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Published Articles

The Volume 11, No 2, June 2006

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The Effect of Ball Waviness on Nonlinear Vibration Associated with Rolling Element Bearings

S. P. Harsha, C. Nataraj, P. K. Kankar


An analytical model was developed to investigate the nonlinear vibrations of a rotor bearing system due to ball waviness. In the analytical formulation the contacts between the balls and the races are modelled as nonlinear springs, whose stiffnesses are obtained by using Hertzian elastic contact deformation theory. The governing differential equations of motion are obtained by using Lagrange's equations. The implicit type of numerical integration technique Newmark-beta with Newton-Raphson method is used to solve the nonlinear differential equations iteratively. A computer program was developed to simulate the effect of ball waviness. The formulation predicts the discrete spectra with specific frequency components for each order of ball waviness. Numerical results obtained from the simulation are compared with those of prior researchers.

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Higher-Periodic and Aperiodic Stick-Slip Dynamics in a Friction Damper

Naresh K. Chandiramani


A single-degree-of-freedom friction wedge damper suspension model is considered with nonlinear stiffness and viscous damping included. The response to harmonic excitations is obtained for the Coulomb, slip-rate, and smooth friction formulations. Since the convergence towards steady state is slow, especially for higher excitation frequencies, a modified shooting method is used when seeking periodic responses for the discontinuous friction models. The concept of stationary flow curves is used for stick-slip analysis when considering the discontinuous friction models. The coexistence of multiple stick regions - which could further complicate the dynamics - is obtained as a direct and nontrivial consequence of the stiffness nonlinearity. Results depicting stick displacements versus forcing amplitude, forcing frequency, wedge angle, and sideframe angle, are reported. The stick displacements are higher when a softening stiffness nonlinearity is present. Period multiplications, demultiplications, and chaotic responses are obtained when nonlinear stiffness (cubic) and viscous damping are considered and the track excitation frequency is varied. The present analysis, with nonlinearities included, could serve as a more realistic plant model in compensator design for real-time control.

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Nonlinear Dynamics for an Electromechanical Integrated Electrostatic Harmonic Drive

Lizhong Xu, Cuirong Zhu and Lei Qin


In this paper, an electromechanically coupled dynamic equation for an integrated electrostatic harmonic drive is transformed into a balance equation for static displacement and a dynamic equation for dynamic displacement. By defining the electric field force in Fourier series form, static and dynamic electric field forces are presented. Mode functions and nonlinear dynamic equations for the drive system are introduced. Using Lindstent?s perturbation method, nonlinear free vibration, forced response far from and near to natural frequency are discussed, respectively. Results show: The natural frequencies of the nonlinear drive system are smaller than those of the linear system. As the nonlinear parameter increases, the natural frequencies of the drive system decrease and the vibrating magnitude increases. When the exciting frequency is far from the natural frequency of the drive system, periodic times of the forced responses are all identical for different modes. When the exciting frequency is near to the natural frequency of the drive system, periodic times of the forced responses are different for different modes. The initial voltage and the clearance between the flexible ring and the outer stator have an obvious influence on the natural frequency and vibrating magnitude of the nonlinear drive system.

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Parametric Analysis of Ring Gear Structure Vibration Modes

Romil P. Tanna, Teik C. Lim


A comprehensive parametric analysis to quantify the sensitivity of planetary ring gear structural modes to variations in its geometrical parameters is performed by applying the dynamic finite element method. Specifically, the effects of the rim thickness to radius ratio, rim thickness to width ratio, helix angle, number of teeth, number of splines, and spline height and angular width are examined. The eigen-solutions reveal four classes of commonly occurring radial inextensional, extensional, out-of-plane bending and torsional ring modes. Of these four groups of modes, the radial inextensional and out-of-plane bending, frequencies are generally lower and fall more within the undesirable gear whine spectrum caused by the transmission error excitation. The predicted natural frequency sensitivities to the ring gear design parameters of interest are compiled graphically. The resulting trends are useful in identifying more robust designs with a lesser tendency towards the generation and transmissibility of excessive gear noise levels.

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