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Chapter 3Forced Vibrations of SDOF
Dr.-Ing. Azmi Mohamed Yusof
Faculty of Mechanical Engineering
At the end of this chapter students will be able to state, derive and apply
the fundamental principle of vibration involving:-
Forced vibration of undamped system with harmonic excitation
Forced vibration of damped system with harmonic excitation
Various applications, specifically the system with rotating unbalanced
and method for vibration isolation
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Course outcome
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Undamped system with harmonic oscillation
Consider the simplest vibrating system as shown infig. (a)
The time varying external force is given by
cos
Applying Newtons second law for the FBD shown in
fig. (b),
cos
or
cos (1)
Where is the natural frequency of the mass, while is the frequency of the external force
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=0
=0
The solution to this second order, linear, nonhomogeneous ODE containcomplementary solution , and particular solution
---- (2)
The complementary solution :
0
cos sin ---- (3)
Let assume the particular solution as :
cos ---- (4)
Obtaining the velocity and acceleration term for eqn. 4 gives
cos
sin
cos
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Substituting the above equations into eqn. (1)
cos
cos
cos
;
From eqn. (2), , the final solution gives
cos sin
cos
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The complementary solution defines the freevibration (see figure (a)) and it will typically dampen
out.
Therefore xc is referred to as transient
The particular solution describes the force vibration
caused by the applied load (fig. (b))
There resultant vibration is shown in fig. c
As the free vibration will in time dampen out, the
remaining vibration will be the force vibration
Thus, is called steady state (see fig. (d)).
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We need to determine the constant D & E by using
initial conditions.
Suppose 0 and 0 , then wehave
and
By substituting into equation 1
cos
sin
cos
If the static deflection of the mass due to force Fo is
given by
, the maximum amplitude can be
written as,
1
1
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The ratio, is known as the magnification factor
(M)
M The ratio of the amplitude of vibration to the
amplitude of zero frequency deflection
The plot
or Magnification factorversus the
frequency ratior
is shown in fig. (a)
The asymptote occurs at r =1, thus the system
response can be of three types
When 0 1
When
= 1
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Case 1, when 0 natural frequency
of forced vibration
The amplitude of forced response > the static deflection
The harmonic response of the system xp(t) is in phase with the external
force. See the plots as shown in the figure
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External force excitation System response
Case 2, when > 1
The natural frequency of the free vibration
response < natural frequency of forced
vibration
cos , where
and has opposite sign, 180o out ofphase
When
, then X 0 : the response at
very high frequency is close to zero.
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External force excitation
System response
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External force excitation
System response
Case 3, when
= 1
The condition when = n is called resonance
Amplitude of oscillation increases linearly with
time or infinite
Undesirable condition which may harm overall
system
The equation of motion is
The response of the system at resonance is:
Example
A weight of 50N is suspended from a spring of stiffness 5000 N/m and is
subjected to a harmonic force of amplitude 40N and frequency of 4 Hz.
Determine
a) the extension of the spring due to suspended spring
b) the static displacement of the spring due to the maximum applied
force
c) the amplitude of the forced motion of the weight
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Solution
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Example
A 50 kg mass is hanging from a spring of stiffness 5 x 104 N/m. A
harmonic force of magnitude 100 N and frequency 100 rad/s is applied to
the system. Determine
a) the amplitude of the forced response
b) the natural frequency of the system
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Solution
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Example
A reciprocating pump, having a mass of 68 kg, is mounted at the middle of a
steel plate of thickness 1 cm, width 50 cm, and length 250 cm, clamped along
two edges as shown in the figure below. During operation of the pump, the plate
is subjected to a harmonic force, 220 cos62.832 N. Determine theamplitude of vibration of the plate. (Take the plate equivalent stiffness of the
beam as
, and Youngs modulus of the beam as 200 Gpa).
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Solution
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Forced vibration with viscous damping
Consider a viscously damped SDOF spring-mass
system as shown in fig. (a).
Suppose the external force applied is in the form of
sin
Using Newtons second law of motion for FBD shown in
fig. (b),
cos or
cos
The particular solution can be assumed in the form
cos sin The velocity and acceleration terms become,
sin
cos
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Substitute these results into equation 1,
cos sin cos
Extracting the algebraic equation involving term A and B,
and 0
Dividing both equations with k
1 2 and 2 1 0
Where
,
and
Solve the above equation for A and B yield,
,
The result for a steady state equation
1 21 cos 2 sin
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It can also be written in a generalized form
Introducing the magnification factor M,
Where the phase angle is:
The plot for M versus r, and versus r are shown in the figure
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Finally, The complete solution is given by
cos cos
and can be determined from the initial conditions 0 0
See textbook page 275 for complete solution (long equation).
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Example
A SDOF spring-mass-damper system is subjected to a harmonic force.The amplitude is found to be 25 mm at resonance and 10 mm at a
frequency 0.75 times the resonant frequency. Determine the damping
ratio for the system.
Solution
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Example
A SDOF damped system is composed of a mass of 10 kg, a spring
having a constant of 2000 N/m, and a dashpot with damping constant of
50 Ns/m. The mass of the system is acted on by a harmonic force F = Fo
sin t having a maximum value of 250 N and a frequency of 5 Hz.
Determine the complete solution for the motion of the mass.
Solution
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Example
A 25 kg mass is mounted on an oscillator pad whose stiffness is 5 x 105
N/m. When the system is subjected to a harmonic excitation of
magnitude 300 N and frequency 100 rad/s, the phase different between
the excitation and the steady state response is 25o. Determine:
a) the damping ratio of the isolator pad
b) the isolator pads maximum deflection due to this excitation
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Response due to harmonic motion ofthe base/support motion
Sometime the base or support of a
spring-mass-damper-system undergoes
harmonic motion.
Example of this situation is the car
moving on bumpy road as shown in the
figure.
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Let y(t) denote the displacement of the base and x(t) the displacement of
the mass
The system is simplified in the figure shown. The subsequent analysis is
based on the FBD shown. The net elongation of the spring is
The relative velocity between two ends of the damper is
The equation of motion according to the FBD is obtained as:
k 0
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Rearrange the equation
If sin, then cos sin
If sin and - and substitute into eqn. 1 then sin
where , and
Recall our steady state response for a spring-mass-damper system
sin
sin
Therefore,
sin
sin
Where
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The amplitude of oscillation is obtained as
or
Rearrange the equation,
If Z is the relative displacement between the mass and the base, then
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The maximum force transmitted to the base is given by
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Example
The simple model of motor vehicle
is shown in the figure, that can
vibrate in the vertical direction
while traveling over a rough road.
The vehicle has a mass of 1200
kg. The suspension system has a
spring constant of 400 kN/m and a
damping ratio of 0.5. If the vehicle
speed is 20 km/h, determine the
displacement amplitude of the
vehicle. The road surface varies
sinusoidally with an amplitude of Y= 0.05m and a wavelength of 6m.
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Solution
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Class exercise
A 50 kg mass is attached to the base through a spring in parallel with a
damper as shown in fig. below. The base undergoes a harmonic
excitation of y(t) = 0.20 sin 30t. The stiffness of the spring is 30000 N/m
and the damping constant is 200 Ns/m. Determine a) the amplitude of the
masss absolute displacement, b) the amplitude of its displacement
relative to its base [ans: a) 0.38m; b) 0.56m]
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m
k c
y(t) = 0.20 sin 30t
x(t)
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Solution
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Class exercise
A racing car is modeled as a SDOF damped system vibrating in the
vertical direction. The elevation of the road is assumed to vary
sinusoidally. The distance from peak to through is 0.2 m and the distance
between peaks is 70m. The natural frequency of the system is 2 Hz and
the damping ratio of the damper is 0.15. Determine the amplitude of
vibration of the racing car at a speed of 120 km/h. [ans: 1.06 m]
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Solution
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Rotating unbalanced
Unbalanced forces in many rotating
mechanism is the common sources of
vibration excitation.
Consider the rotating component is
mounted in bearing and rotates
counterclockwise.
Variables:
M = total mass of the system
m = eccentric mass located at e from the
center of rotation.
x = the displacement of the machine in
vertical direction
= angular speed
t = time
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The component of the displacement of the eccentric mass in vertical
direction is given by:
sin
the acceleration is obtained as
sin
The total inertia forces of the machine
Applying Newtons second law of motion
and using equation 3 yield,
sin
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Comparing this equation with fundamental equation
We obtain that
Recall the steady state solution for the system
sin , where
,
and
Finally the steady state response is obtained as:
/
1 2
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Basic damped forced equation Equation 4
sin
sin
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This equation can also be written as
1 2sin
Thus the amplitude of vibration is obtained as
1 2
Peak deflection of the mass M at resonance is given from
; thus
The force transmitted to the base
1 2
1 2sin
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Example
An electric generator weighing 981 N and operating at 600 rpm is
mounted on four parallel springs of stiffness 5000 N/m each. Determine
the maximum permissible unbalance in order to limit the steady state
deflection to 6 mm peak-to-peak.
Solution
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Example
An electric motor of mass M, mounted on an elastic
foundation, is found to vibrate with a deflection of
0.15 m at resonance. It is known that the unbalanced
mass of the motor is 8% of the mass of the rotor due
to manufacturing tolerances used, and the damping
ratio of the foundation is 0.025, Determine:
a) the eccentricity or radial location of the
unbalanced mass (e).
b) the peak deflection of the motor when the
frequency ratio varies from resonance
c) the additional mass to be added uniformly to the
motor if the deflection of the motor is to be reduced
to 0.1 m.
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Solution
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Vibration isolation and force transmissibility
To minimize excessive vibration excitation that may
contribute to system failure, an isolator is installed to
support the structure
The vibration isolator is typically designed on the
machine with flexible support
Good isolator design must consider proper selection of
the stiffness and damping coefficients.
Consider a spring-mass- damper system (fig.(a)) with
external force applied in the form of sin
The FBD for the system and the velocity triangle isshown in fig. (b) and (c).
The force transmitted to the support can be written as
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sin
sin
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Transmissibility, is defined as the ratio of the transmitted force to thatof the disturbing force
When the damping is negligible, then
1
1
To reduce the amplitude X without changing Td,
Isolated mass m can be mounted on larger mass M
The stiffness k must be increased to keep the ratio
constant
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M
mk
The plot Td versus r is shown in the figure below
At region Td >1 for r < 1.41, the amplitude of transmitted force is greater
than the amplitude of applied force.
For r < 1.41, the transmitted force to the support can be reduced by
increasing the damping factor.
Vibration isolation is best accomplished by an isolator composed only of
spring elements for which r > 1.41 with no damping element used in the
system.
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Transmiss
ibility,
Td
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Example
A machine of mass 100 kg is mounted on springs anddamper as shown in the figure. The total spring stiffness is
50,000 N/m and the damping factor is 0.20. A harmonic
force, F = 200 sin 13.2t acts on the mass. Determine:-
a) the amplitude of the motion of the machine
b) its phase with respect to the existing force
c) the transmissibility
d) the maximum dynamic force transmitted to the foundation
e) the maximum velocity of the motion
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Solution
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