Objective: To calculate the polar moment of inertia of an assembly and using the result to predict the periodic time of a trifilar suspension of the 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:

where

Ig m k

I g = mk 2 = inertia in kg.m2 about the mass centre = mass in kg = 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 h = local mass in kg = 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 = ∑ I g + mh 2

(

)

mr 2 2

The polar moments of inertia for some standard solids are: Cylindrical solid Circular tube Square hollow section

I Cylinder =

I tube =

I sq =

( r : radius of cylinder)

m 2 ro + ri 2 2

(

)

( ri and ro : inside and outside radius)

m 2 ( ao + ai2 ) 6

( ai and

ao

: inside and outside length)

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.

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1

φ

L

φ

1 2

φ

θ

O 3 2

θ

3

θ

Ø600

Figure 1. Trifilar suspension From Figure 1, the equation of motion is: I d 2θ mgR 2 + θ =0 dt 2 L (1)

Comparing this to the standard equation (2nd order differential equation) for Simple Harmonic Motion (SHM), d2y + ω2 y = 0 dx 2

(2)

the frequency ω in radians/sec and the period T in seconds can be calculated by:

ω=

and

mgR 2 LI

(3) (4)

T = 2π

LI mgR 2

Assuming the general solution for the equation (1) is θ = θ sin (ωt ) , solve the differential equation (1) to obtain equation (3) and use frequency ω = 2πf to obtain equation (4). Show the derivation in your report.

Experiment:

A circular plywood platform, as shown in Figure 2, has three solid masses located as shown with reference to the centre of the platform. Using a spreadsheet or otherwise devise a tabular method for calculating the polar moment of inertia of the platform alone and the assembly. Measure the length of the chains supporting the platform. In your case, use apparatus A or the alternative B, and the three radii to be used are: R1 = ___________ mm for hollow square section; Page 2/4

R2 R3

= ___________ mm for cylinder; = ___________ mm for the circular tube.

R3 t R1 R2

A

Figure 2. Assembly details Using the result of your calculation of inertia to predict the periodic time of the SHM for both the platform and the assembly. Assemble the masses on the platform as specified above and obtain an experimental value for the period. Repeat the experiment for the platform alone. Compare calculated values and experimental data, and explain the discrepancies. Discuss and quantify sources of errors.

Discussions: Compare the experimental time and calculated time t from equation 4. Determine the %error and identify and explain the sources of error. Plot a graph of time vs linearity of the graphs.

I for experimental and calculate data. Comment on the m

Apparatus Data: Set A Set B Mass (kg) Dimensions (mm) Mass (kg) Circular platform 2.0 Ø 600 2.7 Cylinder 6.82 Ø126 5.6 Tube 2.196 78 I/D, 1.29 98 O/D Square Section 2.503 A=100 2.37 t=6

Dimensions (mm) Ø 600 Ø129 87 I/D, 102 O/D A=100 t=6.5

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Density of mild steel, ρ steel = 7,800kg / m 3 .

Report format:

The report for the labs must have the following sections: Introduction – give a background to the subject and experiment...