UNIVERSITY OF TABUK

FACULTY OF ENGINEERING
MECHANICAL ENGINEERING Lab
DEPARTMENT Manual
Practical Experiment Instructions
Sheet
Exp. Title Study of free and forced vibrations EXP. NO. 2
Mechanical Fall, 2017
Vibration Lab. Date: / /2017

Name: _________ I.D.#:________ Section#:____ Group #: __

Objectives:

1. Determine the resonance frequency with inactive Dynamic absorber and

to compare this with theoretical values.

2. Determine the damping ratio of the system.

3. To observe the phase shift of the system with inactive dynamic absorber.

Requirement :

Vibration frame, Springs, damping, mass, exciter motor, dynamic absorber

Description apparatus:

The module vibration frame consists of a number of rigid vertical and horizontal members each having
profiled grooves running along their length. Within these grooves are fitted special profiled nuts, which allow all the
component parts of the module to be fitted into position. The core components of the module are (figure 1):
1. Rigid steel beam
2. Beam pivot
3. Motor exciter
4. Spring
5. Vertical support
6. Damping tank
7. Damper/mass support

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 LVDT Sensor Motor Exciter
sENSER Spring

Rigid Steel Beam

Damping Tank

Figure 1.

Rigid beam:
The rigid steel beam has the following dimensions and mass:
Dimensions (nominal) : 25.4*12*840 mm3, Weight=2.101 Kg.
The beam fits into a beam pivot arrangement that is attached to the inner vertical member of the frame. The beam pivot
restricts the movement of the beam to One-Degree-of-Freedom (1DOF), in the vertical plane of the vibration Frame.
Spring:
As the free end of the rigid steel beam is suspended as spring, the spring is attached to the beam using a special clamp.
The clamp can be adjusted so that the spring can be moved along the beam and fixed into position. The spring dimensions
as follows: Outside diameter= 31.75 mm; Wire diameter= 3.25mm; Free length=152.4 mm.
Damper:
This comprises a rigid stem that is attached to the rigid steel beam via the stem holder. The location along the beam can
be adjusted using the simple clamping mechanism of the stem holder.
Motor Exciter:
The motor exciter is rigidly attached to the beam and can be adjusted to be positioned anywhere along the beams main
length. The motor exciter consists of a horizontal motor onto which is attached an 18-tooth pulley wheel. The component
suspended below the motor is a bearing block, which contains a ground steel shaft and two precision bearings. Attached
onto the shaft at either side of the bearing block are two to72 tooth aluminum pulley wheels. A pulley belt runs between
the 18-tooth and 72-tooth pulley wheels. This gives a gear ratio of 1:4. Every revolution of the motor, the eccentric mass
revolves 0.25 turns.
Attached to each of the 72-tooth pulley wheels are two eccentric masses. The eccentricity produced coming from the
fact that a hole has been introduced into each mass. As the eccentric masses rotate they create a force on the beam, which
increases with rotation frequency. The force is directed downwards onto the beam when the hole is at top dead centre.
This force is the disturbing force for the rigid beam and you will see that as the masses, that rotate the beam, are disturbed
vertically up and down.

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The speed of rotation of the eccentric masses is controlled by the speed controller unit, which itself is powered from the
tachometer unit. The tachometer unit is powered using 240 Vac/50 Hz single phase live, neutral and earth. The
tachometer then supplies 24 Vdc to the speed controller, which in turn supplies 24 Vdc to the motor.
The frequency of rotation of the eccentric masses is observed from the digital speed display. It is displayed in units of
Hz. A proximity sensor attached to the back of the motor exciter is used to generate the signal for the tachometer. As the
hole in the rear eccentric mass passes the proximity sensor a signal pulse is generated and fed back to the tachometer.
The signal pulse is the shape of a square wave.
The position of the motor exciter can be adjusted along the length of the beam by means of slackening the clamping
screws and clamps, which surround the beam. The overall mass of the motor exciter including cables, is: 5.105 Kg.
Tachometer:
The tachometer unit displays the excitation frequency of the input force to the beam and powers the speed controller
and has an output for the proximity sensor.
When the tachometer unit is powered up, the speed controller unit can also function.
Tachometer Operation:
The tachometer has a power to switch on its front panel. In its ‘0’ position the unit is off and in its ‘1’ position the unit
is on. The front panel also has a tachometer display, which displays the excitation frequency of the input force to the
beam with 2 decimal place resolution.
On the rear panel of the tachometer are the following.
1. Power lead input socket.
2. Circuit breaker (RCCB). If this trips, then, the unit is either faulty or an external fault has occurred. The trip
should always be in the ‘ON’ position in order for the unit to operate.
3. Speed controller output. This socket allows the plug from the speed controller unit to be inserted and carries the
24 Vdc supply voltage to the speed controller for operating the motor exciter.
4. Tachometer sensor output. The proximity sensor from the motor exciter is plugged into this socket so that the
tachometer can receive its signal.
5. LVDT out socket. This allows the signal from the LVDT to be passed onto an oscilloscope or another signal
monitoring device.
6. The socket is a BNC style as used on oscilloscopes.
7. Trigger out. This socket again has a BNC socket and is used to pass the proximity sensor signal from the LVDT
to an oscilloscope for triggering.
8. All other sockets are not connected.
Speed Controller:
The speed controller is powered from the tachometer when connected to this unit. It has a rotary potentiometer in its
front panel which when turned increases the speed of the motor exciter. This potentiometer has fine control and you will
notice that for some of the early rotations at each end the travel the motor will not rotate. This is just a function of the
potentiometer and controller combination. Once the motor starts you will notice how fine the control is. The rear panel
of the speed controller has the input socket for the motor exciter to connect to.

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Speed Controller operation:
The rear panel of the speed controller has a wire protruding from it. This should be connected to the speed controller
socket on the rear panel of the tachometer. The remaining socket (labeled ‘Motor In 24V dc’) has the plug from the
motor exciter fitted. This then allows the motor to be powered and operated. The front panel has a multi-turn rotary
potentiometer fitted. When the potentiometer is rotated clockwise the motor exciter will increase in speed and when
rotated anti-clockwise the motor exciter will reduce in speed until it stops. The speed controller will not function unless
the tachometer is powered, turned on and connected to the speed controller.You will notice that when the potentiometer
is rotated clockwise the motor exciter does not start to rotate for a short period. This is a totally normal and a continual
slow rotation of the pot in clockwise direction that will eventually start the motor exciter.
LVDT sensor:
The LVDT and the tachometer are not supplied as standards with the vibration frame

Theory:
Free vibration:
The beam is assumed to be rigid, so that it may considered as an SDof system. Its deflection from the horizontal
position of equilibrium may be expressed either by the angle of deflection from the horizontal, or by the vertical
displacement of the tip, as shown in figure 2.

Figure 2

The solutions for free and forced vibration are independent of which of the two degrees of freedom is selected.
We begin with the solution based upon the Principle of Conservation of Energy. Let the angular displacement be
represented by  we know that the SDof system will have one natural frequency, so that we may assume that  takes
d
the form   A sin( ) . The kinetic energy of a body having of inertia I about an axis and angular speed about that
dt
axis is given by :
 d 
2

I 
KE  
dt 
(1)
2
The maximum kinetic energy of the beam is therefore given by :

I  A 
2

KEMAX  (2)
2

4
[Show that the moment of inertia of a uniform rigid beam of length L and mass per unit length m around an axis at one
m L3
end is given by I  ].
3
The total moment of inertia of the system is the sum of that of the beam plus that of the exciter mass Mv. If the center
of mass of the exciter system is located at a distance  from the axis of beam rotation, its moment of inertia is given
by M v  2 .
The potential energy is stored in the spring the end of which undergoes a vertical displacement x, which equals L 
for angular displacements very much less that 1 radian. Hence,we can use Eq.2 to write :
K  LA
2

PEMAX  (3)
2
Equating the maximum kinetic and potential energies gives the natural frequency of free vibration as :
K L2
0  (4)
 M v L2 
 3   Mv 
2

 
Where the total mass of the beam M b  m L .
The alternative form of solution is based upon N2LM which states that the rotational moment of inertia of a body
times its angular acceleration equals the moment acting on the body. It can be written as:
d 2
I2
 T  K L2 (5)
dt
Where T is the external moment applied to the rod, for which the free vibration equals zero.

Forced vibration of the Damping beam :

The equation of angular motion of the beam system excited by an applied moment and damped by a viscous damper
attached to the beam at a distance s from the axis rotation is :
d 2 d
I 2  T  K L2  C s 2 (6)
dt dt
Where C is the viscous damping coefficient. The moment applied by an external force F acting vertically at a
distance z from the axis of rotation is given by T=Fz. Comparison of this equation with the equation of forced
damping mass spring system reveals the following equivalences :
 x I M T F KL2 K C s2 C
Assuming that the system behaves linearly and the applied moment is simple harmonic in the form T  R e jt , the
R
solution is analogous to that for the damped mass-spring system. The normalized angular displacement is , the
KL2
Cs 2 K
damping ratio is give by   , and the rest of the equations are the same, with 0 
2 I 0 M
Note on the Excitation Force in this Experiment
In this experiment, the excitation force F on the beam is proportional to  2 because the rotating discs each contain a
cut-out hole that is offset from the axis of rotation, the force is not independent of frequency as was assumed in the
foregoing analysis. Therefore, it will be necessary to normalize the measured response values to account for this
factor. The force applied to the beam by the exciter is given by :
F  2 2M hr (7)

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Where r is the radial distance from the axis of rotation to the center of the cut-outs and Mh is the missing of each
hole.

Measurement of system damping (damping ratio):

The theory is too lengthy to be included here, but the result shows that the amplitude of free vibration following an
initial displacement from equilibrium decays at a rate given by:
Ae w0t (8)

Where t is the time elapsed since the cessation of excitation, provided that the damping is less than critical. The
2 2n
period of free vibration is equal to . Over the duration n periods, the fractional in amplitude is given by e ,
0
which can be determined from a storage oscilloscope trace.
Thus, fractional reduction =
dn
 e 2 n (9)
d0
Where :
dn= amplitude after n complete cycles
d0= amplitude of 1st cycle
n= number of cylces
 = damping ratio

Experimental procedure:

A. Experiment 1 : Determination of the Resonance Frequency

The objective of this experiment is to determine through experimental running, the natural frequency of the damping
forced vibrations and compare it with the theoretical values.
1. Using the potentiometer on the front of the speed controller, we vary the exciter speed (excitation frequency)
slowly through its available range and observe the variation of the amplitude of the beam response. A clear
increase in response (resonance) should be observed within a small speed range. Also, by overlaying the
proximity sensor and beam response traces check that the phase between the two signals varies rapidly with the
frequency in this range. By viewing the approximate frequency at which resonance occurs, it is possible to fine
tune the range of frequencies over which it is useful to take results so that you are not taking results with minimal
output.
2. Increase the speed range by suitable increments, we say 0.1 Hz (6 rpm). Allow the vibration to stabilize after
each speed change. At each speed, we record the tachometer frequency (Hz), and the peak-to-peak the amplitude
of the response signal. We record these values in table 1.

Table1. Amplitude of the beam response

Excitation Beam Response ( Peak to Peak), mv
Frequency Normalised
(Hz) (inactive DA) (inactive DA)

The peak to peak output can be seen below. It is the complete amplitude of the LVDT signal and will be given in mV.

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 1. As indicated in the theory section in the manual ‘Theory of the beam vibration experiment’, the inertia force
applied to the beam by the exciter is proportional to the square of frequency. Therefore, divide the peak to
peak response amplitudes by the square of the excitation frequencies for the listed values. Complete the last
column in table 1 (normalized).
2. Plot the corrected (normalized) beam response amplitude (y-axis) against the excitation frequency for the
listed values.
3. Plot the corrected (normalized) beam response amplitude (y-axis) against the excitation frequency (x-axis) in
Hz [If tachometer display in rpm then conversion is tachometer rate (rpm)/60]
4. Identify the resonance frequency from the response peak.

B. Experiment 2 : phase shift

The objective of this experiment is to observe the phase shift of the system.
1. Keep the system set up as in part above.
2. Having viewed the system response from part of the experiment you know whereabouts resonance occurs for
the apparatus set up from part.
3. While doing this monitor, the position of a leading edge of the proximity sensor signal with the peak of the
closet beam response (LVDT signal), refer to diagram above. Continue to monitor this pairing while going
through resonance and out of resonance. You should see a change (a phase change).
4. Bring the back to the lowest excitation frequency. Record in table 2 the excitation frequency (column 1) and
the time difference (columne2) between the leading edge of the proximity sensor and the peak of the beam
response.
5. Continue recording while increasing the excitation frequency through the range of frequencies used in part 1.
Record all the results in table 2.
Table 2. Phase shift of the exciter motor.
Excitation t (ms) t (s ) Periodic time, Phase Phase
T (S) Angle Angle
Frequency,  Refer to
(Rads) (deg)
(Hz) diagram

1. The image below illustrates how the phase of the response relative to the force may be estimated from the overlaid
traces of the proximity sensor and LVDT signals.

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 t T

Proximity Sensor LVDT

Signal signal

Figure 3. Diagram of the phase shift.

From the diagram:

t 
   (2 ( ))  ( ) (10)
T 2
Where:
 Phase angle (degrees or radians)
t = time difference between proximity sensor signal and LVDT signal (Seconds)
T=Periodic time of excitation (Hz).
As a guide, the phase difference between the response and the tachometer signals should be close to zero at frequencies
well below the resonance frequency, close to -90 ° at resonance and close to – 180 above the resonance frequency. These
may be reversed if the positive sense of the LVDT and the excitation force are not the same. In this case, a correction of
180° is necessary.
From table 2 complete the remaining columns as follows:
 Plot a graph of phase angle in degrees (y-axis) against excitation frequency (x-axis).
 Compare the excitation frequency relating to 90 degrees phase angle with the resonance frequency obtain from
part1. How close do they match? What does this tell you?
C. Experiment 3 : Damping ratio

The objective of this experiment is to determine the damping ratio of the system without from phase curve and from
phase curve:
Without phase curve:
Using the monitoring system chosen, capture and store the beam displacement by displacing the tip of the beam by a
few millimeters and releasing it. You may find that a number of attempts are required to ensure that the displacement
captured and stored is sufficient to take readings. Try to achieve upwards of 10 full cycles. Determine the damping
ration of the system.
From phase curve:
Using the following equations:

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 2 
tan   (11) ,
(1  2 )

(1  2 )
  tan  (12)
2
Where :
 = phase angle (radians)
  damping ratio


 (13)
 0

 = Excitation frequency, Hz
0 = resonant frequency, Hz

1. Choose an excitation frequency from table 2a and enter this into column 1 of table 3
2. Calculate column 3.
3. For the value of excitation frequency used in step 7 above, read off the associated value of phase angle  from
table 4 and write this into column 4 of table 2.
4. Calculate column 5 in table 3, remembering that the value of  should be in radians.
5. Using column 6, calculate the product of the product of column 3 and column 5. This will give you the damping
ratio.
6. Repeat for other values of excitation frequency from table 2.
7. Compare these values with the calculated value obtained by experiment.

Table 3: Damping ratio

Excitation  (1   2 ) Phase tan  Damping

Frequency, 0 2 angle,  (rads) ratio 
 (Hz) (rads)

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 Beam Response ( Peak to
Excitatio Peak), mv
n Normalise
Frequenc d
y (Hz) (inactive (inactive
DA) DA)
3.1
3.2
3.3
3.4
3.5

3.6
3.7
3.8
3.9

4
4.1
4.2
4.3
4.4
4.5

4.6

4.7

4.75

4.8

4.85

4.9

5.05

5.1

5.15

5.2

5.25

5.3

5.4

5.5

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Excitation t (ms) t (s ) Periodic Phase Phase
time, T (S) Angle Angle
Frequency,  Refer to
(Rads) (deg)
(Hz) diagram

3.1
3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.1
4.2

4.3

4.4

4.5
4.6

4.7

4.75
4.8

4.85

4.9
5

5.05

5.1

5.15

5.2

5.25

5.3

5.4

5.5

11