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Harmonic Oscillations and Resonances in 3-D Nonlinear Dynamical System
Usama H. Hegazy1,
, Mousa A. ALshawish1
1Department of Mathematics, Faculty of Science, Al-Azhar University, Gaza, Palestine
Abstract
This paper is concerned with the three dimensional motion of a nonlinear dynamical system. The motion is described by nonlinear partial differential equation, which is converted by Galerkin method to three dimensional ordinary differential equations. The three dimensional differential equations, under the influence of external forces, are solved analytically and numerically by the multiple time scales perturbation technique and the Runge-Kutta fourth order method. Phase plane technique and frequency response equations are used to investigate the stability of the system and the effects of the parameters of the system, respectively.
Keywords: Galerkin method, resonances, nonlinearities
Copyright © 2016 Science and Education Publishing. All Rights Reserved.Cite this article:
- Usama H. Hegazy, Mousa A. ALshawish. Harmonic Oscillations and Resonances in 3-D Nonlinear Dynamical System. International Journal of Partial Differential Equations and Applications. Vol. 4, No. 1, 2016, pp 7-15. http://pubs.sciepub.com/ijpdea/4/1/2
- Hegazy, Usama H., and Mousa A. ALshawish. "Harmonic Oscillations and Resonances in 3-D Nonlinear Dynamical System." International Journal of Partial Differential Equations and Applications 4.1 (2016): 7-15.
- Hegazy, U. H. , & ALshawish, M. A. (2016). Harmonic Oscillations and Resonances in 3-D Nonlinear Dynamical System. International Journal of Partial Differential Equations and Applications, 4(1), 7-15.
- Hegazy, Usama H., and Mousa A. ALshawish. "Harmonic Oscillations and Resonances in 3-D Nonlinear Dynamical System." International Journal of Partial Differential Equations and Applications 4, no. 1 (2016): 7-15.
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At a glance: Figures
1. Introduction
Problems involving nonlinear differential equations are extremely diverse, and methods of solutions or analysis are problem dependent. Nonlinear systems are interesting for engineers, physicists and mathematicians because most physical systems are nonlinear in nature. The sub-combination internal resonance of a uniform cantilever beam of varying orientation with a tip mass under vertical base excitation is studied. The Euler–Bernoulli theory slender beam was used to derive the governing nonlinear partial differential equation [1]. The dynamic stability of a moving string in three-dimensional vibration is investigated [2]. Three nonlinear integro-differential equations of motion are studied and the analysis is focused on the case of primary resonance of the first in-plane flexural mode when its frequency is approximately twice the frequency of the first out-of-plane flexural-torsional mode [3]. The method of multiple time scales is applied to investigate the response of nonlinear mechanical systems with internal and external resonances. The stability of vibrating systems is investigated by applying both the frequency response equation and the phase plane methods. The numerical solutions are focused on both the effects of the different parameters and the behavior of the system at the considered resonance cases [4, 5]. The nonlinear characteristics in the large amplitude three-dimensional free vibrations of inclined sagged elastic cables are investigated [6]. The nonlinear forced vibration of a plate-cavity system is analytically studied. Galerkin method is used to derive coupled nonlinear equations of the system. In order to solve the nonlinear equations of plate-cavity system, multiple scales method is employed. Closed form expressions are obtained for the frequency-amplitude relationship in different resonance conditions [7]. The steady-state periodic response of the forced vibration for an axially moving viscoelastic beam in the supercritical speed range is studied [8]. For this motion, the model is cast in the standard form of continuous gyroscopic systems. Internal Various approximate analytical methods are developed for obtaining solutions for strongly nonlinear differential equations in a complex function. The methods of harmonic balance, Krylov-Bogoliubov and elliptic perturbation are utilized [9]. The problem of suppressing the vibrations of a hinged-hinged flexible beam when subjected to external harmonic and parametric excitations is considered and studied. The multiple scale perturbation method is applied to obtain a first-order approximate solution. The equilibrium curves for various controller parameters are plotted. The stability of the steady state solution is investigated using frequency-response equations. The approximate solution was numerically verified. It is found that all predictions from analytical solutions were in good agreement with the numerical simulations [10].
2. Equations of Motion
The nonlinear partial differential equation governing the flexural deflection 
of the beam, subject to harmonic axial excitation
, is given by [11,12][, ]
 | (1) |
under the following boundary conditions:
 | (2) |
Equation (1) can be converted to a three dimensional nonlinear ordinary differential equations applying the method of Galerkins and using the following expression
 | (3) |
into equation (1). Then we have
 | (4) |
 | (5) |
 | (6) |
3. Perturbation Solution
The method of multiple scales is applied to determine an approximate solution for the differential equations (4-6). Assuming that g, h and k are in the forms
 | (7) |
Where
,
The time derivatives are written as
 | (8) |
Where
 | (9) |
Substituting eqs. (7-9) into eqs. (4-6) and equating coefficients of same powers of ε yields:
 | (10) |
 | (11) |
 | (12) |
 | (13) |
 | (14) |
 | (15) |
The solution of eqs. (10-12) is expressed as
 | (16) |
 | (17) |
 | (18) |
Where
are complex functions in
Substituting eqs. (16-18) into eqs (13-15),we get
 | (19) |
 | (20) |
 | (21) |
Where
denotes a complex conjugate of the preceding term.
The general solution of eqs. (19-20) can be written in the following form
 | (22) |
 | (23) |
 | (24) |
From eqs. (22-24) the following resonance cases are extracted:
• Internal Resonance:
1. 
,
2. 
,
• External Resonance:
a. Primary resonance
1. 
,
2. 
,
3. 
,
b. Simultaneous resonance
1. 
.
4. Stability Analysis
We shall consider the resonance case Ω ≈
when ω1 ≈ ω2 ≈ ω3. Using the detuning parameter σ, the resonance case are expressed as
 | (25) |
Substituting eq. (25) into eqs. (19-21) and eliminating terms that produce secular term then performing some algebraic manipulations, we obtain the following modulation equations:
 | (26) |
 | (27) |
 | (28) |
Letting
 |
 |
 |
where 
are functions of 
Separating real and imaginary parts gives the following six equations governing the amplitude and phase modulations
 | (29) |
 | (30) |
 | (31) |
 | (32) |
 | (33) |
 | (34) |
where
 |
The steady-state solutions of eqs. (29-34) are obtained by setting 
into eqs. (29-34). This results in the following nonlinear algebraic equations, which are called the frequency response equations:
 | (35) |
 | (36) |
 | (37) |
The coefficients 
(1,2,...,15) , are given in Appendix.
5. Numerical Results and Discussions
In this section, the Runge-Kutta fourth order method is applied to determine the numerical time series solutions (t, g), (t, h), and (t, k)and the phase planes (g, v), (h, v), (k, v), respectively, for the three modes of the nonlinear system (4-6). Moreover, the fixed points of the model is obtained by solving the frequency response equations (35-37) numerically.
5.1. Time-response SolutionA non-resonant time response and the phase plane of the three modes of vibration of the system is shown in Figure 1. In Figure 2, different resonance cases are investigated and an approximate percentage of increase, if exists, in maximum steady-state amplitude compared to that in the non-resonant case is indicated.
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(a) Internal resonance cases
 |
 |
(b) External resonance cases
(1) primary resonance:
 |
(2) Simultaneous resonances
 |
The numerical results are presented graphically in Figs. (3-5) as the amplitudes 
against the detuning parameters 
for different values of other parameters. Each curve in these figures consists of two branches. Considering Figure 3(a) as basic case to compare with, it can be seen from Figure 3(b), (c) that the steady-state amplitude 
decreases as each of 
are increased but in Figure 3(e) the steady- state amplitude 
increases as each of 
increases .Whereas the frequency response curves in Figure 3(h) are shifted to the right as 
increases.
Considering Figure 4(a) as basic case to compare with, it can be seen from Figure 4(c) that the steady-state amplitude 
increases as each of
are increasing. But in Figures 4(b),(c), (d) and(i), the steady- state amplitude 
decreases as each of 
are increased. In Figure 4(h) the curves are shifted to the right as 
increases. Whereas, the frequency response curve are bent to right as
varies from negative to positive values, showing hardening nonlinearity effect, Figure 4(f), (g).
Considering Figure 5(a) as basic case to compare with, it can be seen From Figures 5(b),(c) that the steady-state amplitude 
increases as each of
are increased. The nonlinearity effect of 
is shown in Figure 5(f), (g), whereas the curves are being shifted in Figure 5(d).
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, 
6. Conclusions
We have studied the analytic and numerical solutions of three dimensional nonlinear differential equations that describe the oscillations of abeam subjected to external forces. The multiple scales method and Runge-Kutta fourth order numerical method are utilized to investigate the system behavior and its stability. All possible resonance cases were be extracted and effect of different parameters on system behavior at resonant condition were studied. We may conclude the following:
(1) The steady-state amplitude of the first mode increases as each of the external force amplitude 
and nonlinear coefficients 
are increased.
(2) The steady-state amplitude of the first mode decreases as each of the linear damping coefficient 
and the nonlinear coefficient 
and the natural frequency 
are increased.
(3) The steady-state amplitude of the second and third mode increase as the external force amplitude 
increases.
(4) The steady-state amplitude of the second mode decreases as each of the linear damping coefficient
and nonlinear coefficient
and the second mode amplitude 
and the natural frequency 
are increased.
(5) The steady-state amplitude of the third mode increases as each of the external force amplitude 
and the first mode amplitude 
are increased.
Nomenclature
: Natural frequencies of the system
: Small dimensionless perturbation parameter
: Linear damping coefficients
: Nonlinear parameters
: Excitation force amplitude
: Excitation frequency
: Differential operators
: Radius of gyration of cross-section area
: Mass per unit length of beam
: Young's modulus
: Moment of inertia
: Length of beam
: Time
: Real coefficients
: Fast time scale
: Slow time scale
: Non- dimensional quantities
: Perturbation variables expansion
: Phase angles of the polar forms
: Complex valued quantities
: Complex conjugate for preceding terms at the same equation
: Detuning parameter
: Steady-state amplitudes
References
| [1] | Yaman, M, “Analysis of subcombination internal resonances in a non- linear cantilever beam of varying orientation with tip mass,” Int J Non-Linear Mech, 58. 22-29. 2014. | ||
 In article | View Article | ||
| [2] | Huang, J., Fung, R. and Lin, C, “Dynamic stability of a moving string undergoing three-dimensional vibration,” Int J Mech Sci, 37 (2). 145-160. 1995. | ||
| In article | View Article | ||
| [3] | Pai, P.F. and Nayfeh, A.H, “Three-Dimensional nonlinear vibrations of composite beams-III. Chordwise excitations,” Nonlinear Dyn, 2. 137-156. 1991. | ||
| In article | View Article | ||
| [4] | Hegazy, U.H., 3:1 Internal resonance of a string-beam coupled system with cubic nonlinearities, Commun Nonlinear Sci Numer Simul, 15, 4219-4229, 2010. | ||
| In article | View Article | ||
| [5] | Hegazy, U.H, “Nonlinear vibrations of a thin plate under simultaneous internal and external resonances,” J Vib Acoust, 132. 051004. 9 pages. 2010. | ||
| In article | |||
| [6] | Srinil, N., Rega, R. and Chucheepsaku, S, “Large amplitude three-dimensional free vibrations of inclined sagged elastic cables,” Nonlinear Dyn, 33. 129-154. 2003. | ||
| In article | View Article | ||
| [7] | Sadri, M. and Younesian, D, “Nonlinear harmonic vibration analysis of a plate-cavity system,” Nonlinear Dyn, 74. 1267-1279. 2013. | ||
| In article | View Article | ||
| [8] | Zhang, G., Ding, H., Chen, L. and Yang, S, “Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds,” J Sound Vibr, 331 (7). 1612-1623. 2012. | ||
| In article | View Article | ||
| [9] | Cveticanin, L, “Analytical methods for solving strongly non-linear differential equations,” J Sound Vibr, 214 (2). 325-338. 1998. | ||
| In article | View Article | ||
| [10] | Hegazy, U.H, “Single-Mode response and control of a hinged-hinged flexible beam,” Arch Appl Mech, 79. 335-345. 2009. | ||
| In article | View Article | ||
Appendix
The coefficients presented in (35), are as follows :
 |
The coefficients presented in (36), are as follows:
The coefficients presented in (37), are as follows :
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