# Trifilar Suspension

Summary

The polar moment of inertia for an assembly of solid objects was calculated using the trifilar suspension apparatus. The periodic time for the experimental and theoretical results were analysed and compared in order to study the relationship between the mass moment of inertia and the mass of an assembly.

Table of Contents

1. Introduction – page 3

2. Theory – page 4 - 7

3. Apparatus – page 8

4. Procedure – page 9

5. Results – page 10 - 11

6. Discussion – page 12 - 13

7. Conclusion – page 14

8. References – page 14

Introduction

The moment of inertia I is a measure of the resistance of a body to angular acceleration [1]. An important factor as the resulting moment governs the analysis of rotational dynamics with an equation of the form M=I∝ which defines a relationship between several properties including angular acceleration and torque [2]. The polar moment of inertia is the measure of a body’s resistance to torsion and is used to calculate the angular displacement and periodic time of the body under simple harmonic motion [3]. The moment of inertia of any mechanical component that will encounter rotational motion must be analysed as part of the design phase. From the complex assembly of a steam turbine to the simplicity of a flywheel, the periodic time for a component can be compared with other prototypes in order to find the most efficient assembly before going into production. The trifilar suspension is an assembly that is used to determine the moment of inertia of a body about an axis passing through the body’s mass centre, perpendicular to the plane of motion [4]. Loading the assembly with various objects and with an understanding of the parallel axis theorem, it is possible to determine the total moment of inertia for the entire assembly.

Theory

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: Ig=mk2

WhereIg = inertia in Kgm2 about the mass centre

m = mass in Kg

k = radius of gyration about the 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 Wherem = local mass in Kg

h = 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: I= (Ig+ mh2)

The polar moments of inertia for some standard solids are:

Cylindrical solid| ICylinder= mr22 (r : radius of cylinder)| Circular tube| ITube= m2(ro2+ ri2) (ri and ro : inside and outside radius)| Square hollow section| ISquare= m6(ao2+ ai2) (ai and ao : inside and outside length)| Table 1: Polar moment of inertia for some standard solids

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. Ø600

Ø

Ø

Ø

L

θ

x

θ

θ

1

2

3

Figure 1: Trifilar suspension

IAssembly=m0R022+ m1(a02+a12)6+ m1R12+ m2r22+ m2R22+ m3r02+ri22+ m3R32

From Figure 1, ϴ is the angle between the radius of the circular platform R and the tangential reference line x sinϴ=xR

Since ϴ is very small sinϴ= ϴ = tanϴ so

ϴ= xL (1)

Considering a free body diagram

ϴ

mg

F

mg

Figure 2: Free body diagram

tanϴ=ϴ= Fmg (2)

Substituting equation (1) into (2) and rearranging for the force F F= mgxL (3)

Understanding the standard equation for torque, FR=I∝

Wherex=Rϴ

∝ = d2ϴdt2

-mgϴR2L=Id2ϴdt2 (4)

The equation of motion for figure 1 is:

Id2ϴdt2+ mgR2Lϴ=0 (5)

Comparing this to the standard equation (2nd order differential equation) for Simple Harmonic Motion d2ydx2+ ω2y=0, the frequency ω in radians/sec and the period T in seconds can be calculated. Assuming the general...

References: [1] R.C. Hibbeler “Engineering Mechanics – Dynamics” Tenth Edition p377

[2] http://en.wikipedia.org/wiki/Moment_of_inertia

[3] http://en.wikipedia.org/wiki/Polar_moment_of_inertia

[4] R.C. Hibbeler “Engineering Mechanics – Dynamics” Tenth Edition p378

[5] R.C. Hibbeler “Engineering Mechanics – Dynamics” Tenth Edition p378

Trifilar Suspension Dynamics Laboratory sheet

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