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.

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.

Multimode Dynamics of Inextensional Beams on the Elastic Foundation With Two-to-One Internal Resonances

2013 ◽  
Vol 80 (6) ◽  
Author(s):  
Lianhua Wang ◽  
Jianjun Ma ◽  
Minghui Yang ◽  
Lifeng Li ◽  
Yueyu Zhao
Keyword(s):  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Continuation Methods ◽  
Nonlinear Phenomena ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Nonlinear Responses ◽  
Methods Numerical

The modal interactions and nonlinear responses of inextensional beams resting on elastic foundations with two-to-one internal resonances are investigated and the primary resonance excitations are considered. The multimode discretization and the method of multiple scales are applied to obtain the modulation equations. The equilibrium and dynamic solutions of the modulation equations are examined by the Newton–Raphson, shooting, and continuation methods. Numerical simulations are performed to investigate the chaotic dynamics of the beam. It is shown that the nonlinear responses may undergo different bifurcations and exhibit rich nonlinear phenomena. Finally, the effects of the foundation models on the nonlinear interactions of the beam are examined.

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 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.

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.

Nonlinear Forced Vibration Analysis of Smart Curved CNTs Conveying Fluid

2021 ◽  
Author(s):  
Vahid Mohamadhashemi ◽  
Amir Jalali ◽  
Habib Ahmadi
Keyword(s):  
Carbon Nanotube ◽  
Velocity Vector ◽  
Multiple Scales ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Maximum Amplitude ◽  
Beam Theory ◽  
Physical Parameters ◽  
Conveying Fluid

In this study, the nonlinear vibration of a curved carbon nanotube conveying fluid is analyzed. The nanotube is assumed to be covered by a piezoelectric layer and the Euler–Bernoulli beam theory is employed to establish the governing equations of motion. The influence of carbon nanotube curvature on structural modeling and fluid velocity vector is considered and the slip boundary conditions of CNT conveying fluid are included. The mathematical modeling of the structure is developed using Hamilton’s principle and then, the Galerkin procedure is employed to discretize the equation of motion. Furthermore, the frequency response of the system is extracted by applying the multiple scales method of perturbation. Finally, a comprehensive study is carried out on the primary resonance and piezoelectric-based parametric resonance of the system. It is shown that consideration of nanotube curvature may lead to an increase in nonlinearity. Implementing the fluid velocity vector in which nanotube curvature is included highly affects the maximum amplitude of the response and should not be ignored. Furthermore, different system parameters have evident impacts on the behavior of the system and therefore, selecting the reasonable geometrical and physical parameters of the system can be very useful to achieve a favorable response.

scholarly journals On the Application of the Multiple Scales Method on Electrostatically Actuated Resonators

2019 ◽  
Vol 14 (4) ◽  
Author(s):  
Saad Ilyas ◽  
Feras K. Alfosail ◽  
Mohammad I. Younis
Keyword(s):  
Analytical Solution ◽  
Multiple Scales ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Equation Of Motion ◽  
Higher Order ◽  
Resonance Excitation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Electrostatically Actuated

We investigate modeling the dynamics of an electrostatically actuated resonator using the perturbation method of multiple time scales (MTS). First, we discuss two approaches to treat the nonlinear parallel-plate electrostatic force in the equation of motion and their impact on the application of MTS: expanding the force in Taylor series and multiplying both sides of the equation with the denominator of the forcing term. Considering a spring–mass–damper system excited electrostatically near primary resonance, it is concluded that, with consistent truncation of higher-order terms, both techniques yield same modulation equations. Then, we consider the problem of an electrostatically actuated resonator under simultaneous superharmonic and primary resonance excitation and derive a comprehensive analytical solution using MTS. The results of the analytical solution are compared against the numerical results obtained by long-time integration of the equation of motion. It is demonstrated that along with the direct excitation components at the excitation frequency and twice of that, higher-order parametric terms should also be included. Finally, the contributions of primary and superharmonic resonance toward the overall response of the resonator are examined.

Load Transfer Control for a Gantry Crane with Arbitrary Delay Constraints

2002 ◽  
Vol 8 (2) ◽  
pp. 135-158 ◽  
Author(s):  
Paolo Dadone ◽  
Hugh F. Vanlandingham
Keyword(s):  
Load Transfer ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Gantry Crane ◽  
Phase Plane Analysis ◽  
Plane Analysis ◽  
Nonlinear Approach ◽  
Linearized Model ◽  
Nonlinear Equations Of Motion ◽  
Quadratic And Cubic Nonlinearities

This paper describes a method to move the load of a gantry crane to a desired position in the presence of known, but arbitrary, motion-inversion delays as well as cart acceleration constraints. The method idea is based on a phase-plane analysis of the linearized model. In order to limit residual pendulation at the goal position, the method is extended to account for quadratic and cubic nonlinearities. The method of multiple scales is used to determine an approximate solution to the nonlinear equations of motion, thus providing a more accurate measure of the frequency of the oscillations. The nonlinear approach is very successful in limiting residual oscillations to very small values (less than 1 degree of amplitude), offering a reduction, with respect to the linear case, of as much as two orders of magnitude. Finally, this method offers a rationale for the future development of a controller for suppression of load oscillations in ship-mounted cranes in the presence of arbitrary delays.

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