The Magneto-Elastic Internal Resonances of Rectangular Conductive Thin Plate With Different Size Ratios

2017 ◽  
Vol 34 (5) ◽  
pp. 711-723
Author(s):  
J. Li ◽  
Y. D. Hu ◽  
Y. N. Wang
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Thin Plate ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Transverse Magnetic ◽  
Phase Portraits ◽  
Internal Resonances ◽  
Basic Equations

AbstractBased on the basic equations of electromagnetic elastic motion and the expression of electromagnetic force, the electromagnetic vibration equation of the rectangular thin plate in transverse magnetic field is obtained. For a rectangular plate with one side fixed and three other sides simply supported, its time variable and space variable are separated by the method of Galerkin, and the two-degree-of-freedom nonlinear Duffing vibration differential equations are proposed. The method of multiple scales is adopted to solve the model equations and obtain four first-order ordinary differential equations governing modulation of the amplitudes and phase angles involved via 1:1 or 1:3 internal resonances with different size ratios. With a numerical example, the time history response diagrams, phase portraits and 3-dimension responses of two order modal amplitudes are respectively captured. And the effects of initial values, thickness and magnetic field intensities on internal resonance characteristics are discussed respectively. The results also present obvious characteristics of typical nonlinear internal resonance in this paper.

scholarly journals Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Bamadev Sahoo ◽  
L. N. Panda ◽  
G. Pohit
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Mean Value ◽  
Phase Portraits ◽  
Axially Moving Beam ◽  
Internal Resonances ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

Internal Resonance and Nonlinear Vibrations of Eccentric Rotating Composite Laminated Circular Cylindrical Shell

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Yufei Zhang
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Internal Resonances

Abstract This paper is focused on the internal resonances and nonlinear vibrations of an eccentric rotating composite laminated circular cylindrical shell subjected to the lateral excitation and the parametric excitation. Based on Love thin shear deformation theory, the nonlinear partial differential equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton’s principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The excitation conditions of the internal resonance is found through the Campbell diagram, and the effects of eccentricity ratio and geometric papameters on the internal resonance of the eccentric rotating system are studied. Then, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Finally, we study the nonlinear vibrations of the eccentric rotating composite laminated circular cylindrical shell systems.

Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances

1997 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Walter Lacarbonara ◽  
Char-Ming Chin
Keyword(s):  
Boundary Conditions ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Buckling Mode ◽  
Stable Mode ◽  
One To One ◽  
Nonlinear Normal

Abstract Nonlinear normal modes of a buckled beam about its first buckling mode shape are investigated. Fixed-fixed boundary conditions are considered. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate expressions for the nonlinear normal modes are obtained by applying the method of multiple scales to the governing integro-partial-differential equation and boundary conditions. Curves displaying variation of the amplitude with the internal resonance detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess either one stable mode, or three stable normal modes, or two stable and one unstable normal modes. On the other hand, for a one-to-one internal resonance between the first and second modes, two nonlinear normal modes exist. The two nonlinear modes are either neutrally stable or unstable. In the case of one-to-one resonance between the third and fourth modes, two neutrally stable, nonlinear normal modes exist.

Multimode Interactions in Suspended Cables

2002 ◽  
Vol 8 (3) ◽  
pp. 337-387 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat ◽  
Char-Ming Chin ◽  
Walter Lacarbonara
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Internal Resonances ◽  
Out Of Plane ◽  
Space Formulation ◽  
Suspended Cables ◽  
One To One ◽  
Plane Mode

We investigate the nonlinear nonplanar responses of suspended cables to external excitations. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The sag-to-span ratio of the cable considered is such that the natural frequency of the first symmetric in-plane mode is at first crossover. Hence, the first symmetric in-plane mode is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and, simultaneously, in a two-to-one internal resonance with the first symmetric out-of-plane mode. Under these resonance conditions, we analyze the response when the first symmetric in-plane mode is harmonically excited at primary resonance. First, we express the two governing equations of motion as four first-order (i.e., state-space formulation) partial-differential equations. Then, we directly apply the methods of multiple scales and reconstitution to determine a second-order uniform asymptotic expansion of the solution, including the modulation equations governing the dynamics of the phases and amplitudes of the interacting modes. Then, we investigate the behavior of the equilibrium and dynamic solutions as the forcing amplitude and resonance detunings are slowly varied and determine the bifurcations they may undergo.

Nonlinear Vibrations of a Thin Plate Under Simultaneous Internal and External Resonances

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Usama H. Hegazy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Thin Plate ◽  
Nonlinear Response ◽  
Phase Plane ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
State Response ◽  
The Stability ◽  
Parametric And External Excitations

The dynamic behavior of a rectangular thin plate under parametric and external excitations is investigated. The motion of the thin plate is modeled by coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought by applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency-response function and the phase-plane methods. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. Moreover, the chaotic motion of the thin plate is found by numerical simulation.

NONLINEAR MAGNETOELASTIC VIBRATION EQUATIONS AND RESONANCE ANALYSIS OF A CURRENT-CONDUCTING THIN PLATE

2008 ◽  
Vol 08 (04) ◽  
pp. 597-613 ◽  
Author(s):  
YUDA HU ◽  
JING LI
Keyword(s):  
Magnetic Field ◽  
Frequency Response ◽  
Thin Plate ◽  
Transverse Magnetic Field ◽  
Steady Motion ◽  
Transverse Magnetic ◽  
Approximate Analytic Solution ◽  
Solution Stability ◽  
Amplitude Frequency Response ◽  
Basic Set

Based on the Maxwell equations, the electromagnetic constitutive relations and boundary condition, the electrodynamic equation and the electromagnetic force expressions in an electromagnetic field are derived. Using the principle of virtual work, the basic set of equations for nonlinear electromagnetic elasticity vibration expressed by the displacement of a thin plate in a longitudinal and a transverse magnetic field is obtained, respectively. In addition, we study the nonlinear principal resonance and the solution stability of a thin plate with two opposite sides simply supported and subjected to a mechanical live load and in a constant transverse magnetic field. By the method of multiple scales, the amplitude frequency response equation and the approximate analytic solution in steady motion are also derived. According to the characteristic of singularity and the Lyapunov stability theory, the stability of the solution is analyzed and the critical condition of stability is determined. Finally, by means of numerical calculations, the amplitude frequency response curves, time history response plots and phase charts of the magnetoelasticity vibration are obtained.

scholarly journals One-to-One and Three-to-One Internal Resonances in MEMS Shallow Arches

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Hassen M. Ouakad ◽  
Hamid M. Sedighi ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Microelectromechanical Systems ◽  
Multiple Scales ◽  
Internal Resonances ◽  
Interesting Phenomenon ◽  
Shallow Arches ◽  
Energy Harvesters ◽  
Modal Coupling ◽  
One To One ◽  
Direct Attack

The nonlinear modal coupling between the vibration modes of an arch-shaped microstructure is an interesting phenomenon, which may have desirable features for numerous applications, such as vibration-based energy harvesters. This work presents an investigation into the potential nonlinear internal resonances of a microelectromechanical systems (MEMS) arch when excited by static (DC) and dynamic (AC) electric forces. The influences of initial rise and midplane stretching are considered. The cases of one-to-one and three-to-one internal resonances are studied using the method of multiple scales and the direct attack of the partial differential equation of motion. It is shown that for certain initial rises, it is possible to activate a three-to-one internal resonance between the first and third symmetric modes. Also, using an antisymmetric half-electrode actuation, a one-to-one internal resonance between the first symmetric and the second antisymmetric modes is demonstrated. These results can shed light on such interactions that are commonly found on micro and nanostructures, such as carbon nanotubes.

scholarly journals Nonlinear Forced Vibration of a Viscoelastic Buckled Beam with 2 : 1 Internal Resonance

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Liu-Yang Xiong ◽  
Guo-Ce Zhang ◽  
Hu Ding ◽  
Li-Qun Chen
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Mode Shapes ◽  
Analytical Approximations ◽  
Buckled Beam ◽  
Validity Range ◽  
The Galerkin Method

Nonlinear dynamics of a viscoelastic buckled beam subjected to primary resonance in the presence of internal resonance is investigated for the first time. For appropriate choice of system parameters, the natural frequency of the second mode is approximately twice that of the first providing the condition for 2 : 1 internal resonance. The ordinary differential equations of the two mode shapes are established using the Galerkin method. The problem is replaced by two coupled second-order differential equations with quadratic and cubic nonlinearities. The multiple scales method is applied to derive the modulation-phase equations. Steady-state solutions of the system as well as their stability are examined. The frequency-amplitude curves exhibit the steady-state response in the directly excited and indirectly excited modes due to modal interaction. The double-jump, the saturation phenomenon, and the nonperiodic region phenomena are observed illustrating the influence of internal resonance. The validity range of the analytical approximations is assessed by comparing the analytical approximate results with a numerical solution by the Runge-Kutta method. The unstable regions in the internal resonance are explored via numerical simulations.

Electromagnetic Field in the Chaotic Motion of Thin Plate

2010 ◽  
Vol 163-167 ◽  
pp. 3114-3117
Author(s):  
Li Jing Liu ◽  
Zhi Ren Wang ◽  
Jin Feng Lv
Keyword(s):  
Magnetic Field ◽  
Electromagnetic Field ◽  
Mechanical Loading ◽  
Thin Plate ◽  
Chaotic Motion ◽  
Critical Condition ◽  
Melnikov Method ◽  
Transverse Magnetic ◽  
Smale Horseshoe ◽  
Vibration Equation

Based on the vibration equation of beam plate, under mechanical loading in a uniform transverse magnetic field, the vibration equation of the conductive beam plate is reduced to two cases ,which is no-pertubation system and pertubation system. For pertubation system ,n-order harmonic orbit is given by means of the Melnikov method. Finally, the critical condition of chaos phenomena is given in the transformation of Smale horseshoe.

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