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.

Nonlinear Vibrations of a Beam-Spring-Large Mass System

2017 ◽  
Author(s):  
Mohammad A. Bukhari ◽  
Oumar R. Barry
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Large Mass ◽  
Mode Shapes ◽  
Resonance Excitation ◽  
Response Curves ◽  
Mass System ◽  
Spring System

This paper presents the nonlinear vibration of a simply supported Euler-Bernoulli beam with a mass-spring system subjected to a primary resonance excitation. The nonlinearity is due to the mid-plane stretching and cubic spring stiffness. The equations of motion and the boundary conditions are derived using Hamiltons principle. The nonlinear system of equations are solved using the method of multiple scales. Explicit expressions are obtained for the mode shapes, natural frequencies, nonlinear frequencies, and frequency response curves. The validity of the results is demonstrated via comparison with results in the literature. Exact natural frequencies are obtained for different locations, rotational inertias, and masses.

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.

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 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.

Investigation on Nonlinear Trends of Composite Laminated Plate

2012 ◽  
Author(s):  
Wei Zhang ◽  
Jianen Chen ◽  
Qian Wang ◽  
Min Sun
Keyword(s):  
Multiple Scales ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Laminated Plates ◽  
Laminated Plate ◽  
Response Curves ◽  
Composite Laminated Plate ◽  
Nonlinear Trend ◽  
Nonlinear Trends

The nonlinear trends of composite laminated plates are investigated. The governing equations of motion for the plate are derived with the von Karman strain-displacement relations for the geometric nonlinearity and the Reddy’s third-order shear deformation plate theory. The four dimensional nonlinear averaged equations with the case of 1/2-subharmonic resonance and principal parametric resonance for the first mode and primary resonance for the second mode are obtained by applying the method of multiple scales. The frequency-response curves are analyzed under consideration of strongly coupled of two modes. The influences of the coefficients in dynamic equations and the detuning parameters on the nonlinear trend are studied, and the results indicate that the composite laminated plate may have different trends of nonlinearity under aforementioned resonance conditions. The sweep experiment is conducted to find the softening and hardening nonlinearity. The different trends are obtained when the excitation amplitude is 1.2g. The spectrums of the different stages of the test show that the change of the nonlinear trend may be caused from the sub-harmonic resonance in this test.

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 Cubic Velocity Feedback Control of High-Amplitude Vibration of a Nonlinear Plant to a Primary Resonance Excitation

2007 ◽  
Vol 14 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Li Jun ◽  
Shen Rongying ◽  
Hua Hongxing
Keyword(s):  
Feedback Control ◽  
Multiple Scales ◽  
Primary Resonance ◽  
High Amplitude ◽  
Resonance Excitation ◽  
Feedback Gain ◽  
Response Curves ◽  
Power Requirement ◽  
Velocity Feedback ◽  
Nonlinear Plant

High-amplitude response suppression of the primary resonance of a nonlinear plant under cubic velocity feedback control is investigated. By means of the multiple scales method, two equations on the amplitude and phase of the response of the nonlinear system are obtained and the force-response and frequency-response curves are shown. The stability analyses for the open- and closed-loop responses of the system are carried out and the performance of the control strategy is investigated. The instantaneous power requirement of the control law is also examined. It can be demonstrated that appropriate choice for the feedback gain can greatly reduce the response amplitude of the primary resonance and completely eliminate the multiple responses. Finally the perturbation solutions are verified with numerical simulations.

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.

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.

Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

2017 ◽  
Vol 9 (6) ◽  
pp. 1485-1505
Author(s):  
Lingchang Meng ◽  
Fengming Li
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Laminated Plates ◽  
Laminated Plate ◽  
Composite Laminated Plate ◽  
Vibration Localization ◽  
Composite Laminated Plates ◽  
Localization Phenomenon ◽  
Harmonic Resonance

AbstractThe nonlinear transverse vibrations of ordered and disordered two-dimensional (2D) two-span composite laminated plates are studied. Based on the von Karman's large deformation theory, the equations of motion of each-span composite laminated plate are formulated using Hamilton's principle, and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin's method. The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales. The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out. The effects of the disorder ratio and ply angle on the two different resonances are analyzed. From the numerical results, it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon, and with the increase of the disorder ratio, the vibration localization phenomenon will become more obvious. Moreover, the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration, and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.

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