Mm203 labmanual

2003 Dublin City University
MECHANICS OF MACHINES 1
LABORATORY MANUAL
2nd
Year – Semester 4
Computer Aided Mechanical and Manufacturing Engineering, CAM2
Medical Mechanical Engineering, MEDM2
Mechatronic Engineering, ME2
Laboratory Co-ordinator
Dr. Paul Young
Room: S374
Phone: 01 700-8216
E-mail: Paul.Young@dcu.ie

MM203 - Mechanics of Machines 1
Practical Manual
2004 Paul Young, Dublin City University 2
Demonstrators
Paul Young
Julfikar Haider
Technician
Chris Crouch
Experiments
1. Gears
2. Clutch
3. Planar Mechanisms
4. Gyroscope
5. Static & Dynamic Balance
Safety Notices
A) Eating, drinking and smoking are prohibited in the
laboratories at all times.
B) Extreme care must be exercised operating lab
equipment. Examine equipment before use, and ask
demonstrators for assistance when necessary. Use
the lab handouts as a guide for start up and shut
down procedures.

Practical Manual
Introduction
There are five practicals available for this module, although each student will only
do four. As a result, at each session, all five will be available. Laboratory manuals
are only available online. It is the responsibility of the student to download the
laboratory sheet and bring it with them. Hardcopy is essential.
Students must purchase a hardback laboratory book with graph paper and each
report will be written within the 3 hour scheduled session. The books must be left
in the classroom at the end of the session and will be graded before the next
scheduled session.
Report Guidelines
Comprehensive guidelines for reports were given in MM101 & MM102, and those
still apply here. The first page of each report should be a title page and should
include page numbers. Contents should be listed in the same manner as a text
book. Take all rough measurements on paper first and keep the book for the
report only.
As outlined in the lectures, laboratory reports should be kept relatively short, but
should include some essential components:
• Introduction - a brief description of the subject matter
• Aims/Goals - bullet point statements of what you intend to achieve
• Procedure - Just putting 'as per handout' is not quite enough. Describe the
procedure in minimalist terms and include any relevant diagrams,
particularly those which assist in understanding the measurements
recorded. It is important to clarify any methods which are not explicitly
stated in the instructions.
• Results - The information recorded and, if not in the procedure, any relevant
information about the manner in which ambiguous results were decided in a
consistent fashion reported. (i.e. enough information for others to repeat the
experiment and arrive at the same results)
• Calculations - Any subsequent calculations based on the results
• Discussion - A discussion of the contents of the results and calculations. If
there are three graphs, then some comment on each graph is necessary.
• Conclusions - These should relate directly to the goals/aims. Bullet point
format is fine.
There is enough information in the handouts to complete the experiments. Any
apparent lack of information may be overcome through consulting reference books
in the library, asking the demonstrators or discussion with the lecturer.

Practical Manual
Brief Description
The numbers shown here relate to the experiment number in the timetable.
1. Gears
Using a simple configurable arrangement this laboratory seeks to provide an
understanding of simple and compound gear trains before gaining experience with
the calculation of gear ratios for epicyclic gear systems.
2. Clutch
Using a simple arrangement of axial load and applied torque the limits of friction
are determined for three different clutches. The readings are the used to validate
the theory.
3. Planar Mechanisms
This laboratory uses linkages attached to drawing boards to demonstrate the
kinematic properties of three mechanisms - Scotch Yoke, Elliptic Trammel and
Single/Double Toggle
4. Gyroscope
This experiment demonstrates the relationship between precession and the
gyroscopic couple. Using a series of balance readings the relationship between
the speeds of rotation/precession and the gyroscopic couple is investigated.
5. Static & Dynamic Balancing
Static and dynamic imbalance/balance are demonstrated initially in a simplified
arrangement. By determining the out-of-balance moment for each of four masses
an analytic solution to balancing the system is undertaken. Proof of the solution is
found through application of the solution on the apparatus.

Practical Manual
Gear Train Experiment
Aims
To gain an understanding of the speed ratios and directions of rotation of simple
and compound gear trains
Equipment
The apparatus consists of four spur gears, a frame, three axle pins and a vernier
callipers. In addition, several spacers and washers are included to allow alignment
of the gears and ensure tight meshing.
Figure G1 – Gear Train Apparatus
Procedure
Initial Inspection
The four gears should be inspected and the measurements of the pitch circle
diameter and the number of teeth taken for each gear. These can then be used to
determine the module of this gear system. The pitch circle is shown in the figure
below where it can be seen that the two pitch circles intersect at a single point.
The module is the Pitch Circle Diameter divided by the number of teeth. Record
your results in table G1 below:

Practical Manual
Label Description
DG Pitch Circle Diameter Gear
DP Pitch Circle Diameter Pinion
DRG Root Circle Diameter Gear
DRP Root Circle Diameter Pinion
DOG Outside Circle Diameter Gear
DOP Outside Circle Diameter Pinion
hk Working Depth
ht Whole Depth
a Addendum
b Dedendum
c Clearance
p Circular Pitch (measured along
Pitch Circle)
t Tooth Thickness (measured
along Pitch Circle)
C Centre Distance
Figure G2: Diagram of Gear Train
Table G1: Gear Measurements
Gear # Diameter (mm) # of teeth Module (mm)
1
2
3
4
Convert the module to equivalent teeth/inch and using the information in table G2
of metric and imperial gear sizes determine whether the gears are metric or
imperial. Hint: 1 inch = 25.4 mm
Table G2: Standard Metric & Imperial Modules
Metric Imperial Metric Imperial
Module
mm
Equivalent
Diametral Pitch
teeth/inch
Closest Standard
Imperial Gear
teeth/inch
Module
mm
Equivalent
Diametral Pitch
teeth/inch
Closest Standard
Imperial Gear
teeth/inch
0.3 84.667 80 4 6.350 6
0.4 63.500 64 5 5.080 5
0.5 50.800 48 6 4.233 4
0.8 31.750 32 8 3.175 3
1 25.400 24 10 2.540 2.5
1.25 20.320 20 12 2.117 2
1.5 16.933 16 16 1.588 1.5
2 12.700 12 20 1.270 1.25
2.5 10.160 10 25 1.016 1
3 8.467 8

Practical Manual
Simple Gear Train - Part 1
The simplest gear train involves only two gears. Select any two gears and fix them
to the arm using the pivots provided. Mark the mesh point on both gears. Rotate
the gear (not the pinion) anticlockwise one revolution and count the number of
revolutions (including any fractional rotation) and direction of the pinion. Note the
gear ratio for this pair of gears and repeat using the other two gears.
Figure G3
Table G3: Two Gear System Results
Gear Rotation Pinion Rotation
Assy
Teeth
Gear
Teeth
Pinion Dir Revs Dir Revs
Tooth
Ratio
Rotation
Ratio
1
Anti-
Clock 1
2
Anti-
Clock 1
Simple Gear Train - Part 2
Using the last pair of gears from Part 1 above, insert a third gear as an 'idler'
between the two gears and ensure a good meshing between the gears. Mark the
mesh points and again rotate the gear through one anticlockwise rotation. Count
the rotations and direction of the final gear. Replace the idler gear with the
remaining gear and repeat the experiment.
Table G4: Idler Gear System Results
Gear Idler Pinion
Assembly
TeethGear
TeethIdler
TeethPinion
Dir Revs Dir Revs Dir Revs
ToothRatio
Gear:Pinion
RotationRatio
Gear:Pinion
1
Anti-
Clock 1
2
Anti-
Clock 1
Note the effect of the insertion of the idler gear and the change in size of the idler
gear on the overall ratio.

Practical Manual
Compound Gear Train
The term compound gear train refers to systems that involve four or more gears
where at least two gears rotate together on the same shaft. The gear train should
be built as shown in the diagram below. The two smaller gears can be fixed
together with a pin forming the central compound idler. Place this with the smaller
gear uppermost first.
Assembly 1 Assembly 2
Figure G4: Compound Gear System
As before, rotate the Input gear through one anticlockwise revolution noting the
direction and rotation of the other gears. Rearrange the meshing of the gears so
that the larger gear on the idler is uppermost as shown below. Rotate the input
gear again and repeat the observations. Determine the gear ratios between the
input and output gears from these observations.
Table G5: Compound Gear System Results
Gear Compound Pinion
Assembly
ToothRatio
Gear:Compound
ToothRatio
Compound:Pinion
Dir Revs Dir Revs Dir Revs
RevRatio
Gear:Pinion
1 Anti-Clock 1
2 Anti-Clock 1
Again note how the gear ratios are affected by the introduction of the compound
gear and the effect of changing the gear on the system. Find the relationship
between the tooth rations and the rev ratio.
Epicyclic Gear System
The epicyclic gear system consists of at least three gears – sun gear, planet gear
& ring gear - and a rotating carrier (or link). The example system allows you to see
the operation of a complete assembly – a mechanism at the back allows either the
link or the ring gear to be fixed and the handwheel drives the sun gear. An
epicyclic gear system allows concentric input and output shafts. There are three
possible combinations of input and output available … sun : link (lock ring), sun :

Practical Manual
ring (lock link), ring : link (lock sun). Note the direction of rotation of each
gear pair on the sample system with the link locked … the sun & planet behave as
seen from the previous part of the experiment, but the internal teeth on the ring
gear mean that it rotates in the same direction as the planet gear.
Link
Sun Planet
Ring
The behaviour of the individual gears may be simulated using the same equipment
as before. The arm on which the gears are mounted can rotate simulating the link.
Any gear positioned at the centre of rotation will operate as a sun gear. A gear
meshing with the sun gear will behave like a planet gear. Simulation of the ring
gear is achieved by placing a third gear on the arm and holding it so that it cannot
rotate. The goal of this is to gain an understanding of the relationships between
the movement of the gears so that the relationship between the movement of the
sun and the link for the supplied example system may be calculated.
Step 1 – Effect of Rotating Link
Place the second largest gear on the shaft at the end of the arm to act as the sun
gear. Add the smallest gear so that it meshes with the sun acting as a planet gear.
The largest gear should be placed to mesh with the planet gear. Mark all the gears
so that their orientation is clearly visible. Unlock the arm from the frame and,
ensuring that the sun gear rotates with the arm rotate the arm through one
revolution anticlockwise. Stop the movement at quarter cycles to help keep track
of the movement. Record the rotation in the first row of table G6.
Step 2 – Effect of Fixing Link
Fix the arm and rotate the sun gear. Note how the system behaves and the
relationship between rotation of the gears and their teeth.
Step 3 – Simulation of Locked Gear
To simulate the operation of the epicyclic system with the ring gear locked the
outer gear must be prevented from rotating. Holding this gear in the same
orientation (with respect to the room) by hand while rotating the link through one
revolution. Record the movement of the other two gears in the second row of table
G6. This gives the gear ratio between the link and the sun gear.

Practical Manual
Table G6: Epicyclic Gear system
Arm Sun Gear Planet Gear Ring Gear
Revs Direction Revs Direction Revs Direction Revs Direction
1 Anti-Clockwise 0 none
1 Anti-Clockwise 0 none
Analysis of Example System
There is a fully assembled sample system on the bench. The gear ratio in an
epicyclic gear system may be analysed using the following steps.
1. Rotate whole system together one revolution anti-clockwise
2. Fix the Link and rotate the gear which is normally fixed one
revolution clockwise recording the movement of the other gears
3. Add the movements from steps 1 & 2
Example:
In an epicyclic gear system the sun gear has 30 teeth, the planet gear has 20
teeth and the ring gear has 70 teeth. Calculate the gear ratio between the Ring
and the Link with the sun gear fixed.
Revolutions
Procedure Sun Planet Ring Link
1 Rotate whole system 1 rev anticlockwise +1 +1 +1 +1
2 Fix Link and Rotate Sun Gear 1 rev clockwise -1 +
20
30 +
70
30 0
3 Sum steps 1 & 2 0 +2½ +1 7
3
+1
The gear ratio between the ring and the link is +1 7
3
:+1
Count the number of teeth on the example system and calculate the ratio between
the sun and the link with the ring gear fixed.

Practical Manual
Clutch Plate Experiment
Aim
The purpose of the experiment is to verify the application of the expression for
maximum torque
T = µWRmean
Where T = torque, µ = coefficient of friction, W = axial load & Rmean is
the mean radius
and in particular that
a) torque at slip is proportional to the normal force and
b) the torque at slip is proportional to radius of the friction ring
Apparatus
A wall bracket with a stationary horizontal plate supports three rings of brake
friction material concentrically to a central ball bearing that locates a rotating
upper plate of aluminium alloy. The rings are held by pairs of pegs that fit holes on
a diameter of each ring. They lift off the pegs, and are meant to be used one at a
time.
The periphery of the upper plate is grooved to guide two cords over a pair of
diametrically opposed pulleys and thence down to two equal load hangers. This
system, when loaded with equal weights, produces a pure torsional drive for the
plate that acts as a turntable on the friction rings. By loading the turntable a
normal force or pressure acts between the upper plate and the ring in use.
Table 1 - Weight Set for Clutch Experiment
No.
Weight (N)
No.
Weight (N)
No.
Weight (N)
No.
Weight (N)
4 50 2 10 4 2 2 0.5
2 20 2 5 2 1
Procedure
The apparatus must be kept free of oil or grease (the bearings are sealed), and the
turntable must NEVER be put in place without a friction ring in position to keep the
turntable off the ring location pegs.

Practical Manual
Figure C1: Key Dimensions for Clutch Apparatus
1. Check the mass of the turntable using the weighing scales: MT kg.
2. Calculate the self load due to gravity of the turntable: WT Newtons
3. Check the mass of the hangers using the weighing scales: MH kg.
4. Calculate the self load due to gravity of the hangers: WH Newtons
5. Measure the mean diameter at which the torque is applied: D mm.
6. Measure the internal and outer diameters of each of the friction rings while the
turntable is removed (push the central locating spigot upward to help lift it).
Ring
Inner Diameter
I.D. (mm)
Outer Diameter
O.D. (mm)
Mean Radius
Rmean = (OD+ID)/4
A: Small
B: Medium
C: Large
7. Remove the two larger friction rings, leaving the smallest (A) in position, and
replace the turntable with the torque cords taken over their pulleys. Hook on
the load hangers. With only the self-weight of the turntable as its load add
equal value weights to the load hangers until, with a slight start, the turntable
very slowly revolves at a constant speed. Record the load per hanger in the
table on the following page. Repeat the procedure adding increments of 50N
onto the turntable up to 200N.

Practical Manual
8. Unload and remove the turntable. Exchange the friction ring for the next larger
one and repeat the above procedure. Finally, use the largest friction ring in a
similar way.
Results A: Small B: Medium C: Large
Rmean
Nominal
Load
Actual W:
Self Load +
Added Load
Load on 1
Pulley, L =
WH + WA
Torque
L x D
Load on 1
Pulley, L =
WH + WA
Torque
L x D
Load on 1
Pulley, L =
WH + WA
Torque
L x D
Self-Load
50 N
100 N
150 N
200 N
9. At the end of the experiment, position all three friction rings on the stationary
plate and replace the turntable.
On a single graph plot the Torque vs Load for each ring. Note the nature of the
graphs and comment on the relationship between the three sets of readings.
On a second graph plot Torque vs Rmean for a load of 150 Newtons on the friction
rings. Again comment on the form of the graph and its relation to the theory.
Final Challenge:
From your graphs calculate the coefficient of friction between the clutch material
and the underside of the turntable and suggest why using the second graph for this
calculation might improve the result.

Practical Manual
Planar Mechanisms Experiment
Introduction
The motion of complex assemblies is determined by the configuration of links and
joints. Here we have reduced the complexity by limiting the movements to be
planar and controlling the scope of the links and the joints. Using the three
configurations supplied the operation of rotational and sliding joints may be
examined and explored. Care should be taken in the execution of the experiment
as the equipment is designed to disassemble easily, allowing for insertion of the
drawing paper. The information recorded during the experiment will be used as the
basis for for calculations later on, so errors in the 'operational' element of this
laboratory will affect the end results and possibly marks.
The design and use of machines demands a knowledge of the overall motion and
the relative motion of all parts of the machine. Another issue is the concept of how
specific motions can be generated. In some cases it is self evident, but in others
some ingenious mechanisms only careful analysis will show the precise nature of
movements.
Machines are made up from a number of parts connected together in various ways
to produce the required movements. Two parts of a machine with are in contact,
and which undergo relative motion are called a pair. The types of relative motion
commonly required are sliding, turning (rotation) or screwing. The pairs which
permit these motions are known as lower pairs. All other motions (partial turning,
partial sliding) are called higher pairs.
Goals of the Experiment
• Understand the movement of interconnected rotating links
• Understand the behaviour of pinned and sliding joints
• Practice velocity and acceleration diagrams
• To investigate the loci of pairs on a selection of mechanisms.
Two mechanisms are available on the drawing boards -
• Slotted Link (Scotch Yoke)
• Single & Double Action Toggle

Practical Manual
Slotted Link (Scotch Yoke)
Introduction
The slotted frame A is fixed and also carries the fixed axis 0 of the revolving link
OP. The slotted link BC contains the slider pinned to the revolving link at P. As OP
revolves the element BC reciprocates in the slotted frame with a pure simple
harmonic motion whose amplitude is governed by the radius OP. Two positions for
P at 50 and 25 mm radii are provided. This mechanism is always used where it is
required to produce perfect simple harmonic motion in a line. One application is for
driving the paddle of a wave making machine in hydrological models. The link OP
is commonly part of a complete disc which then acts as a flywheel to even out
variations in the forces and driving torque.
Figure PM1 - Slotted Link Apparatus
Procedure
To plot the locus of Q it will be necessary to use a base line related to the angular
position of OP. To assist in this P is on a circular protractor with a centre 0. Place a
sheet of graph paper under the link ensuring, by eye, that the graoh paper is
aligned with the movement of the end of the link. The vertical lines are now
perpendicular to OQ. Choose a line outside the range of travel of the point Q and
set out points along this line, at a scale of 1 cm = 20°, providing points at 20°
intervals from 0 to 80° and then from 90° to 350° with a final point at 360°.

Practical Manual
With the protractor 0° on the line OQ (Q at its leftmost position) and P pinned at 50
mm radius mark the position of the point Q at the intersections of the line visible
through Q and the lines representing 0° and 360°. Advance the protractor by 20°
and mark the intersection of the position of Q and the 20° line.
Repeat this procedure every 20° up to the 80° point and then advance 10° to 90°.
Carry on as before plotting the locus or Q as the link OP revolves.
Repeat the whole procedure with P at 25 mm radius.
Using a French curve, or by eye, draw in the two loci of P through the plotted
points. The loci should be cosine curves. Add a base line through the 90° and 270°
points and check the symmetry of the loci and the amplitudes. If we assume that
the disk is rotating at a constant speed of 1 rad/s mark this base line in seconds.
Reporting
Include the graph, or a copy, in your log book. Comment on your observations and
indicate how you would generate a sine wave from the equipment. Find the slope
of the line at 90° and comment on your finding. If the general equation for a cosine
function is y = x Cos(θ - φ), where φ is a constant and θ the angle of rotation,
describe how the equipment could be used to generate this graph.

Practical Manual
Single & Double Action Toggle
Introduction
In the single toggle mechanism Figure PM3 the links AB and BC are pivoted at B to
the connecting rod BD which is operated by a crank OD. The link CB pivots about
the fixed point C, and A is attached to a sliding member constrained to move in a
straight line. Any force applied to BD by the crank OD is greatly magnified at A due
to the relatively slight movement produced at A compared with that of D. The force
at A attains maximum value when the links AB and BC are in line. The pivot B does
not travel beyond the line of action drawn between A and C. Consequently its use
is normally restricted to machines where only a single stroke is required to
complete the operation.
Figure PM3 - Single Action Toggle
Where two strokes are required it is usual to employ the double action mechanism
shown in Figure PM4. The location of the crank is such that the links AB and BC
are straightened before the crank has completed one half of its revolution. Thus,
after the first stroke the links are carried beyond the centre line and, as the crank
continues to revolve, a second working stroke is produced. The first and second
strokes may be of equal or unequal lengths dependent upon the position of the
crank relative to the line of the straightened toggle. Nevertheless the rightmost
travel of A will be the same for the two strokes.

Practical Manual
Figure PM4 - Double Action Toggle
As the maximum force coincides with the limit of travel, this kind of mechanism
could be used for upsetting the head of a nail on round wire rod, or in stone
crushing or embossing machines.
Objectives
• To determine the motion of point A on the mechanism as link OD rotates at
constant angular velocity.
• To understand space and velocity diagrams
• To compare mathematical, graphical and experimental answers
Procedure
The crank OD revolves on a removable peg which can be located in one of two
holes in the drawing board. For accuracy a 360o
disk is provided to determine the
orientation of the crank, the 0o
position is in line with the crank when the point A is
at the left end of its travel. Place the peg in the lower hole insert a sheet of A3
paper and position the crank on this centre using the shorter radius OD1 = 40 mm.
Mark the position of Point C and draw the loci of A, B and D1 for one revolution of
the crank. Divide the right hand locus of D1 into approximately 30° intervals and for
each position of D1 mark and number the corresponding positions of B and A. Then
find the position of the mechanism at the mid-travel of A.
Remove the A3 paper carefully and insert and orient a piece of graph paper such
that point A tracks along one of the short lines. Clamp the paper carefully in place.
Insert the disk marked in 10o
angles under the crank and orient it carefully - the
zero degree line should be visible through the hole at point D. Mark the graph

Practical Manual
paper along the long axis (outside the travel of point A) at 20°/10mm. With the
crank at 0° mark the position of A on the graph paper at the 0° and 360° lines.
Repeat this at 20° intervals (if required use 10° or 30° increments) to mark the
intermediate positions of point A on the graph paper. Draw the curve for the locus
of A.
Move the crank centre peg to the upper hole and change the crank radius to OD2 =
65 mm. Insert a new A3 sheet and draw the loci of A, B and D2 for one revolution
of the crank. Mark and number sufficient corresponding positions of D2, B and A to
show how the mechanism performs. Replace the A3 sheet, mark and draw the
position of the mechanism with the crank in the 50° position. Remove the A3
paper and insert a new sheet of graph paper, aligning it as before and marking the
long axis in degrees (10mm = 20°). As before use your judgement to mark enough
points to allow you to draw the graph showing the position of A with respect to the
crank angle.
Analysis
• If the crank is rotating anti-clockwise at 30 rpm.
• Mark the long axis of the graph of A vs θ with the appropriate time in
seconds.
• Determine from the graph the velocity of the point A at the 50o
position.
• From the space diagram, using graph paper draw the velocity diagram
for the mechanism, with the crank at the 50o
position
Reporting
Discuss the differences in the two mechanisms and the reasons for the difference.
Explain how the system magnifies the force available at point A.
The two strokes of the double toggle have different amplitudes. Determine those
amplitudes and suggest where the centre O should be positioned for equal strokes
of A? What further modification is needed to produce two equal strokes of A of the
same length as the single toggle did?
Include all graphs and diagrams in your report.

Practical Manual
Gyroscope
Introduction
A gyroscope is made up of a rotor spinning about an axis. Any attempt to change
the axis of rotation requires work as it is an attempt to change the angular
momentum. As long as the system keeps spinning the orientation of the axis of
rotation will remain constant unless an outside force or couple is applied. The rotor
is often kept spinning by driving it with an electric motor. Mounting the spinning
rotor in gimbals which isolate it from any external couple results in a system
capable of providing a constant reference direction which is used for example as a
gyro-compass in moving vehicles. Alternatively, fixing the axis of rotation with
respect to the vehicle generates forces and couples at the mounting points when
the vehicle accelerates. These can form the inputs of an electronic system which
controls actuators on the vehicle to provide gyro-stabilizers
Goals
• Investigate the direction of the gyroscopic couple
• Determination of the Moment of Inertia of the Rotor
• Investigate the relationship between the gyroscopic couple, angular
velocity of the rotor and the precession of the axis of rotation
Background
Whenever the direction of the axis of rotation of a body is changed gyroscopic
action occurs. The angular momentum of the rotating body causes the axis of
rotation to remain in the same direction so long as no external couple acts on the
body. If a couple is applied about an axis normal to the axis of rotation, a torque
reaction is produced which results in the axis of rotation turning in the plane in
which lie the axis of rotation and the axis of the applied couple. This movement is
termed Precession. Conversely, if the axis is precessed a couple will result about
the orthogonal axis. This is the Gyroscopic Couple. It should be noted that, for the
same rate of precession, the Applied Couple (CA) and the Gyroscopic Couple (CR)
will have the same magnitude but the opposite sense.
The diagram below shows the directions involved. In this case the axis of rotation
is the Z axis, and the couple is applied about the X axis. The resultant motion of
the axis of rotation therefore lies in the X-Z plane, indicated on the diagram as a
rotation about the Y axis.

Practical Manual
Important Comments:
CA : Applied Couple
ω : Angular Velocity
Ω : Rate of Precession
ω >> Ω
Figure GE1
When the axes are orthogonal as shown above, it should be noted that the
direction of the precession does not change over time. If this was not the case
then the movement of the axis of precession would also generate a gyroscopic
couple. The equipment used in this experiment has been designed to ensure that
there is no change in the axis of precession.
Procedure
The equipment is covered by a plexiglass dome for your safety. Before
removing the dome ensure that all controls are switched off and that the
system is at rest. After any adjustments, the cover must be replaced and
securely fastened before the controls are switched on.
Failure to observe this rule will result in zero marks being awarded to the
group for this laboratory, a penalty which will affect the final mark for this
module.
1. Determination of the Direction of the Gyroscopic Couple
1. Ensure that the rotor assembly is balanced about the pivot point on the
bracket. If necessary move the balance weight on the torque arm by
slackening the retaining screw. Take care to re-tighten the screw in the new
position. Replace the safety cover.

Practical Manual
2. Set the motor driving the rotor running and check to ensure that the system
is still balanced and that no precession of the rotor occurs. Note the
direction of rotation of the rotor.
3. Set the precession motor running. The balance of the rotor should be
affected by this action. Note the direction of the precession and whether the
torque arm rises or fall. The arm will move in the same direction as the
gyroscopic couple.
4. Using the plugs on the front of the equipment reverse the polarity of the
rotor motor and repeat steps 2 & 3. Then reverse the polarity of the
precession motor and repeat. Finally change the polarity of the rotor motor
back to its original direction and repeat.
5. Draw schematics of the gyroscope similar to that shown below and indicate
the relationships derived from these observations.
Figure GE2
2. Determination of the Moment of Inertia of the Armature
1. Fold out the support arm attached to the casing.
2. Hang the spare armature from the arm using the wires.
3. Carefully twist the assembly about its vertical axis, taking care not to move
the axis too much and ensuring that the wires are kept taut, by about 10o
and then release. The armature will start to twist about the vertical axis
passing the equilibrium position and then oscillating about that position.
4. Use the stopwatch to determine the time for (say) 50 oscillations and
calculate the periodic time, T. Repeat until you are satisfied that you have a
good result.
5. Measure the length of the wires, l, and the distance between them, d.
6. Given that the mass of the armature is 1.09kg, calculate the moment of
inertia of the rotor using the equation below:

Practical Manual
l
d
mg
T
I
44
2
2
2
π
=
where m is the mass of the armature and g is the acceleration due to
gravity.
3. Determination of the relationship between the angular velocity of the
rotor, the Gyroscopic Couple and the Rate of Precession
In this part you will vary two control parameters, Gyroscopic Torque (CR) and Rotor
Speed (ω), and measure the resultant rate of Precession (Ω). The values for the
control parameters are given in the table below:
For each combination of the input parameters:
1. Add the mass onto the end of the torque arm. Calculate the torque on the
system and the direction. This is the Applied Couple, CA.
2. Connect the rotor and precession motor controllers so that the gyroscope
will tend to raise the torque arm.
3. Set the rotor speed using the indicator.
4. Vary the rate of precession until the torque arm is level (the scribed line on
the arm should line up with the stripes on the bracket). In this state the
Gyroscopic Couple is equal to the Applied Couple.
5. Measure the rate of precession by timing a suitable number of revolutions of
the assembly (total measuring time should be at least 30 seconds). Record
the rate of precession.

Practical Manual
Table GE1
Balance
Mass
Couple
CR
Nominal
Rotor
Speed
ω
Actual Rotor Speed
ω
Rate of Precession
Ω
1/ Ω
g Nm RPM RPM rad/s rev/s Rad/s s
3,000
2,500
2,000
1,500
1,000
100
250
3,000
2,500
2,000
1,500
1,000
150
700
3,000
2,500
2,000
1,500
250
1,000
3,000
2,500
2,000
350
1,500

Practical Manual
Analysis of the Results
The results obtained from these three experiments can now be used to verify the
theoretical relationship between the gyroscopic couple, the angular momentum and
the rate of precession.
CR = Iω X Ω
For each of the masses applied to the torque bar, the balance position represents
a constant gyroscopic couple, CR. Since the moment of inertia, I, is constant then
for a particular mass the equation can be rewritten as
CR/I = K = ω X Ω
where K is a constant.
Since ω and Ω are perpendicular then
ω = K/Ω or ω = Kx(1/Ω)
This means that for a given mass the graph of the reciprocal of the rate of
precession (1/Ω) against the rotor velocity (ω) should be a straight line.
1. Using a single graph plot 1/Ω vs ω for each mass.
2. Using the information from this graph, plot CR vs Ω for ω = 300 rad/s.
3. Again from the first graph plot CR vs ω for Ω = 2.5 rad/s.
Report
In your report compare the experimental findings with the theory.
From graph 1 estimate a value for I and compare this with the results from part 2.
Draw a Free Body Diagram of the armature in part 2. If possible show how the
restoring torque returning the armature to the equilibrium is a function of the angle
of rotation.
Explain any sources of errors and estimate their magnitude.

Practical Manual
Static & Dynamic Balance Experiment
Introduction
Shafts which revolve at high speeds must be carefully balanced if they are not to
be a source of vibration. If the shaft is only just out of balance and revolves slowly
the vibration may merely be a nuisance but catastrophic failure can occur at high
speeds even if the imbalance is small.
For example if the front wheel of a car is slightly out of balance this may be felt as
a vibration of the steering wheel. However if the wheel is seriously out of balance,
control of the car may be difficult and the wheel bearings and suspension will wear
rapidly, especially if the frequency of vibration coincides with any of the natural
frequencies of the system. These problems can be avoided if a small mass is
placed at a carefully determined point on the wheel rim.
It is even more important to ensure that the shaft and rotors of gas turbine engines
are very accurately balanced, since they may rotate at speeds between 15,000 and
50,000 rev/min. At such speeds even slight imbalance can cause vibration and
rapid deterioration of the bearings leading to catastrophic failure of the engine.
It is not enough to place the balancing mass such that the shaft will remain in any
stationary position, i.e. static balance. When the shaft rotates, periodic centrifugal
forces may be developed which give rise to vibration. The shaft has to be
balanced both statically and dynamically.
Usually, shafts are balanced on a machine which tells the operator exactly where
he should either place a balancing mass or remove material. The apparatus
requires the student to balance a shaft by calculation or by using a graphical
technique, and then to assess the accuracy of his results by setting up and running
a motor driven shaft. The shaft is deliberately made out of balance by clamping
four blocks to it, the student being required to find the positions of the third and/or
fourth blocks necessary to statically and dynamically balance the shaft.
Aims
The goals of this experiment are:
• To gain an understanding of the meaning of the terms static and dynamic
balance.
• To demonstrate the dynamic balancing of an unbalanced shaft using two
added eccentric masses.
Theory
A shaft with masses mounted on it can be both statically and dynamically
balanced. If it is statically balanced, it will stay in any angular position without
rotating. If it is dynamically balanced, it can be rotated at any speed without

Practical Manual
vibration. It will be shown that if a shaft is dynamically balanced it is automatically
in static balance, but the reverse is not necessarily true.
Static Balance
Figure SDB1 shows a simple situation where two masses are mounted on a shaft.
If the shaft is to be statically balanced, the moment due to weight of mass (1)
tending to rotate the shaft anti-clockwise must equal that of mass (2) trying to turn
the shaft in the opposite direction.
Figure SDB 1 - Static Balance
Hence for static balance,
m1r1cosα1 = m2r2cosα2
The same principle holds if there are more than two masses mounted on the shaft,
as shown in figure SDB2.
Figure SDB 2 - Static Balance of Three masses
The moments tending to turn the shaft due to the out of balance masses are:-

Practical Manual
Mass Moment Direction
1 m1.g.r1cosα1 Anticlockwise
2 m2.g.r2cosα2 Clockwise
3 m3.g.r3cosα3 Clockwise
For static balance,
m1r1cosα1 = m2r2cosα2 + m3r3cosα3
In general the values of m, r and α have to be chosen such that the shaft is in
balance. However, for this experiment the product W.r can be measured directly
for each mass and only the angular positions have to be determined for static
balance.
If the angular positions of two of the masses are fixed, the position of the third can
be found either by trigonometry or by drawing. The latter technique uses the idea
that moments can be represented by vectors as shown in figure SDB3(a). The
moment vector has a length proportional to the product mr and is drawn parallel to
the direction of the mass from the centre of rotation.
Figure SDB 3 - (a) MR Vectors from Centre to Mass (b) Rearrangement to
show Closed Polygon
For static balance the triangle of moments must close and the direction of the
unknown moment is chosen accordingly. If there are more than three masses, the
moment figure is a closed polygon as shown in Figure SDB3(b). The order in
which the vectors are drawn does not matter, as indicated by the two examples on
the figure.
If on drawing the closing vector, its direction is opposite to the assumed position of
that mass, the position of the mass must be reversed for balance.
Dynamic Balance
The masses are subjected to centrifugal forces when the shaft is rotating. Two
conditions must be satisfied if the shaft is not to vibrate as it rotates:
a) There must be no out of balance centrifugal force trying to deflect the shaft.
b) There must be no out of balance moment or couple trying to twist the axis of
the shaft.

Practical Manual
If these conditions are not fulfilled, the shaft is not dynamically balanced.
Figure SDB 4 - Dynamic Out-of-Balance for a Two Mass System
Applying condition a) to the shaft shown in Figure SDB.4 gives:
F1 = F2
The centrifugal force is mrω2
Therefore:
m1r1ω2
= m2r2ω2
Since the angular velocity, ω, is common to both sides then for dynamic balance
m1r1 = m2r2
This is the same result for the static balance of the shaft. Therefore if a shaft is
dynamically balanced it will also be statically balanced.
The second condition is satisfied by taking moments about some convenient datum
such as one of the bearings.
Thus,
a1F1 = a2F2
For this simple case where m1 and m2 are diametrically opposite and F1 = F2
(condition a) then dynamic balance can only be achieved by having a1 = a2 which
means that the two masses must be mounted at the same point on the shaft.
Unlike static balancing where the position of the masses along the shaft is not
important, the dynamic twisting moments on the shaft have to be eliminated by
placing the masses in carefully calculated positions. If the shaft is statically
balanced it does not follow that it is also dynamically balanced.

Practical Manual
In order for static balance to be achieved the sum of the vectors representing the
couple due to each rotor must form a closed polygon. In the case where there are
three rotors, the simplest arrangement to give balance is shown in
Figure SDB 5 - Dynamically Balanced Shaft with 3 Eccentric Masses
In this case, it is clear from the first requirement that:
F2 = 2 x F1
The second criterion then says that:
a2F2 = a1F1 + a3F1
2a2F1 = a1F1 + a3F1
a2 = (a1 + a3)/2
or that the eccentric mass in the middle has twice the m.r value of the two masses
on either side and is equidistant from both masses.
The general case, where the eccentric masses differ on each rotor and the
directions are not exactly opposite is shown in Figure SDB6.

Practical Manual
Figure SDB 6 - General Case for Three Out-of-Balance Masses
The method for balancing such shafts requires the addition of two extra eccentric
masses to the system at locations chosen by the engineer. These masses are
determined in many ways. One method will be outlined in the lectures, while this
experiment shows a method that uses preset out of balance forces and determines
the orientation and position of the masses relative to the eccentric masses already
on the shaft.
Equipment:
The Static and Dynamic Balance Equipment consists of a shaft mounted on a plate
isolated from the base by rubber bushes. A motor, also attached to the plate, may
be used to drive the shaft using a belt. Four identical eccentric mass blocks are
provided, together with four individual inserts which may be used to alter the
imbalance on each block. An extension shaft and pulley are stored on the base
and used in conjunction with the string/buckets and ball bearings to determine the
imbalance associated with each block. Two hexagonal keys (Allen keys) are
provided to clamp the blocks on the shaft and the inserts into the blocks. Two
guides, one on the pulley at the end of the shaft and one on the plate, are used to
measure the relative angle and position along the shaft of each block. At either
side of the plate two clamps allow the plate to be locked to the base. The plate
should be clamped for the static parts of the experiment. If the motor is in use then
the clamps should be released and secured away from the plate by tightening the
screws.

Practical Manual
Figure SDB 7 - Eccentric Masses and Axial Positioning
Experimental Procedure:
The experiment is divided into three parts:
1. Observation of the phenomenon
2. Calculation of a solution to a posed problem
3. Implementation of the solution
Demonstration of Static and Dynamic Balance
For this part of the experiment the four blocks will be used without the inserts. This
provides four identical eccentric masses for use in demonstration of static and
dynamic balance.
Step 1 - Static Imbalance
Lock the plate to the base. Attach one of the blocks securely to the shaft near the
pulley. The mark on the protractor should align with zero when the block is
positioned against the guide. Slide the guide clear of the block. Rotate the shaft
and observe the behaviour. Record your observations.
Step 2 - Static Balance
Attach a second block near the opposite end of the shaft. Align this block such that
the protractor reads 180o
. Rotate the shaft and observe the behaviour. Record
this and compare with step 1.
Step 3 - Dynamic Imbalance
Release the plate and secure the clamps clear of the plate. Attach the belt
between the motor and the pulley on the end of the shaft. Ensure that all loose
components are removed from the equipment and then place the safety cover over
the motor and shaft. Switch on the motor controller and the motor. Slowly
increase the speed of the motor and observe the behaviour of the shaft and plate.

Practical Manual
Record your observations. Switch off both the motor & controller and allow the
shaft to come to rest before removing cover.
Step 3 - Dynamic Balance
There are three steps to this part of the experiment
1. Preparation of the imbalances
Clamp the plate and remove the blocks from the shaft. Insert the four
circular imbalances into the blocks and clamp them securely. Attach one of
the blocks to the shaft with the protractor reading 0o
. Remove the drive belt
from the motor and attach the pulley extension to the shaft so that the pulley
overhangs the end of the bench. Loop the string and buckets around the
pulley so that there is no slip. Add ball bearings to one bucket until the
protractor reads 90o
. At this point the moment due to the bearings is equal
to the eccentric moment of the block. Record the number of bearings and
repeat for the other three blocks.
2. Calculation of the solution
The problem for solution must be posed carefully if a satisfactory solution is
to be found in the time available. The relative axial position and angular
orientation of the two largest eccentricities should be selected first and will
represent the dynamic imbalance to be corrected. An axial spacing of 100
mm and relative angle of 150o
provide a reasonable starting point. The
main limitations are the total length of the shaft and the thickness of each
block. The solution is calculated graphically in two parts. Initially, the static
balance of the system is obtained by
a. Draw vectors representing the two imbalances set above. These
have a length proportional to the number of ball bearings and a
direction relative to the angular orientation. The vector should go
from the centre of the shaft along the block. The drawing should like
Figure SDB 3b with the m1r1 and m2r2 vectors only.
b. On the graph, knowing the lengths of the other two imbalances,
complete the four sided closed polygon. Record the angular
orientation of the two balancing vectors.
The axial position of the two balancing masses needs to be calculated next.
In this case, it is simplest to take the largest eccentricity as the reference
axial location, eliminating it from this part of the calculation.
c. On a new diagram draw a vector representing the axial turning
moment of the second eccentricity. The length is proportional to the
number of ball bearings and the axial distance from the reference.
d. Complete the closed triangle using the directions for the balancing
eccentricities found in b. From the scale calculate the axial distance
associated with each eccentricity.
Assemble all the information into a table indicating the eccentricity (mr) of
each block, its axial location (l) and its angular orientation (θ).

Practical Manual
Figure SDB 8 - Graphical Solution to Dynamic Balance
3. Test the result
Carefully attach the blocks to the shaft at the locations and orientations in
the table. Remove the pulley from the system and reattach the motor drive
belt. Release the platform clamps and secure them to the base. Put safety
cover in place and run motor. Record whether dynamic balance has been
achieved and if necessary revise calculations.
Reporting
Your report should give a detailed description of your observations and include all
rough work and calculations. The graphs should be clearly labelled and neat.

Mm203 labmanual

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