Large-Amplitude Vibrations of Thin Panels

2006 ◽  
Author(s):  
M. Amabili
Keyword(s):  
Boundary Conditions ◽  
Lyapunov Exponents ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Time Integration ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Different Boundary Conditions ◽  
Model C

Geometrically nonlinear vibrations of circular cylindrical panels with different boundary conditions and subjected to harmonic excitation are numerically investigated. The Donnell's nonlinear strain-displacement relationships are used to describe geometric nonlinearity; in-plane inertia is taken into account. Different boundary conditions are studied and the results are compared; for all of them zero normal displacements at the edges are assumed. In particular, three models are considered in order to investigate the effect of different boundary conditions: Model A for free in-plane displacement orthogonal to the edges, elastic distributed springs tangential to the edges and free rotation; Model B for classical simply supported edges; Model C for fixed edges and distributed rotational springs at the edges. Clamped edges are obtained with the Model C for very high value of the stiffness of rotational springs. The nonlinear equations of motion are obtained by the Lagrange multi-mode approach, and are studied by using the code AUTO based on pseudo-arclength continuation method. Convergence of the solution with the number of generalized coordinates is numerically verified. Complex nonlinear dynamics is also investigated by using bifurcation diagrams from direct time integration and calculation of the Lyapunov exponents and the Lyapunov dimension. Interesting phenomena such as (i) subharmonic response, (ii) period doubling bifurcations, (iii) chaotic behavior and (iv) hyper-chaos with four positive Lyapunov exponents have been observed.

Effect of Boundary Conditions on Nonlinear Vibrations of Circular Cylindrical Panels

2006 ◽  
Vol 74 (4) ◽  
pp. 645-657 ◽  
Author(s):  
M. Amabili
Keyword(s):  
Boundary Conditions ◽  
Lyapunov Exponents ◽  
Nonlinear Vibrations ◽  
Chaotic Behavior ◽  
Continuation Method ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Different Boundary Conditions ◽  
Cylindrical Panels ◽  
Model C

Geometrically nonlinear vibrations of circular cylindrical panels with different boundary conditions and subjected to harmonic excitation are numerically investigated. The Donnell’s nonlinear strain–displacement relationships are used to describe geometric nonlinearity; in-plane inertia is taken into account. Different boundary conditions are studied and the results are compared; for all of them zero normal displacements at the edges are assumed. In particular, three models are considered in order to investigate the effect of different boundary conditions: Model A for free in-plane displacement orthogonal to the edges, elastic distributed springs tangential to the edges and free rotation; Model B for classical simply supported edges; and Model C for fixed edges and distributed rotational springs at the edges. Clamped edges are obtained with Model C for the very high value of the stiffness of rotational springs. The nonlinear equations of motion are obtained by the Lagrange multimode approach, and are studied by using the code AUTO based on the pseudo-arclength continuation method. Convergence of the solution with the number of generalized coordinates is numerically verified. Complex nonlinear dynamics is also investigated by using bifurcation diagrams from direct time integration and calculation of the Lyapunov exponents and the Lyapunov dimension. Interesting phenomena such as (i) subharmonic response; (ii) period doubling bifurcations; (iii) chaotic behavior; and (iv) hyper-chaos with four positive Lyapunov exponents have been observed.

Nonlinear Vibrations of Doubly Curved Shallow Shells

2004 ◽  
Author(s):  
M. Amabili
Keyword(s):  
Chaotic Behavior ◽  
Continuation Method ◽  
Time Integration ◽  
Elastic Strain Energy ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Internal Resonances ◽  
Shallow Shells ◽  
Non Linear ◽  
Doubly Curved

Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular boundary, simply supported at the four edges and subjected to harmonic excitation normal to the surface in the spectral neighbourhood of the fundamental mode are investigated. Two different non-linear strain-displacement relationships, from the Donnell’s and Novozhilov’s shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometric imperfections are taken into account. The solution is obtained by Lagrangian approach. The non-linear equations of motion are studied by using (i) a code based on arclength continuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio among their natural frequencies, giving rise to internal resonances, is discussed. Shell stability under dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behavior have been observed.

Nonlinear Vibrations of Circular Cylindrical Shells With Different Boundary Conditions

2009 ◽  
Author(s):  
M. Amabili ◽  
Ye. Kurylov
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Vibrations ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Continuation Method ◽  
Elastic Strain Energy ◽  
Variable Boundary ◽  
Different Boundary Conditions ◽  
Simply Supported ◽  
Circular Cylindrical Shells

Large-amplitude nonlinear vibrations of circular cylindrical shells with different boundary conditions are investigated. The Sanders-Koiter nonlinear shell theory, which includes shear deformation, is used to calculate the elastic strain energy. Shell’s displacement fields (longitudinal, circumferential and radial) are expanded by means of a double mixed series: harmonic functions for the circumferential variable; Chebyshev polynomials for the longitudinal variable. Boundary conditions for both simply supported and clamped-clamped shells are exactly satisfied. The Lagrangian approach is applied to obtain a system of nonlinear ordinary differential equations. Different expansions involving from 14 to 34 generalized co-ordinates, associated with natural modes of both simply supported and clamped-clamped shells are used to study the convergence of the solution. The nonlinear equations of motion are studied by using arclength continuation method and bifurcation analysis. Numerical responses obtained in the spectral neighborhood of the lowest natural frequency are compared with the results available in literature.

A Nonlinear Shear Deformable Nanoplate Model Including Surface Effects for Large Amplitude Vibrations of Rectangular Nanoplates with Various Boundary Conditions

2015 ◽  
Vol 07 (05) ◽  
pp. 1550076 ◽  
Author(s):  
Reza Ansari ◽  
Mostafa Faghih Shojaei ◽  
Vahid Mohammadi ◽  
Raheb Gholami ◽  
Mohammad Ali Darabi
Keyword(s):  
Boundary Conditions ◽  
Surface Stress ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Free Vibrations ◽  
Model Parameters ◽  
Geometrically Nonlinear ◽  
Residual Surface Stress ◽  
Various Boundary Conditions ◽  
Shear Deformable

In this paper, a geometrically nonlinear first-order shear deformable nanoplate model is developed to investigate the size-dependent geometrically nonlinear free vibrations of rectangular nanoplates considering surface stress effects. For this purpose, according to the Gurtin–Murdoch elasticity theory and Hamilton's principle, the governing equations of motion and associated boundary conditions of nanoplates are derived first. Afterwards, the set of obtained nonlinear equations is discretized using the generalized differential quadrature (GDQ) method and then solved by a numerical Galerkin scheme and pseudo arc-length continuation method. Finally, the effects of important model parameters including surface elastic modulus, residual surface stress, surface density, thickness and boundary conditions on the vibration characteristics of rectangular nanoplates are thoroughly investigated. It is found that with the increase of the thickness, nanoplates can experience different vibrational behavior depending on the type of boundary conditions.

Comparison of Different Shell Theories for Large-Amplitude Vibrations of Circular Cylindrical Shells

2002 ◽  
Author(s):  
M. Amabili
Keyword(s):  
Large Amplitude ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Continuation Method ◽  
Elastic Strain Energy ◽  
Harmonic Excitation ◽  
Lagrange Equations ◽  
Simply Supported ◽  
Nonlinear Shell ◽  
Circular Cylindrical Shells

Large-amplitude (geometrically nonlinear) vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. The Lagrange equations of motion are obtained by an energy approach, retaining damping through Rayleigh’s dissipation function. Four different nonlinear shell theories, namely Donnell’s, Sanders-Koiter, Flu¨gge-Lur’e-Byrne and Novozhilov’s theories, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories. Numerical responses obtained by using these four nonlinear shell theories are also compared to results obtained by Galerkin approach, used to discretise Donnell’s nonlinear shallow-shell equation of motion. A validation of calculations by comparison to experimental results is also performed. Boundary conditions for simply supported shells are exactly satisfied. Different expansions involving from 14 to 48 generalized coordinates, associated to natural modes of simply supported shells, are used. The nonlinear equations of motion are studied by using a code based on arclength continuation method that allows bifurcation analysis.

An exact solution for free vibration of thin functionally graded rectangular plates

2010 ◽  
Vol 225 (3) ◽  
pp. 526-536 ◽  
Author(s):  
A Hasani Baferani ◽  
A R Saidi ◽  
E Jomehzadeh
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Functionally Graded Plates ◽  
Different Boundary Conditions

The aim of this article is to find an exact analytical solution for free vibration characteristics of thin functionally graded rectangular plates with different boundary conditions. The governing equations of motion are obtained based on the classical plate theory. Using an analytical method, three partial differential equations of motion are reformulated into two new decoupled equations. Based on the Navier solution, a closed-form solution is presented for natural frequencies of functionally graded simply supported rectangular plates. Then, considering Levy-type solution, natural frequencies of functionally graded plates are presented for various boundary conditions. Three mode shapes of a functionally graded rectangular plate are also presented for different boundary conditions. In addition, the effects of aspect ratio, thickness—length ratio, power law index, and boundary conditions on the vibration characteristics of functionally graded rectangular plates are discussed in details. Finally, it has been shown that the effects of in-plane displacements on natural frequencies of functionally graded plates under different boundary conditions have been studied.

scholarly journals Thermal Effects on Nonlinear Vibrations of an Axially Moving Beam with an Intermediate Spring-Mass Support

2013 ◽  
Vol 20 (3) ◽  
pp. 385-399 ◽  
Author(s):  
Siavash Kazemirad ◽  
Mergen H. Ghayesh ◽  
Marco Amabili
Keyword(s):  
Thermal Loading ◽  
Nonlinear Vibrations ◽  
Equations Of Motion ◽  
Time Integration ◽  
Harmonic Excitation ◽  
Axially Moving Beam ◽  
Moving Beam ◽  
Axially Moving ◽  
The Galerkin Method ◽  
Continuation Technique

The thermo-mechanical nonlinear vibrations and stability of a hinged-hinged axially moving beam, additionally supported by a nonlinear spring-mass support are examined via two numerical techniques. The system is subjected to a transverse harmonic excitation force as well as a thermal loading. Hamilton's principle is employed to derive the equations of motion; it is discretized into a multi-degree-freedom system by means of the Galerkin method. The steady state resonant response of the system for both cases with and without an internal resonance between the first two modes is examined via the pseudo-arclength continuation technique. In the second method, direct time integration is employed to construct bifurcation diagrams of Poincaré maps of the system.

Nonlinear Planar Dynamics of Fluid-Conveying Cantilevered Extensible Pipes

2013 ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Michael P. Païdoussis ◽  
Marco Amabili
Keyword(s):  
Flow Velocity ◽  
Trivial Solution ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Time Integration ◽  
Axial Direction ◽  
Transverse Displacement ◽  
Lagrange Equations ◽  
Critical Flow Velocity ◽  
The Stability

This paper for the first time investigates the nonlinear planar dynamics of a cantilevered extensible pipe conveying fluid; the centreline of the pipe is considered to be extensible resulting in coupled longitudinal and transverse equations of motion; specifically, the kinetic and potential energies are obtained in terms of longitudinal and transverse displacements and then the extended version of the Lagrange equations for systems containing non-material volumes is employed to derive the equations of motion. Direct time integration along with the pseudo-arclength continuation method are employed to solve the discretized equations of motion. Bifurcation diagrams of the system are constructed as the flow velocity is increased as the bifurcation parameter. As opposed to the case of an inextensible pipe, an extensible pipe elongates in the axial direction as the flow velocity is increased from zero. At the critical flow velocity, the stability of the system is lost via a supercritical Hopf bifurcation, emerging from the trivial solution for the transverse displacement and non-trivial solution for the longitudinal displacement and leading to a flutter.

scholarly journals Circuit Implementation of a Modified Chaotic System with Hyperbolic Sine Nonlinearities Using Bi-Color LED

Technologies ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 15
Author(s):  
Christos K. Volos ◽  
Lazaros Moysis ◽  
George D. Roumelas ◽  
Aggelos Giakoumis ◽  
Hector E. Nistazakis ◽  
...  
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Original System ◽  
Dynamical Analysis ◽  
Phase Portraits ◽  
Circuit Realization ◽  
Hyperbolic Sine ◽  
Route To Chaos ◽  
Sine Functions

In this paper, a chaotic three dimansional dynamical system is proposed, that is a modification of the system in Volos et al. (2017). The new system has two hyperbolic sine nonlinear terms, as opposed to the original system that only included one, in order to optimize system’s chaotic behavior, which is confirmed by the calculation of the maximal Lyapunov exponents and Kaplan-Yorke dimension. The system is experimentally realized, using Bi-color LEDs to emulate the hyperbolic sine functions. An extended dynamical analysis is then performed, by computing numerically the system’s bifurcation and continuation diagrams, Lyapunov exponents and phase portraits, and comparing the numerical simulations with the circuit simulations. A series of interesting phenomena are unmasked, like period doubling route to chaos, coexisting attractors and antimonotonicity, which are all verified from the circuit realization of the system. Hence, the circuit setup accurately emulates the chaotic dynamics of the proposed system.

Chaotic Behavior of an Inelastic Beam

1998 ◽  
Vol 14 (4) ◽  
pp. 217-221
Author(s):  
Jiin-Po Yeh
Keyword(s):  
Lyapunov Exponent ◽  
Autocorrelation Function ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Time History ◽  
Harmonic Excitation ◽  
Largest Lyapunov Exponent ◽  
Bifurcation Phenomenon ◽  
Data Points ◽  
The Relationship

ABSTRACTThe dynamical system considered in this paper is an inelastic beam whose supports are subjected to a harmonic excitation. This paper first explores whether the system has chaotic motion. The appearance of the irregular time history, strange attractor on the Poincaré map as well as period-doubling bifurcation phenomenon strongly indicates that chaos indeed exist in this system. After finding the chaos phenomenon, this paper continues to investigate the relationship between the decay time of the autocorrelation function and the largest Lyapunov exponent. The Poincaré mapping points are chosen to be the sampled function of the discrete autocorrelation function. It's found that a power model of regression analysis can fit with good accuracy the data points, which are composed of the mapping times for the autocorrelation to decay into the square of the mean of the Poincaré points and the corresponding largest Lyapunov exponent.

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