Structure Parameters and Displacement Equations of Goldberg Mechanisms With the Rank More Than sixStructure Parameters and Displacement Equations of Goldberg Mechanisms With the Rank More Than Six

Abstract

An infinite number of N-bar Goldberg mechanisms with the rank more than six are proposed. A single closed-loop eight-bar mechanism is first developed by removing the common side links of adjacent loops from the five-loop Bennett mechanism in which all loop frames are collinear. An analysis of trajectory circles of the links shows that the mechanism developed maintains the original constraint conditions and kinematic characteristics, and that its degree of freedom (DOF) is also one. This novel single-loop mechanism is named the eight-bar Goldberg mechanism because of its similar constraint characteristics with the five-bar and six-bar Goldberg mechanisms. The eight displacement parameters of this mechanism are analyzed by using the interior angles of adjacent links, and the five cosine equations for the tangent product of the stagger angles between adjacent links are established in together with one angular displacement equation of odd links and one angular displacement equation of even links. This novel mechanism has one DOF, eight displacement parameters and seven independent displacement equations. The loop rank of the eight-bar Goldberg mechanism is seven, which revises the existing mechanism theory of single closed-loop mechanisms with the rank no more than six. Similarly, a nine-bar Goldberg mechanism is constructed, and its nine displacement parameters are analyzed. This mechanism has eight independent displacement equations and the loop rank is eight, proving that the mechanisms with the rank more than six are not rare. Then, according to the characteristics of the independent displacement equations of the eight-bar and nine-bar Goldberg mechanisms, the N (N>7) displacement parameters of N-bar Goldberg mechanisms are analyzed, and the (N–1) independent displacement equations are deduced with odd and even Ns. It is proved that the loop rank of N-bar Goldberg mechanisms is (N–1), so there exist an infinite number of mechanisms with the rank more than six