Vibration of rings on a general elastic foundation

The free vibration eigensolutions of a thin ring on a general elastic foundation are obtained by perturbation and Galerkin analyses. Natural frequencies and vibration modes are determined as closed-form expressions for a ring having a circumferentially varying foundation of very general description. The elastic foundation consists of two orthogonal distributed springs oriented at an arbitrary inclination angle. The foundation stiffnesses vary circumferentially. The simple eigensolution expressions explicitly show the parameter dependencies, lead to natural frequency splitting rules for degenerate unperturbed eigenvalues at both first and second orders of perturbation, and identify which nodal diameter Fourier components contaminate a given n nodal diameter base mode of the free ring. Discrete spring supports are treated as a special case where the natural frequencies are determined by five parameters: nondimensional spring stiffness, stiffness angle, support angle, number of springs, and location of the springs. The predicted effects of these parameters on the natural frequencies are verified numerically. As an application and as the motivating problem for the study, the natural frequencies and vibration modes of a ring gear used in helicopter planetary gears with unequally spaced planets are investigated.

[1]  F. M. Detinko Free vibration of a thick ring on multiple supports , 1989 .

[2]  R. Parker,et al.  Vibration and Coupling Phenomena in Asymmetric Disk-Spindle Systems , 1996 .

[3]  Jonathan A. Wickert,et al.  MEASUREMENT AND ANALYSIS OF MODULATED DOUBLET MODE RESPONSE IN MOCK BLADED DISKS , 2002 .

[4]  Jonathan A. Wickert,et al.  Response of modulated doublet modes to travelling wave excitation , 2001 .

[5]  Robert G. Parker,et al.  Exact Perturbation for the Vibration of Almost Annular or Circular Plates , 1996 .

[6]  Singiresu S. Rao,et al.  In-Plane Flexural Vibrations of Circular Rings , 1969 .

[7]  A. K. Mallik,et al.  Free vibration of thin circular rings on periodic radial supports , 1977 .

[8]  R. H. Badgley,et al.  Reduction of Vibration and Noise Generated by Planetary Ring Gears in Helicopter Aircraft Transmissions , 1973 .

[9]  C.H.J. Fox,et al.  A simple theory for the analysis and correction of frequency splitting in slightly imperfect rings , 1990 .

[10]  Donald R. Houser,et al.  Mathematical models used in gear dynamics—A review , 1988 .

[11]  M. Savage,et al.  Effects of rim thickness on spur gear bending stress , 1991 .

[12]  Robert G. Parker,et al.  Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration , 1999 .

[13]  Jonathan A. Wickert,et al.  Spatial Modulation of Repeated Vibration Modes in Rotationally Periodic Structures , 2000 .

[14]  C. D. Mote,et al.  Exact Boundary Condition Perturbation Solutions in Eigenvalue Problems , 1996 .

[15]  C. D. Mote,et al.  Vibration and parametric excitation in asymmetric circular plates under moving loads , 1987 .

[16]  B. L. Dhoopar,et al.  Free vibration of circular rings on radial supports , 1979 .

[17]  T. Y. Yang,et al.  Natural frequencies and modes of rings that deviate from perfect axisymmetry , 1986 .

[18]  T. J. McDaniel Dynamics of Circular Periodic Structures , 1971 .

[19]  Ahmet Kahraman,et al.  A deformable body dynamic analysis of planetary gears with thin rims , 2003 .

[20]  Robert G. Parker,et al.  Exact boundary condition perturbation for eigensolutions of the wave equation , 1998 .

[21]  V. R. Murthy,et al.  Dynamic characteristics of stiffened rings by transfer matrix approach , 1975 .

[22]  K. B. Sahay,et al.  Vibration of a stiffened ring considered as a cyclic structure , 1972 .

[23]  J.-G. Tseng,et al.  On the Vibration of Bolted Plate and Flange Assemblies , 1994 .