Determination of static and kinematic determinacy of pin-jointed assemblies using rigid-body displacements as primary unknown variables

Abstract

The traditional methods taking nodal displacements as primary unknown variables are only applicable to the determination of the static and kinematic determinacy of pin-bar assemblies. For other types of pin-jointed assemblies, such as reciprocal configurations, traditional methods are no longer valid for determining their static and kinematic determinacy. In order to tackle the problem, this paper proposes a new generic method by taking the rigid-body displacements of each element as primary unknown variables. On the basis of the ‘dislocations’ compatibility at the articulated points in the configurations, the compatibility equations are derived and then kinematic matrix is obtained. Meanwhile, the forces at the articulated points of the elements are also chosen as primary unknown variables, and then the equilibrium equations at the reference point in the elements are derived to give the associated equilibrium matrix. From matrix computations for the obtained kinematic matrix and equilibrium matrix, the static and kinematic determinacy of the configurations can be determined. Furthermore, the geometric nonlinearity of the articulated mechanisms is considered, and then the exact kinematic governing equations and higher-order displacement compatibility conditions are established. Finally, several numerical examples are employed to demonstrate the effectiveness of the proposed new generic method for determining the static and kinematic determinacy. The results from the proposed method are then examined by those from traditional methods and finite element analyses. From the results for numerical examples, the first-order infinitesimal mechanisms occur in the planar reciprocal configurations assembled from regular triangles and the Archimedes tiling, while the curved reciprocal configurations with the same topologies have no redundant restraints and are thus geometrically stable.