of radius . The motion is .
iii) 1 critically damped , repeated roots.
The motion is .
n
n
oscillatory decay
s
aperiodic decay
2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System
iv) 1 overdamped , 1 ( ) 12 2
two negative real roots going to 0 and .
The motion is .
constant, vary
The symmetric roots will be far from origin
n making an angle cos with – axis. −1 ( ) ζ x
2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
Determination of M-B-K
• Mass – directly measure the weight or deduced from
frequency of oscillation
• Spring – from measures of the force and deflection
or deduced from frequency of oscillation
• Damper – deduced from the decrementing responseCh. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
Ex. 1 A given system of unknown mass m and spring k
was observed to oscillate harmonically in free
vibration with T
n = 2π x10-2 s. When a mass M =
0.9 kg was added to the system, the new period
rose to 2.5π x10-2 s. Determine the system
parameters m and k.Ch. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
Ex. 2 A connecting rod of mass m = 3×10-3 kg and IC =
0.432×10-4 kgm2 is suspended on a knife edge
about the upper inner surface of a wrist-pin
bearing, as shown in the figure. When disturbed
slightly, the rod was observed to oscillate
harmonically with ω n = 6 rad/s. Determine the
distance h between the support and the C.M.Ch. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
3 10 kg, 0.432 10 kgm , 6 rad/s
0.2, 0.072 m
Radius of gyration 0.12 m must be greater than
the longest length of the object. 0.072 m
∴ =Ch. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
Ex. 3 A disk of mass m and radius R rolls w/o slip
while restrained by a dashpot with coefficient of
viscous damping c in parallel with a spring of
stiffness k. Derive the differential equation for
the displacement x(t) of the disk mass center C
and determine the viscous damping factor ζ
and the frequency ω n of undamped oscillation.Ch. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
 Ch. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
Ex. 4 Calculate the frequency of damped oscillation
of the system for the values m = 1750 kg,
c = 3500 Ns/m, k = 7×105 N/m, a = 1.25 m, and
b = 2.5 m. Determine the value of critical damping.Ch. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
 Ch. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
Ex. 5 A projectile of mass m = 10 kg traveling with
v = 50 m/s strikes and becomes embedded in a
massless board supported by a spring stiffness
k = 6.4×104 N/m in parallel with a dashpot of
c = 400 Ns/m. Determine the time required for
the board to reach the max displacement and
the value of max displacement.Ch. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio
( ) 7 0.4447 m =Ch. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System
2.3 Natural Frequency, Damping Ratio

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