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Dynamics of Mechanical Systems
Inertia and Efficiency Laboratory
1 Overview
The objectives of this laboratory are to examine some very
common mechanical drive components, and hence to answer the
following questions:
· How efficient is a typical geared transmission system?
· How do gearing and efficiency affect the apparent inertia of a
geared system as observed at (i.e. referred to) one of the shafts?
The learning objectives are more generic:
· To give experience of the kinematic equations relating
displacement, velocity, acceleration and time of travel of a

particle.
· To give experience of applying Newton’s second law to linear
and rotational systems.
· To introduce the concept of mechanical power and its
relationship to torque and angular velocity.
The completed question sheet must be submitted to the
laboratory demonstrator at the end of the lab, and is worth 6%
of module mark.
Please fill in the sheet neatly (initially in pencil, perhaps, then
in ink once correct!) as you will be handing it in with the
remainder of your report.
Note: it is a matter of Departmental policy that students do not
undertake laboratories unless they are equipped with safety
shoes (and laboratory coat). The reasons for this policy are
apparent from the present lab, where descending masses are
involved, and could cause injury if they run out of control.
Safety shoes therefore MUST be worn.
Also, keep fingers clear of rotating parts, whether guarded or
not, taking particular care when winding the cord onto the
capstans. In particular, do not touch (or try to stop) the flywheel
when it is rotating rapidly. Do not move the rig around on the
bench – if its position needs changing, please ask the lab
supervisor.

1
2 Mechanical efficiency, inertia and gearing
2.1 Theory
2.1.1 Kinematics: motion in a straight line
The motion of a particle in a straight line under constant
acceleration is described by the following equations:
2
where s is the distance travelled by the particle during time t, u
is the initial velocity of the particle, v is its final velocity, and a
is the acceleration of the particle.
To think about: which one of these equations will you need to
use to calculate the acceleration of a mass as it accelerates from
rest to cover a distance s in time t? (Hint: note that u is zero
while v is both unknown and irrelevant. You will need to
rearrange one of the above equations to obtain a in terms of s
and t).
2.2 Kinematics: gears and similar devices

If two meshing gears1 have numbers of teeth N1 and N2 and are
connected to the input and output shafts respectively, then the
gear ratio n is said to be the ratio of the input rotational angle
to the output rotational angle (and angular velocity and angular
acceleration), see Fig. 1:
N
2
1
1
1
N

1
2
2
2
Note that n is sometimes expressed as a ratio N2:N1 in
simplified form e.g. 3:1. A ratio of n > 1 implies a reduction

ratio (output
rotates slower than input), but note that in the present case n<1
Fig. 1 as the output (driving the flywheel) rotates faster than the
input.
1 The same mathematics applies to toothed belts, roller chains
etc.; similar analysis also applies to vee belts and similar
friction-based drive systems, but there is no concept of
“teeth” in such a situation and ratios tend to be approximate.
2
If the geared system is perfectly efficient, the torques L1 and
L2 on the two shafts scale inversely with the ratio of speeds;
however, practical geared systems are imperfect and have an
ced “eta”), so in practice the
relationship is
L
N
2
L

N
1
2 1
If a compound gear system is used (as in the present case), the
overall ratio is the product of the ratios of each pair of gears, so
for Fig. 2:
N
N
2 1
N N
4

1 4
2.3 Kinematics: Tangential drives
A cable or cord under tension T, if wrapped around a capstan of
radius r, exerts a torque L:
Fig. 2
The same cable or cord, if it is moving along its
length with velocity v and accelerating with acceleration a,
v
,
a

r
r
2.4 Dynamics: Newton’s second law for linear and rotational
motion
Newton's second law states that a mass, m, experiences an
acceleration, a, proportional to the size of the net force, F,
applied to it, given by:
In the present experiment, the mass m descends with
acceleration a (assumed constant); the gravitational force mg
acts on it (downwards, of course!) and the tension inthe cord T
partially opposes this by providing a smaller upward force.
Within your calculations you will needto write down the
expression for Newton’s second law involving T, m, g and a and
hence express T in terms of m, g and a.
It can also be shown (quoted here without proof) that an
equation similar to F=ma can be written for rotational motion
about a fixed axis:

T
Turn this into a free body diagram to get expression for T in
terms of m, g and a
J
Fig. 3
3

where L is the torque applied to a body, J is its moment of
inertia about the axis of
acceleration resulting from the torque (see Fig. 3).
4
32
2.5 Gearing and referred inertias
If an object (e.g. a flywheel) with moment of inertia J about its
axis of rotation is driven via a gear train (either simple or
compound) with gear ratio n, then the apparent moment of
inertia J’ of the system as observed at the input shaft (i.e. the
moment of inertia referred to the input shaft) can be shown to

be:
J
n
2
For a simple gear train driving a flywheel with a moment
of inertia, J, the rotational equation of motion with
respect to the output shaft is:
L
2

2
and (for a perfectly efficient geared system) the
rotational equation of motion with respect to the input
shaft is:
1
1
Fig. 4

2.6 Experiment 1: Efficiency
32 T
Flywheel
Upper
capstan
64 T
32 T 48 T
Lower capstan
Fig. 5
4

1. Make a note of the number on the piece of apparatus by
recording it in this box
2. Check that the equipment is firmly clamped to the bench
using the G clamp and board provided; this should already have
been done for you.
3. The lower capstan has a long length of cord attached to it
which will initially be slack. Ensure there is a 300g mass
attached to the cord, then by spinning the flywheel manually
and guiding the cord, wind it neatly onto the capstan so that it
does not build up. With care you should be able to wind it so
that it takes up most of the width in a single layer without
crossing over itself. Wind it up until the mass is about 150mm
above the surface of the bench, then one of the team should hold
the flywheel to avoid the weight unwinding the cord (refer to
Fig. 5).
4. Hold the small red (upper) capstan with 100g (consisting of
the hanger only) hanging from the cord. Mount the upper
capstan on the top shaft but do not yet tighten the grubscrew.
5. Rotate the upper capstan on the shaft so that the 100g mass is
just clear of the bench, then tighten the grub screw to keep it in
place on the shaft. It should now be possible to let go of the
flywheel without the system turning.
6. On the lower capstan (300g) mass, start gently adding 10g
masses one at a time, each time gently starting the rotation of
the flywheel. Keep adding these masses until the lower mass
just creeps down slowly without stalling or accelerating. This
corresponds to the variousfrictional losses and inefficiencies
being overcome. If necessary, gently wind the flywheel back so
that the masses return to their starting positions. Note the mass
on the lower capstan hanger required to maintain continuous

movement, and enter your observations in the table below.
Record sufficient repeat readings that you can estimate an error
in you result and calculate an average value.
7. Repeat step 6 starting with mass combinations of 600g
/200g, 900g/300g, 1200g/400g. In the last case, you will
probably find it convenient to add an extra 100g mass to make
1300g rather than adding a lot of 10g masses.
8. Note down any problems you experienced, and your actions,
for the report.
Mass on upper
Mass on lower
Repeat measurements
capstan (g)
capstan (g)

5
To process your results, plot the load on the lower capstan
required to overcome losses against the load on the upper
capstan. Draw the best straight line through the points, noting
that it will not quite go through the origin. Note: thediameters
of the upper and lower capstans are equal so torque is
directlyrelated to load.
1500
1400
1300
1200
1100
1000
900
Load on

800
lower
700
capstan
(g)
600
500
400
300
200
100

0
0 50 100 150 200 250 300 350 400 Load on upper capstan (g)
From your graph estimate:
· the ratio of torques on the lower to upper capstans required to
cause rotation (using the gradient of the graph)
· the efficiency of the geared system (knowing the gear ratio
and the ratio of the torques)
· the mass (and hence the torque) required at the lower capstan
to cause rotation (i.e. to overcome bearing friction) when there
is no load applied to the system (using the intercept of the
graph).
6
2.7 Experiment 2: Inertia of a geared system
The apparatus used in this experiment is as shown in Fig. 6.
1. Using the metre rule provided, measure the height of the
bench top above the floor.
2. Using the dial callipers, measure the diameter of the lower
capstan with and without the cord wrapped on it. Average the
readings to obtain the effective (pitch circle) diameter of the
cord on the drum, and divide this value by 2 to obtain the

effective radius.
32 T
Flywheel
64 T
32 T
48 T
Lower
capstan
Fig. 6.
3. Measure the diameter and length of the flywheel. Note also
that the density of 316 stainless steel is 8000 kg/m3.

4. Hold the flywheel to stop it rotating, then loosen the
grubscrew holding the upper capstan and remove it and the mass
hanging from it. Gently guide the other mass so it comes to rest
on the bench just below the lower capstan.
5. Remove masses from the hanger so that the total mass is
200g.
6. By rotating the flywheel by hand, adjust the position of the
mass so that it is hanging freely with its base just level with the
bench top. (In practice, it is easiest to wind the system until the
mass is just supported, then hold the flywheel stationary as you
slide the mass off the edge of the bench. If the mass catches on
the edge of the bench, ask the supervisor or demonstrator to
adjust the apparatus. DO NOT attempt to adjust theposition of
the rig yourself).
7. While simultaneously starting the stopwatch, release the
flywheel so that the mass gradually accelerates towards the
floor, speeding up the geared system from rest. Time the
interval from releasing the flywheel to the mass reaching the
floor, and make a note of this time. Record your observations in
the table of time vs. mass given below.
8. Do not brake the flywheel manually to stop it – let it stop of
its own accord as follows. After the mass touches the ground, it
will
“rebound” due to the inertia of the system and the cord will
start to rewind onto the capstan. Allow this to occur twice (so
that the cord rewinds again on the front of the capstan), catch
the stationary flywheel as the top of the second rebound is
reached, then wind the mass back up the rest of the way,
ensuring that the cord winds neatly onto the lower capstan
without crossing or building-up.

9. Repeat steps 6 - 8 for masses of 300g, 400g, 500g and 600g.
7
Mass on cord
Time of travel to
Repeat measurements
(g)
floor (s)
To process your results, set up a table similar to the following

and the torque L applied by the cord to the capstan. If at all
possible, complete at least one line of the table, relating to the
heaviest mass (600g) and get the demonstrator to check it, so
you are clear in your minds about how to complete the table.
Col
1
2
3
4
5
6
7
8
m
m
mg
t
a

T
L
(g)
(kg)
(N)
(s)
(ms-2)
(N)
(rad s-2)
(arising from
cord tension)
(Nm)

600
Before you start processing your results, be very clear which
formulae youwill be using to convert:
· Time of travel t of the mass over distance s to give the
acceleration a
· The acceleration a of mass m, along with its self-weight mg,
to give the tension in the cord
· Tension in the cord to the resulting torque on the lower
capstan

· Acceleration of the mass (and of the cord) to give the angular
acceleration of the lower capstan
You are strongly advised to insert these formulae into the table
above and check them with the demonstrator before proceeding
with your calculations. Perform the line of the calculations
relating to 600 g load withinthe laboratory session so you can
be confident of what you are doing.
Hence (after completing the laboratory session) finish the table
gradient of this line
8
which is the experimentally-measured apparent inertia of the
system referred to the lower capstan.
Also (after the laboratory session) perform a theoretical
calculation of the apparent moment of inertia of the system
referred to the lower capstan, using a simple calculation which
considers only the moment of inertia of the flywheel (calculated
from its dimensions and density, which is 8000 kg/m3) and the
gear ratio connecting it to the capstan. What could have
contributed todifferences between the experimental and
theoretically-calculated values of referred inertia.

α (m/sec2)
9
3 Report
3.1 Report writing
A template for this report is available to download from
Moodle.
The report should contain the following:
(i) Heading Sheet– fill in the date of the experiment, the author

of the report and the date of the report.
No summary, methodology or theory sections are required for
this report.
(ii) Results and Calculations –see template and information
below
(iii) Discussion - see template and information below
This section should be written as a continuous piece of text so
that it draws out the conclusions from the results and answers
the discussion
points listed below. A numbered discussion listing, i.e. a series
of answers to the discussion points with no linking text, will
lose marks.
3.2 Results and calculations
The results section of the lab report should contain the
following:
measurements and an average, taken for experiment 1:
Efficiency. (7 marks)
repeated readings (5marks)
for the torque ratios, the gear efficiency and the mass required
to cause rotation when the mass on the upper capstan is zero.
(10 marks)

repeat measurements and an average, for experiment 2: Inertia
of a geared system (7 marks)
h error bars estimated from your
repeated readings (5marks)
measured values (mass and t) (5 marks)
each mass (5 marks)
10
lower capstan, J (3 marks)
referred to the lower capstan. (3 marks)
3.3 Discussion points
The following points should be addressed in the discussion in
addition to any other relevant points you may wish to make.
Your discussion should not have numbered answers and should
be written as continuous prose covering these points, the quality

of your style in this section will be assessed and be worth 5
marks.
1. Have the objectives been achieved? (3 marks)
2. Has the apparatus proved to be satisfactory? Discuss any
difficulties you had in taking the measurements and how you
dealt with them (3 marks)
4. What experimental uncertainties/errors are there and how
could they be eliminated or minimised? Consider the
uncertainties in both the repeat measurements of the mass
required to set the lower capstan in motion, and in timing the
mass travelling to the floor. (10 marks)
5. Compared with the results using a theoretically-calculated
gear ratio (based upon the numbers of teeth etc.) which of the
following quantities are scaled exactly (i.e. according to an
exact ratio) by a set of real gears: torque, position, speed,
acceleration? Why?. (12 marks)
6. How does the measured ratio L/α referred to the axis of the
long
capstan compare with the theoretical calculations of referred
inertia? With
reference to your answer to 5), explain the discrepancy. (12
marks)
Bear in mind that a discussion should be a piece of continuous
prose, not a series of answers to questions. You should back up
statements with evidence drawn from your data, so be factual
and specific, any statements that are sweeping and
unsubstantiated would lead to a low mark.

NB Any books consulted should be referenced in a section
entitled “References”at the end of the report. The format (for a
book) should be as follows:
[Author(s), title of book, publisher, place of publisher, year,
page(s) or chapter(s), ISBN]
3.4 Presentation
In industry presentation is very important and reports should be
as neat as possible and laid out well. For that reason marks are
given for good presentation. Tables should have clear headings,
including the units of the quantities and all graphs should be
clearly labelled. It is not necessary that a
11
graph should fill the whole of an A4 sheet. Try to select a
balance between accuracy of reading and clarity. Plot the
independent variable along the x-axis. If the graph includes
more than one curve then use different symbols for each and
include a key to the side of the graph. As your submission will
be electronic you will need to produce graphs electronically,
excel is available on all University teaching computers.

r5.pdfr6.pdfInertiaOverall.docxDynamics of.docx

12
Template_MM1DMS.docx
Module Name:Dynamics of mechanical systems
Module code: MM1DMS
Laboratory Name:Inertia and Efficiency Laboratory
Date of Laboratory:
Date Report Due:
Name of Student:
Student ID:
Section
Marks available for Section

Marks Awarded
Results –Tables
19
Results – Graphs
10
Results – Equations and calculations
21
Discussion – are objectives covered
3
Discussion – was equipment satisfactory
3
Discussion – Errors
10
Discussion – Comparison of results to the gears ratio
12
Discussion – Variation between experimental and calculated
inertia
12
Communication – presentation, clarity of report &
appropriateness of structure of report
10
Total
100

Information in italic blue writing should be removed from this
document before submission and is provided simply to describe
what you should include in each section.
Results and calculations
This section should be written in a manner that presents your
results succinctly covering the elements described below. Make
sure that you explain and link each of the sections. The report
should make sense to a reader who had not carried out the
experiment themselves. Tables and calculations without
explanation will receive a low mark.
It is important that your calculations and results are clearly and
neatly presented so that maximum credit can be obtained. Do
not include the rough tables and graphs produced during
lab.Calculations need to be fully documented and explained,
give one specimen set of numerical values only.
The results section of the lab report should contain the
following:
· A table containing all the readings, including repeat
measurements and an average, taken for experiment 1:
Efficiency. (7 marks)
· A graph of these resultswith error bars estimated from your
repeated readings (5 marks)
· Detail the equations used and report your calculated values for
the torque ratios, the gear efficiency and the mass required to
cause rotation when the mass on the upper capstan is zero. (10
marks)
· A table containing the mass and time values, including repeat
measurements and an average, for experiment 2: Inertia of a
geared system (7 marks)
· A graph of these resultswith error bars estimated from your
repeated readings (5 marks)

· Give the equations required to calculate α and L from your
measured values (mass and t) (5 marks)
· A table containing the calculated values of a, T, α and L for
each mass (5 marks)
· Give the experimental value for the referred inertia on the
lower capstan, J (3 marks)
· Perform a theoretical calculation for the moment of inertia
referred to the lower capstan. (3 marks)Discussion
The following points should be addressed in the discussion in
addition to any other relevant points you may wish to make.
Your discussion should not have numbered answers and should
be written as continuous prose covering these points, the quality
of your style in this section will be assessed and be worth 5
marks.
1. Have the objectives been achieved? (3 marks)
2. Has the apparatus proved to be satisfactory? Discuss any
difficulties you had in taking the measurements and how you
dealt with them (3 marks)
3. What experimental uncertainties/errors are there and how
could they be eliminated or minimised? Consider the
uncertainties in both the repeat measurements of the mass
required to set the lower capstan in motion, and in timing the
mass travelling to the floor. How do they impact on your final
results (10 marks)
4. Compared with the results using a theoretically-calculated
gear ratio
(based upon the numbers of teeth etc.) which of the following
quantities
are scaled exactly (i.e. according to an exact ratio) by a set of
real gears:
torque, position, speed, acceleration? Why?. (12 marks)
5. How does the measured ratio L/α referred to the axis of the
long

capstan compare with the theoretical calculations of referred
inertia? With
reference to your answer to 5), explain the discrepancy. (12
marks)
Bear in mind that a discussion should be a piece of continuous
prose, not a series of answers to questions. You should back up
statements with evidence drawn from your data, so be factual
and specific, any statements that are sweeping and
unsubstantiated would lead to a low mark.
Be concise and do not labour the obvious. Make a list of the
points to be made before you start writing. A rambling,
digressive discussion is wasteful of your time and that of the
reader. The discussion should not normally exceed one page
You will be expected to comment on the level of agreement
between theory and experiment. There will be limits to the
range and quality of your results, so do not read more into them
than is warranted. Develop a critical approach and aim to give
a coherent argument.
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