Magneto-Aeroelastic Internal Resonances of a Rotating Circular Plate Based on Gyroscopic Systems Decoupling

2020 ◽  
pp. 2150010
Author(s):  
Wenqiang Li ◽  
Yuda Hu
Keyword(s):  
Circular Plate ◽  
Multiple Scales ◽  
Period Doubling ◽  
Coefficient Matrix ◽  
Transverse Displacement ◽  
Internal Resonances ◽  
Resonance Amplitude ◽  
Steady State Responses ◽  
Nonlinear Modulation ◽  
Gyroscopic Systems

In this paper, a principal and 3:1 internal resonance of an edge-clamped conductive circular plate rotating in air-magnetic environment is investigated, where the electromagnetic force expressions and a simple empirical aerodynamic model are used in modelling. Based on the transverse displacement assumption with a combination of two degenerate linearized modes, the 2 degree of freedom (2-DOF) magneto-aeroelastic asymmetric nonlinear gyroscopic systems are derived by utilizing the Galerkin approach. The method of multiple scales and the solvable condition for the coefficient matrix of the gyroscopic systems is employed to decouple the 2-DOF nonlinear gyroscopic systems and achieve the nonlinear modulation equations of the steady state responses. The numerical results for the relationship between two eigenfrequencies verify the existence of 3:1 internal resonances of the circular plate rotating in air-magnetic field. In addition, the resonance amplitude varying with detuning parameters and excitation amplitudes are plotted under principal and 3:1 internal resonances, respectively. It is found that the system may lose stability generated by a Hopf bifurcation, which may finally evolve into chaotic state through period-doubling bifurcation of intermittent periodic responses.

Two-to-One Internal Resonances in Buckled Beams

1995 ◽  
Author(s):  
Wayne Kreider ◽  
Ali H. Nayfeh ◽  
Char-Ming Chin
Keyword(s):  
Multiple Scales ◽  
Primary Resonance ◽  
Period Doubling ◽  
Equilibrium Solutions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Buckled Beams ◽  
Second Mode ◽  
Quadratic Nonlinearities ◽  
Period Doubling Bifurcations

Abstract The vibrations of buckled beams with two-to-one internal resonances (ω2 ≈ 2ω1) about a static buckled position are analyzed. General boundary conditions and harmonic excitations (frequency Ω) in both the transverse and axial directions are considered. The analysis assumes a unimodal static buckled deflection, considers quadratic nonlinearities only, and determines the amplitude and phase modulation equations via the method of multiple scales. The following specific cases are treated: Ω ≈ 2ω2, Ω ≈ ω1 + ω2, and Ω ≈ ω1. From the modulation equations for a primary resonance of the second mode (i.e., Ω ≈ ω2), one-mode and two-mode stable equilibrium solutions are found in addition to dynamic solutions caused by Hopf bifurcations. In the region of dynamic solutions, a variety of phenomena are documented, including period-doubling bifurcations, intermittency, chaos, and crises.

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

1997 ◽  
Author(s):  
Char-Ming Chin ◽  
Ali H. Nayfeh
Keyword(s):  
Natural Frequency ◽  
Parametric Resonance ◽  
Axial Load ◽  
Multiple Scales ◽  
Period Doubling ◽  
Equilibrium Solutions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Modal Damping ◽  
Second Mode

Abstract The nonlinear planar response of a hinged-clamped beam to a parametric excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact via a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of principal parametric resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of parametric resonance of the first mode, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single-, and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. In the region of dynamic solutions, some phenomena are documented, including period-doubling bifurcations and blue-sky catastrophes.

scholarly journals Magnetoelastic Principal Parametric Resonance of a Rotating Electroconductive Circular Plate

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Zhe Li ◽  
Yu-da Hu ◽  
Jing Li
Keyword(s):  
Magnetic Field ◽  
Circular Plate ◽  
Parametric Resonance ◽  
Magnetic Induction ◽  
Multiple Scales ◽  
Resonance Amplitude ◽  
Thin Circular Plate ◽  
Detuning Parameter ◽  
Principal Parametric Resonance ◽  
Magnetic Induction Intensity

Nonlinear principal parametric resonance and stability are investigated for rotating circular plate subjected to parametric excitation resulting from the time-varying speed in the magnetic field. According to the conductive rotating thin circular plate in magnetic field, the magnetoelastic parametric vibration equations of a conductive rotating thin circular plate are deduced by the use of Hamilton principle with the expressions of kinetic energy and strain energy. The axisymmetric parameter vibration differential equation of the variable-velocity rotating circular plate is obtained through the application of Galerkin integral method. Then, the method of multiple scales is applied to derive the nonlinear principal parametric resonance amplitude-frequency equation. The stability and the critical condition of stability of the plate are discussed. The influences of detuning parameter, rotation rate, and magnetic induction intensity are investigated on the principal parametric resonance behavior. The result shows that stable and unstable solutions exist when detuning parameter is negative, and the resonance amplitude can be weakened by changing the magnetic induction intensity.

scholarly journals Modeling and 1 : 1 Internal Resonance Analysis of Cable-Stayed Shallow Arches

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Jiangen Lv ◽  
Zhicheng Yang ◽  
Xuebin Chen ◽  
Quanke Wu ◽  
Xiaoxia Zeng
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Period Doubling ◽  
Dynamic Responses ◽  
Shallow Arch ◽  
Response Curves ◽  
Internal Resonances ◽  
Essential Information ◽  
Modulation Equations ◽  
Inclined Angle

In this paper, an analytical model of a cable-stayed shallow arch is developed in order to investigate the 1 : 1 internal resonance between modes of a cable and a shallow arch. Integrodifferential equations with quadratic and cubic nonlinearities are used to model the in-plane motion of a simple cable-stayed shallow arch. Nonlinear dynamic responses of a cable-stayed shallow arch subjected to external excitations with simultaneous 1 : 1 internal resonances are investigated. Firstly, the Galerkin method is used to discretize the governing nonlinear integral-partial-differential equations. Secondly, the multiple scales method (MSM) is used to derive the modulation equations of the system under external excitation of the shallow arch. Thirdly, the equilibrium, the periodic, and the chaotic solutions of the modulation equations are also analyzed in detail. The frequency- and force-response curves are obtained by using the Newton–Raphson method in conjunction with the pseudoarclength path-following algorithm. The cascades of period-doubling bifurcations leading to chaos are obtained by applying numerical simulations. Finally, the effects of key parameters on the responses are examined, such as initial tension, inclined angle of the cable, and rise and inclined angle of shallow arch. The comprehensive numerical results and research findings will provide essential information for the safety evaluation of cable-supported structures that have widely been used in civil engineering.

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.

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.

Nonlinear vibrations of a thin isotropic circular plate subjected to moving point loads

2020 ◽  
pp. 095440622097615
Author(s):  
Amit K Rai ◽  
Shakti S Gupta
Keyword(s):  
Boundary Conditions ◽  
Frequency Response ◽  
Circular Plate ◽  
Nonlinear Vibrations ◽  
Moving Load ◽  
Harmonic Balance Method ◽  
Angular Speed ◽  
Transverse Displacement ◽  
Governing Equations ◽  
Rotating Load

Here, we have studied the linear and nonlinear vibrations of a thin circular plate subjected to circularly, radially, and spirally moving transverse point loads. We follow Kirchoff’s theory and then incorporate von Kármán nonlinearity and employ Hamilton’s principle to obtain the governing equations and the associated boundary conditions. We solve the governing equations for the simply-supported and clamped boundary conditions using the mode summation method. Using the harmonic balance method for frequency response and Runge-Kutta method for time response, we solve the resulting coupled and cubic nonlinear ordinary differential equations. We show that the resonance instability due to a circularly moving load can be avoided by splitting it into multiple loads rotating at the same radius and angular speed. With the increasing magnitude of the rotating load, the frequency response of the transverse displacement shows jumps and modal interaction. The transverse response collected at the centre of the plate shows subharmonics of the axisymmetric frequencies only. The spectrum of the linear response due to spirally moving load contains axisymmetric frequencies, the angular speed of the load, their combination, and superharmonics of axisymmetric frequencies.

Response of a Stationary, Damped, Circular Plate Under a Rotating Slider Bearing System

1993 ◽  
Vol 115 (1) ◽  
pp. 65-69 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Circular Plate ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Slider Bearing ◽  
Numerical Examples ◽  
Elastic Circular Plate ◽  
Parametric Resonances ◽  
Supercritical Speed ◽  
Bearing System

This paper is to demonstrate that axisymmetric plate damping will suppress unbounded response of a stationary, elastic, circular plate excited by a rotating slider. Use of the method of multiple scales shows that the axisymmetric plate damping will suppress parametric resonances excited by slider stiffness and slider inertia at supercritical speed. In addition, the plate damping will increase the onset speed above which slider damping destabilizes the elastic circular plate. Moreover, numerical examples show that the plate damping could stabilize the plate/slider system at discrete rotation speeds above the onset speed.

Bio-MEMS Circular Plate Sensors Under Electrostatic Hard Excitations: Frequency Response of Superharmonic Resonance of Fourth Order

2020 ◽  
Author(s):  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Frequency Response ◽  
Circular Plate ◽  
Fourth Order ◽  
Multiple Scales ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Amplitude Frequency Response ◽  
Elastic Circular Plate ◽  
Modes Of Vibration ◽  
The Difference

Abstract This paper investigates the frequency-amplitude response of electrostatically actuated Bio-MEMS clamped circular plates under superharmonic resonance of fourth order. The system consists of an elastic circular plate parallel to a ground plate. An AC voltage between the two plates will lead to vibrations of the elastic plate. Method of Multiple Scales, and Reduced Order Model with two modes of vibration are the two methods used in this work. The two methods show similar amplitude-frequency response, with an agreement in the low amplitudes. The difference between the two methods can be seen for larger amplitudes. The effects of voltage and damping on the amplitude-frequency response are reported. The steady-state amplitudes in the resonant zone increase with the increase of voltage and with the decrease of damping.

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