University of Basrah
College of Engineering
Mechanical Engineering Department
Vibration Laboratory
Experiments of
Mechanical
Vibration
Laboratory
Prepared by
Mr. Jaafar Khalaf Ali
Mr. Ali Hassan Abdelali
2007-2008
Copyrights © 2008 College of Engineering-University of Basrah
Mechanical Vibration Laboratory
Introduction
This booklet is dedicated for those student having mechanical vibration courses in their
studies including, but not limited to, students of the Fourth stage in the Department of
Mechanical Engineering. It contains several experiments to help in understanding and
testing some vibration applications starting from the simplest oscillatory motion
represented by the simple pendulum, moving through mass-spring system, torsional
undamped and damped vibration, forced vibration, two-degree of freedom system and
finally whirling of shafts and Dunkerley's Equation.
Based on the guides and catalogues provided by the TecQuipment (TQ) Company,
manufacturer of experimental devices, and also some other theoretical references, the
provided experiments were prepared carefully to ensure simplicity and avoid confusion.
Some misprints in the equations mentioned in TQ guides were avoided by returning to
textbooks and derivation of these equations from basic concepts.
Regards
Preparation Staff
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Mechanical Vibration Laboratory
Experiment No. 1 Simple Pendulum
Aim of the experiment
1. Validation of simple pendulum theory.
2. Estimation of gravitational acceleration, g.
1. Introduction
A pendulum is an object that is attached to a pivot point so it can swing freely. This object is
subject to a restoring force that will accelerate it toward an equilibrium position. When the
pendulum is displaced from its place of rest, the restoring force will cause the pendulum to
oscillate about the equilibrium position. In other words, a weight attached to a string swings
back and forth.
A basic example is the simple gravity pendulum or bob pendulum. This is a weight (or bob) on
the end of a massless string, which, when given an initial push, will swing back and forth under
the influence of gravity over its central (lowest) point.
The regular motion of pendulums can be used for time keeping, and pendulums are used to
regulate pendulum clocks. A simple pendulum is an ideality involving these two assumptions:
•
•
The rod/string/cable on which the bob is swinging is massless and always remains taut.
Motion occurs in a plane.
2. Theory of Simple Pendulum
Under the above assumptions, the equation of motion of
simple pendulum can be written as (see Figure 1):
Iθ + mg sinθ l = 0 assuming small angle θ ,
ml 2θ + mg θ l = 0
θ
l
(1)
lθ + g θ = 0
mg sin θ
Where l : the length of string in meter
θ : swing angle in radian
I : second moment of inertia about pivot in kg.m2
From the equation of motion, one can find the natural
frequency as follows:
ωn =
2π
τ
=
τ = 2π
g
l
mg
Figure-1 Simple Pendulum
or in other words;
g
l
, hence
(2)
l
g
2
Mechanical Vibration Laboratory
Where ωn is the natural frequency in rad/sec and τ is the time of one cycle (period) in seconds.
From the above equations, it is clear that the natural frequency is a function of the string length
and does not depend on the mass of the pendulum. From eqs (2), one can find g as follows:
 l 
g = 4π 2  2 
τ 
(3)
3. Apparatus and Tools
Pendulum Accessory B1
The apparatus and tools used in this
experiment can be listed as follows:
•
•
•
•
•
Universal vibration rig
Pendulum accessory, on which
simple pendulums are attached.
Two balls, steel and wooden,
attached to flexible strings of
variable lengths to form the
simple pendulums.
Measuring tape to measure the
length of string.
Stop watch to measure the time
required by a number of complete
oscillations.
Wooden ball
Steel ball
Universal Vibration Apparatus
Figure 2 Apparatus used in the experiment
4. Procedure of the Experiment
1) Set the string length for both balls at 30 cm by using the measuring tape, the length is
measured from the end of fixing nut to the center of the ball. Displace both balls to the
same level and release them at the same time. Record your notices about the phase and
duration of oscillation for both balls.
2) Consider now the steel ball, change the length of string to a number of values 15, 22, 30,
37, 42, 50 and 56 cm and measure the time required by complete 30 oscillations.
Evaluate the time of one period by dividing the measured time by 30.
3) Repeat the procedure in point (2) to the wooden ball and record the results as in table 1.
2
4) Evaluate g for both balls from eq. (3) using the best value of l/τ which can be found by
linear curve fitting.
String
length (m)
Time of 30 complete oscillations
Steel
Wooden
Periodic time τ (sec)
Steel
Wooden
15
22
30
.
.
Table-1 Experimental Readings
5. Discussion
1) Compare the values of g for both balls and state the reasons of difference.
2) Discuss the sources of inaccuracies and state how we can reduce errors in the experiment.
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Mechanical Vibration Laboratory
Experiment No. 2 Mass-Spring Systems
Aim of the experiment
1. Verification of simple mass-spring system theory
2. Estimation of stiffness factor k for a spring
3. Estimation of gravitational acceleration g
1. Introduction
Helical or coil springs are commonly used in wide variety of mechanical systems. Their basic
work is to produce a force which is proportional to the deflection or vise versa. Figure-1 shows a
typical force-deflection diagram for a helical spring. In the linear region of this diagram, the
relation between force and deflection obeys Hook's Law:
∆F = k ∆x
Where k is called stiffness of the spring (N/m). The
reciprocal of k is called deflection coefficient which x
is the deflection introduced by a unit force. If a mass
is attached to one end of a spring while the other
end is fixed, the resulting system is called simple
mass-spring which oscillates harmonically
according the following equation (neglecting all
types of damping forces);
(1)
∆x
∆F
Force
Figure-1 Force-Deflection Plot
m x + k x = 0
(2)
Where m is the mass in kg. The natural frequency in rad/sec and the periodic time of oscillation
are given by;
k
m
ωn =
⇒ τ = 2π
(3)
m
k
2. Apparatus
C1
Figure-2 shows the required set-up for the
experiment. The main frame is the universal
vibration apparatus. Suspend any one end of the
spring supplied from the upper adjustable
assembly (C1) and clamp to the top member of
the portal frame. To the lower end of the spring
is bolted a rod and integral platform (C3) onto
which masses of 0.4 kg each can be added. The
rod passes through a brass guide bush, fixed to
an adjustable plate (C2), which attaches to the
lower member. A depth gauge is supplied,
which can be used to measure deflection when
applying masses or force to the spring.
4
Spring
Slide
gauge
C3
C2
Figure-2 Apparatus of the experiment
Mechanical Vibration Laboratory
3. Procedure of the Experiment
Part A: Static Deflection
1) Carefully adjust the bush guide to ensure it is directly below the top anchorage point to
reduce the friction. Friction can be minimized by applying grease or oil to the bush.
2) Adjust the depth gauge so it can measure full stroke of the spring by pulling its wire
adequately. Record the initial reading as the reference reading.
3) Add weights in incremental fashion with increment 400g and record the corresponding
deflection ∆x . Record the results in a table as follows;
Mass (gr)
Deflection (mm)
0.0
0.0
800
1200
1600
…..
3600
4000
Now, from eq. (1) , one can write
x
(4)
m
From the constructed table, we can obtain a relation between g and k using the mean values
of deflection and mass;
x
g=k
(5)
m
Part B: Oscillatory Motion
mg = k x ⇒ g = k
1) Record the mass of the spring and the mass of the platform C3.
2) With only platform C3 attached to the spring, pull down the compound and release it to
introduce oscillatory motion. Measure the time required by 20 oscillations.
3) Add masses incrementally keeping in mind that the mass of the platform C3 should be
considered, and measure the time of 20 oscillations. Record your readings as shown in
the following table;
Mass (gr)
1585
2385
-
Time of 30 Oscillations
τ
τ2
From eq. (3) , one can set the following relation ;
m
k = 4π 2 2
τ
(6)
By linear curve fitting between the mass m and τ2, find the best value of the slope and,
hence, the value of k. Returning back to eq. (5), find the value of g. Find also the value of the
effective mass for the spring (theoretically one third of the spring mass) from the intercept of
the best line with m-axis.
4. Discussion
1) Discuss the sources of inaccuracies for both parts of the experiment and state how we
can reduce errors.
2) Compare the value of effective mass with the theoretical one stating the reasons of
difference if exists.
3) Mention 4 typical examples for the usage of springs describing their importance.
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Mechanical Vibration Laboratory
Exp. No. 3 Torsional Vibration
Aim of the experiment
1. Estimation of the moment of inertia for a wheel.
2. Estimation of the damping coefficient.
1. Introduction
Twisting or torsional springs are commonly used in the industry to produce moment against
angular displacement. One the most important applications of twisting springs is in the
suspension system of cars. The equation of motion for a wheel attached to the free end of a
twisting spring, as shown in Figure-1, is given by:
I θ + K θ = 0
(1)
Where I s the moment of inertia for the wheel (kg.m2)
θ is the angular displacement
K is the rotational flexibility factor
GJ
(2)
L
Where G is the modulus of rigidity (shear modulus) of the shaft
material in N/m2 (80 GN/m2 for steel)
J is the polar moment of cross-sectional area for the shaft in m4
L is the effective length of the shaft in meter.
From the above, one can find the natural frequency as follows;
K =
ωn =
GJ
IL
which leads to τ = 2π
IL
GJ
L
(3)
Considering the damping effect on the rotational vibration, the
equation of motion will be written as;
θ
Figure-1 Torsional Spring
I θ + C θ + K θ = 0
(4)
Where C is the rotational damping factor (N.m.sec). Introducing the critical damping factor Cc
which is given by 2Iωn, the ratio of damping factor to the critical one is given the damping
coefficient ζ ;
ζ =
C
Cc
(5)
The damping coefficient can experimentally be estimated by measuring the logarithmic
decrement δ.
2. Apparatus
The main apparatus of the experiment is the universal vibration rig as shown in Figure-2. For
the part of undamped torsional vibration, the inertia is provided by a heavy wheel of 254mm
diameter, marked as H2. To the wheel is attached a chuck designed to accept shafts of different
diameter. A sliding block, I1, carries another chuck identical to the one attached to the wheel.
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Mechanical Vibration Laboratory
This block can be moved along a guide to change the effective length of the shaft which passes
through both chucks to produce the
K1
rotational flexibility.
For the part of damped vibration,
there is a vertical shaft gripped at its
upper end by a chuck attached to a
bracket K1, while its lower end is
attached to a heavy wheel K3 with
conical lower end. There is a
transparent container under the wheel
containing damping oil denoted as K4.
This container can be lowered and
raised by means of a knob, allowing
the contact area between the oil and
conical section of the wheel to vary.
This variation will reflect variable
damping effect on the system. The
oscillation can be traced by warping a
paper around the drum located above
the wheel by means of a pen which is
attached to holding arm. The later is
allowed to move downward slowly by
a means of dashpot fixed to the frame,
K2.
H2
I1
K2
K3
K4
Figure-2 Apparatus of the experiment
3. Procedure of the Experiment
Part A: Undamped Vibration
1. Pass the shaft through the bracket center hole so that it enters the chuck on the wheel and
then tighten it.
2. Move the bracket along slotted base until the distance between the jaws of the chuck
corresponds to the required effective length L. tighten the chuck on the bracket.
3. Displace the wheel angularly and measure the time of 20 complete oscillations.
4. Repeat the process for different values of L and record the readings as in Table-1.
5. By finding the best value of τ2 /L using curve fitting, the value of moment of inertia can
be evaluated as follows:
I =
GJ τ 2
4π 2 L
No.
1
2
3
4
5
6
(6)
L (cm)
10
20
25
30
35
45
Time of 20 Oscillations
Period τ
Table-1 Readings for Part A
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τ2
Mechanical Vibration Laboratory
Part B: Damped Vibration
1. Fill the container K4 with oil so that the level is 10cm from the top. Fix a graph paper on
the specified drum.
2. Adjust the knob so that the conical section of the wheel is dipped in the oil and apply
oscillatory motion to the wheel. Plot the damped motion by letting the pen fall down, the
plot looks like a decremented sine wave. Measure two different peaks on the plot,
denoted by x0 and xn , separated by n complete oscillations, ,
3. Repeat the procedure for different levels of dipping. You can record your reading as in
the following table;
No.
x0
xn
n
4. For each case, evaluate the logarithmic decrement from the following equation;
x
1
ln 0
n
xn
Hence, damping coefficient ζ can be found from the following identity;
δ=
δ=
2πζ
(7)
(8)
1− ζ 2
4. Discussion
1) Discuss the sources of inaccuracies for both parts of the experiment.
2) Mention another method to estimate the moment of inertia for a wheel and compare with
the method of this experiment.
3) Give some practical applications for the torsional vibration.
4) Discuss the effect of increasing oil viscosity on the damping factor.
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Mechanical Vibration Laboratory
Exp. No. 4 Forced Vibration with Negligible Damping
Aim of the Experiment:
1. Estimation of the natural frequency for a rigid body-spring system.
2. Verification of resonance condition.
1. Introduction
When external forces act on a vibrating system during its motion, it is termed Forced Vibration.
Under this condition, the system will tend to vibrate at its own natural frequency superimposed
upon the frequency of the exciting force. After a short time, the system will vibrate at the
frequency of the exciting force only, regardless of the initial conditions or natural frequency of
the system. The later case is termed steady state vibration. In fact, most of vibrational
phenomena present in life are categorized under forced vibration. When the excitation frequency
is very close to the natural frequency of the system, vibration amplitude will be very large and
damping will be necessary to maintain the amplitude at a certain level. The later case is called
"resonance" and it is very dangerous upon mechanical and structural parts. Thus, care must be
taken when designing a mechanical system by selecting proper natural frequency that is
sufficiently spaced from the exciting frequency.
2. Theory
Let's consider the system shown in
Figure-1, consisting of:
(1) A beam AB of length L and
mass m, freely pivoted at the
left end A and considered
sensibly rigid.
(2) A spring of stiffness k attached
to the beam at the point C.
(3) A motor with out-of-balance
disks attached to the beam at
point D, M is the mass of the
combined part (motor and
disks).
Figure-1 Forced Vibration
The equation of angular motion is given by:
IA
d 2θ
+ (k L 2θ )L 2 = ( F0 sin ωt ) L1
dt 2
(1)
Where:
θ : Angular displacement of the beam,
F0 : Maximum value of excitation force,
ω : Angular velocity of rotation for the disk,
IA : The moment of inertia of the system about point A;
mL2
I A ≈ ML +
3
2
1
Eq. (1) can be re-written as:
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(2)
Mechanical Vibration Laboratory
d 2θ
+ bθ = A sin ωt
dt 2
Where;
kL 2
b= 2 ,
IA
A=
(3)
F0 L1
IA
The steady state angular displacement is given by:
θ=
A
sin ωt
b − ω2
(4)
A
b − ω2
(5)
and the maximum amplitude is:
θ max =
i.e., resonance occurs when b -ω2 = 0, or in other words when ω = b .
Note that in practical circumstances, the amplitude may be very large but doesn't become
infinite due to small amount of damping that is always present in any system.
3. Apparatus of the Experiment
The apparatus for this experiment is
shown in Figure-2. It consists of a
rectangular beam D6, supported at
one end by a pin pivoted in ball
bearings which are located in a fixed
housing. The other end of the beam
is supported by a spring of known
stiffness bolted to the bracket C1
which is attached to the upper frame.
This bracket enables fine adjustment
of the spring, thus raising and
lowering the end of the beam.
The DC motor rigidly bolts to the
beam with additional masses placed
on the platform attached. Two outof-balance disks on the output shaft
of the belt-driven unit (D4) provide
the exciting force. The exciting
frequency can be adjusted by means
of the speed control unit. The safety
stop assembly (D5) limits the beam
movement for safety reasons.
Figure-2 Apparatus for the Experiment
The chart recorder (D7) fits to the right-hand vertical member of the frame and provides the
means of obtaining a trace for the vibration. The recorder unit consists of a slowly rotating drum
driven by a synchronous motor, operated from auxiliary supply on the speed control unit. A role
of recording paper is adjacent to the drum and is wound round the drum so that the paper is
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Mechanical Vibration Laboratory
driven at a constant speed. A felt-tipped pen fits to the free end of the beam; means are provided
so that the pen just touches the paper. By switching on the motor, we can obtain a trace showing
the oscillatory motion of the beam free end.
If the amplitude of vibration near to the resonance condition is too large, we can introduce
extra damping into the system by fitting the dashpot assembly (parts D2, D3 and D9) near to the
pivoted end of the beam.
4. Experimental Procedure and Calculations
1. First, plug the electrical lead from the synchronous motor into the auxiliary socket on the
exciter motor and speed control. Adjust the handwheel of bracket C1 so that the beam is
horizontal and bring the chart recorder into a position where the pen just touches the
recording paper.
2. Switch on the speed control unit and adjust the knob of speed so that the amplitude of
oscillation is large enough when the exciter motor mid-way between the spring and
pivot. Adjust the location of exciter to obtain largest amplitude.
3. Bring the pen into contact with the paper and record 30 cycles or more. Then measure
the length of the trace corresponding to 30 oscillations, d1.
4. Stop the exciter motor, then measure the speed of paper by measuring the length of the
trace corresponding to −for example− 20 seconds, d2. You can use stop watch for timing.
The speed of paper v = d2 / 20 (cm/sec).
5. Calculate the total time for 30 oscillation, T, by dividing d1 from step (3) by v from step
1 30
(4). The cyclic time is then t = T / 30, and the experimental frequency is f exp = =
t T
6. Record the distance of the exciter motor from the pivot, L1, the distance of the spring,
L2, and the length of the beam, L. Measure also the width and thickness of the beam to
calculate its mass, m, from the product of volume by density of the steel 7800 kg/m3.
b
Take M = 2.4 kg and k = 950 N/m. Evaluate the theoretical frequency f theo =
.
2π
5. Discussion
1) Compare between the theoretical and experimental frequencies obtained in the experiment
and state the reasons of difference if exist.
2) State the effect of resonance and how we can avoid it.
3) What are the factors affecting the natural frequency of a system?
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Mechanical Vibration Laboratory
Exp. No. 5 Two Degree of Freedom Torsional Vibration
Aim of the Experiment:
1. Estimation of the natural frequency for two rotor system.
2. Comparison of the theoretical and experimental frequencies.
1. Introduction
The degree of freedom of a system refers to the
number of vibrating objects or parts such that each part
has its own displacement. Consider the system shown
in Figure-1. Two wheels are connected by a shaft of
rotational stiffness K. The equations of motion for this
system can be written as:
θ2
θ1
K
I1
I2
Figure-1 Two Degree Rotor System
I 1θ1 + K (θ1 − θ 2 ) = 0
I θ + K (θ − θ ) = 0
2 2
2
(1)
(2)
1
Where θ1, θ2 are the angular displacements for wheels, I1, I2 are the moments of inertia and K is
GJ
the rotational stiffness, K =
, where L is the effective length of the shaft. The above two
L
equation may be written as:
K − ω 2 I 1
− K  θ1   0
(3)

  =  
K − ω 2 I 2  θ 2   0
 −K
ω can be found by equating the determinant of the system matrix to zero, where two values will
be obtained, either ω=0 or ω =
K (I1 + I 2 )
I 1I 2
, from which one can find the periodic time as
follows:
τ = 2π
LI 1I 2
GJ ( I 1 + I 2 )
(4)
2. Apparatus
The apparatus of this experiment is the universal
vibration rig used in experiment No. 3. It is shown
in Figure-2. With the bracket (I1) replaced by a
second wheel (H1) which is free to rotate on ball
bearing fixed to the left frame. Both wheels have
chucks fitted for use with shafts of different
diameters. It is not possible to vary the effective
length of the shaft; therefore a number of shafts of
different diameters are to be used. Other tools
required are measuring tape, stop watch and square
key used to release and tighten the chucks.
H1
H2
Figure-2 Apparatus of the experiment
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Mechanical Vibration Laboratory
3. Procedure of the Experiment
1. Use wheels of known moment of inertia. You can return to the results of experiment No.
3 to find the moment of inertia for the wheel H2, while the moment of inertia for H1 can
be found using the same procedure in experiment No. 3.
2. Measure the effective length of the shaft between the ends of the two chucks, measure
also shaft diameter and calculate the periodic time theoretically from eq. (4) above. Use
G= 80 GN/m2 and J =
π
d4.
32
3. Rotate each wheel through a small angle in opposite direction and then release. Measure
the time required for 20 complete oscillations using the stop watch. Calculate the
periodic time by dividing the total time by 20.
4. Replace the shaft by another one of different diameter and repeat steps 2 and 3. Arrange
your readings as follows;
Shaft diameter
(mm)
I1 (kg.m2)
I2 (kg.m2)
Time of 20
Oscillations
Exp. Period τ
Theo. Period
4. Discussion
1) Compare the values of theoretical and experimental periodic times for each rotor and
explain the reasons of difference if exist.
2) What does the first frequency ω1 = 0 means in physical sense?
3) Find the mode shapes for the system in the experiment and find the location of the node
theoretically. How can we find the node (non-moving section of the shaft)
experimentally?
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Mechanical Vibration Laboratory
Exp. No. 6 Whirling of Shafts
Aim of the Experiment:
1. Verification of Whirling theory.
2. Verification of Dunkerley's Equation.
1. Introduction
For any rotating shaft, a certain speed exists at which violent instability occurs. The shaft
suffers excessive deflection and bows, a phenomenon known as whirling. If this critical speed of
whirling is maintained (called First Critical speed), then the resulting amplitude becomes
sufficient to cause buckling and failure. However, if the speed is rapidly increased before such
effects occur, then the shaft is seen to re-stabilize and run true again until another specific speed
is encountered where a double bow is produced as shown in Figure-1. The second speed is
called "Second Critical".
Whirling speed depends primarily on the stiffness of the shaft and mass distribution (as will
be seen later). When the shaft is loaded, the whirling speed will be shifted due to the effect of
the new mass. Dunkerley set the equation that relates the overall whirling frequency with critical
frequencies introduced by the shaft and load individually. This equation is valid for any number
of loads.
Studying whirling of shaft is of great important due to
huge number of applications in various fields. For
example, all rotating machinery involve shafts with
rotating parts such as rotors in electrical motors,
impellers in pumps, blades in turbines ….etc. On the
other hand, Dunkerley's Equation is found to be useful
not only in studying whirling of loaded shafts, but also
in structural analysis and frequency response testing.
Mode 1
Mode 2
Figure-1 Modes of Whirling
2. Theory
The critical frequency for a shaft may be obtained from the fundamental frequency of a beam
subjected to a transverse vibration;
f =λ
EIg
wL4
(1)
Where
f : critical frequency in Hz
E : Young's modulus
I : Second moment of area of the shaft; I =
π
d4
64
w : Weight per unit length of the shaft
λ : Constant dependant upon the fixing conditions and mode and can be found from the
following table;
Type of support
Simply supported
Supported-Fixed
Fixed-Fixed
λ1 (first mode)
1.573
2.459
3.75
λ2 (second mode)
6.3
7.96
8.82
For a shaft loaded with a number of disks as shown in Figure-2, the first critical frequency for
the system can be found from Dunkerley's Equation as follows;
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Mechanical Vibration Laboratory
1
f
2
=
1
f s2
+
1
f 12
+
1
f
2
2
+
1
f 32
+ ......
(2)
Where
f : critical for the system as a whole
Disk 2
Disk 1
Disk 3
fs : critical speed of the shaft alone (first
critical calculated from eq. (1))
f1, f2, f3 : critical speeds due to attaching
disk 1, 2 and 3 individually without the
effect of other masses.
L
Figure-2 Shaft Loaded with Three Disks
3. Apparatus of the Experiment
This is TecQuipment TM1 Whirling of Shafts Apparatus shown in Figure-3. The shaft is
located in the kinematic coupling and either the fixed or free type end bearing. Several shafts of
various lengths and diameters are available.
Double Universal
Joint
Kinematic Coupling
Assembly
Nylon Bushes
(Guards)
Sliding End Support
Bearing
Motor
Frame
Figure-3 Whirling of Shafts Apparatus
The kinematic coupling and sliding end bearings have been designed to allow the shaft
movement in a longitudinal direction. The sliding end bearing is interchangeable to allow the
selection of support type between directionally fixed and free support. A movable part is
provided as a part of the kinematic coupling which allows the selection of support type. When
this part moved away from the coupling, the resulting support will be directionally free.
The shaft is driven by a DC motor capable of providing 6000 RPM through the kinematic
coupling which possesses double universal joint. The motor speed is controlled by TQ E3
control unit. In order to maintain the amplitude of vibration within specific limits, two nylon
guards are provided and are adjustable along the length of the apparatus. The sliding end bearing
may be moved to enable various shaft lengths to be selected.
In addition, four disks are supplied to providing loading to the shaft. These disks can be fitted to
the 7mm diameter shaft. Two of them are of equal mass at 300g, the third has a mass of 400g.
Additionally, a stroboscope is used to measure the rotational speed and also to observe the shaft
configuration during whirling. This stroboscope may be synchronized through a trigger signal
provided by TQ TM1 apparatus.
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Mechanical Vibration Laboratory
4. Experimental Procedure and Calculations
Part A: Whirling of shafts without loading
1. Attach a shaft of known diameter and length to the apparatus.
2. Select simply supported configuration by moving out the sliding part of the kinematic
coupling and using the free support at the other end. Calculate the theoretical first and
second whirling speeds from eq. (1). The density of shaft material is 8200 kg/m3, and
Young's modulus is 207 GN/m2.
3. Switch on the speed control unit and adjust the speed carefully until obtaining the largest
amplitude of whirling. Read the speed on the stroboscope and observe the shaft in the
first mode, it should contain a single bow. Increase the speed slowly until you obtain the
second mode and record the rotational speed. Observe the shaft in the second mode.
4. Change the support type to fixed-supported and then to fixed-fixed and repeat steps 2
and 3.
5. Replace the shaft with another one of different diameter and repeat the above steps.
Record the results as in a table as below:
No.
1
2
Shaft
Diam.
(mm)
3
7
Shaft
Length
(m)
Simply supported
Theo.
Exp.
Supported-Fixed
Theo.
Exp.
Fixed-Fixed
Theo.
Exp.
Part B: Whirling of loaded shafts
1. Use the 7mm shaft in simply-supported configuration. Attach the first disk of 400g midway between the two supports.
2. Switch on the speed control unit and adjust the speed carefully until you obtain whirling
condition. Record the whirling frequency of the system f.
3. Calculate the critical frequency for the first disk alone, f1 , from the following equation:
1
f
2
=
1
f
2
s
+
1
f 12
(3)
Where fs is the whirling frequency for the shaft alone in the simply-supported configuration
and can be taken from Part A.
4. Remove disk No. 1 and attach disk No. 2 (300g) at 0.25L from the motor-side support
and repeat the above procedure to calculate f2 for the second disk alone.
5. Attach disk No. 3 alone at 0.75L from the motor-side support and repeat the procedure to
calculate f3 for the third disk alone.
6. Attach all the three disks at the same positions and run the DC motor to find the critical
frequency for the combination. Verify that eq. (2) is satisfied. Arrange your reading as in
the table below:
No. Loading
System critical freq. f
(as measured)
16
Shaft critical freq. fs
Disk freq. fi
i =1, 2, 3
Mechanical Vibration Laboratory
1
2
3
4
Disk 1 alone
Disk 2 alone
Disk 3 alone
All disks
(From part A)
(From part A)
(From part A)
(From part A)
___
5. Discussion
1) Compare the values of theoretical and experimental frequencies for Part A and state the
reasons of differences if exist.
2) For Part B, compare the value of the observed critical frequency for the combined system
with that one calculated from eq. (2). Is Dunkerley's Equation satisfied?
3) Explain how we can avoid critical frequencies in the manufacturing of rotating
machinery.
17