Minimization of Shaking Force and Shaking Moment in Rotating Mechanical System

This chapter presents an analytical procedure on the balancing of rotating mechanical system. The rotating mass system is said to be balanced, if the inertial forces and inertial moments induced due to unbalanced masses of the rotating system vanish. The dynamic performance is significantly affected by unbalanced forces and moments. To completely balance a rotating mechanical system, it must be balanced statically and dynamically. In this study, minimization of shaking moment along with shaking force is examined to balance rotating mechanical systems completely. The generalized expression for shaking moment as well as shaking force in the rotating mechanical system is written with Newton-Euler equations of motion. Balancing problem is defined as an optimization problem to minimize shaking moment along with shaking force simultaneously. The shaking moment along with shaking force in rotating mechanical system is minimized by placement of correction mass at proper locations in at least two balancing planes separated by some distance. The static balancing is achieved when the inertia forces of the rotating mechanical system are counterbalanced by the centrifugal forces of balancing weights, resulting in no shaking force exerted on the frame. Dynamic balancing occurs when the inertial force as well as moment produced by the inertial forces is counterbalanced simultaneously by the centrifugal forces of the balancing weights, resulting in no shaking moment and shaking force acting on the frame. Using an optimization algorithm in the MATLAB environment, the values of corrective masses and their appropriate positions on the respective balancing planes are determined. Numerical examples of a mechanical system with rigid bodies coupled to a shaft, rotating in parallel planes, demonstrate the efficacy of the suggested methodology. The results highlight the effectiveness of the optimization method, showcasing higher computational efficiency, and a significant reduction in both shaking moment and shaking force.