Mechanical EngineeringSchool of Engineering and Physical Sciences

UNDERGRADUATE REPORT

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MODULE No: B58ED_2008-2009)

MODULE TITLE: Mechanical Engineering Science

ASSIGNMENT TITLE: Laboratory Report : Trifilar Suspension

Lecturer: Dr. Hisham

Year: 2 Term: 2 Session: 2

Grade: Name: HITESH PATHAK

Registration Number: 071286471

Marker’s Initials: Term Address:

Email: Hp35@hw.ac.uk

Submission Date:

Group/Group Members:(If applicable)

GROUP B

1. 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.

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

2. 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,

∑ += )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

6

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.

Fig

ure 3 Dimensions of Trifilar Suspension

Figure 2 Schematic Diagram of the Trifilar Suspension Setup

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

Rxθs inθ == 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,

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]

Ø600

Ø Ø Ø

L

θØ

θ

θ

1

2

3

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.

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

.

2. Results and Calculations

3.1 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

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 8.8 0.73 0.09 0.80 0.011

Cylindrical TubeCircular 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

3.2 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

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.

Table1 Recorded and Calculated Values (3 sets of trials)

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 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

3. 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.