Experimental Analysis of MDOF Forced Vibration System - WSU Report

This report presents an experimental study on a Multi-degree of freedom forced vibration system, focusing on measuring un-damped natural frequencies. The system consists of a four-floor structure mounted on springs, and the vibration is induced by an external force. The methodology involves using a multi-degree of freedom test rig, electromagnetic shaker, and accelerometers to measure the acceleration of each floor. Matrix iteration and Holzer methods are employed to determine the natural frequencies and mode shapes. The report details the equipment used, the experimental method, and the calculations performed, along with the results obtained from the experiment. The study also covers the theoretical background of vibration, including free and forced vibrations, damped and un-damped vibrations, and resonance, and a literature review of the development of vibration analysis.

Multi degree of freedom forced vibration system
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Abstract
In this report an experimental study is done on a Multi-degree of freedom of forced
vibration, in which the vibration is due to the external force, the vibration is a fundamental
phenomenon in engineering system, which occurs when a body is displaced from its mean
position, due to the conversion of strain energy into kinetic energy and followed by the
conversion of kinetic energy stored inside the body into strain energy. In this study a Multi-
degree of freedom test rig is used with a structure consisting of four floor which is mounted on a
spring, the final goal of this study is to measure the un-damped natural frequencies that occurs in
the system. Matrix iteration method and Holzer method are used to find the natural frequency for
each floor of the rig and to find the mode shapes after the measurement of natural frequencies.
The experimental setup, equipment used and the final calculations and the outcome of the
experiments are discussed.

Table of Contents
Introduction......................................................................................................................................4
Literature review..............................................................................................................................5
Methodology....................................................................................................................................6
Equipment used...........................................................................................................................6
Experimental method...................................................................................................................7
Results..............................................................................................................................................8
Discussion......................................................................................................................................12
Conclusion.....................................................................................................................................12
References......................................................................................................................................14

Introduction
Vibration in our day to day life is inevitable it may be, a vibration caused by the washing
machines, a vibration caused by an engine while running, vibration of buildings due to earth
quakes, vibration caused to human body while exercising, etc…. When a body is displaced from
its equilibrium position by the application of any external forces it exhibits vibration, due to its
elasticity as the internal forces of the body in the form of strain energy are present in the body.
As the external force is applied the strain energy begin to build up inside the body and when it is
released, these internal energy in the form of strain tries to bring back the body to its original
position, as the object reaches its equilibrium position the total strain energy is converted into
kinetic energy due to which the body continuous to move in the opposite direction, now the
conversion of kinetic energy into strain energy cycle repeats due to which the vibratory motion
occur.
Some of the important types of vibrations are Free or natural vibrations which occurs
when no external forces acts upon the body, after providing with initial displacement the body is
under free vibration, it is also called as natural vibration and the frequency of the free vibration is
known as free frequency or natural frequency, there are three types of free vibration are
longitudinal vibration, which occurs when a body is set into vibration longitudinally or in a up
and down motion, which might be seen in springs commonly, transverse vibration which occurs
when a body is set into vibration in a transverse direction or in a to and fro motion, in which the
object is said to move in a direction perpendicular to the axis of the body, in this case bending
stresses are introduced in the object, and finally the torsional vibration, which occurs when the
object moves in a circular motion about its axis, which are generally seen in shafts, when the
shafts are set into rotation they exhibit torsional vibration there are many other important factors
also included in the torsional vibration such as critical speed of shaft or the whirling speed of
shaft, in case of torsional vibration the object experiences torsional shear stresses.
Forced vibration occurs when the vibration of the body is due to some external forces, in
other word the tendency of an object to force another object to set into vibrational motion is
known as the forced vibration, for example when a tuning fork is set into vibration by hitting it
with a rubber mallet, the sound produced is very low and inaudible on the other hand if the
tuning fork is set into vibration by colliding it upon the glass particle, then the glass forces the air

surrounding it into vibrational motion and thus the amplitude of the wave is increased, this
clearly explains the concept of forced vibration.
There is another final classification of vibrations known as Damped vibrations and un-
damped vibrations, the main difference between them is that in case of the damped vibrations
there is a reduction in amplitude across every cycle of vibration, which means that the vibrating
object loses its energy to the surrounding over a period of time, whereas in case of un-damped
vibrations there will not be any reduction of amplitude over the period of time, which means that
the energy of the vibrating object will not be dissipated to the surroundings over the period of
time.There is an importantphenomenon known as resonance which occurs when the frequency of
the external force is same as that of the natural vibrations.
Literature review
At 1874, Dr. G. P. de Laval an engineer invented a milk separator machine which worked
at speed between 6000 to 10000 RPM, he also introduced a steam turbine which worked on
higher speeds than any other machines at that time, which paved a way for the development of
vibrational analysis in machineries. During 1910, Mechanical vibrations became an engineering
discipline with the publication of W. Hort’s book titled “Technische Swingungslhere” which is
widely accepted as a text book for the vibrational engineers, at 1928 fopple solved the problems
developed by den Hartog under the formation of critical speeds and anticipated whirl. The
general theory of system instability was developed by liapounovin in 1907. A paper titled
“Vibration Tolerance” was published at 1939 which was done by T.C. Rathbone. During the year
1938, Jarvis and mills showed tha association between the relative speed of brake circle and the
wheel speed with respect to self-invigorated vibration point of view, it was demonstrated that the
change the vibrations which are started by the contact of the brake in the auto , can lower the
braking instability of the auto. Early days the vibration testing instruments were very large and
they were highly expensive to afford, also they are immobile they cannot be transferred from one
place to another. In 1948, Art Crawford Company built a vibration analyzer with their own
design and idea which can perform phase analysis, it was very famous during that time and it
paved its way towards the foundation of (IRD) International Research and Development
Corporation in 1952. Until late 1970 s, the Fourier analysis technique was used with various
filter such as heterodyne filters, swept filters or fixed filters. During 1980 s several industries

began to implement its condition monitoring programs. After the emergence of personal
computers, which revolutionized the industry with its microprocessors which can perform
several tasks within quick time, several developments in the field of vibration is emerging in the
study of vibrations, the various factors involved, the various types of vibrations, the
characteristics of those vibrations, the advantages and the disadvantages, the mode of
transmission of the vibration, damped and un-damped vibrations, free and forced vibrations as
such many concepts are developed and still under research.
Methodology
Equipment used
The equipment used in this study to find the frequencies of the Multi degree of freedom forced
vibration is the Tower rig which is show in the figure, the Tower rig consists of a structure with
four floors with equal height difference between them, and the entire structure is isolated from
the surface by mounting it on springs which supports the rig, the springs also places the entire
structure under forced vibration at different frequency. To provide the rig system with the
required frequency an amplifier is used. The oscillating force required for the multi degree
freedom system test rig is provided by an electromagnetic shaker, an electromagnetic shaker
generates dynamic forces in order to achieve structural excitation by the help of permanent
magnets and electrical energy. In order to measure the acceleration of the structure under each
floor, accelerometers are mounted on each floors in such a way that it can be viewed on each
floors. The dimensions of the test rig structure is 25 x 150 x 200 mm (millimeters) The material
used to build the rig is mild steel and the mass density of mild steel is 7850 kg/m3.
The regulating state of system is-
m ¨x +c ˙x+ kx=F
Where-
F = external force acting on system
m = mass of system

k = spring stiffness
c = Damping coefficient
Figure: The frequency measurement equipment for multi-degree of freedom system
Experimental method
Initially the preliminary setup and tests were conducted, measuring the accuracy of the
accelerometers, amplifiers and the electromagnetic shaker are done and the required adjustments
are done on the systems, the electromagnetic shaker is turned on and maintained at required level
the multi degree of freedom test rig system which is mounted on four springs is set into vibration
through. The system consists of four springs, four masses and four dampers. The signal generator
is turned on and tuned to capture the oscillation that is being produced on the structure the

oscillation was observed to be in the range of 1 to 30 hertz (Hz). The acceleration reading that
shown on the accelerometers are noted.
The main goal of the experiment is to determine the input forces and the subsequent
output accelerations which would assist in obtaining the frequency of the system.
The frequency can be found using the matrix iteration method, and the mode shapes are
found using the Holzer method.
Results
After successfully conducting the experiment the final results are obtained by following
calculations to obtain the frequencies of the system and the model shape. There are some
assumption which are made initially to carry out the calculation they are, the springs are linear,
the masses which are used for the experiment are rigid masses, there is no friction between the
parts that is the frictional losses are negligible, there is no dampening of the structure, the system
has four degrees of freedom.
Methods such as Newtonian and holzer’s methods are used to measure the mode shapes
and the natural frequencies of the multiple degree of freedom system. The structure is considered
to be as a cantilever beam and the problems is solved for the cantilever beam structure with the
formulation and basic governing equations which are used for the cantilever beams, the masses
and the deflection along the structure is shown in the figure.

Multi-Degree of Freedom System converted as cantilever beam
M 1
d2 y1
d t 2 + ( k1 + k2 ) y1−k2 y2 =0
M 1
d2 y1
d t2 +k1 y1+ k2 y1 −k2 y2=0
Now as the cantilever deflection is as follows-
Deflection, δ= w l2
12 EI Thus,
This equation of motion can be solved using Newton’s method as-
−ℵ [ M ] { y }+ [ K ] { y } =0
−ℵ [ M 1 0 0
0 M 2 0
0 0 M 3 ][ y 1
y 2
y 3 ]+
[k1 +k2 −k2 0
−k2 k2+k3 −k3
0 −k3 k3 ] [ y 1
y 2
y 3 ]=0
[ k1 +k2 −M 1 ℵ −k2 0
−k2 k2 +k 3−M 2 ℵ −k3
0 −k 3 k 3−M 3 ℵ ] [ y 1
y 2
y 3 ] =0
Solving this gives-
ℵ = k1
2 M 1
The value of lander is found using the trial methods
The natural frequencies will be found while plotting the graph which is shown below,

It is clear that the values w1, w2, w3, w4 shows the natural frequencies of the system
The conversion of the frequency to hertz can be accomplished by using the following equation,
Hz = √ wn
2 Π (hertz)
The matrix iteration can also be used to find the natural frequencies and the mode shape with the
characteristic equation obtained at the step 1 and the stiffness form from the previous equation
we get,
1
ℵ { y }= [ K ] [ M ] { y }
[ 0.4 0.4 0.4
0.4 0.4 0.4
0.4 0.4 0.4 ]∗
[ 3.1 0 0
0 3.1 0
0 0 3.1 ] + [ 0.5
−0.5
0.5 ] =
[ 0.32
0.33
0.32 ][ 0.49
0.5
0.5 ]
The lambda value can be obtained by multiplying the inverse of the stiffness matrix with the
stiffness matrix, which gives the mode shape as well as the natural frequency for the system.

Holzer’s method
The holzer’s method is used to find the mode shape of the multi degree of
freedom forced vibration structure after finding the natural frequencies of the
system the method is similar to frequency scanning by giving different input values
to the lambda we can find the required variables and the mode shape for the multi
degree of freedom forced vibration system is given in the figure.
Free body diagram of holzer system
M d2 x
d t 2 +Kx=0

Figure: Mode shapes for multi degree of freedom system
Discussion
We can clearly see that the vibration of the multi degree of freedom system is completely
due to the electromagnetic shaker which when shakes the whole system, the experiment is
carried out thus it is very much important to make sure that there are no problems in the shaker
which is used for the experiment it is suggested to use a new set of equipment for the experiment
and also to calibrate each and every components such as accelerometer, the signal generator,
masses, springs and the mounting every time before the experiment is carried out and to separate
the system from the surroundings, we can obtain more accurate results when the surroundings
are maintained at optimal level and the equipment used are of greater precision and accuracy.
Going for high accuracy equipment are greatly suggested for the perfect outcome of the
experiment.
Conclusion
In this system the natural frequencies and the modal shapes of the multi degree of
freedom system are successfully found by using the methods of matrix iteration method, holzer’s
method and the Newtonian method, with proper experimental setup which consisted of, the
Tower rig consisting a structure with four floors with equal height difference between them, the
entire structure was isolated from the surface by mounting it on springs which supports the rig.
The oscillating force required for the multi degree freedom system test rig is provided by an
electromagnetic shaker, an electromagnetic shaker generates dynamic forces in order to achieve
structural excitation by the help of permanent magnets and electrical energy. In order to measure
the acceleration of the structure under each floor, accelerometers are mounted on each floors in
such a way that it can be viewed on each floors. The frequency of the multi degree of freedom
forced vibration system is measured using amplifiers, thus then the experiment was conducted by
turning on the electromagnetic shaker which when shakes the entire structure, the oscillation of
the structure was found by the amplifiers and the acceleration at each floor is found using
accelerometers which were mounted on each floors. These results which are all obtained assisted
in finding the natural frequencies of the system and the modal shape.