Theory

The moment of inertia of solid objects can be calculated by integrating the second moment of inertia of mass about a selected axis. The general formula used for inertia is:

Where

= moment of inertia in kgm2 about the mass centre

= mass in kg

= radius of gyration in m about the mass centre

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

Where m = 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:

The polar moments of inertia for some standard solids are:

Cylindrical solid:

Circular Tube:

Square hollow 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

Image taken from - http://vision.hw.ac.uk/courses/1/B58EC_2011-2012/

θ can be calculated using the formula :

as θ is a very small angle

Differentiating the equation with respect to time, t, gives :

And differentiating again gives:

Next it can be said that :

Where F = Force required to moved the plate distance x in N

mg = weight of the circular plate

We can now rearrange this equation to give:

From this it can then be said that:

By substituting,

and

into the previous equation it simplifies to give:

...............equation of motion

Comparing this to the standard equation (2nd order differential equation) for Simple Harmonic Motion (SHM):

By comparing these two equations, the frequency ω in rad/s and the period T in s can be calculated. Assuming that the general solution to the derived equation of motion is:

This means that the derived equation of motion can be rearranged to derive an equation for the frequency ω:

By taking the previous equation and substituting it into the following:

Gives:

This means that we can now calculate a theoretical value for the periodical time and compare it to the experimental results.

Equipment

The Experiment required the following equipment:

Trifilar suspension assembly (already set up in the lab)

Wooden metre stick, accuracy

1mm

Stopclock, accuracy

0.01s

3 solid objects of know geometry and mass

Mild steel cylinder

Mild steel tube

Mild steel square section

This experiment will require at least three attendants, one to displace the platform, one to record the time of oscillations and the final to record the results.

Hanger holding suspension chains

The Suspension was set up as shown in the diagrams below:

Suspension plate that objects will rest on

Chains connect suspension plate to hanger

The diagram below shows the set up of the objects on the suspension plate when the objects are offset from the centre of the platform, which will be the first set up for the experiment

Cylinder

Square section

Tube

Circular Platform

This diagram shows the plate with objects concentric to the platform which will be the second set up for the experiment

Tube

Circular Platform

Square section

Cylinder

The third set up for the experiment will simply be the platform on its own with no objects.

Experiment Data

Density of mild steel, 7,800kg /m3

Table taken from http://vision.hw.ac.uk/courses/1/B58EC_2011-2012

Length, L, of chains (m) that attach the platform to the hanger (All chains should be equal in length, making the platform level) = 1.930m

Total mass, M, of platform and objects (kg) = 2 + 6.82 + 2.196 + 2.503 = 13.519 kg

Procedure

The first step of the experiment is to inspect the set up of the trifilar suspension, making sure the chains are tight and the platform is level and free to move without obstruction.

Next the length of the chains connecting the suspension to the hanger must be measured using the metre stick.

Now the object must be placed on the circular platform as discussed before at a set distance from the centre of the platform.

Place the metre stick below the platform along the plane of its tangent, making sure it is clearly visible, this step is in place so that the platform can displaced by an equal, measured amount each time.

Reset the stopclock so that it is ready to record the time

Displace the platform by the measured distance, sighting it against the metre stick.

Release the platform, starting the stopclock at the same moment

Stop the stopclock at the point where the platform has made two oscillations and record time.

Repeat steps 7 and 8 about ten times in order to obtain more accurate results by use of an average.

The next step is to repeat the experiment but this time using a clear platform (no object)

Record times for two oscillations of the cleared platform

Again in order to gain a wider picture of what is going on in the trifilar suspension place the weight back on the platform but this keeping them concentric to the platform and repeat the experiment

Record times of two oscillations for concentric object set up.

Tabulate results and use them to generate graphs in order to make clear comparisons and discussion.

Results

Table 1 – Platform with objects at 150mm from centre

Table 2 –Platform with objects arranged concentric

Table 3 – Platform without any objects

Calculations

Calculations of I values for objects and platform:

Calculation of Itotal for objects set at distance 150mm from centre of platform:

Calculated periodic time, T, for objects placed 150mm from centre:

Calculation of Itotal for objects placed concentric to platform (s):

Calculated periodic time, T, for objects placed concentric to platform (s):

Calculated periodic time, T, for platform without objects (s):

Results/Discussion

Table 4 – Showing calculated results and experimental results

Graph 1

The above graph shows the comparison of the calculated periodic time against the experimental periodic time, where calculated results were obtained by formula discussed in the theory section and experimental results recorded during the trifilar suspension experiment. The graph shows a small error in the experimental results for the concentric set up.

Graph 2 shows the relationship between the results, both recorded and calculated, and the ratio

. From the graph the linear relationship, with positive slope, can clearly be seen

Errors

As you can see the results obtained from the experiment are relatively close to the values of T calculated using the theory. These discrepancies could be explained by any combination of the following:

Human error involved in measurement of distances and lengths as these were only measured using a simple metre stick.

Human error involved with measuring the time using a stopclock, as starting and stopping the clock at the precise time would be very difficult considering human reaction times

Inaccuracies associated with equipment,

1mm with the metre stick and

0.01s with the stopclock

Platform experience movement in the horizontal plane which would affect results as not all energy goes to rotational movement

Human error involved in making sure the platform is displaced by exactly the same distance each time

Human error involved in sighting when the platform has completed an oscillation.

Objects were not held in place during the experiment, so again movement on the

horizontal plane may have occurred which would again affect the results.

Masses of the objects may not be precise as they were taken as given rather than measured on the actual day.

Conclusion

The trifilar suspension proves there is a relationship between the polar moment of inertia and the equation for periodic time, T, derived from the simple harmonic motion of the assembly. By completing the experiment it is clear to see that periodic time, of both calculated and experimental, are directly proportional to the ratio

if experimental errors are taken into account. If the experiment was to be undertaken again, the following steps could be taken to increase the accuracy of the results:

Take more results for the periodic time for all set ups (being restricted by time we were forced to only take three readings for both concentric and no object set ups.

Devise a way to ensure the platform is displaced by the same amount each time, this could be done by use of a small post and visible mark on the platform, making sure both are aligned correctly before release.

Use more and different arrangements of the solid objects to gain an even wider picture of the workings of the assembly.

References

1 -

2 - R.C. Hibbeler “Engineering Mechanics – Dynamics” Tenth Edition

3- The course lab book was used to provide a template of the report, using the theory and experimental data provided