Optimal synthesis of function generating spherical and RSSR mechanisms

Abstract * —This paper deals with the design of spherical and RSSR mechanisms generating an input/output (I/O) function. To achieve this objective, the proposed method uses a dimensional synthesis technique based on local optimization. Exact-gradient determination is used in the minimization of the objective function. The multibody system is described by means of constraint equations which are used to carry out the necessary differentiation. Due to the kinematic characteristics of this kind of mechanisms, the technique developed here provides a valuable tool for obtaining the optimal solution to fulfil the I/O requirements. In this way, four-bar spherical mechanisms together with RSSR linkages are optimized. Three examples are shown to illustrate the application of the method. Keywords: optimal design, dimensional synthesis, 3D linkages I. Introduction Spherical and RSSR mechanisms are spatial linkages that can be very useful to obtain I/O relationships between skewed and intersecting shafts (see Fig. 1). One of the main advantages of these mechanisms is that they can be used to replace gear sectors [3], [4]. Obviously, gear arrangements have many advantages when constant velocity ratios are demanded, however, their substitution by linkages could provide some improvements under certain circumstances. For instance, it is very usual to demand a nonlinear I/O relationship in the design of machines. In this case, noncircular gears could be used, although that may result in very costly systems, and it is not always possible to achieve feasible solutions. Furthermore, gears cannot generate special motions such as crank-rocker or double-rocker, which can be easily obtained by using linkages. Another disadvantage of using gears is that they need larger size than linkages when the I/O shafts are too far from each other. Finally, sometimes during the machine operation it is necessary that the transmission angle (the relative angle between input and output shafts) undergoes variations with a minimal alteration in the velocity ratio. Gear arrangements cannot allow this kind of alterations whereas linkages can provide significant variations with small changes in the kinematic transmission. In spherical linkages (see Fig. 1a) the motion of any link is constrained to remain in a spherical surface. In this case, the input and output shafts intersect each other, this