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TECHNICAL PAPERS

# Nonstationary Vibration of a Fully Flexible Parallel Kinematic Machine

[+] Author and Article Information
Zili Zhou

Department of Mechanical Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada

Chris K. Mechefske

Department of Mechanical Engineering, Queen’s University, Kingston, ON K7L 3N6, Canadachrism@me.queensu.ca

Fengfeng Xi

Department of Aerospace Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada

J. Vib. Acoust 129(5), 623-630 (Mar 19, 2007) (8 pages) doi:10.1115/1.2748477 History: Received September 13, 2006; Revised March 19, 2007

## Abstract

This paper studies the problem of the nonstationary vibration of a fully flexible parallel kinematic machine (PKM) that has flexibilities both in links and in joints. In the stationary case, the PKM was treated as a varying structure and the natural frequencies and mode shapes changed with the changes in the PKM configuration, without consideration of the PKM nominal motion. In the nonstationary case as studied in this paper, the nominal motion is included to investigate how it would affect the natural frequencies and mode shapes. To do so, a nonstationary model is developed using the elasto-dynamics method. First, a kinematic model is built based on rigid links and ideal joints, which is used to solve the PKM nominal motion. Second, the kinetic model is developed considering the flexibilities in the links and joints. In this case, the vibration equations would contain the Coriolis and gyroscopic damping matrix and the tangential and normal stiffening matrix, which are the terms resulting from the nominal motion. The instantaneous eigensolutions are obtained from the nonstationary eigenequations. The results show that (i) the slider velocity affects the instantaneous natural frequencies more than the slider acceleration; and (ii) the nominal motion has an effect on the system eigencharacteristics (e.g., the nonstationary frequencies can be higher or lower than the stationary ones) but the effect is small in an absolute amount (within $2.1Hz$ in natural frequencies presented at set nominal motions of the studied PKM prototype). This is because the extra inertial force from the nominal motion is always much smaller than the stiffness force in the system bodies as long as the bodies are made of hard material. The method presented is more convenient to use for the multibody system with flexible joints than other methods.

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## Figures

Figure 1

The PKM prototype

Figure 2

The PKM model

Figure 3

Position vector decomposition

Figure 5

Comparison of stationary and nonstationary natural frequencies

Figure 6

Comparison of stationary and nonstationary mode shapes

Figure 4

Flexible joint model

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