by Brian Callahan
January 2004
Described below is the method we use for determining the mass moment of inertia (MMOI) of our dynamometer inertia wheels1. The method is very accurate and simple to execute. It is also quite general in nature, and so may be applied to any solid object.
The part is suspended by two, three, or four wires. It is then set into torsional oscillation, and the time period of oscillation is measured.
Please refer to Figure 4. Knowing the weight of the test piece W, the vertical distance from the part to the suspension points L0, the radius at the suspension points R1, the radius at the part R2, and the period of oscillation t, the moment of inertia J is calculated from
| (1) |
Synopsis:
Step 1: Weigh the part.
Step 2: Connect the part to the suspension wires. Measure R1, R2, and L0.
Step 3: Start the part oscillating. Keep the oscillation amplitude less than 5 degrees. (This takes a light touch.)
Step 4: Measure the time it takes to oscillate 100 cycles, and divide by 100. The quotient is t.
Step 5: Calculate the inertia using equation 1.
That's it! Once set up, the method is simple to execute, and the only instruments needed are a stopwatch and a precision scale. If interested in the nitty-gritty, please feel free to read further.
As the part rotates, the wires move in a pendulum motion, and the part raises vertically. By solving the equation for potential energy and kinetic energy, the expression for J is derived in terms of the weight of the part, the period of oscillation, and the geometry of the wire suspension.
Notice that the number of wires is irrelevant. Choose two, three, or four depending on the part geometry. Circular parts like inertia wheels work best using three.
Within the constraints given below, the frequency of oscillation is independent of amplitude. So, timing many consecutive oscillations and then dividing by the number of oscillations greatly increases the accuracy of the time period measurement.
Rather than connecting the wires directly to the part, the part may sit in a tray, and the assembly tested. The MMOI of the tray is measured in isolation and subtracted from the total. This is necessary for inertia wheels because they have no features on the outside diameter for attaching the suspension wires.
The wires must have negligible stiffness in bending, and hence develop negligible strain energy. Using safety wire with a diameter of 0.032 in and choosing an L0 of at least 10 in makes the length to diameter ratio large, so this assumption is easily satisfied.
The suspension platform and the part must both be horizontal.
The angle of oscillation must be small. Otherwise, the simplifying assumption of pure sinusoidal motion used in the derivation would be invalid, and the full pendulum motion equations would have to be used. Limiting the angle of oscillation to 5 degrees will satisfy this requirement.
The suspension wires must be at least six times as long as the larger of R1 or R2. This is also to satisfy the simplifying assumption of sinusoidal motion instead of full pendulum motion.
All variables in the equations are assumed to be pure quantities. That is, units are unspecified. British units may be used, but only with the appropriate multiplication factors. If SI units (Système International d'Unités) are used, no factors are necessary, and the result will have units of mass·length2. For the kg-m-s system, this would be kg m2.
Accuracy of the weight measurement affects the accuracy of the MMOI directly. Therefore, most cheapo food scales are woefully inadequate. Use a scale that is accurate to 0.1% at least. Also note that a weight measurement is better than a mass measurement for this technique, because it avoids the necessity to measure the local gravitational acceleration.
Accuracy of the radii and length measurements also directly affect the accuracy of the MMOI, this time in a squared relationship. Be aware of the exact points of connection at the part and at the overhead suspension structure. Usually, the wires are connected to some sort of through-hole. My strategy is to position the points of connection such that the wire is pulled across an edge. For example, if R2 > R1, then I loop the wire over the outside edge of the hole in the part, and under the inside edge of the hole in the suspension structure, as shown in Figure 4.
It takes a light touch to start the oscillation. The part will tend to swing in both horizontal axes if you're not careful. Be patient and be delicate.
Counting to 100 is not as easy as it sounds! This is especially true if the oscillation period is excessively slow or excessively fast. I count out loud to help with this. Your eyes will also go fuzzy. Remember to blink. :)
As indicated above, this technique is general and not limited to inertia wheels. Indeed, it is not limited to symmetrical parts. Other examples for which this technique could be useful:
· engine flywheels
· connecting rods
· propellers
· the whole boat - in the yaw, pitch, and roll axes
1The British Internal Combustion Engine Research Association (1958), Handbook of Torsional Vibration, Cambridge University Press, London, UK.