Three-Dimensional Free Vibration Analysis of Gears with Variable Thickness Using the Chebyshev–Ritz Method

To reflect vibration more comprehensively and to satisfy the machining demand for high-order frequencies, we presented a three-dimensional free vibration analysis of gears with variable thickness using the Chebyshev–Ritz method based on three-dimensional elasticity theory. We derived the eigenvalue...

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Bibliographic Details
Published inMathematical problems in engineering Vol. 2018; no. 2018; pp. 1 - 11
Main Authors Qin, Hui-bin, Wang, Shi-ying, Lv, Ming, Li, Yi, Fu, Jun-fan
Format Journal Article
LanguageEnglish
Published Cairo, Egypt Hindawi Publishing Corporation 01.01.2018
Hindawi
Hindawi Limited
Subjects
Algebra
Annular plates
Chebyshev approximation
Eigenvalues
Elasticity
Engineering
Finite element analysis
Finite element method
Free vibration
Gears
Machining
Mathematical analysis
Polynomials
Resonant frequencies
Ritz method
Theory
Three dimensional analysis
Variable thickness
Vibration analysis
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Summary:To reflect vibration more comprehensively and to satisfy the machining demand for high-order frequencies, we presented a three-dimensional free vibration analysis of gears with variable thickness using the Chebyshev–Ritz method based on three-dimensional elasticity theory. We derived the eigenvalue equations. We divided the gear model into three annular parts along the locations of the step variations, and the admissible function was a Ritz series that consisted of a Chebyshev polynomial multiplying boundary function. The convergence study demonstrated the high accuracy of the present method. We used a hammering method for a modal experiment to test two annular plates and one gear’s eigenfrequencies in a completely free condition. We also applied the finite element method to solve the eigenfrequencies. Through a comparative analysis of the frequencies obtained by these three methods, we found that the results achieved by the Chebyshev–Ritz method were close to those obtained from the experiment and finite element method. The relative errors of four sets of data were greater than 4%, and the errors of the other 48 sets were less than 4%. Thus, it was feasible to use the Chebyshev–Ritz method to solve the eigenfrequencies of gears with variable thickness.
ISSN:1024-123X
1563-5147
DOI:10.1155/2018/9684154