The Nonlinear Response of a Simply Supported Rectangular Metallic Plate to Transverse Harmonic Excitation

2000 ◽  
Vol 67 (3) ◽  
pp. 621-626 ◽  
Author(s):  
O. Elbeyli and ◽  
G. Anlas
Keyword(s):  
Critical Points ◽  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Stability Of Solutions ◽  
Metallic Plate ◽  
Simply Supported

In this study, the nonlinear response of a simply supported metallic rectangular plate subject to transverse harmonic excitations is analyzed using the method of multiple scales. Stability of solutions, critical points, types of bifurcation in the presence of a one-to-one internal resonance, together with primary resonance, are determined. [S0021-8936(00)00603-6]

Nonlinear Vibrations of a Beam-Spring-Mass System

1994 ◽  
Vol 116 (4) ◽  
pp. 433-439 ◽  
Author(s):  
M. Pakdemirli ◽  
A. H. Nayfeh
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Primary Resonance ◽  
Fast Time ◽  
Lagrange Equations ◽  
Response Curves ◽  
Mass System

The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.

Multimodal Resonances in Suspended Cables via a Direct Perturbation Approach

1997 ◽  
Author(s):  
Giuseppe Rega ◽  
Walter Lacarbonara ◽  
Ali H. Nayfeh ◽  
Char-Ming Chin
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
A Priori ◽  
Three Dimensional ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Finite Amplitude ◽  
Perturbation Approach ◽  
Plane Symmetric ◽  
Complex Valued

Abstract We analyze the nonlinear three–dimensional response of an elastic suspended cable with small sag-to-span ratio to a harmonic excitation. We investigate the case of primary resonance of the first in-plane symmetric mode when it is involved in a one–to–one internal resonance with the first antisymmetric planar and nonplanar modes and a two–to–one internal resonance with the first symmetric nonplanar mode. We apply the method of multiple scales directly to the governing two integro–partial–differential equations and associated boundary conditions with no a priori assumption on the shape of the motion. The result is a system of four coupled nonlinear complex–valued equations describing the modulation of the amplitudes and phases of the four interacting modes. The spatial-temporal corrections to the displacement field at higher orders show that the solution is not separable in space and time. Prelimary comparisons with a companion Galerkin-type discretized model show that the latter must be used with some care in studying finite–amplitude motions of cables.

Effect of Two-to-One Internal Resonances in a Nonlinearly Restrained Fluid Valve

1999 ◽  
Vol 121 (1) ◽  
pp. 59-63 ◽  
Author(s):  
G. Anlas¸
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Distributed Parameter ◽  
Parameter System ◽  
Response Curves ◽  
Internal Resonances ◽  
Relief Valve ◽  
Steady State Solutions

The effect of two-to-one internal resonances on the nonlinear response of a pressure relief valve is studied. The fluid valve is modeled as a distributed parameter system at one end and nonlinearly restrained at the other. The method of multiple scales is used to solve the system of partial differential equation and boundary conditions. Frequency-response curves are presented for the primary resonance of either mode in the presence of a two-to-one internal resonance. Stability of the steady-state solutions is investigated. Parameters of the system leading to two-to-one internal resonances are tabulated.

scholarly journals Accuracy Improvement of the Method of Multiple Scales for Nonlinear Vibration Analyses of Continuous Systems with Quadratic and Cubic Nonlinearities

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Akira Abe
Keyword(s):  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Continuous Systems ◽  
Driving Frequency ◽  
Accuracy Improvement ◽  
Definition Of ◽  
Quadratic And Cubic Nonlinearities

This paper proposes an accuracy improvement of the method of multiple scales (MMSs) for nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. As an example, we treat a shallow suspended cable subjected to a harmonic excitation, and investigate the primary resonance of the th in-plane mode () in which and are the driving and natural frequencies, respectively. The application of Galerkin's procedure to the equation of motion yields nonlinear ordinary differential equations with quadratic and cubic nonlinear terms. The steady-state responses are obtained by using the discretization approach of the MMS in which the definition of the detuning parameter, expressing the relationship between the natural frequency and the driving frequency, is changed in an attempt to improve the accuracy of the solutions. The validity of the solutions is discussed by comparing them with solutions of the direct approach of the MMS and the finite difference method.

Nonlinear Response of a Buckled Beam to a Three-to-One Internal Resonance

2003 ◽  
Author(s):  
Samir A. Emam ◽  
Ali H. Nayfeh
Keyword(s):  
Periodic Orbits ◽  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Experimental Results ◽  
Buckled Beam ◽  
Stability And Bifurcations ◽  
The Stability ◽  
Multi Mode

We investigate the nonlinear response of a clamped-clamped buckled beam to a three-to-one internal resonance between the first and third modes when one of them is externally excited. To examine whether the first and third modes are nonlinearly coupled, we use the method of multiple scales to directly attack the partial-differential equation and associated boundary conditions and obtain the equations governing the modulation of their amplitudes and phases. We find that the two modes are nonlinearly coupled. To investigate the large-amplitude dynamics, we use a multi-mode Galerkin discretization to obtain a reduced-order model of the problem. We use a shooting method to compute periodic orbits of the discretized equations and Floquet theory to investigate the stability and bifurcations of these periodic orbits. We note an energy transfer from the first mode, which is externally excited by a primary resonance, to the third mode. We obtain preliminary experimental results of the energy exchange between the first and third modes as a result of a three-to-one internal resonance. More experimental results are being generated.

scholarly journals Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

2018 ◽  
Vol 14 (1) ◽  
Author(s):  
Feras K. Alfosail ◽  
Amal Z. Hajjaj ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Chaotic Motions ◽  
Different Types ◽  
Quadratic Nonlinearities ◽  
Multiple Scales Perturbation Method ◽  
Good Agreement ◽  
Perturbation Solutions

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

Reduced Order Model for MEMS Cantilever Resonators Under Soft AC Voltage Near Natural Frequency

2012 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Israel Martinez
Keyword(s):  
Natural Frequency ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
First Order ◽  
Electrostatically Actuated

The nonlinear response of an electrostatically actuated cantilever beam microresonator is investigated. The AC voltage is of frequency near resonator’s natural frequency. A first order fringe correction of the electrostatic force and viscous damping are included in the model. The dynamics of the resonator is investigated using the Reduced Order Model (ROM) method, based on Galerkin procedure. Steady-state motions are found. Numerical results for the uniform microresonator are compared with those obtained via the Method of Multiple Scales (MMS).

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 Vibration of Thin Elastic Plates, Part 2: Internal Resonance by Amplitude-Incremental Finite Element

1984 ◽  
Vol 51 (4) ◽  
pp. 845-851 ◽  
Author(s):  
S. L. Lau ◽  
Y. K. Cheung ◽  
S. Y. Wu
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Internal Resonance ◽  
Harmonic Excitation ◽  
Plate Element ◽  
Elastic Plates ◽  
Simply Supported ◽  
Speed Up ◽  
Numerical Process ◽  
Thin Elastic Plates

The simple amplitude-incremental triangular plate element derived in Part 1 of this paper is applied to treat the large-amplitude periodic vibrations of thin elastic plates with existence of internal resonance. A simply supported rectangular plate with immovable edges (b/a = 1.5) and having linear frequencies ω13 = 3.45 ω11 is selected as a typical example. The frequency response of free vibration as well as forced vibration under harmonic excitation are computed. To the best knowledge of the authors, these very interesting results for such plate problems have not appeared in literature previously. Some special considerations to simplify and to speed up the numerical process are also discussed.

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