Structural Dynamics of Parallel Kinematic Machines
By David S. Hardage
December 2000
Chairman: Dr. Gloria J. Wiens
Major Department: Mechanical Engineering
ABSTRACT
Throughout recent years, the evolution of machine tools has been facilitated by technological advances in manufacturing methods and changing design philosophies, and has led to a number of innovative parallel machine architectures that deviate from those of the conventional serial mechanism. However, recent speculation has suggested that for any conceivable mechanism pose, there is at least one direction in which the stiffness at the end effector is less than the stiffness of one of the links. The purpose of this thesis is to present a methodology for modeling the structural dynamics of parallel kinematic machines that can be used to evaluate machine performance to guide the design process, and ultimately to improve control schemes.
The purpose of this research is to develop a methodology for identifying the structural dynamic parameters of PKMs. The scope includes the derivation of an analytical model for simulating the vibration response and modal parameter configuration dependencies of a PKM. This model combines rigid-body and flexible-body dynamics using a component mode approach, and yields a set of linear ordinary differential equations (ODEs) that can be used to explore several of the issues associated with the structural dynamics of PKMs. Specifically, modes, mode shapes, and modal stiffnesses can be correlated with variations of the platform position and orientation within the workspace by solving this system of equations for the free vibration response. The accuracy of these results cam be verified through experimental modal testing and analysis. A detailed derivation of the equations of motion for a type of parallel manipulator, in which the kinematic chains that connect the platform and the base are composed of two distinct links, is presented. The platform is modeled as a rigid body and the links are modeled as flexible bodies. Component mode synthesis techniques are used to relate the boundary conditions of the links to the motion of the platform. The Lagrangian approach is used to develop the equations of motion from expressions for the potential and kinetic energy of the system. The result is a linear system of equations in six degrees of freedom.
This derivation is extended to illustrate how, with a few modifications, it can be used to perform an analysis of the Stewart platform mechanism. The theory is compared with the results of modal testing performed on the Hexel Tornado 2000 located at Sandia National Laboratories in Albuquerque, New Mexico. It is shown that the theory is able to track the first two fundamental modes with reasonable accuracy. A number of table are given that compare resonant frequencies and mode shapes. In addition, several plots of theoretical mode shapes are given.
Finally, the dependency of resonant frequency and modal stiffness on machine configuration is investigated and followed by a discussion of a few of the issues that are related to these modal parameters. The trends are illustrated in a number of graphs. Observations based on the visual inspection of these graphs is included.
It is shown that the variation of the frequency and stiffness parameters is dependent on machine configuration, and tends to reflect certain characteristics of the mechanism geometry, such as symmetry of the joint arrangements and symmetry due to platform orientations. In general, it was determined that stiffness and frequency increase as the distance between the platform and the base decreases, and decrease as the platform begins to approach the boundaries of the workspace. In confirming the research of others, it was shown that the theoretical stiffness of a single strut is often less than the theoretical stiffness of the machine in a given directions. It was also seen that bending of the links can play a significant role in the dynamics of PKMs.