Mechanical EngineeringSchool of Engineering and Physical Sciences

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MODULE No: B5.1PZ_2006-2007

MODULE TITLE: Mechanical Engineering Science

ASSIGNMENT TITLE: Laboratory Report 2: Trifilar Suspension

Lecturer: Dr.

Year: 2 Term: 2 Session: 2

Grade: Name: JK

Registration Number: XXXXXXXX

Marker’s Initials: SZ Term Address:

Email: XXXXXXXX

Submission Date: 17/02/08

Group/Group Members:(If applicable)

GROUP A X

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Index

Page no.

Aim 3

Summary 3

1. Introduction 3

2. Theory/Aim 4

3. Experimental Methods 8

3.1 Procedure 8

3.2 Apparatus/Equipment 9

4. Results and Calculations 10

4.1 Data 10

4.2 Calculations 11

5. Discussion 12

6. Conclusion 15

7. References 15

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Aim

To calculate the polar mass moment of inertia of an assembly and using the result to predict the periodic time of a triflar suspension of the assembly. Using this assembly, further physical understanding of mass moment of inertia will also be demonstrated using bodies with known mass and geometry.

Summary

The moment of inertia of rigid bodies is calculated using the triflar suspension arrangement. To evaluate

the slope, the derived formulae and the given data are used. The recorded time periods for the actual

and theoretical values are analyzed and compared to study the relationship. All of these values agreed

within the estimated experimental errors.

Introduction

Moment of inertia, also called mass moment of inertia or the angular mass, is the rotational analogue

mass. That is, it is the inertia of a rigid rotating body with respect to its rotation. The moment of inertia

plays much the same role in rotational dynamics as mass does in basic dynamics, determining the

relationship between angular momentum and angular velocity, torque and angular acceleration, and

several other quantities. While a simple scalar treatment of the moment of inertia suffices for many

situations, a more advanced tensor treatment allow s the analysis of such complicated systems as

spinning tops and gyroscope motion [1].

The moment of inertia of an object about a given axis describes how difficult it is to change its angular

motion about that axis. Mass moment of inertia of a mechanical component plays an essential role

whenever a dynamic analysis is considered important for the design [1].

A trifilar suspension is a type of assembly that makes use of free torsional oscillation. It is used to

determine the moments of inertia of a body about an axis passing through its mass centre. Trifilar

suspensions are commonly used for school workshop experiments. [2] Figure below displays a standard

trifilar suspension arrangement.

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Figure 1 displays a schematic of a standard trifilar suspension arrangement.

Theory/Aim

In the experiment the periodic time is measured and compared with the theoretical periodic time. The

periodic rotation will be calculated using the calculated mass moment of inertia and the derived equations

for the theoretical time period. The polar mass moment of inertia was calculated using the Parallel Axis

Theorem. Different properties of the equipment, variations and actual measurements will affect the

results. Figure below displays a schematic diagram of Trifilar suspension.

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Figure 2 Schematic Diagram of the Trifilar Suspension Setup

2.1 Formulae

Equations that will calculate polar moment of inertia and periodic rotation are needed.

The moment of inertia of a solid object is obtained by integrating the second moment of mass about a particular axis. The general formula for inertia is [3]:

2g mkI =

Where,Ig is the inertia in kgm2 about the mass centrem is the mass in kgk is the radius of gyration about mass centre in m

In order to calculate the inertia of an assembly, the local inertia Ig needs to be increased by an amount mh2

Where,m is the local mass in kg.h is the distance between parallel axis passing through the local mass centre and the mass centre for the overall assembly.

The Parallel Axis Theory has to be applied to every component of the assembly. Thus,

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∑ += )mh(II 2g

The polar moments of inertia for some standard solids are:

Cylindrical solid

2

mrI

2

0 =

Circular tube)r(r

2

mI 2

i2

0tube +=

Square hollow section)a(a

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mI 2

i2

0sq.section +=

An assembly of three solid masses on a circular platform is suspended from three chains to form a trifilar suspension. For small oscillations about a vertical axis, the periodic time is related to the Moment of Inertia.

Figure 3 Dimensions of Trifilar Suspension

From Figure [3.1],

θ is the angle between the radius and the tangential reference line. Therefore by using the equation,

Rxθsinθ == Since θ is a very small angle

Where, R is the Radius of the circular platform.

Differentiating θ gives, dt

dθω =

Then differentiating again gives 2

2

dt

θdα =

Now,

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Ø600

Ø Ø Ø

Lθ

Ø

θ

θ

1

2

3

mg

F

L

xsinθ ==

>>> L

xmgF =

Using the standard equation for Torque, IαFR =

Hence

IαRL

xmgFR −=

=

where Rθx = and 2

2

dt

θdα =

After simplification the equation becomes

L

mgθR

dt

θdI

2

2

2

=

−

[1]

Equation for the 2nd order differential SHM is taken as

0xωdx

yd 22

2

=+[2]

Therefore, by drawing comparisons between Equation [1] and Equation [2], an equation for the angular velocity ω can be derived.

Generalizing the theoretical aspect of the experiment, w can be calculated using Integration.

( )ωtθsinθ = >> ( )ωtθωcos

dt

dθ = >>

( )ωtsinθωdt

θd 22

2

=

Putting this in Equation [1], an equation for the angular velocity can be determined.

Therefore, simplifying Equation [1] using the value for the angular acceleration the equation becomes

( )( ) 0L

mgRωtωsinI

2

=+−

This simplifies further to ( )

L

mgRωI

22 =

Therefore the angular velocity LI

mgRω

2

=

The time period is inversely proportional to the angular velocity and hence can be calculated to compare with the experimental time period.

Using the equation ω

2πT =

,

The theoretical periodic time can be calculated in terms of the mass and the moment of inertia.

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Hence 2mgR

LI2πT =

[3]

Where,

I is the Polar Moment of InertiaL is the Vertical length of the Trifilar suspensionm is the Mass of the shapes placed on the Circular platform R is the Radius of the Circular platform

Experimental Method

3.1 Procedure

The equipment to be used in the experiment is prepared. The trifiliar suspension is observed before

performing the experiment. The reference tangential line was drawn on the paper to make it correspond

with the tip of the circular platform. The Trifilar suspension was then rotated to check for any irregular

rotational movement. The stopwatch was calibrated for the zero error and the timer was adjusted to zero.

When everything was set, the pointer was kept on the reference line and the circular platform was allowed

to complete one oscillation. Nothing was placed on the platform. The time for the oscillation is note. The

time for the oscillation was noted. This process was repeated three times and the average periodic time

reading was taken into consideration. The tube was kept next on the circular platform at the centre of

gravity point and the procedure will be repeated 2 more times.

A solid mass was then positioned over the centre of mass of the circular platform and steps 3 and 4 were repeated.

Then, all the masses (solid mass, square section and cylinder) were placed on the circular platform. All the masses were positioned at equal distances from the centre and steps 3 and 4 were repeated again.

After getting the required time period for the oscillations, the height of the trifilar suspension was

measured using a measuring tape.

After the test it is now possible to calculate the periodic time using the data from the tables and derived

formulae and compare with the theoretical values

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3.2 Apparatus/Equipments:

The trifilar suspension is used while conducting the experiment. The main apparatus consisted of circular

plywood attached and hooked onto a hanger via chains.

The chains were tight to keep the base of the trifilar suspension as flat as possible. The chains are held

from the top by means of a hangar, on which the chains are hooked and joined.

To determine the mass moments of inertia a trifilar pendulum will be created using different type of solid

masses or weights. The stop watch is used to record the time of the oscillations.

The distance of the solid masses to the centre of the circular plywood is measured using a ruler and the

chain that supports the circular plywood was measured using a measuring tape.

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Figure 4 shows the set up of apparatus/equipment:

Figure 5 shows the masses arrangement on the platform.

Results and Calculations

4.1 Data

Technical Data:

Circular Platform Weight: 2 kg Diameter: 600 mm

Cylinder Weight (mild steel) Weight: 6.8 kg Diameter: 126 mm

Circular Hollow Tube (mild steel) Weight: 2.2 kg Diameter(inner): 78 mm Diameter(outer): 98 mm

Square Section (mild steel) Weight: 2.5 kg Area: 100 mm Thickness: 6 mm

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Trifilar String Length: 2.12m Trifilar Base Radius: 0.33m

Table 1 below shows the recorded time and mass for each load. After doing the necessary calculations,

the theoretical data’s were displayed in a table. The readings were compared to draw a possible trend.

Calculations were then used to plot a graph between the experimental and the tabulated data.

Load Mass

(kg)

Experimental

Time (sec)

Polar Moment Theoretical

TimeCircular Platform

Cylindrical Tube

8.8 0.73 0.09 0.80 0.011

Circular Platform

Cylindrical Tube

Hollow Circular Tube

Square Hallow Section

13.5 1.38 0.38 1.65 0.028

Circular Platform 2 1.77 0.09 2.06 0.045

4.2 Calculations

Mass Moment of Area about the centroid of the weights is calculated.

Circular Platform 222

0 09.02

3.02

2kgm

mrI =×==

Hollow Cylinder Weight ( ) ( ) 2222200 0043.0039.0049.0

2

196.2

2kgmrr

mI i =+=+=

Square Hollow Weight ( ) ( ) 2222200 0019.0047.005.0

2

503.2

6kgmaa

mI i =+=+=

Using the derived equation, 22

mgR

LIT π=

T =0.80 sec

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Table1 Recorded and Calculated Values (3 sets of trials)

Discussion

Moment of inertia is the name given to rotational inertia, the rotational analogue of mass for linear motion.

It appears in the relationships for the dynamics of rotational motion. The moment of inertia must be

specified with respect to a chosen axis of rotation.

The Parallel axis theorem was used to calculate the moment of inertia of the rigid bodies used during the

experiment.

In the experiment the Periodic Time and ratio of mI are discussed and compared by the graph and

tabulated results in the previous sections.

As for the measured and theoretical periodic time, both quantities are also proportional to each other

which are proved by the calculations and the graph trends.

The slopes was plotted and compared and it was almost constant throughout the experiment.

The results showed that by comparing each test, errors made during the experiment can be analyzed and

support the discussion effectively.

Graph:

The graph below displays the comparison between the Theoretical and Measured Periodic times. Graph

1 show the trend observed when the values for the trials were plotted against time. The graph shows a

linear relationship and the gradient of the slope is positive which shows that this is a positive slope. The

experimental time was calculated using the trifilar suspension and the 3 set of weights by rotating the

circular platform while the experimental time was calculated using the theory of moment of inertia and the

parallel axis theorem.

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Graph 1 Theoretical and Measured Time Chart

Graph 2 shows the comparison between the experimental with calculated time with the ratio of mI .

The graph shows a linear relationship and the gradient of the slope is positive which shows that this is a

positive slope which shows the directly proportionality of the ratio to the Experimental time.

Graph 2 Measured Periodic Time Relationship

Graph 3 shows the comparison between the calculated time with the ratio of mI .The theoretical slope.

The graph shows a linear relationship and the gradient of the slope is positive which shows that this is a

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positive slope which shows the direct proportionality of the ratio to the calculated time. The graph shows

that there are small errors in the second set of measurements.

Graph 3 Calculated Periodic Time Relationship

Error analysis:

The error percentage could be around 10% because there’s a very small difference between the actual

and ideal values we got.

Sources of experimental error:

• Measurements/Readings accuracy (stopwatch)

• The start of the oscillation was not exactly according to the drawn tangential path.

• Room temperature and pressure

• The stability of the apparatus and equipments

• Calculations

Resolution to experimental errors:

• Avoid measurement/readings errors (stopwatch)

• Wear proper lab clothing’s to ensure safety and protection.

• Masses should be firmly held

• Set room temperature

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Conclusion

The moment of inertia of rigid bodies is calculated using the triflar suspension arrangement.

The experimental periodic time is measured and compared with the calculated theoretical time. The

periodic rotation will be calculated using the calculated mass moment of inertia and the derived equations

for the theoretical time period.

The Theories of Parallel Axis and Moment of Inertia are used to calculate and compare the experimental

and theoretical readings.

After analyzing the experimental and theoretical results the test period for both theoretical and

Experimental times respectively were directly proportional to the ratio of mI .

It can be concluded that the theoretical time calculated was similar to the experimental time measured.

This shows that the lab experiment is accurate.

The experiment is successful though there are small possible errors in the experiment. All of these

values agreed within the estimated experimental errors. To improve the accuracy of the result the

experiment should be performed carefully and the instruction should be followed.

References

1. Mass Moment of Inertia(2008): http://en.wikipedia.org/wiki/Moment_of_inertia

2. Trifilar Suspension(2008):http://www.eng.uwo.ca/mme385y/Experiments/MME

3. Course Handouts(2008): Trifilar suspension

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