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.

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.

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.

scholarly journals Modeling and 1 : 1 Internal Resonance Analysis of Cable-Stayed Shallow Arches

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Jiangen Lv ◽  
Zhicheng Yang ◽  
Xuebin Chen ◽  
Quanke Wu ◽  
Xiaoxia Zeng
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Period Doubling ◽  
Dynamic Responses ◽  
Shallow Arch ◽  
Response Curves ◽  
Internal Resonances ◽  
Essential Information ◽  
Modulation Equations ◽  
Inclined Angle

In this paper, an analytical model of a cable-stayed shallow arch is developed in order to investigate the 1 : 1 internal resonance between modes of a cable and a shallow arch. Integrodifferential equations with quadratic and cubic nonlinearities are used to model the in-plane motion of a simple cable-stayed shallow arch. Nonlinear dynamic responses of a cable-stayed shallow arch subjected to external excitations with simultaneous 1 : 1 internal resonances are investigated. Firstly, the Galerkin method is used to discretize the governing nonlinear integral-partial-differential equations. Secondly, the multiple scales method (MSM) is used to derive the modulation equations of the system under external excitation of the shallow arch. Thirdly, the equilibrium, the periodic, and the chaotic solutions of the modulation equations are also analyzed in detail. The frequency- and force-response curves are obtained by using the Newton–Raphson method in conjunction with the pseudoarclength path-following algorithm. The cascades of period-doubling bifurcations leading to chaos are obtained by applying numerical simulations. Finally, the effects of key parameters on the responses are examined, such as initial tension, inclined angle of the cable, and rise and inclined angle of shallow arch. The comprehensive numerical results and research findings will provide essential information for the safety evaluation of cable-supported structures that have widely been used in civil engineering.

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.

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.

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.

PARAMETRICALLY EXCITED NONLINEAR TWO-DEGREE-OF-FREEDOM SYSTEMS WITH NONSEMISIMPLE ONE-TO-ONE RESONANCE

1995 ◽  
Vol 05 (03) ◽  
pp. 725-740 ◽  
Author(s):  
C. CHIN ◽  
A.H. NAYFEH
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Degree Of Freedom ◽  
Parametrically Excited ◽  
Parametric Resonances ◽  
Quadratic Nonlinearities ◽  
One To One ◽  
Two Degree Of Freedom

The response of a parametrically excited two-degree-of-freedom system with quadratic and cubic nonlinearities and a nonsemisimple one-to-one internal resonance is investigated. The method of multiple scales is used to derive four first-order differential equations governing the modulation of the amplitudes and phases of the two modes for the cases of fundamental and principal parametric resonances. Bifurcation analysis of the case of fundamental parametric resonance reveals that the quadratic nonlinearities qualitatively change the response of the system. They change the pitchfork bifurcation to a transcritical bifurcation. Cyclic-fold, Hopf bifurcations of the nontrivial constant solutions, and period-doubling sequences leading to chaos are induced by these quadratic terms. The effects of quadratic nonlinearities for the case of principal parametric resonance are discussed.

scholarly journals Response of a Parametrically Excited System to a Nonstationary Excitation

1995 ◽  
Vol 1 (1) ◽  
pp. 57-73 ◽  
Author(s):  
Harold L. Neal ◽  
Ali H. Nayfeh
Keyword(s):  
Computer Simulations ◽  
Governing Equation ◽  
Digital Computer ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Saddle Points ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Analog Computer ◽  
Modulation Equations

The response of a single-degree-of-freedom system to a nonstationary excitation is investigated by using the method of multiple scales as well as analog- and digital-computer simulations. The unexcited system has one focus and two saddle points. The system can be used to model rolling of ships in head or follower seas. The method of multiple scales is used to derive equations governing the modulation of the amplitude and phase of the response. The modulation equations are used to find the stationary solutions and their stability. The response to nonstationary excitations is found by integrating the original governing equation as well as the modulation equations. There is good agreement between the results of both approaches. For some frequency and amplitude sweeps, the nonstationary response found from integrating the original governing equation exhibits behaviors that are analogous to symmetry-breaking bifurcations, period-doubling bi furcations, chaos, and unboundedness present in the stationary case. The maximum response amplitude and the excitation amplitude or frequency at which the response becomes unbounded are found as functions of the sweep rate. The results of the digital-computer simulations are verified with an analog computer.

Magneto-Aeroelastic Internal Resonances of a Rotating Circular Plate Based on Gyroscopic Systems Decoupling

2020 ◽  
pp. 2150010
Author(s):  
Wenqiang Li ◽  
Yuda Hu
Keyword(s):  
Circular Plate ◽  
Multiple Scales ◽  
Period Doubling ◽  
Coefficient Matrix ◽  
Transverse Displacement ◽  
Internal Resonances ◽  
Resonance Amplitude ◽  
Steady State Responses ◽  
Nonlinear Modulation ◽  
Gyroscopic Systems

In this paper, a principal and 3:1 internal resonance of an edge-clamped conductive circular plate rotating in air-magnetic environment is investigated, where the electromagnetic force expressions and a simple empirical aerodynamic model are used in modelling. Based on the transverse displacement assumption with a combination of two degenerate linearized modes, the 2 degree of freedom (2-DOF) magneto-aeroelastic asymmetric nonlinear gyroscopic systems are derived by utilizing the Galerkin approach. The method of multiple scales and the solvable condition for the coefficient matrix of the gyroscopic systems is employed to decouple the 2-DOF nonlinear gyroscopic systems and achieve the nonlinear modulation equations of the steady state responses. The numerical results for the relationship between two eigenfrequencies verify the existence of 3:1 internal resonances of the circular plate rotating in air-magnetic field. In addition, the resonance amplitude varying with detuning parameters and excitation amplitudes are plotted under principal and 3:1 internal resonances, respectively. It is found that the system may lose stability generated by a Hopf bifurcation, which may finally evolve into chaotic state through period-doubling bifurcation of intermittent periodic responses.

THE RESPONSE OF A NONLINEAR SYSTEM WITH A NONSEMISIMPLE ONE-TO-ONE RESONANCE TO A COMBINATION PARAMETRIC RESONANCE

1995 ◽  
Vol 05 (04) ◽  
pp. 971-982 ◽  
Author(s):  
C. CHIN ◽  
A. H. NAYFEH ◽  
D. T. MOOK
Keyword(s):  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Power Spectra ◽  
Period Doubling ◽  
Supersonic Stream ◽  
Equilibrium Solutions ◽  
Shooting Technique ◽  
First Order ◽  
Period Doubling Bifurcations ◽  
Combination Parametric Resonance

The Galerkin procedure is used to discretize the nonlinear partial differential equation and boundary conditions governing the flutter of a simply supported panel in a supersonic stream. These equations have repeated natural frequencies at the onset of flutter. The method of multiple scales is used to derive five first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the excited modes. Then, the modulation equations are used to calculate the equilibrium solutions and their stability, and hence to identify the excitation parameters that suppress flutter and those that lead to complex motions. 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. The complex motions are characterized by using phase planes, power spectra, Lyapunov exponents, and dimensions.

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