Angular Mass is a Matrix:    I

Inertia is an everyday experience for most of us.  It is tough to get out of bed in the morning and get started.  As Newton said, an object at rest wants to remain at rest.  To engineers however, qualitative descriptions are not enough - we must quantify everything.  Mass is the quantification of inertia.  Mass measures the resistance of an object to a change in its state of linear motion.  Thus mass measures the linear inertia of an object.

Hey but not all motions are linear!  More often than not we find ourselves going around in circles and making little headway.  What about rotational motion?  Certainly it also exhibits inertia.  It is a challenge to get things started spinning. It is also a challenge to stop something from spinning once it has begun.  Thus we also need to quantify rotational inertia.  Thus we need a measure of rotational or angular mass.

It would be great if every object had a single unique rotational mass just like every object has a single unique mass.  Unfortunately life is not this simple.  Consider two identical pivoted rods.  In each case let the pivot axis be perpendicular to the length of the rod.  If one of them is pivoted at its center point while the other is pivoted at an end point, it is much easier (takes less torque or moment) to initiate spinning of the one pivoted at its middle.  Try it and see.  Thus our first depressing conclusion is that rotational mass of an object depends upon the point designated as the pivot point of the object.  Thus any measure of rotational mass must indicate the reference or pivot point.  If that was the end of the bad news, we could be fairly happy.  However the ease of spinning up an object also depends on the orientation of the object relative to the spin axis.  Consider again two slender rods, both with pivot axis through their center.  In the first the pivot axis direction is perpendicular to the length of the rod.  In the second rod the pivot axis direction is parallel to the length of the rod.  You will see it is much easier (takes less torque or moment) to spin up the rod along an axis parallel to the rod.  Thus any measure of rotational mass must designate not only a reference point but also a reference direction.  More bad news.  But it gets worse!  Any automobile owner is familiar with the cost of balancing ones tires.  This includes both static and dynamic balancing.  What is dynamic balancing?  Dynamic balancing is necessitated by the observation that even if you make sure the mass center of the tire is along the axle axis, rotation of the tire can still cause excessive vibration.  In particular a spin of a dynamically unbalanced tire about one axis can actually induce moments about an axis perpendicular to the spin axis.  Oh my!  Thus rotation about one axis can produce coupling effects about other axes.  Thus rotational mass must also specify the coupling attributes of the object.  Thus a rotational mass must not only designate a reference point, and a reference direction, but must also specify a second direction relative to coupling effects.  Oof!

Based on the above discussion it is not surprising that the specification of rotational mass for an object about a particular point requires six parameters - three to specify the rotational mass about each of three orthogonal axes through the point, and three to specify coupling effects for each of the three choices of two axes out of the three orthogonal axes.  These six parameters are typically arranged into a three by three matrix and called the mass inertia tensor about the particular point (the more intuitive term angular mass matrix is also used).  The inertia tensor about a particular point is denoted I and its components are shown below.  The convention of representing a three by three matrix as an underlined bold character was first introduced in the discussion of the rotation matrix.  This introduction occurred during the discussion of the time derivatives of vectors and different observers.

        |    I_xx    -I_xy     -I_zx    |
I =   |   -I_xy     I_yy     -I_yz   |
        |   -I_zx    -I_yz      I_zz    |

Note that the inertia tensor (angular mass matrix) is symmetric, the elements above the diagonal are equal to the elements below the diagonal.  Thus as indicated the inertia tensor contains but 6 distinct numbers. Appendix B, beginning on page 665 of our text book discusses the inertia tensor and rotational inertia in some detail.  This web page will focus on some of the more important issues.

Elements of the inertia tensor are defined as integrals over the body.  In order to evaluate the inertia tensor about some point O of a body we establish an axis system with origin at that point O.  Denoting the vector from that origin to any other point in the object expressed relative to the established axes as (x, y, z), the elements of the inertia tensor about point O are defined as the following mass integrals:

            |    I_xx = ⁽₎  (y²+z²) dm       -I_xy = - ⁽₎ x y dm            -I_zx = - ⁽₎ z x dm      |
I_O =     |  -I_xy = - ⁽₎ x y dm           I_yy = ⁽₎  (z²+x²) dm         -I_yz = - ⁽₎  y z dm     |
            |  -I_zx = - ⁽₎ z x dm             -I_yz = - ⁽₎  y z dm           I_zz = ⁽₎  (x²+y²) dm   |

The diagonal terms of the matrix are called mass moments of inertia.  We are familiar with I_zz from our investigation of planar rigid body motion.  The off-diagonal terms are call mass products of inertia.  These are the animals that arise in the coupling effects associated with dynamic balancing.  Note that in performing the above integrals, the differential mass, dm, is normally converted to a density, r, multiplied by a differential volume dV.  The integral is then converted to a triple integral with the details depending upon whether cartesian, cylindrical, spherical or some other coordinates are used.  At this point a crucial observation must be made.  If the angular mass matrix is to really characterize a body, then it must be a set of numbers that don't change as the body moves.  What this means is that the axes used in evaluating the angular mass matrix (inertia tensor) must be attached to the body and moving with it.  This becomes of critical importance in using the angular mass matrix.  You are forced to use axes attached to the body.  Typically these axes will be rotating.  Great care must then be taken if any time derivatives are required because the axes involved are themselves rotating.  The impact of observer rotation on derivatives has been discussed in some detail in an earlier page on time derivatives of vectors.

The values for these inertia tensor integrals for a number of simple shapes are included in your text in Appendix D on pages 713 to 716.  The values for a uniform rectangular parallelepiped of mass M are given below .  We will let the reference point be the mass center G.  The three axes will be parallel to the edges of the parallelepiped.  If the edge length parallel to the x axis is b, the edge length parallel to the y axis is a, and the edge length parallel to the z axis is L, then:

           |   1/12 M { a² + L² }                0                             0                  |
I_G =    |           0                      1/12 M { L² + b² }               0                  |
           |           0                                   0                 1/12 M { b² + a² }   |

Note as is always the case for symmetrical shapes, the off-diagonal terms (products of inertia) are zero.  Knowing the inertia tensor for the mass center is great, but what if we want to know the angular mass matrix about some other point other than the mass center?  Do we have to do the integrals again?  Oof!  Fortunately not.  If we know the angular mass matrix about the mass center for any shaped body, then we can calculate the angular mass matrix about any other point.  Calling the other point A and letting its location relative to the mass center be specified by the vector (X, Y, Z) leads to the following result for the inertia tensor (angular mass matrix) about the point A.  Note that the components (X, Y, Z) of the relative position vector must be measured relative to the same body-fixed axes within which the inertia tensor relative to the mass center was evaluated.  Note the use in the following of a subscript G to indicate the original angular mass matrix components about the mass center.

             |    I_Gxx  +   M (Y²+Z²)             -I_Gxy  -  M X Y                   -I_Gzx  -  M Z X             |
I_A =      |      -I_Gxy  -  M X Y               I_Gyy  +   M (Z²+X²)              -I_Gyz  -  M Y Z             |
             |      -I_Gzx  -  M Z X                  -I_Gyz  -  M Y Z                I_Gzz  +   M (X²+Y²)         |
 

The relationship given above between the angular mass matrix at the mass center and the angular mass matrix at some other point is known as the transfer theorem or the parallel axis theorem.  It is very useful in dealing with composite bodies or in dealing with bodies with pivot points at locations other than the mass center.

In summary, rotational inertia is complicated!  Angular mass is a matrix that depends upon the reference point of the object.  Elements of the angular mass matrix about the mass center are tabulated for common shapes.  Elements of the angular mass matrix for more complicated shapes can either be evaluated using integration or from superposition of simpler shapes.  Simple rules exist for how the angular mass matrix changes as the reference point changes.  And we haven't even talked about what happens if you rotate the axes used in evaluating the angular mass matrix.  Nor have we even mentioned the principal axes of inertia.  Life is too short!  Despite the complexity of rotational inertia, the engineers plight is clear.  Things spin, thus we must be able to evaluate angular mass matrices (shucks).  As practice makes perfect, the consideration of an example will be most useful.  Click here to get some practice in evaluating inertia tensors.
 
Copyright (1998) by Nels Madsen.  Last Updated: February 28, 1998