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.

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]

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.

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.

A study of the nonlinear primary resonances of a micro-system under electrostatic and piezoelectric excitations

2014 ◽  
Vol 229 (10) ◽  
pp. 1904-1917 ◽  
Author(s):  
Ali K Hoshiar ◽  
Hamed Raeisifard
Keyword(s):  
Multiple Scales ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Nonlinear Effects ◽  
Resonance Excitation ◽  
Smart Device ◽  
Perturbation Approach ◽  
Softening Effect ◽  
Electrostatically Actuated ◽  
The Galerkin Method

In this article, a nonlinear analysis for a micro-system under electrostatic and piezoelectric excitations is presented. The micro-system beam is assumed as an elastic Euler-Bernoulli beam with clamped-free end conditions. The dynamic equations of this model have been derived by using the Hamilton method and considering the nonlinear inertia, curvature, piezoelectric and electrostatic terms. The static and dynamic solutions have been achieved by using the Galerkin method and the multiple-scales perturbation approach, respectively. The results are compared with numerical and other existing experimental results. By studying the primary resonance excitation, the effects of different parameters such as geometry, material, and excitations voltage on the system’s softening and hardening behaviors are evaluated. In an electrostatically actuated micro-system, it was showed that the nonlinear behavior occurs in frequency response as softening effect. In this paper, it is demonstrated that by applying a suitable piezoelectric DC voltage, this nonlinear effects can be controlled and altered to a linear domain. This model can be used to design a nano- or micro-scale smart device.

Nonlinear Forced Vibration Analysis of Fluid Conveying Nanotubes Under Electromagnetic Actuation

2014 ◽  
Author(s):  
Zia Saadatnia ◽  
Ebrahim Esmailzadeh ◽  
Davood Younesian
Keyword(s):  
Governing Equation ◽  
Forced Vibration ◽  
Multiple Scales ◽  
Fluid Velocity ◽  
Fluid Motion ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Electromagnetic Excitation ◽  
Electromagnetic Actuation ◽  
Nonlinear Forced Vibration

Nonlinear forced vibration of fluid-conveying nanotubes based on Euler-Bernoulli beam theory under electromagnetic actuation is studied. The nanotube is modeled as cantilever type beam and the effects of fluid motion and external harmonic excitation are considered in the governing equation of the structure vibration. The Galerkin procedure is applied in order to discretize the governing equation of vibration of the system. The well-known multiple scales method is utilized to investigate the primary resonance in the forced vibration of nanotubes. The effects of various parameters, namely, fluid velocity, position of applied force, aspect ratio and electromagnetic excitation on the primary resonance of the system are fully investigated. It is revealed that the electromagnetic excitation is highly influential on the frequency response of the considered system.

scholarly journals The Study of Primary and Internal Resonance on 3D Free-Free Double-Section Beam

2020 ◽  
Author(s):  
Yi-Ren Wang ◽  
Yun-Shuo Chang
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Perturbation Technique ◽  
Primary Resonance ◽  
Space Technology ◽  
Nonlinear Properties ◽  
Wide Range ◽  
Engineering Problems ◽  
Time Marching ◽  
Free Beam

This work investigates the primary resonance and internal resonance of a double-section beam with cubic nonlinearities. This model can be applied in a wide range of engineering problems, such as rocket and missile structures. Even space technology has been developed for decades; several nonlinear properties deserve further study, especially, for the internal resonance. The method of multiple scales (a perturbation technique) is employed to analyze this nonlinear problem. This study focuses on finding the forcing conditions of this 3D double-section beam to trigger the often-ignored internal resonance or prime resonance in rocket structures. A primary resonance is found on a uniform free-free beam at certain flight speed. The three-to-one internal resonance of the double-section beam occurs within the first and the second modes in the diameter ratio of 1/0.75 with the length ratio of 0.33 or 0.51. The semi-analytical results are verified by the time marching numerical method.

The Global Attractor of the 3D Complex GL Equation and the Estimate of its Dimensionality

2012 ◽  
Vol 588-589 ◽  
pp. 2051-2054
Author(s):  
Ben Tu Li ◽  
Zhi Chao Yuan
Keyword(s):  
Global Attractor ◽  
Nonlinear Term ◽  
Dimensional Space ◽  
A Priori ◽  
Three Dimensional ◽  
A Priori Estimation ◽  
Complex Valued ◽  
Three Dimensional Space

In three-dimensional space, by using the method of a priori estimation, we have studied the complex-valued GL equation, which has the 2 -th power of the nonlinear term. We have proved that the existence of the global attractor of this equation with the problem of period and border value, and we have studied the dimensionality of Hausdorff and fractal of the global attractor.

The Response and Instability of Cross-Rope Suspension Towers Under Harmonic Excitation

2017 ◽  
Vol 17 (10) ◽  
pp. 1750124 ◽  
Author(s):  
Zhitao Yan ◽  
Yu Zhu ◽  
Yi You ◽  
Jing Wang
Keyword(s):  
Transmission Lines ◽  
Multiple Scales ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Steady State Solution ◽  
Harmonic Excitation ◽  
Wind Load ◽  
Dynamic Force ◽  
Harmonic Load ◽  
Straight Beam

The galloping or vortex-induced vibration of transmission lines will lead to a periodic excitations to the masts of the cross-rope suspension tower (CRST). The mast of the CRST is modeled as a straight beam with an elastic support subjected to a pulsating axial force on the top, which will change the stiffness of the mast, thereby resulting in produce harmonic excitation and instability. The dynamic characteristics of the system are investigated, which show that the bending frequency of the CRST decreases linearly with increase in axial static load, while it increases nonlinearly with the increase in boundary stiffness. Then, the method of multiple scales is adopted to analyze the vibration. It is found that the wind load on the mast brings primary resonance, but has no effects on instability. In addition, the steady state solution of the primary resonance is obtained by the polar form of the reduced amplitude modulation equations (RAMEs), with the effects of the following parameters on the vibration amplitude of the mast studied: the prestressing load in the guy, magnitude of the dynamic force, detuning parameter and wind load. Finally, the instability regions of two cases ([Formula: see text] near [Formula: see text] and [Formula: see text] near [Formula: see text]) are studied by the Cartesian form of the RAMEs, with focus on the influence of the axial harmonic load produced by the galloping of the transmission lines on the instability area. It is observed that the magnitude of excitation frequency of the dynamic force in the range of instability region becomes larger until the spring stiffness is increased up to a certain value.

NONLINEAR RESPONSES OF A STRING-BEAM COUPLED SYSTEM WITH FOUR-DEGREES-OF-FREEDOM AND MULTIPLE PARAMETRIC EXCITATIONS

2009 ◽  
Vol 19 (01) ◽  
pp. 225-243 ◽  
Author(s):  
D. X. CAO ◽  
W. ZHANG
Keyword(s):  
Nonlinear Dynamic ◽  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Coupled System ◽  
Dynamic Responses ◽  
Governing Equations ◽  
Chaotic Motions

The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string — principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string — is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions.

The Consistency between MLP Method and Modified Method of Multiple Scales for Strong Nonlinear Primary Resonance of Duffing Equation Subject to Harmonic Excitation in Complex Number Field

2014 ◽  
Vol 472 ◽  
pp. 62-68
Author(s):  
Zhi’an Yang ◽  
Xi Li ◽  
Jia Jia Meng
Keyword(s):  
Frequency Response ◽  
Number Field ◽  
Complex Number ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Duffing Equation ◽  
Modified Method ◽  
Amplitude Frequency Response ◽  
Complex Number Field

With changing strong nonlinear Duffing equation subject to harmonic excitation with damping in complex number field as an object, the amplitude frequency response equation of primary resonance of the system is obtained through parametric transformation with application of MLP method and modified method of multiple scales. In different approximation solution forms and different time scales, the two methods lead to the same amplitude frequency response equation. Thus, the two methods are mutually verifiable. Numerical analysis shows that for the strong nonlinear Duffing equation with damping in complex number field, the nonlinear stiffness coefficient is more than zero and the amplitude frequency response curve of primary resonance leans to the left, which is different from the weak nonlinear Duffing equation.Chinese books catalog: O321

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