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Educational
PRODUCTS
CONTENTS
1
INTRODUCTION
1
2
GENERAL DESCRIPTION OF THE APPARATUS
Portal Frame
Speed Control Unit and Excitor Motor
List of Components
3
4
4
4
3
EXPERIMENTS
Experiment 1:
Experiment 2:
Experiment 3:
Experiment 4:
5
5
6
8
Simple Pendulum
Compound Pendulum
Centre of Percussion
Determination of the Acceleration due to Gravity by
means of a Kater Pendulum
Experiment 5: Bifilar Suspension
Experiment 6: Mass-Spring Systems
Experiment 7: Torsional Oscillations of a Single Rotor
Experiment 8: Torsional Oscillations of a Single Rotor with Viscous
Damping
Experiment 9: Torsional Oscillations of a Two Rotor System
Experiment 10: Transverse Vibration of a Beam with One or More
Bodies Attached
Experiment 11: Undamped Vibration Absorber
Experiment 12: Forced Vibration of a Rigid Body - Spring System with
Negligible Damping
-Spring
System
14: Forced Damped Vibration of a Rigid Body -Spring
10
11
12
14
16
19
20
24
25
Experiment 13: Free Damped Vibrations of a Rigid Body
Experiment
System
4
REFERENCES
APPENDIX EXCITOR MOTOR AND SPEED CONTROL
27
29
31
A-1
SECTION 1: INTRODUCTION
The TQ Universal Vibration Apparatus enables
studentsto perform a comprehensiverangeof vibration
experimentswith the minimum amount of assembly
time andthe maximumadaptability.
The experimentslead the studentthroughthe basics
of vibration theory by, initially, very simple
experiments which make way for those of a more
extensivenatureas experimentalaptitudeincreases.
Although the policy of the experimentsis to give the
student a general insight into experimental methods,
there hasbeensomeattemptto evoke further study and
critical appreciationby questionsposed at the end of
someof the tasks.
This manual primarily give details of the apparatus
required and the experimentaltechniquesinvolved for
eachexperimentin turn. Eachexperimentstartswith an
'Introduction' dealing with the purposeand basictheory
involved. further sections detail the apparatus and
experimental method with reference to diagrams
includedin the text.
finally, the form of calculationsand resultsis given,
followed by any 'further Considerations'which may be
significant.
SECTION 2: GENERAL DESCRIPTION OF THE APPARATUS
Figure1 TM16universalvibrationapparatus
TQ Universal Vibration
Portal Frame
List of Components
The apparatusshown in Figure I, consistsof a basic
portal frame, robustly constructedfrom square,rolled
hollow section, vertical uprights and double channel
horizontal members.The frame mountsOIl four castors
for easeof mobility.
Screwjacks allow the weight of the frame to transfer
to the floor during experiments,which enablesthe entire
rig to be levelled prior to the experimentalwork and
guaranteesrigidity.
The frame has been fully machined so as to be
adaptable to accept all the listed experiments. An
attractivewoodenstoragecupboardis titted at the front,
which housesall the componentswhen not in use.
Speed Control Unit and Exciter Motor
TM18f ~RECI'IOH MOTOR '~EED UHIY
--
.-- .--
~
--0
--
0'-
<D>
Figure2 Speedcontrolunit frontpanel layout
A d.c. motor is used for all forced vibrations
experiments powered by a conb"Ol unit. This
combinationcomprisesof a control box and d.c. motor,
whjch provides high precjsion speed conb"Olof the
motor up to 3000 rev/min, jrrespectjve of the normal
load fluctuationsof the motor.
The front panelof the unjt containsa speedconb"Ol,
a fully calibrated speed meter incorporating an
automaric range switchjng device (there being two
ranges:0 - 1500 and 0 - 3000 rev/min), and power
socketsfor:
Mains input;
d.c. motor;
Auxiliary output (either to a stroboscopeor chart
recorder),sometimesmarkeddrum supply.
IP~rt No.
Descrl~
181
Pendulumsub-frame(crosa
82
C2
C3
101;1
D3
D4
D5
Experiment
1 ,2.3.4,5
S.
Godball
S.
baR
K
ndulum
W
ndul~
Sin
ndulum
BiNar8~sion
--
87
~1
.
~
.~
5
jTop adjustinga~1y
lG4JQe~sh~
(Spring)
1
~ing
-
platform
6,12,13.14
6
~
10
I~~...n_~
~~~
1= =,11fY
\ Out-of-balance
I
10
discs
I Beam ~
14
i
~~
\E1
Trunnionmounting with lateral
-
- t2.1~.14
10,11
m
!2t!1
Su
i E3
E5
Su
E6
Contactor
F:11
precision motor speed ~
unit with exciter mdor and
raduated discs
G1
l!:!!
I~
It for micrometer
10,11
Rectan ular section base
r10~3.14
10,11
11
IVlbratio~ 8t1S9rber
I
10,11
Rdor (254mn diamet8f')
.f
Rotor and additionalmassea
~
(16&':!!!dian!8ter) l!1
~
IK3
~
~~
Shaftau
Pe
Rotor8
~r8nt
t
!
8
drum
011
~
c!
8
Table 1
The oil suppliedwith the Universal Vibration Apparatus
is Shell Vitrea oil.
SECTION 3: EXPERIMENTS
Experiment 1: Simple Pendulum
Introduction
One of the simplest examples of free vibration with
negligible dampingis the simple pendulum.The motion
is simple harmonic, and is characterised by the
equation:
dlx
7
--=-8.%
/
The periodic time is:
Procedure
Measureand note the length /, the distance from the
bottom of the chuck to the centre of the ball. Displace
the pendulum through a small angle 9, and allow to
swing freely. Once settled,measurethe time taken for
50 oscillationsandrecordthe periodic time, t.
Repeatthe procedurefor variousvaluesof / for both
the woodenball and the steel ball. Enter the results in
Table 2. Plot a graph for valuesof r againstvaluesof
length/.
t = 2n II
g
Results
In this experiment,the object is to analysethe above
equationfor the periodic time by varying the length of
the pendulum, I, and timing the oscillations. The
independenceof the size of the massof the particle is
demonstrated.
Apparatus
Sub-frame(crossbeam)
Smallwoodenball
Small steelball
lnextensibleflexible cord (not supplied)
Stopwatchor clock (not supplied)
Metre rule (not supplied)
BI
B2
B3
Length
(m)
Time for 50 complete
oscillations
Wood
Steel
Period 't
Steel
-r
Steel
I
0.10
.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Table2 Experiment1 results
Both the steeland woodenballs attachto lengthsof cord
approximatelyone metre long, each of the two cords
suspendingfrom the small chucksat either end of the
sub-frame. You can vary the length by pulling the
thread through the chuck and the hole above the sub.
frame.
8;1
0
0.10
0.20
0.30
0.40
Lengu, of pendulum, I (m)
Figure 3 Graph of -I against I for a simple
pendulum
The graphresultsin a straight line, giving a relationship
betweenr and I of the form:
r=KI
where K is a constant equal to
4~
g
Hence the value of g, the accelerationdue to gravity,
canbe determined.
Figure3
Further Considerations
I. What inaccuracies exist in this method for
calculatinga satisfactoryvalue for g?
2. How can you overcometheseinaccuracies?
fQ Universal Vibration
Experiment 2: Compound Pendulum
Introduction
A rigid body that swings about a fIXed horizontal axis,
shown in figure 5, displacesthrough an angle 9 and is
subjectto a restoringcouplemgh sinO.
Apparatus
The compound pendulum consists of a steel rod of
length 762 mrn and diameter 12.7min. The rod is
supported on the cross member B 1 by an adjustable
knife-edge which, when moved along the rod,
effectively altersthe valueof h discussedabove.
Procedure
Figure 5 Compound pendulum
If anglee is sensiblysmall,the equationof motion
becomes:
d2e
-+
~
~
]e = 0
IA
m = Massof the body;
h
= Distanceof the masscentrefrom the swing
axis;
9 = Angular displacement;
fA = Moment of inertia of the body aboutthe swing
axis.
This is simple harmonicmotion so the constant:
Detennine the location of the centre of gravity of the
rod (midway alongthe rod).
For a given value of LI from one end, tighten the
knife-edge and then suspendthe rod by placing the
knife-edge on the cross beam so that it swings freely
through a small angle without any rotation of the
support.
Once the system is swinging freely measureand
note the time taken for 20 complete oscillations and
recordthe periodictime, 'to
Repeatthe procedurefor differing valuesof LI and
enter the values in Table 3. In order to perfonn further
tests, slackenedoff the knife-edge be and move along
the rod to a new position. It is found that removing the
pendulum from the cross-beam to carry out any
adjustmentsis the easiestmethod.
Results
The expressionfor the periodic time transformsto
rh=~2+~2
g
g
Plot a graph of rh to a baseof ~. Detenninethe slope
of the line g. and from the interceptdetenninek.
!!!!!!. =r02
IA
d 6L
' d"
2K Th" "
an ulepeno Ictlme t=-.
IS gIves:
CD
't = 2nII
~-;;;gh
IA
=IG + m~
(by the parallel axis theorem)
and
In = mJ!
where k is the radius of gyration of the body about axis
through the mass centre parallel to the swing axis.
Therefore,
Table 3
Theoreticalvalue ofk can be found using Routh's Rule
which for a rod of small cross-sectiongives:
.2 =
c
TQ Universal Vibration
Further Considerations
I. Calculatethe length of the simple equivalent
pendulum
for theabovecasewhere
~
=21tH
(simplependulum)
is equalto
for a compoundpendulum.
2.
Find the two valuesof h which satisfythe resulting
quadraticequationgiving equalperiodic times.
~
TQ Universal Vibration
Experiment 3: Centre of Percussion
Introduction
If you subject a compoundpendulum supportedon a
horizontalpivot to an impact force at an arbitrary point,
there will be a horizontal reaction at the pivot. We can
liken this to a cricket bat striking a ball - thereis one
particular point at which the strike occurs, for which
there is no horizontal reaction at the pivot of the
compound pendulum. Such a point is the centre of
percussion.The location of sucha point is the object of
this particularexperiment.
J
Sikiable~t;
Figure 7
Detennine the centre of gravity of the pendulum by
resting the board. with the steel weight at distancey.
&om a knife-edge support. The distance h &om the
knife-edgeof the pendulumto the balancingknife-edge
canthen be detennined(seeFigure 7).
For each position of the steel weight measurethe
time taken for 20 completeoscillations and record the
resultsin Table 4.
From the valuesof t and h in Equation(I) calculate
the valueof kAand comparewith the theoreticalvalues.
Figure 6 illustratesthe apparatus,and consistsof a steel
ball as part of a simple pendulum (B6) and the
rectangularshapedwooden compoundpendulum (B5)
having an adjustablesteel weight slidable in a central
slot. Both are suspendedon steel knife-edgesfrom the
horizontal cross-beam(B 1) at the top of the portal
frame. The simple pendulum is located in a V -groove
whilst the knife-edgeof the compoundpendulumrests
on the flat surfaceof the beam.
Part A: Determining the Centre of
Percussion
Results
Table 4 will indicate the variation of periodic time as
the radius of gyration about the point of suspension
varies.Calculatea theoretical value for k. the radius of
gyration about the centre of gravity, from the
dimensionsof the pendulum.
Test
Number
To fmd the centre of percussion of the compound
pendulum,first determinethe periodic time. From this
the radius of gyration about the pivot axis, kA, can be
found using formula:
t = 21&
1*i 2
k
1m)
.1
2
~
~
5
Part B: The Centre of Percussion in
Relation to the Point of Suspension
Using the results of Part A, show that the centre of
percussion can be at a distance from the point of
suspensionequalto its equivalentlength:
1=
gh
(1)
where
h = Distancefrom the point of suspensionto the
kl
kA
(m)
Table 4
Procedure
cenh'eof gravity and
= k2 + h2 (parallel axis theorem).
Time for 20
oscillations
k2 + h2
h
where
/
k
=
h
=
Length of the equivalentpendulum;
= Radius of gyration about the centre of
gravity;
Distanceof the point of suspensionfrom the
centreof gravity.
TQ Universal Vibration
Procedure
Adjust the length of the simple pendulum(86) so that
the length of the bob from the knife-edgeis equalto the
length of the compoundpendulum. Allow the simple
pendulumto swing, so that the sphericalbob strikesthe
edge of the compoundpendulumat its perigee(lowest
point of its path) and causesthe latter to swing.
By constraininghorizontal movementof the simple
pendulum in its V-groove, the only horizontal
movementpossible is that of the compoundpendulum
resting on its flat support. It may be observedthat no
horizontal movement is produced with the simple
pendulum= I and that for any other values,horizontal
movement is produced. A pencil mark on the crossbeam underthe initial knife-edgeposition may be used
asa referencemark.
TQ Universal Vibration
Experiment 4: Determination of the Acceleration due to Gravity by means of a Kater
(reversible) Pendulum
Introduction
Procedure
The Kater pendulum is a device for accurately
determining accelemtiondue to gravity. It consistsof
two adjustableknife-edgesand an adjustablecylindrical
bob. Arranging their relative positions to give equal
periodic times when suspendedfrom either knife-edge
producestwo simultaneousequations:
tl
= 21t~ ~
1~
g~
't2 = 21t1M~ ~
V-~
Position the knife-edgesa set distanceapart, and then
suspendthe pendulum from one of the knife-edges.
Allow the pendulumto swing ft-eely and measurethe
time taken for 50 oscillations, and from this find the
periodic time t..
Reversethe pendulumand suspendit from the other
knife-edge. By suitable positioning of the cylindrical
bob, obtain the periodic time t2 to be approximately
equalto tl.
Rechecktl and carry out any further adjustmentsto
obtainan equaltime of swing.
Once t2 has beenaltered to be approximatelyequal
to tl, allow the pendulumto swing for 200 oscillations
and notethe times for both tl and t2.
47
and
~
4r
=h]+k2
By arrangement
4r r.+~
-=+
g
2(11.
+~)
r.-~
2(11.
-~)
Apparatus
The apparatusrequiredfor this experimentconsistsof a
pendulum having two adjustable knife-edges and an
adjustable cylindrical bob (B4) suspendedfrom the
hardenedsteelcross-beam(BI). SeeFigure 8.
Figure 9
Find the cenb'eof gravity of the pendulumby balancing
it on a knife-edge and measuring hi and h2, the
respectivedistancesof the knife-edgesfrom the cenb'e
of gravity. The distancebetweenthe two edges is the
lengthof the simple equivalentpendulum,L.
Results
Ih1=O.2om
1't1=
1~.O.30m
112.
1
1
Table 5
4r
g
= J!f...:!~ + J!f...:!..:!ll
2(~+~)
2(~-~)
From which the valueof g is detennined
TQ Universal Vibration
Experiment 5: Bifilar Suspension
Introduction
The bifilar suspensioncan determine the moment of
inertia aboutan axis by suspendingtwo parallel cordsof
equal length through the mass centre of bodies, as
shown in Figure 10. Angular displacementof the body
about the vertical axis through the masscentreG is by
angle8, which is sensiblysmall.
81
Knowing the periodic time and the magnitudesof the
various parameters, the radius of gyration k and
thereforethe valueof I canbe determined.
Apparatus
Figure 10 showsthe apparatusand consistsof a unifonn
rectangularbar B7 suspendedby fine wires from the
small chucksas usedin Experiment I. Drawing the two
wires through the chucks and tightening alters the
lengths of the suspension.The bar is drilled at regular
intervals along its length so that two 1.85kg masses
may be peggedat varying points along it.
Procedure
With the bar is suspendedby the wires, adjust length L
to a convenientsize, and measurethe distancebetween
the wires, b. Displacethe bar through a small angleand
measurethe time taken for 20 complete oscillations.
From this, calculatethe periodictime.
Adjust the length of the wires, L, and measurethe
time taken for a further 20 swings. Increasethe inertia
of the body by placing two massessymmetrically on
either side of the centreline distance x apart, and
repeatingthe procedurefor various valuesof L and the
distancebetween the masses.Calculate the radius of
gyration k of the system as previously outlined.
Tabulatethe resultsin Table 6.
Figure 10
Results
I
Test
number
I
L
(m)
x
J
't
I
k
(m) I (s) I (m)
~Jml/=m~!
(ml)
I (kg)
I (kgmz)
1~
4
Table 6 Results for bifilar suspension
It is instructiveto comparethe value of I obtainedin a
particular test with the value of I detemlined
analytically,using L~mx2.
Figure 11
This equationof the angularmotion is:
,
Id2e
mgb2
~pe
dtL
4£
which may be written as:
"
gb2
9+,--:-,-,9=0
4klL
Themotion is clearly simpleharmonicand the period is:
t = 47t6;- 2L
gb'
where I is moment of inertia about swing axis through
G(J=mf).
Further Considerations
1. Somenoteworthypoints will have arisenas a result
of performing this experiment. Write out your
conclusions.
2. How would the radius of gyration, and hence
moment of inertia, of a body using the bifilar
suspensionbe determined?
TQ Universal Vibration
Experiment8: Mass- Spring Systems
Introduction
A helical spring. deflecting as a result of applied force,
confonns to Hooke's Law (deflection proportional to
deflectingforce).
The graph of force against deflection is a stniigbt
line asshownin Figure 12.
Figure 13
Figure 12
Ax
is the 'deflection coefficient' in
4F
metresper newton.
The reciprocal of this is the stiffness of the spring
and is the force required to produceunit deflection. A
rigid body of massM under elastic restraint,supported
by spring(s).fonDSthe basisof all analysisof vibrations
in mechanical systems.The basic equation is of the
form:
Slope of the line
Mx=-k;x
where k = stiffness of the spring
This clearly simple harmonicmotion of periodic time
t:
Procedure: Part A
Fix the specimenspring to the portal frame, with the
loading platfonn suspendedunderneathand the guide
rod passingthroughthe guide bush.Carefully adjustthe
systemto ensurethat the guide bush is directly below
the top anchoragepoint, since any misalignment will
produceexperimentalerrorsdueto friction. Friction can
be minimisedby usinggreaseor oil aroundthe bush.
Using the gauge measurethe length of the spring
with the platfonn unloaded.Add weights in increments,
taking note of the extensionin Table 7, until reachinga
suitable maximum load. Remove the weights, again
noting the length at each increment, as the system is
unloaded.From thesevaluesdetenninethe mean value
of extensionfor the spring.
M
~.
DefI-.l:tlon x
Loadlnq
Unloadin~
Mean k
Imml
~.
'C-
~7tJ¥
~
...Q!
Q
Apparatus
Figure 13 showsthe requiredset-upfor the experiment.
Suspendanyone of the three helical springs supplied
from the upper adjustableassembly(CI) and clamp to
the top memberof the portal frame.
To the lower end of the spring is bolted a rod and
integralplatform (C3) onto which 0.4 kg massesmay be
added.The rod passesthrougha brassguide bush,fixed
to an adjustableplate (C2), which attachesto the lower
member.A depth gaugeis supplied which, when fitted
to the upper assemblywith its movable stem resting on
the top plate of the guide rod, can be used to measure
deflection,and therebythe stiffness,of a given spring.
-.1!
2.0
M
-1:!.
..1.1.
M4.0
Table 7
Plot a graph for the extension against load, and from
this determinethe spring stiffness,k.
TO Universal Vibration
Procedure: Part B
Add massesto the platfonn in varying increments,pull
down on the platfonn and releaseto produce vertical
vibrations in the system.For each incrementof weight
notethe time taken for 20 completeoscillationsin Table
8, and from this calculatethe periodictime, "to
~
: Meancoil diameter:
Meanwire diameter:
Number of coils:
Time for 20
oscillations
At
k
~
Period't
(6)
,
(a2)
0
1.0
2.0
3.0
4.0
Mass,M (kg)
5.0
6.0
Figure 15 Part B graph
From dIe intercept of dIe line on the M-axis, the
effective massof the spring can be found (m). Compare
the value of m obtained with the generally accepted
value, that is, ~ massof spring. Repeatthe procedure
,
..
.
with the other springs provided.
Table 8
Note that
't2
=
(~)M
The mass of the rod and platform are included in M
above.From Table 8 plot a graph of i againstM and
find the slopeof the graph,g, and the M-axis intercept.
m.
Results
E
.5.
Ic
c
0
~
.
~
0
1.0
2.0
3.0
Mass, M' (kg)
Figure 14 Part A graph
4.0
Further Considerations
1. State your conclusionsin the light of the results
obtained.Hasthe basictheory beenverified?
2. From the experimentsso far perfonned,discussthe
relative merits of each in calculating an accurate
valuefor g. Criteria for your commentsshouldbe:
a. Easeof experimentation;
b. Inherentinaccuracies;
c. Easeof computation.
3. Choosing some typical results, what error is
introduced in calculating g by neglecting the
effectivemassof the spring?
TQ Universal Vibration
Experiment 7: Torsional Oscillations of a Single Rotor
Introduction
This is an exampleof simple harn1onicangularmotion.
The systemis comprisedof a rigid rotor at one end of an
elastic shaft. It is 'torsional vibration' due to the
twisting actionon the shaft.
Analysis of this situation is analogous to the
previousexperiment.Zero replacesdeflection x. and k,
which was stiffness, is now torsional stiffness of the
shaft. The polar moment of inertia of the rotor [,
replacesmassM.
The equation of motion is 19 = - *-6, which is
simple harn1onicmotion. It can be shown that the time
period,'t, is:
't = 2x fJiVOJ
where:
L = Effective length of the shaft;
G= Modulusof rigidity of the materialof the shaft
J = Polarmomentof areaof the shaft section.
Apparatus
For experimentson undampedtorsional vibrations, the
inertia is provided by two heavy rotors, cylindrical in
shape, one 168mm diameter the other 254 mm
diameter.Figure 16 shows the smaller diameter rotor,
H2. The rotor mounts on a short axle, which fits in
either of the vertical membersof the portal frame, and
securesby a knurled knob.
The rotor is fitted with a chuck designedto accept
shaftsof different diameter.An identical chuck rigidly
clampsthe shaft, which is an integral part of a bracket
(II). This is at the sameheight as the flywheel chuck
and adjustable,relative to the baseof the portal frame.
Three steel test shafts are supplied - 3.18,4.76and
6.35 mm in diameter,each965 mm long.
inertia of the smaller rotor. Two pairs of massesare
availableof approximately1800g and 3200 g.
Part A: To Determine the Moment of Inertia
of a Flywheel
Procedure
The moment of inertia of a flywheel (one of the rotors
would be most suitable), can be found experimentally
by the falling weight method. The flywheel mounts as
describedabove so that it can rotate freely on an axle
fitted to one of the vertical membersof the frame.
In the caseof the smallerrotor with addedmasses,it
is necessaryto clamp the rotor in the reversedposition,
since a complete revolution of the whole assemblyis
impossible with the rotor clamped inside the portal
frame.
Attach a body of massm to a length of string. Wind
the string around the circumference of the rotor,
ensuringthat it loops arounda steelpeg projecting from
the rim. Allowed the body to fall through a measured
height h to the groundand recordthe time of descent,I.,
by a stopwatch (not supplied). Note the number of
revolutions,n., during the accelerationperiod.
Find the number of revolutions, n2, and the
correspondingtime, 12,from the instantthe body strikes
the ground to the instantthe rotor comesto rest. Adjust
the length of string so that the string detachesitself as
the body strikes the ground. More than one test should
be performedto obtain averagevaluesfor n., n2and the
times II and 12'
Theory
Apply the basic energy equation: W = M to the two
phasesof the motion of the system.
Accelerationperiod
- TIn, (21t)= !!!..~2-
0]+ mg{O- h]+
2
Decelerationperiod
- Tfn2(27t)=i~
- 0)2
2
:liminating Tf from the above two equationsgives:
~
mgh =
,mv'"+
J~
..
~
II,
from which J canbe calculated
Figure 16
By bolting two pairs of steel arms to each side and
attachingheavymassesat eachend, we can increasethe
Notation
m = massof the falling body (kg).
h = height of fall (m).
v = maximum velocity of the body striking ground
(m/s).
w = correspondingmaximum angular velocity of the
wheel (rad/s).
TQ Universal Vibration
Results
L
{!!!.!!!l
~
~
200
Time for 20
Perlod't
oscillations
~~L
~
~
~
375
450
Table 9
When performing a practical test the value of m should
not be too large, otherwisethe duration of the second
phaseof the motion runs into many minutes.A value of
m equal to about 0.05 kg is suggested.1 comes to
approximately0.18 kgm2
Palt B: Frequency
of Torsional
(Single Rotor System)
Oscillations
Having determined a value for I for a particular rotor by
the method described in Part A using one of the three
shafts, the frequency of torsional oscillations of a single
rotor system can be found experimentally. Compare the
result with theoretjcal prediction.
Procedure
Pass the shaft through the bracket centre hole, so that it
enters the chuck on the flywheel and then tighten. Move
the bracket along the slotted base until the distance
between the jaws of the chuck corresponds to the
required length L. Tighten the chuck on the bracket.
Ensure that the jaws securely grip the shaft. Displace
the rotor (flywheel) angularly and record the time for 20
oscillations.
Vary the distance between the chucks in suitable
increments by sliding the bracket, and record and
tabulate the values of periodic time for the various shaft
lengths.Plota graphof -r againstL.
Further Considerations
1. Using the falling weight method, if the string
wrapped around the axle of the wheel instead of
aroundits rim how would this affect the results?
2. What change(s)in procedurewould be necessaryif
you useda steppedshaft insteadof one of uniform
sectionthroughoutits length?
TQ Universal Vibration
Experiment 8: Torsional
Oscillations
of a Single Rotor with Viscous Damping
Introduction
In this experiment,the effect of including a damperin a
systemundergoingtorsional oscillations is investigated.
The amountof damping in the systemdependson the
extentto which the conical portion of a rotor is exposed
to the viscouseffectsof a gjven oil.
Theory
The equationof the angularmotion is:
oscillation traces on paper wrapped round the drum
mountedabovethe flywheel. Unit K2 consistsof a penholder and pen, which adjust to make proper contact
with the paper;the unit undergoesa conUolleddescent
over the length of the drum by meansof an oil dashpot
clampedto the mainframe.
K1
which may be written
fA)
d2e
dt2'+a"di+b8
where a=-
C
=0
and b=-
J
k
J
The angular displacement is:
e = ce(-a/2t)cos(Pt + ",)
where C and
. are constants.
K2
The periodic time is
t=
2K
P
Measuring amplitudes on the same side of the near
position,the nth oscillation is:
K3
K4
XD
where n is a positive integer correspondingto the
numberof completeoscillationsstartingat a convenient
datum(I = 0).
Figure 18
Apparatus
Figure 18 showsthe apparatus,and consistsof a vertical
shaft gripped at its upper end by a chuck attachedto a
bracket(KI) and by a similar chuck attachedto a heavy
rotor (K3) at its lower end.
The rotor K3 suspendsover a transparentcylindrical
container,K4, containingdampingoil. The oil container
can be raisedor loweredby meansof knurled knobs on
its underside,allowing the contactarea betweenthe oil
in the containerand the conical portion of the rotor to
vary. This effectively varies the damping torque on the
rotor when the latter oscillates. Record damned
You can use various diameter shafts, but due to the
location and necessary fine adjustment of the oil
container the length is restricted to approximately
0.75 m. Measure the angular displacement of the
flywheel by meansof a graduatedscaleon the upperrim
of the rotor. An etchedmarking on the frame servesas a
datumfor the measurementof angulardisplacement.
TQ Universal Vibration
Part A: Determination of Damping
Coefficient
Procedure
Fill the cylindrical container K4 with oil to within
10 mm of the top. Adjust the knobs underneathto level
the oil surfacewith one of the uppergraduationson the
conical portion of the rotor, K3. A depth, d of 175mm
is suggestedfor maximum damping. Details of the
graduationson the rotor are in Figure 19.
Recorda trace of the amplitude of oscillation showing
decayof the vibration due to the damping.The rate of
descentof the pen previously carried out will provide a
suitabletime scale.
From the trace given in Figure 20, measurefive
successiveamplitudesstarting with the initial one (n =
0) andtabulatethe resultsin Table 10below.
Xn
n
(mm)
i1
~
~
~
~
~
Xs=
2
~
~
"
log.I~.
&'.
0
Table 10
Results
Plot a graphof lo8e(xo/ xn) to a baseof n. Confim1that
Selectand fit a suitable shaft, noting the length of the
shaftbetweenthe two insidefacesof the chuck,together
with the diameterof the shaft. Allow the pen to fall, and
measurethe rate of descentof the pen (in mm/second)
by timing the descentof the pen over a fIXed length of
paper,usinga stopwatch.
The system is now ready for recording torsional
oscillations.Raisethe pen to the top of the paperon the
drum and rotate the rotor to an angle of approximately
40° and then release.A trace of the oscillationscan be
obtainedby bringing the pen into contactwith the paper
using the thumbnuton the supportand allowing the pen
to descend.
the dampingis viscous,and that the slope of the line is
equalto (at/2) (the logarithmicdecrement).
The period can be found by timing a convenient
number of oscillations using a stopwatch,whereupon
the constant,a, is detenninedand hencethe value of the
damping coefficient (the torque per unit angular
velocity) in Nm/rad/s-i. The polar momentof inertia of
the rotor is detenninedas in Experiment7.
Part B: Investigation
of how the Damping
Coefficient
depends
on the Depth of
Immersion
of the Rotor in the Oil
Repeat Part A for each oil level as defined by the seven
graduations on the conical portion of the rotor.
The damping coefficient depends on the area A of
the curved surface of the conical portion of the rotor
exposed to viscous damping. This area is equal to 1V/,
where r is the radius of base of core and / is the slant
height equal to
[r2-;-j;i.
Plot a graph of damping coefficient to a base of A
times mean radius.
Results
Tabulatetheseasin Table II
Figure20
TQ Universal Vibration
r
(mm)
-
~
25.0
37.5
~
82.5
~
87.5-
Mean radius
rm (mm)
Area A
(mm2)
A.rm
(mmJ)
~
~
18.75
25.00
~
~
~.75
Table 11 Results of torsional oscillation with viscous damping
~
g
1~
.
'5
0
c
.
~
c
~
!.
.
~
es
~
c
"Q.
E
.
Q
0
100
Dwnplng.,..
200
300
400
L
x eff8Ct1ve (me.n) radius mm x 10
Figure 21
State the probable relationship between the two
parameters.
Perlod'f
(s)
Constanta
Damping
coefficient
TQ Universal Vibration
Experiment
9: Torsional Oscillations of Two Rotor System
Introduction
With the addition of a second rotor, the apparatus
describedin Experiment7B can be usedto investigate
the oscillation of a two rotor system.For sucha system
the periodic time is:
t-
~
chucks fitted for use with shafts of various diameters.
Since both rotors axles are fixed to their respective
vertical members,the length of the shaft may not be
variedbut three shaftsof different diameterare supplied
andthreecombinationsof different inertiasarepossible.
Procedure
~ GJ(/. + /2
where
II = Momentof inertiaof therotor 1;
h = Moment of inertia of rotor 2;
L = Length of the shaftbetweenthe rotors;
G = Modulus of rigidity of the materialof the shaft;
J = Polar second moment of area of the shaft
section.
One of the shaftsclampsbetweenthe two rotors H. and
H2 of predetenninedinertia. Recordthe effective length
of the shaft measuredbetweenthe jaws of the chucks.
Carefully tighten the chucksto ensurethat neither rotor
canslip relative to the shaft.
Rotateeachrotor through a small angle in opposite
directionsand then release.Torsional oscillationsof the
system are thereby set up and the time for 20
oscillationsrecorded.
The periodic time of the systemmay be determined
and comparedwith the theoretical value given by the
fonnula quoted in the introduction. Detennine the
momentsof inertia of the rotors the methoddescribedin
Experiment7.
Results
Polar secondmomentof area J = ~d4
The generally acceptedvalue of G for steel is 82 GPa
and for g 9.81 m/s2.
The apparatus,as in Figure 22, is that of Experiment7,
with the bracket (II) replacedby a secondrotor (HI)
which is free to rotate on a axle fixed to the left-hand
vertical member of portal frame. Both rotors have
Shaft diameter
mm
3.17
~
~
4.76
_6.35
Table 12
11
k~!!!~-
12
-~~
Further Considerations
When oscillating torsionally, the two rotors oscillate
back-to-backabout a non-moving section of the shaft,
called the node.It is instructiveto locatethe position of
the node for a given pair of inertiasand their shaft.This
can be doneby introducinga third (dummy) rotor in the
form of a cardboard disc (of negligible inertia) and
moving it along the shaftto a position where it becomes
fixed in space.
Time for 20
oscillations
-
Period
't
Theoretical value of
period
TQ Universal Vibration
Experiment 10: Transverse Vibration of a Beam with One or More Bodies Attached
Introduction
The frequencyof transversevibrations of a beam with
bodies attached is identical to the critical (whirling)
speed of a shaft of the same stiffness as the beam,
carrying rotors of masseswhich COITespond
to those of
the bodieson the beam.
One has to think in terms of small size rotors,
otherwisegyroscopiceffectsare involved. In the caseof
a beam with just one body attached,the basic theory is
the sameas that in Experiment6. For a beamwith two
or more bodies attached,other methodscan determine
the frequency of free transversevibrations. Examples
are as follows:
Rayleighor energymethod(gives good results);
Dunkerley equation (only approximate,but quite
adequate);
Rigorous(accurate)analysis(arduous);
Experimental analysis, using the equipment
describedbelow, (fairly simple and quick).
Apparatus
The basic apparatusfor dris experimentis in Figure 23.
A bar of steel of rectangular cross-section(E6) is
supportedat eachend by trunnion blocks. The left-hand
support (01) pivots in two baII bearingsin a housing
locatedon the insideface of the vertical fi'amemember.
leadsto the precisionspeedcontrol unit, which appliesa
wjde rangeof exciting frequenciesto the beam.
Clockwise rotation of the control knob on the speed
control unit wjll increasethe speedof dte motor - thus
increasingthe out-of-balancerotating force producedby
the unbalanced discs. As the speed increases as
indicated by the speedmeter on the control unit. the
beam begins to vibrate transversely.Over a discrete
band of frequenciesincreasingly larger amplitudes of
V10ration are produced which reach a peak at a
frequencycorrespondingto the frequencyof free natural
transversevibration of the system,i.e. beamplus added
components.
Part A: Transverse Vibration of a Beam
Procedure
Suspendbodies of different size mass, m, below the
motor. For each massm, adjust the speedcontrol until
the beamvibratesat its naturalfrequency.
In order to determineaccuratelythe exact value on
the speedmeter,it is expedientto takethe beamthrough
the rangeof excessiveamplitudesseveraltimes, noting
the limits of the range. From these, we can locate the
fi'equencyat which the amplitude and resultant noise
appearsgreatest.Recordyour observationsin Table 13.
Mass m
Frequency
((Hz)
kCJ
1
~
x 10
f
Table 13 Tableof resultsfor Experiment10A
Results
A graphof (1/f
in Figure23.
to a baseof m givesa straight line, as
The intercepton the vertical axis is equalto \
addedcomponents.
Natural
frequency
of
the
system,
i.
e. Deam
Natural frequencyof the beamby itself
The right-hand supportconsistsof two roller bearings,
which are fi'ee to move in a guide block locatedon the
inside face.At the centreof the beambolt a small motor
carrying two 'out-of-balance' discs (part of Excitor
Motor and SpeedControl unit). Connectthe motor via
Dunkerley's equatior
is given by:
+
applicableto this situatior and
~
TQ Universal Vibration
HereJi = naturalfrequencyof a corresponding
light
beam with mass m attached. Clearly when
m = 0, Ji = and/=h
00
Evaluate and compare with the theoretical value
obtainedfrom:
fb =
f~
where
L =
F =
Lengthof the beam(m);
Modulus of elasticity of materialof the beam
I =
mo =
(N/m1;
Secondmomentof areaof the beamsection;
Mass of the beam by itself (kg); no masses
attached.
-
~
Also, from the graph, when the systemis not vibrating
(period t = 0) f = and1/
= O. The corresponding
r
00
value of massm is then equalto me,the equivalentmass
of the beam. me= Amo' whereA is a constant.
Determinethe value of A. How doesit comparewith
the generallyacceptedvalue ofO.5?
Further Considerations
We can testthe validity of the Dunkerleyequationin its
more familiar form by moving the motor with out-ofbalancediscs away from the centre of the beam and
attachinga heavy body of known mass at some other
point on the beam. The Dunkerley equation then
becomes:
j
= -j["i;11J:ii,+'~
Theh in this equationcould be the variable parameter
and a graph plotted similar to the one describedabove.
A specialblock for attachingextra massesto the beam
and a suitablevibration generatorof variable frequency
(not supplied with the standardequipment) would be
requiredto performthis additionaltest.
Pan B: Damped Transverse
Beam
Vibration
of a
Introduction
Damping forces are counteractingforces in a vibration
system,which gradually reduce the motion. Damping
occurs in all natural vibrations and may be causedby
Coulomb friction (rubbing between one solid and
another), or viscous resistanceof a fluid as in this
experimenton dampedtransversevibration of a beam
wherea dashpotis use.
Apparatus
This is shown in Figure 23 (the same set up as for
Experiment lOA, but with certain additions). In this
experimentyou will require the amplitude of vibration
and phaseangle.Fit a dashpot(02) and its support(E2)
to the beamto createdamping. Use the contactor(E5)
with its vertically mountedmicrometerto determinethe
amplitude and phase angle very accurately at any
exciting frequency. The electric circuit, of which a
stroboscopeis a part, completes when the contact
element(E5) touchesthe plungerof the micrometer.
Procedure
Allow the speed control unit time to warm-up, then
adjustthe micrometerplunger so that it just touchesthe
contactor.When the stroboscopeswitches to external
stimulus, a discharge occurs on contact. Take the
micrometerreading.Use this value as a datum position
from which valuesof amplitudemay be determined.
Energisethe motor to produce a definite amplitude
at a predetermined frequency. To determine the
amplitude,lower the micrometerheadand then bring up
again to produce contact. It is important that the
stroboscope discharges at a uniform frequency, so
careful adjustment must be made to ensure steady
conditions. At this point. find the amplitude of the
vibration by comparing the new micrometer reading
with that of the original datumposition.
You may also fmd the phaseangle by focusing the
stroboscopeon the graduateddisc on the motor shaft.
Since the stroboscopic discharge should be at a
frequencycorrespondingto the rotational speedof the
motor, the disc may be effectively stopped and the
phaseangle correspondingto the datum mark on the
motor read off. By following this procedurefor a range
of frequencies,you can assessthe effect of dampingby
varying the piston areaof the dashpotand thus altering
the dampingcharacteristicsof the system.
Rotate the two orifice plates inside the dashpot
relative to one another to vary the effective area.
Comparethe resultsobtainedwith thesesettingswith an
undampedcondition (the system minus dashpot).Plot
graphs of amplitude and phase angle against the
frequencyratio, ro/OOn i.e. (excitingfrequency/natural
frequency).
Massm kg
Figure 24 Graph of 1/r against m for the system
Note: At low frequencies,
phaseanglemay not be
obtainable.
Ta Universal Vibration
Results
Motor speed
(rev/mln)
Motor speed
w
w
~
Phase angle
log (O)
Amplitude
x max. (mm)
(rev/mln)
.
.
Phase angle
log (O)
Amplitude
x max. (mm)
~
~
~
~
~
c.!!L
~
eoo
1.QQ...
~
~
~
~
~
121.9.
~
~
~
.~
~
~
~
~
~
~
.~
1075
~
1100
~
JJ.!!!.
~
1400
).l!!l:
1300
~
~
~
~
~
~
~
.~
2500
Table 16
Table 14
Phase angle
Motor speed
(rev/mln)
~
~
..lQQ.
~
~
~
~
~
~
~
~
1055
~
12!§.
~
~
~
~
~
~
~
2500
Table 15
w
log (O)
Amplitude
x max. (mm)
The results, in Tables 14 to 16, show the effect of
increasingdamping on amplitude and phaseangle. For
each damping condition a graph of amplitude against
frequencycan be plotted, from which a value for the
natural frequency for each damping condition can be
found. Typical values obtained in this way are as
follows:
No damping
Light damping
Heavy damping
17.36Hz
17.50Hz
17.58Hz
and from thesevaluesthe frequencyratio can be found
being the exciting frequency/naturalfrequency.Figures
25 and 26 are typical graphs of amplitude and phase
angleplotted againstfrequencyratio.
TQ Universal Vibration
0
Figure 26
0.5
1.0
sg, (ForcinG
1.5
freQuency)
CD. (Natural frequency)
Figure25
2.0
0.5
1.0
1.5
t. (Forclna freQuency)
fn (Natural frequency)
2.0
TQ Universal Vibration
Experiment 11: Undamped Vibration Absorber
Introduction
Excessive vibrations in engineering systems are
generallyundesirableand thereforeavoidedfor the sake
of safetyand comfort. It is possibleto reduceuntoward
amplitudesby attachingto the main vibrating sySteman
auxiliary oscillating system, which could be a simple
mass-springsystem or pendulum. In this experiment,
you will examinethe vibration absorbabilityof a double
cantileversystem.
Apparatus
fixed equidistant from the midpoint of the horizontal
cantilever.The distanceapart of the bodies varies until
the systemis 'tuned'.
Procedure
For a given frequency, the massesof the vibration
absorber are adjustable along their cantilevered leaf
spring so that the energy of vibration tJansmitsto the
absorber and the amplitude of the main (primary)
system,i.e. the motor andbeam,is reducedto zero.
The aim is to detenninethe length I, the distanceof
the centreof eachof the bodiesfrom the midpoint of the
cantilever so that the natural frequency of transverse
vibration of this sub-systemcon-esponds
to the running
speedof the main (primary) system,i.e. the motor and
the beam.
The fonnula for detenniningI is:
f=~~
2n V.m13'
Here
f
=
Natural frequency of the sub (auxiliary)
system;
m = Massof eachof the bodies;
EI = Flexuralrigidity oftbe doublecantilever.
Figure 27
Figure 27 shows the vibration absorber(GI) clamped
below the motor. It comprisestwo bodies of equal mass
TQ Universal Vibration
-
Experiment 12: Forced Vibration of a Rigid Body Spring System with Negligible Damping
Introduction
When external forces act on a system during its
vibratory motion, it is termed forced vibration. Under
conditions of forced vibration, the systemwill tend to
vibrate at its own natural frequencysuperimposedupon
the frequencyof the excitationforce.
Friction and dampingeffects,though only slight are
presentin all vibrating systems;that portion of the total
amplitude not sustained by the external force will
gradually decay. After a short time, the system will
vibrate at the frequency of the excitation force,
regardlessof the initial conditionsor natural frequency
of the system.In this experiment,observeand compare
the natural frequency of the forced vibration of a
rectangularsectionbeamwith the analyticalresults.
Theory
Amplitude:
Resonanceoccurs when b -
<Ii
= o. So the critical
angularvelocity of the motor is given by .Jb.
Note that in practical circumstancesthe amplitude,
althoughit may be very large, doesnot becomeinfmite
becauseof the small amountof dampingthat is always
present.
Apparatus
~1
D&-
.
05.
09,
Figure 28
-Spring!
-:f
D1
02'
The systemis shownin Figure28 and comprisesof:
D3
1. A beamAB, of length b, sensiblyrigid, of massm,
freely pivoted at the left-handend.
2. A spring of stiffi1essS attachedto the beam at the
point C.
3. A motor with out-of-balancediscs attachedto the
beamat D.
.
I'
01
[J6
Weight
t
M = massof the motor including the two discs.
The equation of the angular motion is:
t
'"
-
J5
..
T -&..I... ! S- S.I
Figure 29
I,
~ MLl
+
mLZ
1
the momentof inertia of the systemaboutthe pivot axis,
where:
e = Angular displacementof the beam;
F0 = Maximumvalueof thedisturbingforce;
ro = Angular velocity of rotationto the discs.
The aboveequationreducesto the fonD:
d29
7
+ ho9
= Asinro/
The apparatus shown in Figure 29 consists of a
rectangular beam (D6), supported at one end by a
trunnion pivoted in ball bearings located in a fIXed
housing. The outer end of the beam is supportedby a
helical spring of known stiffnessbolted to the bracket
C I fixed to the top memberof the frame. This bracket
enablesfine adjustmentsof the spring, thus raising and
lowering the end of the beam.
The Excitor Motor and SpeedControl (E II) rigidly
bolts to the beamwith additional massesplacedon the
platform attached. Two out-of-balance discs on the
output shaft of the belt driven unit (04) provide the
forcing motion. The forcing frequencyadjustsby means
of the speedcontrol unit.
TQ Universal VIbration
The chartrecorder(07) fits to the right-handvertical
member of the frame and provides the means of
obtaining a trace of the vibration. The recorder unit
consists of a slowly rotating dnun driven by a
synchronousmotor, operatedfrom auxiliary supply on
the Excitor Motor and Speed Control unit. A roll of
recording paper is adjacentto the dnun and is wound
round the drum so that the paper is driven at a constant
speed.A felt-tipped pen fits to the free end of the beam;
meansare provided for drum adjustmentso that the pen
just touchesthe paper.A small attachableweight guides
the paper vertically downwards. By switching on the
motor, we can obtain a trace showingthe oscillationsof
the end of the beam.
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(part
numbers02, 03 and 09) nearto the pivoted end of the
beam.
Experimental Procedure
First plug the electrical lead from the synchronous
motor into the auxiliary socketon the Excitor Motor and
Speed Control. Adjust the handwheel of bracket C I
until the beamis horizontaland bring the chart recorder
into a position wherethe penjust touchesthe recording
paper.
Switch on the speed control unit so the resuhing
forced vibration causesthe beamto oscillate.It hasbeen
found that a frequency of about 2 Hz is suitable, the
position of the motor can be adjustedaccordingly.The
time for 20 oscillations will then be approximately 10
seconds.The chart recorder can record the number of
cycles performed by the beam in a given time
(calculated,knowing the speedof the paper or, better
still, by visual counting).
Bring the pen into contact with the paper, then
recordthe numberof cyclesand calculatethe cycles per
unit time (i.e. the frequency) of the forced vibration
beam.
You need to known the speedof the paper on the
chart recorder. To obtain this, record a trace for 20
seconds,for example, then measurethe length of the
trace,thuscalculatingthe speedin rnrn/s.
Determine the values of the relevant parametersas
describedin the theory: lengths£., ~ magnitudeof the
massesm and M, also the stiffnessof the spring.
Results and Calculations
Using a stopwatch,time the linear speedof the drum
for 20 vibrations and determinethe time for one cycle
(period of vibration). Using the two different methods
detennine the correspondingfrequency. Calculate the
relevantmomentof inertia.
If
C~
m
m
m
Table 17
The
S
b=-=
/A
Nm-J
kgm2
.,,: It
TQ Univel$8l Vibration
-
Experiment 13: Free Damped Vibrations of a Rigid Body Spring System
Introduction
During vibrations, energy is dissipatedand so a steady
amplitude cannot be maintained without continuous
replacement. Viscous damping in which force is
proportional to velocity affords the simplest
mathematicaltreatment.
A convenient means of measuringthe amount of
damping present is to measurethe rate of decay of
oscillation. This is expressedby the term 'logarithmic
decrement',which is defined as the natural logarithm of
the ratio of successiveamplitudeson the sameside of
the meanposition (seeFigure 19).
In this experiment,the effect of the position of the
dashpotand the correspondingdamping coefficient are
assessedin terms of the logarithmic decrement,
measuredby the decay in amplitudeof a free vibration
of a beam.
Theory
Referringto Figure 29, the disturbing force, FoSin!it is
~
replaced by a damping force cL)
downward. The
dt
equationof the angularmotion becomes:
fAe
= -(c49}L) -
(s~e)~
Xo
=
xI
where:
9+a9+b9=O
Apparatus
The apparatus is as shown in Figure 28 and Figure 29 in
Experiment 12, except that the exciter motor is not
required since only free vibrations are of interest. The
Excitor Motor and Speed Control unit is required in
order to drive the drum on the recorder unit D7. The
system is set vibrating freely by pulling down on the
free end of the beam a short distance (15 - 25 mm) and
releasing. Use the chart recorder to obtain a trace of just
three successive amplitudes on the same side of the
mean position. Vary the damping by moving the
.
ratio
x
-!-
x,
~
~
~
0.25
Table 18 Maximum damping
Log
dec
Log
2
From this the dampingcoefficient, c, the resistingforce
per unit relative velocity canbe determined.
Results
Enter the results in Tables 18 and 19, one relating to
maximum damping(orifice plates in the dashpotset to
give maximum area) and the other to minimum
damping.
x
at
-
Constant
a =-cIJ. and
The theory from now on is identical to that set out in
Experiment8 (the samesymbolsare used).
Amplitude
Procedure
Switch on the speedcontrol unit and connectthe lead
from the motor of recorder unit 07 to the auxiliary
supply socketon the Excitor Motor and SpeedControl
box. Set the dashpotat distanceL, (the distancefrom
the trunnion mounting to the centre of the beamclamp
09), and then pull the beam down a short distance,
underthe point of attachmentof the spring,andrelease.
Bring the recording pen into contactwith the paper
to produce a trace of the decaying amplitude of
vibration and thus produce a trace of the decaying
appliedamplitudeon the chart recorderpaper.
For a given piston area,selectand obtain tracesfor
various valuesof LI. Choosea different piston areaand
repeatthe process.
For eachpiston area and value L" use the trace to
evaluatethe logarithmic decrement.Find the periodic
time of one complete oscillation, 't, in the manner
describedin Experiment12.To recap:
Ln-
which canbe put in the form:
Length L1 (m)
dashpot(D2) and its clampsalongthe beam,and also by
relative rotation of the two orifice plates in the dashpot
to increaseor decreasethe effectiveareaof the piston as
in ExperimentlOB.
.
.!L
Period 't (8)
Constant a
Damping
coeff c
(N/m S.1)
TQ Universal Vibr8tion
Length L1 (m)
Amplitude
ratio
x
-!..
X1
I
x
Log dec Log-!x,
Period 't (s)
Constant a
Damping
coeff c
(HIm s-')
0.10
0.15
Q.~
Table 19 Minimum damping
1:7
.
it
E
MaxImum
piston
aree
I
.
I
Z
C.I
i
S
'A. E
i
~
w
M
t7'
~rea
7
Plot, on the samegraph,valuesof dampingcoefficient c
against L2. Figure 30 shows typical plots. The
logarithmic decrement,hence the damping coefficient
varies accordingto the squareof the distancefrom the
dashpot. Adjusting the position of the dashpoton the
beamproducesany degreeof dampingby consultingthe
graph.This infonnation may be usedin Experiment14.
TQ Universal Vibration
Experiment 14: Forced Damped Vibration of a Rigid Body - Spring System
Introduction
Having establishedthe effect of viscous damping on
free vibrations in the previousexperiment,the effect on
forced vibration is now analysed.To assessthe relative
magnitudeof the forced vibration, use the concept of
'dynamic magnifier'. This is the ratio of the amplitude
of the forced vibration to the deflectionproducedif the
maximum value of the disturbing force F is applied
statically, underthe sameelasticrestraint.
where
m = Mass corresponding to hole in each disc (kg);
r = Radius to centre of hole (m);
(3)
It can be shown that:
Dm
calculatingthe angularvelocity of the discs,(J) rad/s,
from the speedindicatedon the control unit.
Referring to Figure 29, the equationof the angular
motion is:
!_-J 9 = (Fsin(JX)4~_~e)f1-(~9~
which reduces to the standard fonn:
= Asinrot
Only the steady-statemotion is of interesti.e.
=
Asin(CIJt
- 41)
~(b-w2r+<0202
.
= 1-{~/~}
(5)
Angular velocity of discs(rad/s).
Note that the ratio of the rotationalspeedof the discsto
that of the motor is 22:72. Consider this when
{)
!.m!!£090
and in nearly all practical circumstances,damping is
'light', andthereforea is sensiblysmall so
2mrwl (two discs)
9+00+b9
=
(4)
Theory
The out-of-balanceforce is:
C1) =
Dm
(J)
=
Circular frequencyof the forced vibration
~
=
(rad/s);
Circular frequencyof free undampedvibration
(rad/s).
Apparatus
Figure 29 showsthe apparatuswith the dashpotadded
and the addition of an extra item: a plate clampedto the
out-of-balancedisc. The plate holds a piece of circular
paper.
The recording pen fits to pivot (08), which clamps
to the upper memberof the frame and clips above the
frame when not in use. The pen makes a trace of the
locus of the point at any radius on the rotor. Since the
rotor is capable of vertical as well as rotational
movement,you can obtain a trace from which the phase
lag can be determined.
Procedure
Lfm =
90
(2)
where
k
= 84 (torsionalstiffnessof thebeam).
Deflectionsmeasuredarethoseof the endB of the beam
andaregivenbyx = Le so:
The natural frequencyof the system is first found as
describedin Part 4 of Experiment 13, by analysingthe
free vibrations of the system,without the dashpot,from
a traceproducedon the chart recordingunit (D7).
Fit the dashpotunit (D2) at a suitablepoint along the
beam to give a definite degree of damping (as
determinedin Experiment 13). Then rotate the exciter
discs at a very low speedand obtain a datum trace on
the papermountedon the plate attachedto the nearside
disc. Mark the position of the hole in the disc on the
trace.
Increasethe speed of rotation to develop forced
vibration of reasonableamplitudein the beam.Obtain a
second 'dynamic' trace on the paper mounted on the
plate. Also obtain a trace on the chart recorder at the
right-hand end of the beam (as in Experiment 12) in
order to determine the amplitude of the vibrations.
Repeatthe procedurefor different speedsbelow and
above the critical speed to show how the value of
dynamic magnifier varies with frequency for a give
valueof the dampingcoefficient.
TQ Universal Vibration
the exciting force and the resulting vibration, the
dynamic trace displaces along the datum line
correspondingto the out-of-balanceforce. Join up the
points of intersectionof the two tracesand draw a line
through the axis of rotation at right anglesto determine
the phaselag 9 for the various speedsof rotation of the
discs.
Results
Tabulatethe results as shown in Table 20 and 21, the
columns numbering is for the purpose of this
explanationonly.
There are two tables, one for die case of no
damping, the other for a definite degree of damping.
Presenta specimenof calculations in respect of each
table.
Detennine the phase lag from the traces recorded on the
paper as shown in Figure 31. Note that the dynamic
trace displaces relative to the axis of rotation due to the
vibration of the beam. If there is no phase lag between
(I)
Excitor
I
speed
motor
locity
(rad/s)
(rev/min)
(iv)
!!!!l.
w
Amplltude
X- (mm)
Wn
column (iii). Obtain the amplitude column from the
b'aceon the drum recorder07. The correspondingphase
angle lag, is obtained in the manneralready described.
UseEquations(2) and (3) to find the 'Static Deflection';
find Dmusing Equation(4).
~
Ph-.
~v
angle
lag
(O) I
(vi)
Static deflection
~vm
I
(mm)
Dynamic magnifier
(Dm)
~
~
~
~
650
660
m.
~
~
a
Table 20 No damping
I
II
'II
Excitor motor
Angular velocity
speed (rev/min) of disc m (rad/s)
~
560
I
~
I
625
I
m
~
~
zoo
I -900
Table21 Dampino
Jill}
w
w
(Iv)
I
Amplitude
x-
(mm)
I PM" ;~Ie lag
I
(O)
(vII)
(vi)
Static deflection Dynamic magnifier
I
(D",)
(mm)
I
I
Ta Universal Vibration
Plot graphsof Dynamic Magnifier Dmand PhaseAngle
eachto a baseof the ratio oi~. Figures32 and 33 show
typical graphs. Obtain similar results for different
degreesof damping.
180
150
120
C
~
.190
60
30
, ,.
0~-
- ~-
0.85
0.90
I
0.95
1.00
J
1.05
m/m.
Figure 33 Phase lag against ratio oi~
1.10
.
1.15
SECTION 4: REFERENCES
J. Hannahand R.C. Stephens
-
"Mechanics of Machines Advanced Theory and
Examples"
EdwardArnold 1972London
W
.T. Thompson
"Theoryof Vibration"
Allen & Unwin 1981London.
W.W. Seto
"MechanicalVibrations"
SchaumOutline Series,McGraw-HilI, 1964USA.
J.L. Meriam
"Dynamics- 2nd Edition"
JohnWiley 1971USA.
J.P. Den Hartog
"Mechanical Vibrations"
McGraw-Hili 1956 USA.
S.P. Timoshenko
"Vibration Problems in Engineering"
4th Edition, Wiley 1974 W. Sussex, UK.
APPENDIX: EXCITOR MOTOR AND SPEED CONTROL
Operation and Use
A SpeedControl Unit comes as part of the Universal
Vibration Apparatus. The unit provides complete bidirectionaldrive and high precisionspeedcontrol of the
Motorffacho-generator under all nonnal conditions,
using a closed-loopcontrol system.An amplifier detects
any difference betweenthe motor speedand the input
commandvohageset by the 'Set Speed'control on the
front panel. The amplifier drives the motor until the
differenceis approximatelyzero. Provisionsfor the very
high currents for acceleration and deceleration are
automatical. The speed control can maintain speeds
from 3000rev/min down to less than lrev/min when
usedwith the Motorffacho-generator.
The front panel of the unit contains a 'set speed'
control, a fully calibrated speed meter incorporating
automatic range switching, a motor forward/reverse
switch, mainsinput socketand switch, d.c. motor power
socket,external control and an auxiliary oUtput socket
and switch.
Set Speed Control
A ten turn control giving increasing speed with
clockwiserotation.
Speed Meter
The speedmeter has two ranges,0 - 1500rev/min and
0 - 3000 rev/min. Switching between these occurs
automaticallywhen the motor speedincreasesaboveor
decreasesbelow 1500rev/min. The lamps below the
meterindicatesits range.
Motor Forward/ReverseSwitch
You canreversethe direction of rotation of the motor at
any load or speed without damage. It is, however,
recommended that the motor is stopped before
reversing.
External Control
A DIN connectorprovides connectionfor an external
set speed control (I k.o.) which. when connected,
overrides the front panel conb'ol. Connectionsto the
DIN connectorare:
PIN 1 PotentiometerSUpp1y
PIN 2 'earth PIN 3' wiper
Connection
and
Operation
Motor/Tacho-Generator,
TM16f
(F1)
Ensure all switches are in the 'oW position before
proceeding and the motor and/or recorder has been
physically installed as outlined previously.
I.
2.
3.
Connect
the speed controller
to a suitable
mains
supply.
Connect the motor to the socket marked 'd.c.
motor'.
Switch the unit on and adjust ilie motor speed using
ilie 'Set Speed' control.
The unit will
scale.
automatically
switch to the COITeCtspeed
Drum Recorder, TM16d (D7)
I. Switchthe unit off.
2. Connect the drum recorder to the socket marked
either 'auxiliary output' or 'drum supply240 Y'.
3. As (3) above.
4. Switch on the drum recorderwhenrequired
Connecting the Speed Control Unit in
Conjunction with a Stroboscope (not supplied)
I. Switch the unit off.
2. Plug the BNC T-piece into either of the socketson
the portal frame. Connect the 'trigger supply'
socket on the speedcontroller to dte T-piece and
connectthe T-pieceto a stroboscopeusing the BNC
to jack lead.
3. Connectthe leaf contacton the motor to the other
BNC socketon the portal frame via its lead.
4. As (3) above.The stroboscopewill flash onceevery
revolution of d1emotor.
5. Connectthe stroboscopeto a suitablemainssupply.