Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams

1997 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh ◽  
Char-Ming Chin
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiple Scales ◽  
Period Doubling ◽  
Hopf Bifurcations ◽  
Equilibrium Solutions ◽  
Interaction Coefficients ◽  
Partial Differential ◽  
Modulation Equations ◽  
Torsional Inertia

Abstract The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its “exural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to the two integro-partial-differential equations of Crespo da Silva and Glynn. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. The modulation equations exhibit the symmetry property found by Feng and Leal by analytically manipulating the interaction coefficients in the modulation equations obtained by Nayfeh and Pai by applying the method of multiple scales to the governing integro-partial-differential equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.

Bifurcation and Chaos in Externally Excited Circular Cylindrical Shells

1996 ◽  
Vol 63 (3) ◽  
pp. 565-574 ◽  
Author(s):  
Char-Ming Chin ◽  
A. H. Nayfeh
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiple Scales ◽  
Period Doubling ◽  
Single Mode ◽  
Equilibrium Solutions ◽  
Multiple Attractors ◽  
Partial Differential ◽  
Quadratic Nonlinearities ◽  
Mode Solution

The nonlinear response of an infinitely long cylindrical shell to a primary excitation of one of its two orthogonal flexural modes is investigated. The method of multiple scales is used to derive four ordinary differential equations describing the amplitudes and phases of the two orthogonal modes by (a) attacking a two-mode discretization of the governing partial differential equations and (b) directly attacking the partial differential equations. The two-mode discretization results in erroneous solutions because it does not account for the effects of the quadratic nonlinearities. The resulting two sets of modulation equations are used to study the equilibrium and dynamic solutions and their stability and hence show the different bifurcations. The response could be a single-mode solution or a two-mode solution. The equilibrium solutions of the two orthogonal third flexural modes undergo a Hopf bifurcation. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos, multiple attractors, explosive bifurcations, and crises.

Nonlinear forced vibration of rotating composite laminated cylindrical shells under hygrothermal environment

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Xiao Li ◽  
Wentao Jiang ◽  
Xiaochao Chen ◽  
Zhihong Zhou
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiple Scales ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Dynamic Responses ◽  
Partial Differential ◽  
Moisture Concentration ◽  
Hygrothermal Environment ◽  
Laminated Cylindrical Shells

Abstract This work aims to study nonlinear vibration of rotating composite laminated cylindrical shells under hygrothermal environment and radial harmonic excitation. Based on Love’s nonlinear shell theory, and considering the effects of rotation-induced initial hoop tension, centrifugal and Coriolis forces, the nonlinear partial differential equations of the shells are derived by Hamilton’s principle, in which the constitutive relation and material properties of the shells are both hygrothermal-dependent. Then, the Galerkin approach is applied to discrete the nonlinear partial differential equations, and the multiple scales method is adopted to obtain an analytical solution on the dynamic response of the nonlinear shells under primary resonances of forward and backward traveling wave, respectively. The stability of the solution is determined by using the Routh–Hurwitz criterion. Some interesting results on amplitude–frequency relations and nonlinear dynamic responses of the shells are proposed. Special attention is given to the combined effects of temperature and moisture concentration on nonlinear resonance behavior of the shells.

Two-to-One Internal Resonances in Buckled Beams

1995 ◽  
Author(s):  
Wayne Kreider ◽  
Ali H. Nayfeh ◽  
Char-Ming Chin
Keyword(s):  
Multiple Scales ◽  
Primary Resonance ◽  
Period Doubling ◽  
Equilibrium Solutions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Buckled Beams ◽  
Second Mode ◽  
Quadratic Nonlinearities ◽  
Period Doubling Bifurcations

Abstract The vibrations of buckled beams with two-to-one internal resonances (ω2 ≈ 2ω1) about a static buckled position are analyzed. General boundary conditions and harmonic excitations (frequency Ω) in both the transverse and axial directions are considered. The analysis assumes a unimodal static buckled deflection, considers quadratic nonlinearities only, and determines the amplitude and phase modulation equations via the method of multiple scales. The following specific cases are treated: Ω ≈ 2ω2, Ω ≈ ω1 + ω2, and Ω ≈ ω1. From the modulation equations for a primary resonance of the second mode (i.e., Ω ≈ ω2), one-mode and two-mode stable equilibrium solutions are found in addition to dynamic solutions caused by Hopf bifurcations. In the region of dynamic solutions, a variety of phenomena are documented, including period-doubling bifurcations, intermittency, chaos, and crises.

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

1997 ◽  
Author(s):  
Char-Ming Chin ◽  
Ali H. Nayfeh
Keyword(s):  
Natural Frequency ◽  
Parametric Resonance ◽  
Axial Load ◽  
Multiple Scales ◽  
Period Doubling ◽  
Equilibrium Solutions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Modal Damping ◽  
Second Mode

Abstract The nonlinear planar response of a hinged-clamped beam to a parametric excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact via a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of principal parametric resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of parametric resonance of the first mode, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single-, and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. In the region of dynamic solutions, some phenomena are documented, including period-doubling bifurcations and blue-sky catastrophes.

Nonlinear forced vibration of rectangular plates by modified multiple scale method

2021 ◽  
pp. 095440622110527
Author(s):  
Farzaneh Rabiee ◽  
Ali Asghar Jafari
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Forced Vibration ◽  
Multiple Scales ◽  
Plate Theory ◽  
Multiple Scales Method ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Classical Plate Theory ◽  
Nonlinear Forced Vibration

In the present study, the nonlinear forced vibration of a rectangular plate is investigated analytically using modified multiple scales method for the first time. The plate is subjected to transversal harmonic excitation, and the boundary condition is assumed to be simply supported. The von Karman nonlinear strain displacement relations are used. The extended Hamilton principle and classical plate theory are applied to derive the partial differential equations of motions. This research focuses on resonance case with 3:1 internal resonance. By applying Galerkin method, the nonlinear partial differential equations are transformed into time dependent nonlinear ordinary differential equations, which are then solved analytically by modified multiple scales method. This proposed approach is very simple and straightforward. The obtained results are then compared with both the traditional multiple scales method and previous studies, and excellent compatibility is noticed. The effect of some of the main parameters of the system is also examined.

Period doubling and chaos in partial differential equations for thermosolutal convection

Nature ◽  
1983 ◽  
Vol 303 (5919) ◽  
pp. 663-667 ◽  
Author(s):  
D. R. Moore ◽  
J. Toomre ◽  
E. Knobloch ◽  
N. O. Weiss
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Period Doubling ◽  
Thermosolutal Convection ◽  
Partial Differential

EQUILIBRIUM OF TWO POPULATIONS SUBJECT TO CHEMOTAXIS

2004 ◽  
Vol 14 (04) ◽  
pp. 503-533 ◽  
Author(s):  
A. FASANO ◽  
A. MANCINI ◽  
M. PRIMICERIO
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Total Biomass ◽  
Bifurcation Parameter ◽  
Equilibrium Solutions ◽  
Partial Differential ◽  
Chemical Agents ◽  
Two Populations

We consider a system of four partial differential equations modelling the dynamics of two populations interacting via chemical agents. Classes of nontrivial equilibrium solutions are studied and a rescaled total biomass is shown to play the role of a bifurcation parameter.

Nonlinear Mode Localization in Systems Governed by Partial Differential Equations

1996 ◽  
Vol 49 (2) ◽  
pp. 87-99 ◽  
Author(s):  
Alexander F. Vakakis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiple Scales ◽  
Extreme Values ◽  
Nonlinear Partial Differential Equations ◽  
Flexible Structures ◽  
Partial Differential ◽  
Mode Localization ◽  
Infinite Domains ◽  
Nonlinear Mode

The concept of nonlinear normal mode (NNM) is used to study localized oscillations in certain classes of oscillators governed by nonlinear partial differential equations. NNMs are synchronous free oscillations during which all positional coordinates of the system reach their extreme values or pass through the equilibrium position at the same instant of time. Although such motions can be regarded as nonlinear analogs of the linear normal modes of classical vibration theory, not all NNMs are analytic continuations of linear ones. Continuous systems of finite and infinite spatial extent are considered. For periodic assemblies consisting of a finite number of nonlinear structural members, the NNMs are computed asymptotically by solving nonlinear sets of equations possessing regular singular points. Some of the computed NNMs are spatially localized to only a limited number of components of the assembly. The bifurcations giving rise to nonlinear mode localization are examined using the perturbation method of multiple-scales. The implications of nonlinear mode localization on the vibration and shock isolation of periodic flexible structures are discussed. In particular, localized NNMs lead to passive motion confinement of disturbances generated by impulsive loads. Finally, the concept of NNMs is extended to analytically study standing waves with spatially localized envelopes in a class of nonlinear partial differential equations defined over infinite domains. It is shown that NNM-based methodologies can be an effective tool for analyzing such motions.

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