General Solution of the Forced In-Plane Vibration Problem for a Spinning Disk Natalie Baddour & Jean Zu Department of Mechanical and Industrial Engineering University of Toronto Toronto, Ontario Abstract The in-plane problem for a spinning disk is considered. Orthogonality properties of the free-motion mode shapes are derived by recasting the problem in state-space. These orthogonality properties permit a simple formulation to the general solution of the forced in-plane vibration problem.
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Introduction
Vibrations of a spinning disk is an old problem that has attracted the attention of investigators to various aspects of the problem. Spinning disks have numerous applications such as turbines, floppy disks, gyroscopes and circular saws. Both in-plane and transverse vibrations of spinning disks have been investigated in the literature. For linear vibrations these are uncoupled and can be considered separately. In the following, attention will be confined to the in-plane vibrations. In [1], the axisymmetric in-plane vibrations of a solid spinning disk are considered and two types of instability are discussed. Burdess, Wren and Fawcett [2] do not limit themselves to the axisymmetric problem and solve the general problem of in-plane vibrations. The non-symmetric motion has features that are not revealed by the analysis of the symmetric motion. Similarly, Chen and Jhu [3] consider the in-plane vibrations of a spinning annular disk. As previously mentioned, Burdess, Wren and Fawcett [2] solve the general non-symmetric in-plane vibration problem. The free motion mode shapes that are derived in their paper can be used to construct a general solution to the forced vibration problem. Following the example of Wickert and Mote [4], recasting the problem in state space allows for orthogonality properties of the aforementioned mode shapes to be easily derived. Once complete orthogonality properties have been derived, construction of solutions becomes a simple matter. This paper investigates the orthogonality properties of the free motion mode shapes and demonstrates how these properties can be put to use in constructing general solutions to the forced vibration problem.
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Statement of the Problem
Assume that the motion of a particle in the disk only occurs in the plane of the disk and is given by u = (ur , uθ ), where ur and uθ are the radial and tangential displacements respectively. For small displacements, linear stress-strain and linear strain-displacement relationships can be assumed. Furthermore, for a thin disk plane stress conditions can be assumed. The preceding assumptions lead to the following equations of motion : 1 F Lu = u ¨ −Ω2 u+2ΩDu− ˙ ρ
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(1)