Internal Resonances in Microbeam Arrays Subject to Electrodynamical Parametric Excitation

2007 ◽  
Author(s):  
Stefanie Gutschmidt ◽  
Oded Gottlieb
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Nonlinear Effects ◽  
Theoretical Models ◽  
System Stability ◽  
Dynamic Features ◽  
Lumped Mass ◽  
Internal Resonances ◽  
Weakly Nonlinear ◽  
Temporal Interaction

The dynamic response of parametrically excited microbeam arrays is governed by nonlinear effects which directly influence their performance. To date, documented theoretical models consist of lumped-mass systems which do not resolve the spatio-temporal interaction of the individual elements and reproduce measured array response only qualitatively. A consistent nonlinear continuum model is derived using the extended Hamilton’s principle to capture the salient dynamic features of an array of N nonlinearly coupled microbeams. The nonlinear dynamic equations of motion are solved analytically using the asymptotic multiple-scales method for the weakly nonlinear system. Stability analysis of the resulting coexisting solutions enables construction of a comprehensive bifurcation structure for the system. Analytically obtained results for the weakly nonlinear limit of two coupled microbeams are verified numerically.

Numerical Analysis of a Three Element Microbeam Array Subject to Electrodynamical Parametric Excitation

2008 ◽  
Author(s):  
Stefanie Gutschmidt ◽  
Oded Gottlieb
Keyword(s):  
Dynamic Behavior ◽  
Parametric Excitation ◽  
Nearest Neighbor ◽  
Equations Of Motion ◽  
Nonlinear Effects ◽  
Theoretical Models ◽  
System Response ◽  
Lumped Mass ◽  
Internal Resonances ◽  
Temporal Interaction

The dynamic response of parametrically excited microbeam arrays is governed by nonlinear effects which directly influence their performance. To date, documented theoretical models consist of lumped-mass models. While a lumped-mass approach is useful for a qualitative understanding of the system response it does not resolve the spatio-temporal interaction of the individual elements in the array. Thus, we employ a consistent nonlinear continuum model to investigate the nonlinear dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations focus on the behavior of a small size array in its 1:1:1 internal, parametric, and 3:1 internal resonances, which correspond to low, medium and high DC-voltage input, respectively. The dynamic equations of motion are solved numerically. The dynamic behavior of the three beam systems reveals coexisting periodic and aperiodic solutions. Similarities in the comprehensive bifurcation structures of the three beam systems provide insight to the nearest neighbor response of multi-element microbeam arrays subject to electrodynamic parametric excitation.

scholarly journals Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach

2021 ◽  
Vol 11 (20) ◽  
pp. 9486
Author(s):  
Andrea Arena
Keyword(s):  
Nonlinear Dynamic ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Dynamic Features ◽  
Response Curves ◽  
Internal Resonances ◽  
The Third ◽  
Fold Bifurcation

The nonlinear dynamic features of compression roller batteries were investigated together with their nonlinear response to primary resonance excitation and to internal interactions between modes. Starting from a parametric nonlinear model based on a previously developed Lagrangian formulation, asymptotic treatment of the equations of motion was first performed to characterize the nonlinearity of the lowest nonlinear normal modes of the system. They were found to be characterized by a softening nonlinearity associated with the stiffness terms. Subsequently, a direct time integration of the equations of motion was performed to compute the frequency response curves (FRCs) when the system is subjected to direct harmonic excitations causing the primary resonance of the lowest skew-symmetric mode shape. The method of multiple scales was then employed to study the bifurcation behavior and deliver closed-form expressions of the FRCs and of the loci of the fold bifurcation points, which provide the stability regions of the system. Furthermore, conditions for the onset of internal resonances between the lowest roller battery modes were found, and a 2:1 resonance between the third and first modes of the system was investigated in the case of harmonic excitation having a frequency close to the first mode and the third mode, respectively.

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.

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 Transverse Vibrations and 3:1 Internal Resonances of a Beam With Multiple Supports

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
E. Özkaya ◽  
S. M. Bağdatlı ◽  
H. R. Öz
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Mode Shapes ◽  
Response Curves ◽  
Internal Resonances ◽  
Transverse Vibrations ◽  
Second Mode ◽  
Euler Bernoulli Beam ◽  
Correction Terms

In this study, nonlinear transverse vibrations of an Euler–Bernoulli beam with multiple supports are considered. The beam is supported with immovable ends. The immovable end conditions cause stretching of neutral axis and introduce cubic nonlinear terms to the equations of motion. Forcing and damping effects are included in the problem. The general arbitrary number of support case is considered at first, and then 3-, 4-, and 5-support cases are investigated. The method of multiple scales is directly applied to the partial differential equations. Natural frequencies and mode shapes for the linear problem are found. The correction terms are obtained from the last order of expansion. Nonlinear frequencies are calculated and then amplitude and phase modulation figures are presented for different forcing and damping cases. The 3:1 internal resonances are investigated. External excitation frequency is applied to the first mode and responses are calculated for the first or second mode. Frequency-response and force-response curves are drawn.

Nonlinear Free Vibration of a Beam With Off-Axis Attached Mass

2013 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Free Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Beam Theory ◽  
Governing Equations ◽  
Attached Mass ◽  
Lumped Mass ◽  
Nonlinear Free Vibration ◽  
Euler Bernoulli Beam

A model of a clamped-clamped beam with an attached lumped mass is presented in this paper. The system is modeled using the Euler-Bernoulli beam theory. In the models presented in literature, it is assumed that the center of mass of the attached mass is located on the neutral axis of the beam. In this paper, this assumption is relaxed. The governing equations of motion are derived. It has been shown that the off-axis center of mass of the attached mass generates an amplitude-dependent transverse force in the beam, which introduces a quadratic nonlinearity. The nonlinear governing equations of motion are solved using the Multiple Scales method. The nonlinear free vibration frequencies are determined.

Nonlinear Dynamics of Closed Spherical Shells

2005 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Internal Resonances ◽  
Spherical Shells ◽  
Linear Eigenvalue Problem ◽  
Nonlinear Responses ◽  
Nonlinear Equations Of Motion ◽  
Primary Resonance Excitation

We investigate the axisymmetric dynamics of forced closed spherical shells. The nonlinear equations of motion are formulated using a variational approach and surface analysis. First, we revisit the linear eigenvalue problem. Then, using the method of multiple scales, we assess the possibility of the activation of two-to-one internal resonances between the different types of modes. Lastly, we examine the shell’s nonlinear responses to an axisymmetric primary-resonance excitation and analyze their bifurcations.

INTERNAL RESONANCES AND BIFURCATIONS OF AN ARRAY BELOW THE FIRST PULL-IN INSTABILITY

2010 ◽  
Vol 20 (03) ◽  
pp. 605-618 ◽  
Author(s):  
STEFANIE GUTSCHMIDT ◽  
ODED GOTTLIEB
Keyword(s):  
Nearest Neighbor ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Internal Resonances ◽  
Large Size ◽  
Element Array ◽  
Nonlinear Equations Of Motion ◽  
Chaotic Transients ◽  
Nonlinear Continuum ◽  
Continuation Technique

A nonlinear continuum model is used to investigate the dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations concentrate on the region below the array's first pull-in instability, which is shown to be governed by several internal three-to-one and combination resonances. The nonlinear equations of motion for a two-element system are solved using the asymptotic multiple-scales method for the weak nonlinear system. The analytically obtained periodic response of two coupled microbeams is numerically evaluated by a continuation technique and complemented by a numerical analysis of a three-element array which exhibits quasi-periodic responses and lengthy chaotic transients. This study of small-size microbeam arrays serves for design purposes and the understanding of nonlinear nearest-neighbor interactions of medium- and large-size arrays.

Internally Resonant Nonlinear Free Vibrations of Horizontal/Inclined Sagged Cables

2005 ◽  
Author(s):  
G. Rega ◽  
N. Srinil ◽  
S. Chucheepsakul
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Original System ◽  
Nonlinear Normal Modes ◽  
Free Vibrations ◽  
Order Effects ◽  
Internal Resonances ◽  
Closed Form Solutions ◽  
Nonlinear Normal

Internally resonant dynamics in the nonlinear free vibrations of suspended cables are analytically investigated by means of a multi-mode Galerkin-based discretization and second-order multiple scales. Emphasis is placed on planar 2:1 internal resonances. The equations of motion of a general inclined cable model, which account for the dynamic extensibility effects and the system asymmetry due to inclined equilibrium, are considered. By considering higher-order effects due to quadratic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations associated with the resonant nonlinear normal modes reveal the dependence of cable nonlinear response on different resonant and non-resonant modes. Based on the modal convergence properties performed on the resonantly activated cables, the illustrative results provide hints for proper reduced-order model selections from the asymptotic solution. The underlying effects of cable inclination and cable sag are presented. The theoretical predictions are validated by finite difference numerical time laws of the original system equations of motion.

Effects of nonlinear interactions of flexural modes on the performance of a beam autoparametric vibration absorber

2019 ◽  
Vol 26 (7-8) ◽  
pp. 459-474
Author(s):  
Saeed Mahmoudkhani ◽  
Hodjat Soleymani Meymand
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Vibration Absorber ◽  
Primary System ◽  
Lumped Mass ◽  
Response Curves ◽  
Internal Resonances ◽  
The One

The performance of the cantilever beam autoparametric vibration absorber with a lumped mass attached at an arbitrary point on the beam span is investigated. The absorber would have a distinct feature that in addition to the two-to-one internal resonance, the one-to-three and one-to-five internal resonances would also occur between flexural modes of the beam by tuning the mass and position of the lumped mass. Special attention is paid on studying the effect of these resonances on increasing the effectiveness and extending the range of excitation amplitudes at which the autoparametric vibration absorber remains effective. The problem is formulated based on the third-order nonlinear Euler–Bernoulli beam theory, where the assumed-mode method is used for deriving the discretized equations of motion. The numerical continuation method is then applied to obtain the frequency response curves and detect the bifurcation points. The harmonic balance method is also employed for detecting the type of internal resonances between flexural modes by inspecting the frequency response curves corresponding to different harmonics of the response. Parametric studies on the performance of the absorber are conducted by varying the position and mass of the lumped mass, while the frequency ratio of the primary system to the first mode of the beam is kept equal to two. Results indicated that the one-to-five internal resonance is especially responsible for the considerable enhancement of the performance.

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