Fractals Yielded via Time-Varying Nonlinear Equation

2015 ◽  
pp. 287-294
Keyword(s):  
Nonlinear Equation ◽  
Time Varying

The Linear Approximated Equation of Vibration for a Pair of Spur Gears: Theory and Experiment

1992 ◽  
Author(s):  
Yurong Cai ◽  
Teru Hayashi
Keyword(s):  
Analytical Solution ◽  
Nonlinear Equation ◽  
Linear Equation ◽  
Static Load ◽  
Time Varying ◽  
Spur Gears ◽  
Contact Ratio ◽  
Profile Error ◽  
Rotational Vibration ◽  
Set Up

Abstract The nonlinear equation for the rotational vibration of a pair of spur gears has a restriction that the analytical solution of the equation cannot be obtained. In this paper, the linear equation of vibration is derived theoretically and its physical model, i.e. the linear model of vibration is presented. The analytical solution of the linear equation, which is derived by analytical method, agrees well with the numerically calculated result by the nonlinear equation. By analyzing the analytical solution of the linear equation in detail, we clarified the relation between the waveforms of the vibration and the profile error of gear pairs, and also found that the effect of the contact ratio to the vibration is large and complex. The equivalent error, accounting for effects of the static load, the time-varying stiffness and the profile error of gear pairs, is proposed in this paper. It can be considered as promising for evaluating the profile error, because the vibration of gear pairs is excited mainly by the equivalent error. Finally, for confirming the above results, the vibration of two tested gear pairs has been measured by an experimental set-up for this purpose.

The Linear Approximated Equation of Vibration of a Pair of Spur Gears (Theory and Experiment)

1994 ◽  
Vol 116 (2) ◽  
pp. 558-564 ◽  
Author(s):  
Y. Cai ◽  
T. Hayashi
Keyword(s):  
Analytical Solution ◽  
Nonlinear Equation ◽  
Linear Equation ◽  
Static Load ◽  
Time Varying ◽  
Spur Gears ◽  
Contact Ratio ◽  
Profile Error ◽  
Rotational Vibration ◽  
Set Up

The nonlinear equation for the rotational vibration of a pair of spur gears has a restriction that the analytical solution of the equation cannot be obtained. In this paper, the linear equation of vibration is derived theoretically and its physical model, i.e., the linear model of vibration is presented. The analytical solution of the linear equation, which is derived by analytical method, agrees well with the numerically calculated result by the nonlinear equation. By analyzing the analytical solution of the linear equation in detail, we clarified the relation between the waveforms of the vibration and the profile error of gear pairs, and also found that the effect of the contact ratio to the vibration is large and complex. The equivalent error, accounting for effects of the static load, the time-varying stiffness, and the profile error of gear pairs, is proposed in this paper. It can be considered as promising for evaluating the profile error, because the vibration of gear pairs is excited mainly by the equivalent error. Finally, for confirming the above results, the vibration of two tested gear pairs has been measured by an experimental set-up for this purpose.

BIFURCATION OF MULTIPLE LIMIT CYCLES FOR A ROTOR-ACTIVE MAGNETIC BEARINGS SYSTEM WITH TIME-VARYING STIFFNESS

2008 ◽  
Vol 18 (03) ◽  
pp. 755-778 ◽  
Author(s):  
J. LI ◽  
Y. TIAN ◽  
W. ZHANG ◽  
S. F. MIAO
Keyword(s):  
Nonlinear Equation ◽  
Limit Cycles ◽  
Multiple Scales ◽  
Equation Of Motion ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Detection Function ◽  
Single Degree Of Freedom ◽  
Amb System

The bifurcations of multiple limit cycles for a rotor-active magnetic bearings (AMB) system with the time-varying stiffness are considered in this paper. The governing nonlinear equation of motion is established for the rotor-AMB system with single-degree-of-freedom and parametric excitation. Using the method of multiple scales, the governing nonlinear equation of motion is first transformed to the averaged equation, which is in the form of a Z2-symmetric perturbed polynomial Hamiltonian system of degree 5. Then, the bifurcation theory of planar dynamical system and the method of detection function are utilized to analyze the bifurcations of multiple limit cycles of the averaged equation. Four groups of parametric controlling conditions are given to obtain the configurations of compound eyes. It is found that there exist respectively at least 17, 19, 21 and 22 limit cycles in the rotor-AMB system with the time-varying stiffness under the different controlling conditions.

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