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Research Papers

Amplitudes Decay in Different Kinds of Nonlinear Oscillators

[+] Author and Article Information
Mohammad A. Al-Shudeifat

Aerospace Engineering,
Khalifa University of Science,
Technology and Research (KUSTAR),
Abu Dhabi 127788, UAE
e-mail: mohd.shudeifat@kustar.ac.ae

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 7, 2014; final manuscript received November 24, 2014; published online January 27, 2015. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 137(3), 031012 (Jun 01, 2015) (9 pages) Paper No: VIB-14-1381; doi: 10.1115/1.4029288 History: Received October 07, 2014; Revised November 24, 2014; Online January 27, 2015

A transformation is employed to obtain expressions for the decay of the displacement, the velocity, and the energy for various forms of nonlinear oscillators. The equation of motion of the nonlinear oscillator is transformed into a first-order decay term plus an energy term, where this transformed equation can be decoupled into a set of two analytically solvable equations. The decoupled equations can be solved for the decay formulas. Unlike other methods in the literature, this transformation method is directly applied to the equation of motion, and an approximate solution is not required to be known a priori. The method is first applied to a purely nonlinear oscillator with a non-negative, real-power restoring force to obtain the decay formulas. These decay formulas are found to behave similarly to those of a linear oscillator. In addition, these formulas are employed to obtain an accurate formula for the frequency decay. Based on this result, the exact frequency formula given in the literature for this oscillator is generalized by substituting the initial values of the envelopes for the actual initial conditions. By this modification, the formulas for the initial and time-varying frequencies become valid for any combination of the initial displacement and velocity. Furthermore, a generalized nonlinear oscillator for which the transformation is always valid is introduced. From this generalized oscillator, the proposed transformation is applied to analyze various types of oscillators.

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Figures

Grahic Jump Location
Fig. 1

Instantaneous energy obtained via simulation of the system in Eq. (1) and the energy decay envelope obtained from Eq. (9) for a nonlinear oscillator of n = 7/3 in (a) and a linear oscillator of n = 1 in (b)

Grahic Jump Location
Fig. 2

Simulation results for the displacement (a), the velocity (b), the instantaneous energy (c), and the nonlinear force (d) and the associated envelopes in (a)–(c) calculated using the analytical formulas for n = 1/2, α = 0.05  N s/m, β = 1 N/m1/2, x0 = 1 m, and x·0 = 0

Grahic Jump Location
Fig. 3

Simulation results for the displacement (a), the velocity (b), the instantaneous energy (c), and the nonlinear force (d) and the associated envelopes in (a)–(c) calculated using the analytical formulas for n = 4, α = 0.05 N s/m, β = 1 N/m4, x0 = 1 m, and x·0 = 1

Grahic Jump Location
Fig. 4

Comparison of the analytical frequency formula in Eq. (20) with the wavelet transform (a) and simulation (b) for n = 3, α = 0.02 N s/m, β = 1 kN/m3, x(0) = 0.1 m, and x·(0) = 0

Grahic Jump Location
Fig. 5

Comparison of the analytical frequency formula in Eq. (20) with the wavelet transform (a) and simulation (b) for n = 3, α = 0.02 N s/m, β = 10 kN/m3, x(0) = 0, and x·(0) = 1 m/s

Grahic Jump Location
Fig. 6

Comparison of the analytical frequency formula in Eq. (20) with the wavelet transform (a) and simulation (b) for n = 3, α = 0.04 N s/m, β = 10 kN/m3, x(0) = 0, and x·(0) = 1 m/s

Grahic Jump Location
Fig. 7

Frequency versus energy: (a) α = 0.02 N s/m, β = 1 kN/m3, x(0) = 0, and x·(0) = 0.1 m/s and (b) n = 3, α = 0.04 N s/m, β = 10 kN/m3, x(0) = 0, and x·(0) = 1 m/s

Grahic Jump Location
Fig. 8

Comparison of the analytical frequency formula in Eq. (20) with the wavelet transform (a) and simulation and (b) for n = 2, α = 0.02 N s/m, β = 10 N/m2, x(0) = 0, and x·(0) = 1 m/s

Grahic Jump Location
Fig. 9

Comparison of the analytical frequency formula in Eq. (20) with the wavelet transform (a) and simulation and (b) for n = 9/4, α = 0.02 N s/m, β = 10 N/m9/4, x(0) = 0, and x·(0) = 1 m/s

Grahic Jump Location
Fig. 10

Comparison of the analytical frequency formula in Eq. (20) with the wavelet transform for n = 1/2 (a) and n = 1/3 and (b) for α = 0.02 N s/m, β = 10 N/mn, x(0) = 0, and x·(0) = 1 m/s

Grahic Jump Location
Fig. 11

Simulation results for the instantaneous kinetic and potential energies of the system in Eq. (21) for n = 3, α = 0, λ = 1, β = 1 N/m3, k = 1 N/m, x(0) = 0, and x·(0) = 1 m/s for p = 1 (a) and p = 2 (b)

Grahic Jump Location
Fig. 12

Simulation results for the displacement (a) and the velocity and (b) and the corresponding analytical envelopes for n = 2/3, λ = 1, α = 0.2 N s/m, β = 10 N/m2/3, x0 = 1 m, and x·0 = 0

Grahic Jump Location
Fig. 13

Simulation results for the displacement (a) and the velocity (b) and the corresponding analytical envelopes for n = 5/2, λ = 1, α = 0.2 N s/m, β = 50 N/m5/2, x0 = 1 m, and x·0 = 0

Grahic Jump Location
Fig. 14

Simulation results for the displacement (a) and the velocity and (b) and the corresponding analytical envelopes for n = 5/2, λ = 1, α = 0.2 N s/m, β = 20 N/m5/2, x0 = 1  m, and x·0 = 1 m/s

Grahic Jump Location
Fig. 15

Simulation results of the displacements and the corresponding analytical envelopes obtained by Eq. (31) for n = 3, μ = 0.1, β = 10  N/m3, and x·0 = 0 for x0 = 0.1 m (a), x0 = 1 m (b), and x0 = 2 m (c)

Grahic Jump Location
Fig. 16

Simulation results for the displacement (a), the velocity (b), the instantaneous energy (c), and the nonlinear force (d) and the corresponding analytical envelopes in (a)–(c) for the parameter values α = 0.03, k = 0.25, β = 1, x0 = 0, and x·0 = 1 m/s

Grahic Jump Location
Fig. 17

Simulation results for the displacement (a), the velocity (b), the instantaneous energy (c), and the nonlinear force (d) and the corresponding analytical envelopes in (a)–(c) for the parameter values α = 0.03, k = 1, β = -0.3, x0 = 0, and x·0 = 1 m/s

Grahic Jump Location
Fig. 18

Simulation results for the displacement (a), the velocity (b), the instantaneous energy (c), and the nonlinear force (d) and the corresponding analytical envelopes in (a)–(c) for the parameter values α = 0.03, k = -0.2, β = 1, x0 = 0, and x·0 = 2  m/s

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