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The dynamic balancing of mechanisms is an old problem in order to (1) reduce the amplitude of vibration of the frame on which the mechanism is mounted due to the shaking force and shaking moment; and (2) smoothen highly fluctuating driving torque/force needed to obtain nearly constant drive speed. Since any vibration leads to noise, wear, fatigue, etc., in the mechanism, its reduction improves several aspects of mechanical design as well. However, the problem is faced with new challenges, namely, the balancing of shaking force, shaking moment, and input-torque fluctuations together. The shaking force can be eliminated completely by attaching counterweights to the moving links.

The methodology proposed in Chapter 5 to balance planar mechanisms is extended in this chapter for the spatial mechanisms. Spatial mechanisms need more exhaustive treatment compared to its planar counterparts. Both the kinematics and dynamics of the spatial mechanisms are more complex. Regarding the balancing of shaking force and shaking moment in spatial high speed mechanisms/machines, Kaufman and Sandor [103] presented a complete force balancing of spatial mechanisms like RSSR (Revolute-Spherical-Spherical-Revolute) and RSSP (Revolute-Spherical-Spherical-Prismatic).

Mechanical systems are generally classified as open-loop (serial and/or tree type) and closed-loop systems according to their architecture. In this chapter, a dynamic formulation for open-loop systems is proposed to compute constraint wrenches, i.e., the reaction moments and forces at the joints. It is based on the Newton-Euler equations of motion and the concept of the DeNOC matrices. The DeNOC matrices are the logically split natural orthogonal complement (NOC) matrix that was originally proposed in [69].

A mechanical system with closed-loop has a circuit of bodies such that the closed circuit can be traced through the system from one body to another and returns to the original body without traversing any body more than once [6]. Kinematic loops occur in many mechanical systems such as vehicles, spacecrafts, and machines for printing, textile and agricultural purposes. In contrast to open-loop systems, as studied in Chapter 2, closedloop systems tend to possess relatively few degrees of freedom (DOF) compared to the number of connected bodies.