Lumped-parameter modelling of elastically coupled bodies: Derivation of constitutive equations and determination of stiffness matrices

摘要

Modelling of elastically coupled rigid bodies is an important research topic in multibody dynamics. We consider the problem of modelling what can be called flexural joints, where two essentially rigid bodies are coupled by a substantially more elastic body. For modelling general elastic couplings one would like to have generically applicable, lumped parameter, intuitive, Euclidean geometric, accurate models with desirable physical symmetries. The model constitutive equations should be easily and quickly computable. For purely elastic coupling the constitutive equations should be truly energy conservative: the configuration-wrench equations should be derivable from a potential function. Linear and angular momentum should be conserved. Quaternion-based and twist-based modelling methods are presented. The constitutive equations to calculate the configuration-wrench behavior are derived via geometric potential energy functions. Wrenches are computable given the configurations of the rigid bodies and all the stiffness matrices of the compliant element. For an arbitrary pair of elastically coupled rigid bodies there exist coincident, unique points on the bodies known as centers of stiffness at which translation and rotation are minimally coupled. At the center of stiffness there exist three sets of orthonormal principal axes and corresponding principal stiffnesses. These parameters are useful in both analysis and numerical simulation. A finite-element-based method for computing canonical stiffness parameters of elastically coupled rigid bodies is presented. The method is applied to notch hinges and Remote Center of Compliance (RCC) hinges. Standard procedures are presented on how to determine canonical stiffness parameters at the center of stiffness of a spatial compliance. Series of canonical stiffness parameters can be generated automatically using the methods provided. Key program listings are provided which can be used to duplicate the results presented.