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Research Papers

# Nonlinear Oscillation of a Rotor-AMB System With Time Varying Stiffness and Multi-External Excitations

[+] Author and Article Information
M. Kamel

Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt

H. S. Bauomy1

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44511, Egypthany_samih@yahoo.com

1

Corresponding author.

J. Vib. Acoust 131(3), 031009 (Apr 22, 2009) (11 pages) doi:10.1115/1.3085884 History: Received June 13, 2008; Revised December 24, 2008; Published April 22, 2009

## Abstract

The rotor-active magnetic bearing system subjected to a periodically time-varying stiffness having quadratic and cubic nonlinearities is studied and solved. The multiple time scale technique is applied to solve the nonlinear differential equations governing the system up to the second order approximation. All possible resonance cases are deduced at this approximation and some of them are confirmed by applying the Rung–Kutta method. The main attention is focused on the stability of the steady-state solution near the simultaneous principal resonance and the effects of different parameters on the steady-state response. A comparison is made with the available published work.

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## Figures

Figure 1

Nonresonant time response solution at selected values:μ=0.8, ω1=0.2, ω2=3.0, ω=3.9, Ω=1.9, f1i, i=1,2,…,5 are 0.1, 0.05, 0.2, 0.4, 0.1; f2i, i=1,2,…,6 are 0.1, 0.05, 0.35, 0.2, 0.5, 0.4; f3i, i=2,4,6 are 0.2, 0.1, 0.4; Fii=1,2,3 are 0.4, 0.005, 0.004; αi, i=2,3,…,7 are 0.2, 0.2, 0.01, 0.02, 0.05, 0.02 and βi, i=2,3,5,6,7 are 0.2, 0.1, 0.01, 0.05, 0.1, respectively

Figure 2

Time trace of the response of two modes at resonance cases (a) Ω≅ω1, (b) 2ω≅ω1+ω2, and (c) Ω,ω≅ω1, ω2

Figure 3

The basic frequency response curves of the different cases: (1) (a1≠0, a2=0): (ω1=0.2, μ=0.4, α2=0.2, α5=0.02, α6=0.05, f12=0.05, F1=0.5). (2) (a1=0, a2≠0): (ω2=3.0, μ=0.05, β2=0.2, α5=0.004, α6=0.5, f22=0.05, f25=0.5, f36=1.0, F1=4.0). (3) (a1≠0, a2≠0): (ω1=2.0, μ=0.4, α2=1.5, α3=0.04, α5=0.005, α6=0.004, ω2=3.5, f12=0.05, f13=3.0, f15=2.0, F1=5.0), (ω1=0.5, μ=0.05, β2=1.0, β3=0.04, α5=0.04, α6=0.4, ω2=1.0, f22=0.1, f23=1.0, f25=0.2, f34=2.0, f36=0.5, F1=5.0).

Figure 4

Frequency response curves of the first case (a1≠0 and a2=0)

Figure 5

Frequency response curves of the second case (a2≠0 and a1=0)

Figure 6

Frequency response curves of the second case (a2≠0 and a1=0)

Figure 7

Frequency response curves of the third case (a2≠0 and a1≠0)

Figure 8

Frequency response curves of the third case (a2≠0 and a1≠0)

Figure 9

Frequency response curves of the third case (a2≠0 and a1≠0)

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