Analytical and Parametric Study of Damped Vibrations in Gough-Stewart Platforms

Abstract: This work establishes a comprehensive and fully parametric model for the damped vibrations of Gough-Stewart Platforms (GSPs) at symmetric configurations. It is noteworthy that in the literature a complete solution to this problem has not been presented. This work has been carried out in three stages which are the subjects of Papers I, II and III. After the industrial stage of the work which is reflected in Paper I, we have generalized the problem and established a fully parametric model of the damped vibrations of GSPs in Paper II and finally extended the model by taking inertia of the struts into consideration in Paper III. The final model is parametrically developed in terms of all the design variables of the system and can be directly employed for the analysis, optimization and control of GSPs. In this work, we parametrically formulate the kinematic equations which are eventually obtained in the form of a Jacobian matrix. The focus of this work is on the Cartesian-space formulation in which Bryant angles, due to their advantages over Euler angles, are chosen to represent the orientation of the platform. The equations of motion are formulated and linearized based on a Lagrangian dynamics approach where Rayleigh axial damping of the struts is introduced to the system. Inertia, stiffness and damping matrices are parametrically formulated. By introducing the inertia of the struts, the inertia matrix turned out to be quite complicated to formulate. Interestingly, despite its apparent symmetrical geometry, the equivalent inertia matrix is obtained as a non-diagonal matrix. The eigenvectors and damped eigenfrequencies of the system are then parametrically established. In addition, the decoupled vibrations are analytically investigated where it is shown that the consideration of strut inertia may lead to significant changes of the decoupled conditions. Finally, for a reference GSP, the vibrational behavior with respect to different design variables are systematically studied.