Studies Of Some Theoretical Problems On Kinematic Analysis Of Pin-bar Mechanisms

The exact kinematic governing equation of pin-bar mechanisms is established. The movability of pin-bar assemblies is induced mathematically to judge the existence of nonzero root of governing equation for continuous displacement. The number of inextensional deformation modes m>0, resulted from the Equilibrium Matrix Rule, is theoretically clarified to be only the necessary condition for the movability of pin-bar assemblies, but insufficient to determine. Based on analyzing the higher-order terms of compatibility equation, it is found that the assembly is unconditionally movable if the residual elongations caused by first-order inextensional deformation are orthogonal to the vector space consisting of the self-stress modes. The zero singular value of linear compatibility matrix is proved to form an equivalent relationship with the corresponding inextensional deformation mode, and reflects its numeric usability in tracing kinematic bifurcation points. The higher-order compatibility analysis is expounded to be necessary for further investigating the characteristics of bifurcation point and computing consequent kinematic paths.The kinematic analysis of pin-bar mechanisms is discussed from purely geometrical views. Based on the kinematic governing equation, an improved numerical strategy is suggested to calculate the kinematic paths of pin-bar mechanisms. The mathematical property of the limit point in kinematic path is expounded. When a joint in pin-bar mechanism reaches its kinematic limit point, it is proved that the components in all modes of mechanism displacements corresponding to this joint must be zero. This characteristic can be adopted as a rule to determinate and numerically trace the limit point of kinematic path. Further, a special geometrical characteristic of limit configuration is probed and explained theoretically. Two numerical examples are employed to investigate the accuracy and validity of the method put forward in this paper.A numerical strategy is suggested to calculate the kinematic path of loaded pin-bar mechanisms, and be used to analysis the integral lifting construction process of foldable cylindrical latticed shells. Equilibrium conditions of pin-bar mechanisms are derived, and corresponding forces of bars are calculated. A parameter h is introduced to investigate the stability of mechanisms. Then a numerical example, the erection simulation of a seven-bar mechanism, is employed to investigate the validity of the method put forward in this paper. It is found that stability analysis is necessary for this simulation. Several applicable approaches, which are able to affect stability conditions of mechanisms, are analysed.