Problem 6-33 Revisited, the Angular Mass Matrix About A Pivot Point:    I_A

Problem 6-33 on page 440 of our text book involved a plate hinged along one edge.  The figure depicts the plate (green) and three coordinate axes (light blue).  The coordinate axes are to be taken as attached to the plate and moving with it.  The Y axis runs through the center of the plate and along its longest dimension.  The X axis runs through the midpoint of one edge of the plate and is perpendicular to the largest surface of the plate.  The Z axis is in the plane of the plate and is along the shortest edge in the plate plane.  The only permitted movement of the plate is a rotation about this Z axis.  The plate has a length of 0.4 m along the Y axis and a length of 0.25 m along the Z axis.  Its thickness in the X direction is small and will be taken to be zero.  One end of the plate (parallel to the Z axis) is welded to a light axle which is shown in yellow in the figure.  The axle passes through and is supported by two fixed bearings, A and B. These bearings are shown in gray in the figure.  The bearings are each a distance of 0.08 m from the edge of the plate in the -Z and +Z directions respectively.  The bearings permit the axle to rotate freely about the Z axis but prevent any other movement.  The plate is uniform and has a mass of 20 kg.  We can use the edge lengths identified above and the formula quoted in the previous lesson for a rectangular parallelepiped to evaluate the angular mass matrix about the mass center for the plate.
 
 

            |     (1/12) 20 ( 0.4² + 0.25²)             0                      0                 |
I_G =     |                  0               (1/12) 20 ( 0.25² + 0 )           0                 |  kg m²
            |                 0                                    0        (1/12) 20 ( 0 + 0.4²)   |

Evaluating the expressions:

            |   0.371       0                 0      |
I_G =     |      0       0.104              0      |  kg m²
            |      0          0              0.267   |

Of interest will be the angular mass matrix about either one of the hinge points.  We will evaluate the mass matrix about the point A.  The mass center is in the center of the plate.  The vector from the mass center to point A thus runs a distance (1/2) 0.4 m in the negative Y direction, and a distance of (1/2) 0.25 m + 0.08 m in the negative Z direction.  Thus:

r_A/G = -0.2 J - 0.205 K m

We can now use the transfer theorem to evaluate the angular mass matrix about the point A.

            |   0.371       0                 0      |                               |  (0.2²+0.205²)   -0(-0.2)       -(-0.205)0          |
I_A =     |      0       0.104              0      |  kg m²   +   20 kg   |   -0(0.2)         (0.205²+0)      -(-0.2)(-0.205)   |  m²
            |      0          0              0.267   |                               |  -(-0.205)0    -(-0.2)(-0.205)      (0+0.2²)       |

Evaluating:

           |   2.011       0          0        |
I_A =    |      0       0.945    -0.82    |  kg m²
           |      0      -0.82      1.067   |

The above gives the components of the angular mass matrix of the plate about point A.  This angular mass matrix (inertia tensor) has been expressed relative to the X Y Z axes attached to the plate and shown in the figure.  Use of the angular mass matrix normally involves consideration over a time interval, say for example the evaluation of time derivatives.  Under these circumstances if one is to treat the components of the angular mass matrix as constants, then one must consider the X Y and Z axes to be attached to the plate.  Thus the angular mass matrix is expressed in a moving coordinate system and this must be accounted for in any calculations.  In particular in this case while the Z axis never changes direction as the plate moves, clearly both the X and Y axes rotate.  If we need to calculate time derivatives involving the inertia tensor (angular mass matrix), we must account for this rotation of axes.

Another important feature of the above angular mass matrix is the appearance of a non-zero off-diagonal term.  The Y-Z element of the angular mass matrix is non-zero.  Thus we have a coupling between Y and Z rotations,  In particular as the plate rotates about the Z axis, this term will produce consequences in the Y direction.  The exact nature of these consequences will be revealed when you consider the remaining fundamental formulas.

Several homework problems involve the evaluation of angular mass matrices so make sure you can follow the above example!

So we can calculate an angular mass matrix about some weird point.  So what?  That brings us back to our fundamental formulas.  The angular mass is crucial to the calculation of angular momentum and kinetic energy.  Want to know how?  Go back to the fundamental formulas and find out!
 
Copyright (1998) by Nels Madsen.  Last Updated: February 28, 1998