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.

1.
Zhou
,
Z.
, 2005, “
Dynamic Modeling with Eigen-Sensitivity Analysis of a Fully Flexible Parallel Kinematic Machine
,” Ph.D. thesis, Queen’s University, Canada.
2.
Zhou
,
Z.
,
Xi
,
F.
, and
Mechefske
,
C. K.
, 2006, “
Modeling of a Fully Flexible 3PRS Manipulator for Vibration Analysis
,”
ASME J. Mech. Des.
1050-0472,
128
, p.
403
412
.
3.
Winfrey
,
R. C.
, 1971, “
,”
ASME J. Eng. Ind.
0022-0817,
93
, pp.
268
272
.
4.
Midha
,
A.
,
Erdman
,
A. G.
, and
Frohrib
,
D. A.
, 1978, “
Finite Element Approach to Mathematical Modelling of High-Speed Elastic Linkages
,”
Mech. Mach. Theory
0094-114X,
13
, pp.
603
618
.
5.
,
W.
, and
Dubowsky
,
S.
, 1981, “
The Application of Finite Element Methods to the Dynamic Analysis of Flexible Spatial and Coplanar Linkage Systems
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666,
103
, pp.
643
651
.
6.
Song
,
J. O.
, and
Haug
,
E. J.
, 1980, “
Dynamic Analysis of Planar Flexible Mechanisms
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
24
, pp.
359
381
.
7.
Serna
,
M. A.
, and
Bayo
,
E.
, 1989, “
Simple and Efficient Computational Approach for the Forward Dynamics of Elastic Robots
,”
J. Rob. Syst.
0741-2223,
6
, pp.
363
382
.
8.
Shabana
,
A. A.
, and
Wehage
,
R. A.
, 1983, “
A Coordinate Reduction Technique for Transient Analysis of Spatial Substructures with Large Angular Rotations
,”
J. Struct. Mech.
0360-1218,
11
, pp.
401
431
.
9.
Simo
,
J. C.
, and
Vu-Quoc
,
L.
, 1986, “
On the Dynamics of Flexible Beams Under Large Overall Motions—The Planar Case: Part I
,”
ASME J. Appl. Mech.
0021-8936,
53
, pp.
849
854
.
10.
Simo
,
J. C.
, and
Vu-Quoc
,
L.
, 1986, “
On the Dynamic of Flexible Beams Under Large Overall Motions—The Planar Case: Part II
,”
ASME J. Appl. Mech.
0021-8936,
53
, pp.
855
863
.