An analytical game-based design of crank-rocker mechanisms with optimum stroke, time ratio and transmission angle

Abstract

In this study, an analytical approach based on bi-level multi-objective game theory is proposed and implemented for dimensional synthesis of the planar crank-rocker four-bar mechanisms. Two players, namely, Stroke ([Formula: see text]) and Time Ratio ([Formula: see text]) are considered in the level playing the role of a leader. The third player, Transmission Angle ([Formula: see text]), is placed in a lower level operating with the role of the follower’ role. Players’ behavior at upper level is also modeled as a cooperative game. In addition to this, the interaction between upper and lower levels is modeled as a Stackelberg game wherein an analytical approach is given to provide an explicit function of the follower’s Rational Reaction Set (RRS). Such analytical approach of Stackelberg game theory leads to an interesting relationship of length of the coupler and rocker links. It is, therefore, advised that by adopting equal lengths for coupler link and rocker link, one can guaranty the optimality of the transmission angle deviation, ensuring the Grashof condition. Consequently, the stroke, time ratio, and transmission angle criteria are derived explicitly as mathematical functions explaining the links of mechanism’s normalized dimensions. The aforementioned approach coupled with the proposed formulas and charts, greatly simplifies the synthesis of the mechanisms with optimal transmission angle while adopting any desired values of both stroke and time ratio. The theoretical approach proposed are explained with illustrated examples.