Engineering Vibrations
MECH 242
Dr. Nicolas M. Saba
University of Balamand
Department of Mechanical
Engineering
|1 nicolas.saba@balamand.edu.lb
Ex 1.1
A free vibrations test is run to determine the stiffness and damping
properties of an elastic element. A 20-kg block is attached to the
element. The block is displaced 1 cm and released. The resulting
oscillations are monitored with the results shown in Fig. below.
Determine k and c for this element.
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Ex 1.1
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Ex 1.2
A 30-kg block is attached to four identical springs, each of stiffness 23
N/m, placed in parallel. Determine the system's natural frequency in
hertz.
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Ex 1.3
Derive the equation of motion and find the natural frequency of
vibration of each of the systems shown in following Fig.
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Ex 1.4
List three differences between the free vibration response of a system
with Coulomb damping and the free vibration response for a system
with viscous damping whose free vibrations are underdamped.
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Ex 1.4
Solution
Three differences between the systems are:
a. The magnitude of the Coulomb damping has no effect on the
frequency or period of motion while the magnitude of the viscous
damping does affect the frequency ωd and period Td. An increase
in viscous damping leads to a decrease in ωd and an increase in
Td
b. The amplitude of vibration for a system with Coulomb damping
decreases linearly while the amplitude of vibration for a system
with viscous damping decreases exponentially
c. Motion ceases for a system with Coulomb damping when the
amplitude becomes small enough such that the force in the elastic
member is insufficient to overcome static friction, leading to a
permanent displacement from equilibrium. Motion continues
indefinitely for a system with viscous damping.
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Ex 1.5
Determine the number of degrees of freedom necessary for the
analysis of the system of following Fig.
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Ex 1.6
The following data are given for a system with viscous damping:
mass 4 kg, spring constant k = 5 kN/m and the amplitude decreases
to 0.25 of the initial value after five consecutive cycles. Find the
damping coefficient of the damper.
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Ex 1.7
An 80-kg reciprocating machine is placed on a thin, massless beam.
A frequency sweep is run to determine the magnitude of the
machine's rotating unbalance and the beam's equivalent stiffness. As
the speed of the machine is increased, the following is noted:
(a) The steady-state amplitude of the machine at a speed of 65 rad/s
is 7.5 mm.
(b) The maximum steady-state amplitude occurs for. a speed less
than 65 rad/s.
(c) As the speed is greatly increased, the steady-state amplitude
approaches 5 mm.
Assume the system is undamped.
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Ex 1.8
A body of mass 100 kg is suspended by a spring of stiffness of 30
kN/m and dashpot of damping constant 1000 N s/m. Vibration is
excited by a harmonic force of amplitude 80 N and a frequency of 3
Hz. Calculate the amplitude of the displacement for the vibration and
the phase angle between the displacement and the excitation force
using the graphical method.
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Ex 1.9
The figure shows a pendulum which consists of three light arms OA,
OB and OC, rigidly attached to each other and pivoted at O, and of a
mass m which is on the end of the vertical arm OA. The arm OB is
horizontal and attached at its end to a vertical spring having a
stiffness k while the arm OC is vertical and attached at its end to a
horizontally positioned viscous damper whose damping coefficient is
c.
Determine the frequency of small amplitude undamped oscillations of
the pendulum about O and find the critical damping coefficient in
terms of the given parameters.
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Ex 1.9
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Ex 1.10
What is the natural frequency of the 200-kg block of following Fig.?
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Ex 1.11
Part of a structure is modelled by a thin rigid rod of mass m pivoted at
the lower end, and held in the vertical position by two springs, each of
stiffness k, as shown. Find the frequency of small amplitude
oscillation of the rod about the pivot.
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Ex 1.12
A motor generator set of total mass 365 kg is mounted on damped
vibration isolators. It runs at 1450 rev/min, and the unit is observed to
have a vibration amplitude of 0.76 mm. When the unit is disturbed at
standstill, it is found that the resulting damped oscillations have a
frequency of 23 Hz and a decay factor of 15.
Find (a) the coefficient of viscous damping, (b) the magnitude of the
exciting force at the normal running speed, and (c) the amplitude of
the forced vibration if damping were not present.
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Ex 1.13
To determine the amount of damping in a bridge it was set into
vibration in the fundamental mode by dropping a weight on it at centre
span. The observed frequency was 1.5 Hz, and the amplitude was
found to have decreased to 0.9 of the initial maximum after 2 s. The
equivalent mass of the bridge was 105 kg.
Assuming viscous damping and simple harmonic motion, calculate
the damping coefficient, the logarithmic decrement, and the damping
ratio.
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Ex 1.14
A 500-kg vehicle is mounted on springs such that its static deflection
is 1.5 mm. What is the damping coefficient of a viscous damper to be
added to the system in parallel with the springs, such that the system
is critically damped?
Note: The static deflection is the deflection of an isolator that occurs
due to the dead weight load of the mounted equipment
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Ex 1.15
A suspension system is being designed for a 2000-kg vehicle (empty
weight). It is estimated that the maximum added mass from
passengers and cargo is 1000 kg. When the vehicle is empty, its
static deflection is to be 3.1 mm. What is the minimum value of the
damping coefficient such that the vehicle is subject to no more than 5
percent overshoot, empty or full?
Hint: The results of previous experiments show that the damping ratio
must be no smaller than 0.69 to limit the overshoot to 5 percent
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Ex 1.16
The natural frequency of vibration of a person is found to be 5.2 Hz
while standing on a horizontal floor. Assuming damping to be
negligible, determine the following:
a. If the weight of the person is 70 kgf, determine the equivalent
stiffness of his body in the vertical direction.
b. If the floor is subjected to a vertical harmonic vibration of frequency
of 5.3 Hz and amplitude of 0.1 m due to an unbalanced rotating
machine operating on the floor, determine the vertical displacement of
the person.
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Ex 1.17
A lathe can be modeled as an electric motor mounted on a steel
table. The table plus the motor have a mass of 50 kg. The rotating
parts of the lathe have a mass of 5 kg at a distance 0.1 m from the
center. The damping ratio of the system is measured to be = 0.06
and its natural frequency is 7.5 Hz. Calculate the amplitude of the
steady-state displacement of the motor, assuming v = 30 Hz..
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Ex 1.18
For what value of m will resonance occur for the system of following
Fig.
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Ex 1.19
The vehicle with the suspension designed in Problem 1.15
encounters a bump of height 5.5 cm. What is the vehicle's overshoot
if it is carrying 25 kg of fuel, one 80-kg passenger, and 110 kg of
cargo?
Hint: Overshoot for an underdamped system is defined as the
maximum displacement of the system at its first half cycle
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Ex 1.20
An air compressor of 450 kg mass, operates at a constant speed of
1,750 rpm. The rotating parts are well balanced. The reciprocating
parts are of 10 kg. The crank radius is 100 mm. The damper for the
mounting introduces a damping factor ξ = 0.15, (a) specify the springs
for the mounting such that only 20 percent of the unbalance force is
transmitted to the foundation, and (b) determine the amplitude of the
transmitted force.
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Ex 1.21
A 45-kg machine is mounted on four parallel springs each of stiffness
2 x 105 N/m. When the machine operates at 32 Hz, the machine’s
steady-state amplitude is measured as 1.5 mm. What is the
magnitude of the excitation provided to the machine at this speed?
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Ex 1.22
A 110-kg machine is mounted on an elastic foundation of stiffness 2 x
106 N/m. When operating at 150 rad/s, the machine is subject to a
harmonic force of magnitude 1500 N. The steady-state amplitude of
the machine is measured as 1.9 mm. What is the damping ratio of the
foundation?
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Ex 1.23
A 100 kg mass is suspended by a spring of stiffness 30 ×103N/m with
a viscous damping constant of 1000 Ns/m. The mass is initially at rest
and in equilibrium. Calculate the steady-state displacement if the
mass is excited by a harmonic force of 80 N at 3 Hz.
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Ex 1.24
A 120-kg machine is mounted at the midspan of a 1.5 m-long simply
supported beam of a elastic modulus E = 200 X 109 N/m2 and cross-
section moment of inertia I = 1.53 x10-6 m4. An experiment is run on
the system during which the machine is subjected to a harmonic
excitation of magnitude 2000 N at a variety of excitation frequencies.
The largest steady-state amplitude recorded during the experiment is
2.5 mm. Estimate the damping ratio of the system.
Hint: It is noted that for a fixed , the maximum value of
M is
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Ex 1.25
A spring-mass-damper system is subjected to a harmonic force. The
amplitude is found to be 20 mm at resonance and 10 mm at a
frequency 0.75 times the resonant frequency. Find the damping ratio
of the system.
Hint: Small damping
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Ex 1.26
The figure shows a pendulum which consists of a light rigid rod of
length L pivoted to a fixed point at one end and having a mass m
fixed to its other end. A spring of stiffness k is attached as shown, at a
distance a from the pivot. In the position shown the rod is vertical and
the spring is horizontal and unloaded.
Find the frequency of free oscillations of small amplitude in the plane
of the diagram.
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Ex 1.27
The figure shows a diagrammatic end view of one half of a swing-axle
suspension of a motor vehicle which consists of a horizontal half-axle
OA pivoted to the chassis at O, a wheel rotating about the centre line
of the axle, and a spring of stiffness k and a viscous damper with a
damping coefficient c both located vertically between the axle and the
chassis. The mass of the half-axle is m 1 and its radius of gyration
about O is h. The mass of the wheel is m E and it may be regarded
as a thin uniform disc having an external radius r and located at a
horizontal distance s from the pivot O. The spring and the damper are
located at horizontal distances q and p from the pivot O, as shown.
Derive the equation for angular displacement of the axle-wheel
assembly about the pivot O, and obtain from it an expression for the
frequency of damped free oscillations of the assembly. Express this
frequency in terms of the given parameters and the undamped
natural frequency of the assembly.
Hint: m1h2 + m2r2/4 +m2s2
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Ex 1.27
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Ex 1.28
A machine of mass 520 kg produces a vertical disturbing force which
oscillates sinusoidally at a frequency of 25 Hz. The force transmitted
to the floor is to have an amplitude, at this frequency, not more than
0.4 times that of the disturbing force in the machine, and the static
deflection of the machine on its mountings is to be as small as
possible consistent with this.
For this purpose, rubber mountings are to be used, which are
available as units, each of which has a stiffness of 359 kN/m and a
damping coefficient of 2410 N s/m. How many of these units are
needed?
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Ex 1.29
A sketch of a valve and rocker arm system for an internal combustion
engine is given in following Fig. Model the system as a pendulum
attached to a spring and a mass and assume the oil provides viscous
damping in the range of ζ= 0.01.
Determine the equations of motion and calculate an expression for
the natural frequency and the damped natural frequency. Here J is
the rotational inertia of the rocker arm about its pivot point, k is the
stiffness of the valve spring and m is the mass of the valve and stem.
Ignore the mass of the spring.
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Ex 1.29
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Ex 1.30
A new concert hall is to be protected from the ground vibrations from
an adjacent highway by mounting the hall on rubber blocks. The
predominant frequency of the sinusoidal ground vibrations is 40 Hz,
and a motion transmissibility of 0.1 is to be attained at that frequency.
Calculate the static deflection required in the rubber blocks, assuming
that these act as linear, undamped springs.
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Ex 1.31
For the spring-mass-damper system shown to the right, x is
measured from the static equilibrium position, and the surface is
frictionless.
Determine the governing equations of motion;
What is the period of each oscillation in terms of the system
parameters (m, k, c)?
For what value of c is the system critically damped?
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Ex 1.32
The following numerical data apply to a typical viscously-damped
spring-mass system:
m = 30 kg, k = 2 x 105 N/m, F = 100sinvt N.
The maximum steady-state displacement of the system is observed
to be 1.5 mm. Calculate the value of the damping coefficient. Hint:
Assume small damping.
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Ex 1.33
A system has a mass of 225 kg with a spring stiffness of 3.5
× 104 N/m. Calculate the damping coefficient given that the system
has a deflection of 0.7 cm when driven at its natural frequency while
the base amplitude is measured to be 0.3 cm.
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Ex 1.34
A 200-kg turbine operates at speeds between 1000 and 2000 rev/min.
The turbine has a rotating unbalance of 250 g.m.
What is the required stiffness of an undamped isolator such that the
maximum force transmitted to the turbine 's foundation is 1 kN?
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Ex 1.35
Find the natural frequency of vibration of a spring-mass system
arranged on an inclined plane, as shown in Fig.
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