Series expansion method for determination of order of 3-dimensional bar-joint mechanism with arbitrarily inclined hinges

A general method for evaluating the order of infinitesimal mechanism is presented for 3-dimensional bar-joint mechanism, which is defined as assembly of rigid bars connected to nodes by hinges with arbitrarily inclined directions. Compatibility conditions of translations and rotations are derived at bar-ends with respect to the generalized displacements consisting of displacements and rotations of nodes and bars. Degree of kinematic indeterminacy is computed using singular value decomposition of the linearized compatibility matrix. The order of infinitesimal mechanism is determined by the existence condition of higher-order coefficients of the generalized displacements with respect to the path parameter of deformation. The coefficients are obtained in a similar manner as stability analysis of geometrically nonlinear structures, where symbolic computation software package is used for analytical derivation of equations. The detailed procedure of analysis is shown through the numerical examples of a two-bar and a four-bar linkages, and the results are verified using a finite element analysis software.