Welcome to AE Resources
Converted document TRANSFER THEOREM FOR PRODUCTS OF INERTIA

TRANSFER THEOREM FOR PRODUCTS OF INERTIA

To transform the products of inertia to a different reference frame, the transfer theorem for products of inertia is used.
figure images/transfer.png
Figure 1 Arbitrary body with coordinate axes originating at point C and a fixed frame originating at point P.

In this instance, the products of inertia in the reference frame originating at point c, with the axes denoted by a *, is known. When transforming the products of inertia in relation to point C to be in relation to point P, as seen in figure 1↑, the product of inertia, IPxy can be calculated by:
(1) Ipxy  =   − ⌠⌡(xy)dm        =   − ⌠⌡(x* + xcp)(y* + ycp)dm        =   − ⌠⌡(x* + y*)dm − xcpycp⌠⌡dm − xcp⌠⌡y*dm − ycp⌠⌡x*dm      
where xcp and ycp are the respective distances between p and c in respect to the inertial frame of reference. By the virtue of the definition of center of mass, − xcp∫y*dm − ycp∫x*dm disappears. Thus,
Ipxy  =   − ⌠⌡(x*y*)dm − mxcpycp        =  ICxy − mxcpycp      
Similarly, this can be applied to the other products of inertia:
Ipxz  =  ICxz − mxcpzcp       Ipyz  =  ICyz − mycpzcp      
This theorem is useful in converting the products of inertia measured in the inertial frame to a body fixed frame, which simplifies calculations when the body is in motion.