/g91/g93 by the cross products of inertia /g1861/g3052/g3053and /g1861/g3051/g3053, respectively in the last equality of Equation (2a) to account  for pure dynamic imbalance situations and therefore address the general case of arbitrary combined  static and dynamic imbalances. All forces and moments are functions of time. For example, /g2169/g2200should be written as  /g2169/g2200(/g1872), but the time symbol is omitted for readability. From the dynamometer we can read the  exact forces, but the moments will be different due to the height /g507, between the mount interface and the  elevation of the load cells. On the dynamometer, the readings for the moments are given by: /g3420/g1839/g3051 /g1839/g3052/g3424=/g3420/g1839/g3045/g3051 /g1839/g3045/g3052/g3424+/g959 /g3420/g3398/g1858/g3052 +/g1858/g3051/g3424=/g2033/g2870/g3420/g3398/g1861/g3052/g3053 +/g1861/g3051/g3053/g3424+/g4672/g1859 /g2033/g2870+/g959 /g4673 /g3420/g3398/g1858/g3052 +/g1858/g3051/g3424 (2b) The resultant force in the Z direction is independent of time and therefore would not show on the  dynamometer readings since the load cells measure only a change in force. The radial load remains unchanged but the lever arm of the moment is increased by the distance between the mounting surface and the plane of the dynamometer load cells.  Using the expression for the radial load from Equation (1), the magnitude of the radial load is related to  the static imbalance /g80/g72/g3via /g1865/g1857 = /g1865 /g3493/g1876/g2870+/g1877/g2870=/g3495/g1858/g3051/g2870+/g1858 /g3052/g2870=/g1313/g1858/g3045/g1313 /g2033/g2870 (3) In order to address the dynamic imbalance requirement, we use Equation (3). The dynamic imbalance is  related to the imbalance forces and moments as following: The requirements for static and dynamic balancing have to be compared to the values obtained from the  product /g80/g72and /g45/g85/g93computed from Equations (3) and (4) respectively. In addition to the magnitudes of the  static and dynamic imbalances, of practical interest are their corresponding directions. To find the precise  directions, two coordinate systems can be set on the test setup. One is stationary relative to the dynamometer, while the second rotates with the GMI unit. These two coordinate systems share an origin and the z-axis and initially both x-axes and y-axes are co-linear. Since the angle at which GMI has rotated  is known, we can subtract that from the direction of the resultant force to find the physical direction of the imbalance via /g581/g2916= tan/g2879/g2869/g4678f/g2935 f/g2934/g4679/g3398/g581 /g2891/g2897/g2893 (5) /g581/g2897= tan/g2879/g2869/g4678M/g2935/g3398/g959/g942f /g2934 M/g2934+/g959/g942f /g2935/g4679/g3398/g581 /g2891/g2897/g2893 (6)  We have just established that the static and dynamic imbalances form a pair of vectors that may be  plotted in a three-dimensional graph to graphically pinpoint the directions of the imbalances together with the physical locations of the balance masses to visually help the technical team understand which  masses will be most effective.  Optimal Balancing Using Linear Programming  Having determined the levels of static and dynamic imbalances, we now determine the optimal balance  masses that would simultaneously make the radial force and moment on the dynamometer nil. In order for GMI to be statically balanced, its center of mass must reside on the axis of rotation. This can be done 22 221 or                                         (4 ) 1where  a nd.yz y rz r xz x yy z rz r rz rz rr xx zif gJMif fi gJMfiff ii/g90/g90 /g90/g90/g65/g65 /g65 /g65/g16/g16/g173/g189 /g173/g189/g176/g176 /g176/g176 /g167/g183/g32/g32 /g16 /g14 /g39/g174/g190 /g174/g190 /g168/g184/g14/g14 /g169/g185 /g176/g176 /g176 /g176 /g175/g191 /g175/g191 /g16/g16/g173/g189/g173 /g189/g176/g176 /g176/g176 /g167/g183/g32/g32 /g16/g14 /g39/g32 /g32 /g174/g190 /g174/g190 /g168/g184/g14/g14 /g169/g185 /g176/g176 /g176/g176/g175/g191 /g175/g191 307