Mech Vibration Lab

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

1

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


2

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):

0

0

0 sin

2

, angle small assuming

=+

=+

=+

θθ

θθ

θθ θ

gl

lmgml

lmgI

��

��

��

(1)

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:

g

l

l

g

l

gn

πτ

τ

π

ω

2

hence , 2

s;other wordin or

=

=

=

(2)

θ

mg sin θ

mg

l

Figure-1 Simple Pendulum


3

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:

=

2

24τ

πl

g (3)

3. Apparatus and Tools

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.

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.

4) Evaluate g for both balls from eq. (3) using the best value of l/τ2 which can be found by

linear curve fitting.

String

length (m)

Time of 30 complete oscillations Periodic time τ (sec)

Steel Wooden 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.

Steel ball Wooden ball

Universal Vibration Apparatus

Figure 2 Apparatus used in the experiment

Pendulum Accessory B1


4

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:

xkF ∆=∆ (1)

Where k is called stiffness of the spring (N/m). The

reciprocal of k is called deflection coefficient which

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);

0=+ xkxm �� (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

m

kn πτω 2=⇒= (3)

2. Apparatus

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.

Force

x

∆F

∆x

Figure-1 Force-Deflection Plot

C3

Spring

Slide

gauge

Figure-2 Apparatus of the experiment

C2

C1


5


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) 0.0 800 1200 1600 ….. 3600 4000

Deflection (mm) 0.0

Now, from eq. (1) , one can write

m

xkgxkgm =⇒= (4)

From the constructed table, we can obtain a relation between g and k using the mean values

of deflection and mass;

m

xkg = (5)

Part B: Oscillatory Motion

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) Time of 30 Oscillations τ τ2

1585

2385

-

-

-

-

From eq. (3) , one can set the following relation ;

2

24τ

πm

k = (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.


6

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:

0I Kθ θ+ =�� (1)

Where I s the moment of inertia for the wheel (kg.m2)

θ is the angular displacement

K is the rotational flexibility factor

GJK

L= (2)

Where G is the modulus of rigidity (shear modulus) of the shaft

material in N/m2 (80 GN/m

2 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;

n

GJ

I Lω = which leads to 2

I L

GJτ π= (3)

Considering the damping effect on the rotational vibration, the

equation of motion will be written as;

Figure-1 Torsional Spring

0I C Kθ θ θ+ + =�� (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 ;

c

C

Cζ = (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.

θ

L


7

This block can be moved along a guide to change the effective length of the shaft which passes

through both chucks to produce the

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.


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:

2

24

GJI

L

τ

π= (6)

No. L (cm) Time of 20 Oscillations Period τ τ2

1 10

2 20

3 25

4 30

5 35

6 45

Table-1 Readings for Part A

H2

I1

K1

K2

K4

K3



8

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;

01ln

n

x

n xδ = (7)

Hence, damping coefficient ζ can be found from the following identity;

2

2

1

πζδ

ζ=

− (8)

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.


9

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).

The equation of angular motion is given by:

( )2

2 2 0 12( ) sin

A

dI k L L F t L

dt

θθ ω+ = (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;

2

2

13

A

mLI ML≈ + (2)

Eq. (1) can be re-written as:

Figure-1 Forced Vibration


10

2

2sin

db A t

dt

θθ ω+ = (3)

Where; 2

0 12 ,A A

F LkLb A

I I= =

The steady state angular displacement is given by:

2

sinA

tb

θ ωω

=−

(4)

and the maximum amplitude is:

max 2

A

bθ

ω=

− (5)

i.e., resonance occurs when b -ω2 = 0, or in other words when bω = .

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 out-

of-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.

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

Figure-2 Apparatus for the Experiment


11

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

(4). The cyclic time is then t = T / 30, and the experimental frequency is exp

1 30f

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.

Take M = 2.4 kg and k = 950 N/m. Evaluate the theoretical frequency 2

theo

bf

π= .

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?


12

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:

( )

( )1 1 1 2

2 2 2 1

0 (1)

0 (2)

I K

I K

θ θ θ

θ θ θ

+ − =

+ − =

��

��

Where θ1, θ2 are the angular displacements for wheels, I1, I2 are the moments of inertia and K is

the rotational stiffness,GJ

KL

= , where L is the effective length of the shaft. The above two

equation may be written as:

211

222

0

0

K I K

K K I

θω

θω

− − =

− − (3)

ω can be found by equating the determinant of the system matrix to zero, where two values will

be obtained, either ω=0 or ( )1 2

1 2

K I I

I Iω

+= , from which one can find the periodic time as

follows:

( )

1 2

1 2

2LI I

GJ I Iτ π=

+ (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.

I2I1

θ1θ2

K

Figure-1 Two Degree Rotor System

I2I1

θ1θ2

K

Figure-1 Two Degree Rotor System

H2H1


H2H1



13


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 4

32J d

π= .

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.m

2) 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?


14

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.

2. Theory

The critical frequency for a shaft may be obtained from the fundamental frequency of a beam

subjected to a transverse vibration;

4

EIgf

wLλ= (1)

Where

f : critical frequency in Hz

E : Young's modulus

I : Second moment of area of the shaft; 4

64I d

π=

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 λ1 (first mode) λ2 (second mode)

Simply supported 1.573 6.3

Supported-Fixed 2.459 7.96

Fixed-Fixed 3.75 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;

Mode 1

Mode 2

Figure-1 Modes of Whirling

Mode 1

Mode 2

Figure-1 Modes of Whirling


15

2 2 2 2 21 2 3

1 1 1 1 1......

sf f f f f= + + + +

(2)

Where

f : critical for the system as a whole

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.

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.

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.

Disk 1Disk 2 Disk 3

L

Figure-2 Shaft Loaded with Three Disks

Disk 1Disk 2 Disk 3

L

Figure-2 Shaft Loaded with Three Disks

Nylon Bushes

(Guards)

Frame

Motor

Double Universal

Joint

Kinematic Coupling

AssemblySliding End Support

Bearing

Figure-3 Whirling of Shafts Apparatus

Nylon Bushes

(Guards)

Frame

Motor

Double Universal

Joint

Kinematic Coupling

AssemblySliding End Support

Bearing

Figure-3 Whirling of Shafts Apparatus


16

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. Shaft

Diam.

(mm)

Shaft

Length

(m)

Simply supported Supported-Fixed Fixed-Fixed

Theo. Exp. Theo. Exp. Theo. Exp.

1 3

2 7

Part B: Whirling of loaded shafts

1. Use the 7mm shaft in simply-supported configuration. Attach the first disk of 400g mid-

way 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:

2 2 2

1

1 1 1

sf f f= +

(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)

Shaft critical freq. fs Disk freq. fi

i =1, 2, 3


17

1 Disk 1 alone (From part A)



4 All disks (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.

Mech Vibration Lab

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