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HERIOT-WATT UNIVERSITY SCHOOL OF ENGINEERING AND PHYSICAL SCIENCES (MECHANICAL ENGINEERING) MECHANICAL ENGINEERING SCIENCE 3 (B58EC1) DYNAMICS LABORATORY: TRIFILAR SUSPENSION
Objective:
To calculate the polar moment of inertia of an assembly and using the result to predict the periodic time of a trifilar trifi lar 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 i nertia is: I g
where I g m k
=
2
mk
= 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 I g 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
The polar moments of inertia for some standard s tandard solids are: Cylindrical solid
I Cylinder =
Circular tube
I tube
Square hollow section
Isq
=
=
m
m
2
( 6
mr 2
( r : radius of cylinder)
2
(r 2
r 2 )
o + i
2
ao
+
ai ) 2
( r i and
( ai and
ao
r o
: inside and outside radius)
: 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 t he Moment of Inertia.
Page 1/4
1
θ
θ φ
L
φ O
φ 1
3
θ
3
2
2
Ø600
Figure 1. Trifilar suspension From Figure 1, the equation of motion is: I
d 2θ dt 2
+
mgR 2 L
θ = 0
(1)
Comparing this to the standard equation (2 nd order differential equation) for Simple Harmonic Motion (SHM), 2
d y dx
2
+
ω 2 y
=
(2)
0
the frequency ω in radians/sec and the period T in seconds can be calculated by: ω =
and
mgR 2
T = 2π
(3)
LI
LI
(4)
mgR 2
Assuming the general solution for the equation (1) is θ = θ sin (ω t ) , solve the differential equation (1) to obtain equation (3) and use frequency ω = 2π f 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
= ___________ mm for cylinder;
R3
= ___________ 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
I m
for experimental and calculate data. Comment on the
The report for the labs must have the following sections: Introduction – give a background to the subject and experiment Aim/Objective – describe the aim or objective of the experiment Theory – detailed description of the theory and engineering principles. All equations used in the calculations must be shown. Apparatus – a description and diagrams of the apparatus, diagrams must be fully labelled. All variables used in the theory section must be identified. Procedures – a detailed record of the execution of the experiment, that a person could repeat the experiment by reading it. Results – Tabulation of raw readings and the calculated ones. A sample calculation must be shown. Calculated and experimental values must be shown and %errors between them derived. Discussion – discuss on the results obtained. Compare between experimental and calculated results and discuss on the errors. Identify the sources of error and explain with calculations or theory involved. Listing errors without justification is not sufficient and this will not gain any marks. Conclusion – conclude if the aim or objective is achieved. References – Books or publications to support theory and discussion. Lecture notes cannot be used as reference. Identify where the materials are used in your report. Appendix (Optional) – any other information to support the report. Attendance to lab session is compulsory. Any report does not have all the above sections will be rejected and returned, and it will be considered as non-submission. Marks will be deducted for late submission.
