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.

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

1993 ◽  
Author(s):  
H. L. Neal ◽  
A. 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

Abstract 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 bifurcations, 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.

Parametric Stability of Axially Accelerating Viscoelastic Beams With the Recognition of Longitudinally Varying Tensions

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Li-Qun Chen ◽  
You-Qi Tang
Keyword(s):  
Governing Equation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Finite Support ◽  
Governing Equations ◽  
Parametric Stability ◽  
Stability Boundaries ◽  
Axial Acceleration ◽  
The Stability ◽  
Longitudinally Varying Tension

In this paper, the parametric stability of axially accelerating viscoelastic beams is revisited. The effects of the longitudinally varying tension due to the axial acceleration are highlighted, while the tension was approximately assumed to be longitudinally uniform in previous studies. The dependence of the tension on the finite support rigidity is also considered. The generalized Hamilton principle and the Kelvin viscoelastic constitutive relation are applied to establish the governing equations and the associated boundary conditions for coupled planar motion of the beam. The governing equations are linearized into the governing equation in the transverse direction and the expression of the longitudinally varying tension. The method of multiple scales is employed to analyze the parametric stability of transverse motion. The stability boundaries are derived from the solvability conditions and the Routh-Hurwitz criterion for principal and sum resonances. In terms of stability boundaries, the governing equations with or without the longitudinal variance of tension are compared and the effects of the finite support rigidity are also examined. Some numerical examples are presented to demonstrate the effects of the stiffness, the viscosity, and the mean axial speed on the stability boundaries. The differential quadrature scheme is developed to numerically solve the governing equation, and the computational results confirm the outcomes of the method of multiple scales.

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.

Passage of a Rotor Through a Critical Speed

1982 ◽  
Vol 104 (2) ◽  
pp. 370-374 ◽  
Author(s):  
K. Tsuchiya
Keyword(s):  
Computer Simulation ◽  
Asymptotic Expansion ◽  
Digital Computer ◽  
Critical Speed ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Peak Amplitude ◽  
Matched Asymptotic Expansion ◽  
Nonstationary Oscillation ◽  
Pass Through

This paper deals with a nonstationary oscillation of a rotor passing through a critical speed. The analysis is based on the method of multiple scales and the method of matched asymptotic expansion. The peak amplitude of the response and the criteria for the onset of the stalling (inability to pass through the critical speed) are derived. These results are compared with those of digital computer simulation.

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 Pneumatic Artificial Muscle Actuator Under Parametric and Hard Excitations

2021 ◽  
Author(s):  
BHABEN KALITA ◽  
SANTOSHA K. DWIVEDY
Keyword(s):  
Governing Equation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Basin Of Attraction ◽  
Artificial Muscle ◽  
Equation Of Motion ◽  
Phase Portraits ◽  
System Response ◽  
Pneumatic Artificial Muscle ◽  
Stability And Bifurcations

Abstract In this work, a single degree of freedom system consisting of a mass and a Pneumatic Artificial Muscle (PAM) subjected to time varying pressure inside the muscle is considered. The system is subjected to hard excitation and the governing equation of motion is found to be that of a nonlinear forced and parametrically excited system under super- and sub-harmonic resonance conditions. The solution of the nonlinear governing equation of motion is obtained using the method of multiple scales (MMS). The time and frequency response, phase portraits and basin of attraction have been plotted to study the system response along with the stability and bifurcations. Further, the different muscle parameters have been evaluated by performing experiments which are further used for numerically evaluating the system response using the theoretically obtained closed form equations. The responses obtained from the experiments are found to be in good agreement with those obtained from the method of multiple scales. With the help of examples, the procedure to obtain the safe operating range of different system parameters have been illustrated.

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.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

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