3. Partial Shaking Force and Shaking Moment Balancing of Linkages

As was mentioned in Sect. 2.1, the complete shaking force and shaking moment balancing of linkages can only be reached by a considerably complicated design of the initial linkage and by an unavoidable increase of the total mass. This is the reason why in most cases the partial balancing is used in the machinery.

The object of this Chapter is to develop the methods of partial force and moment balancing by means of optimum displacements of counterweighs.

The Sect. 3.1 deals with the shaking moment minimization of fully force balanced planar linkages by transferring the rotation axis of the counterweight mounted on the input crank. Mathematical means for the realization of such a balancing is compact and comprehensible, which allows engineers and technicians to apply the method in short time and without much difficulty.

The similar problems for spatial mechanisms, as well as for balancing by transferring several counterweights are considered in Sects. 3.2 and 3.3.

The balancing method presented in Sect. 3.4 is a symbiosis of two known solutions: approximate shaking force balancing and moment harmonic balancing. To have an identical and simplified mathematical model, the conditions for balancing are formulated by the minimization of the root-mean-square values of the shaking force and shaking moment. Such a solution has two principal advantages: (1) a simple realization, without essential change in the construction of the initial mechanism (i.e., for a shaking force balancing of the linkage, it is enough to add a counterweight mounted on the input crank and for a shaking moment balancing to transfer the axis of rotation of the input link counterweight); (2) a minimization of the shaking moment on the frame of the linkage without increase in the total mass of the counterweights (i.e., only by the use of the counterweight masses designed for the force balancing of the linkage).