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# Vibration Analysis of an Unbalanced Rotating Shaft Due to Ball Waviness

[+] Author Affiliations
S. H. Upadhyay, S. C. Jain, S. P. Harsha

Indian Institute of Technology Roorkee, Roorkee, UL, India

Paper No. DETC2009-86272, pp. 979-986; 8 pages
doi:10.1115/DETC2009-86272
From:
• ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
• Volume 1: 22nd Biennial Conference on Mechanical Vibration and Noise, Parts A and B
• San Diego, California, USA, August 30–September 2, 2009
• Conference Sponsors: Design Engineering Division and Computers in Engineering Division
• ISBN: 978-0-7918-4898-2 | eISBN: 978-0-7918-3856-3

## abstract

This paper presents an analytical model to investigate the nonlinear dynamic analysis of high speed ball bearings due to ball waviness and unbalanced rotor. The Hertzian contact theory is applied to calculate the elastic deflection and nonlinear contact force, while the rotor has translational and angular motions. In the analytical formulation, the contacts between rolling elements and inner/outer races are considered as nonlinear springs. A nonlinear damping formula, correlating the contact damping force with the equivalent contact stiffness and contact deformation rate is used in the derivation. The numerical integration technique Newmark-β with Newton-Raphson method is used to solve the nonlinear differential equations iteratively. The results are presented in the form of FFTs. The formulation predicts the discrete spectra with specific frequency components for each order of ball waviness. Numerical results obtained from the simulation are validated with respect to those of prior results. Due to an unbalanced rotor, the system is bi-periodically excited. In the vibration spectrum, the peak amplitude of vibrations appears at the wave passage frequency (Nw ωroll ), rotational frequency (X) and interaction between wave passage and rotational frequencies.

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