# Determining Mass Moment of Inertia



## Ken I (Sep 26, 2012)

*I came across this method for measuring mass moment of inertia for irregular objects (for regular objects the formulas are pretty straightforward) - I'm in the robot business and I often need to know this for gripper designs - this may be limited by the robot's capability. I have attatched a *.doc file with photos for some grippers I built recently.*



*I thought this might be of interest to members to file away for future reference.*



*DETERMINING MASS MOMENT OF INERTIA - BY TWO FILLAMENT WIRE SUSPENSION*


To actually measure the Mass Moment of Inertia of an irregular object about its center of gravity, a structure can be suspended from a pair of parallel suspension wires of equal length &#8211; preferably equispaced about its center of gravity.

This then behaves as a rotary pendulum which has a near constant time base per oscillation. If the mass of the object is known then the mass moment of inertia can be calculated from this frequency.

The energy in the suspension wires can be ignored if they are long and slender (use the minimum diameter wire/cable that will safely support the load and as long as is practically possible) and the angular motion is small.

The &#8220;windage&#8221; or damping introduced by the viscous damping of the air can also be safely ignored for small motions of heavy objects.

*Step 1 = Weigh object.*

*Step 2 = Suspend from parallel wires, equispaced about CofG.*

*Step 3 = Measure Rotary Pendulum Frequency.*

*Step 4 = Calculate Mass Moment of Inertia.*

The formula uses vn for the average speed in radians per second &#8211; so it is taken that this holds true for complete cycles (constant time base) so the frequency in *Hz* is multiplied by 2p to give radians per second.
The formula :-

*I = m g D2*
*__4 h **v**n**2*

Where
*I* = Mass Moment of Inertia in *kg.m2*
*m* = Mass of object in *kg*
*g* = Acceleration by Gravity (9.806) *m.sec-2*
*D *= Parallel distance between wires in *m*
*h* = Length of Wires in *m*
*v**n* = Average Rate of Rotation (complete cycles) in *Radians.sec-1*
and
*v**n* *= 2 *p* n*
*______ t*

Where

*v**n* = Average Rate of Rotation (complete cycles) in *Radians.sec-1*
*n* = Number of observed oscillations.
*t* = Time taken for observed oscillations in *seconds*

or use the combined formula :-


*I = m g D2 t2 *
*__16 h *p*2** n2 *


Note: The term mass moment is used in preference to &#8220;polar moment of inertia&#8221; which refers to the torsional cross sectional modulus of a structure.

*Step 5 = Re-Calculate Mass Moment of Inertia (for off center rotation only).*

If your suspension wires were equispaced about the center of gravity but the CofG is not the actual center of rotation for the application then an adjustment must be made for this Mass moment being rotated about some other point distant from its actual Centre of Gravity.

Note: When the suspension wires are not equispaced (within reason) about the CofG, the pendulum will nonetheless rotate about its CofG &#8211; then the CofG needs to be determined by calculation via the differing loads between the two wires in balance or by balancing or calculation etc. For symmetrical loads the CofG is readily apparent as in the above case (ignoring the relatively light electrical box etc.)
You can also practically find the CofG by allowing the object to pendulum on the wires. By trial and error (make a mark or a sticker etc.) find the center &#8211; this point will remain in the same place during purely circular oscillations (be carefull not to introduce a &#8220;swing&#8221; component at the same time).

The formula for off CofG rotation:-

*Idisplaced = Icentre + m r2 *

Where
*Idisplaced* = Mass Moment of Inertia in *kg.m2* CofG about radius *r*
*Icentre* = Mass Moment of Inertia in *kg.m2* about CofG
*m* = Mass of object in *kg*
*r* = Radius of CofG from actual rotation centerline in *m*

*Further Comments Re: Mass Moment of Inertia for off center rotation.*

For solid objects at a distance from the center of rotation it is fairly common to ignore the* Icentre *portion of the formula

*Idisplaced = Icentre + m r2 *

And instead only use the *m r2 *component for simplicities sake.

Example a piece of Æ50x63 mild steel with a mass of 1kg has an *Icentre *of only 0.0003125 kg.m2 which if rotated at a radius of 1.0m has a mass moment of 1.0 kg.m2 by the shortcut method which really should 1.0003125 kg.m2 if the entire formula is used.

So for dense objects (such as nuts and bolts) operating at a distance from the center it can be seen that such a shortcut is nonetheless accurate. 

View attachment MASS MOMENT.doc


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