Optimum Balancing of Combined Pitching and Yawing Moments in High-Speed Machinery

Traditional methods of balancing shaking forces and shaking moments in highspeed machinery are well developed and documented. In the case of nonstandard configurations, however, the analytical expressions become complicated and unwieldy. The minimization of pitch and yaw can be a tedious process and difficult to accomplish using conventional techniques. In this study we derive the equations for minimizing any order combined pitching and yawing moments in high-speed machinery by counterweighting the driveshaft or a shaft geared to the driveshaft. The equations, which are algebraic, are given directly as a function of the harmonic coefficients of pitch and yaw and apply to any plane machine configuration. The results yield an optimized configuration having the minimax sum of the amplitudes of pitch and yaw of any order, or the minimax vector sum of pitch and yaw of any order. The concept of balancing efficiency is introduced and the theory is illustrated by numerical examples.