Showing papers on "Multiple-scale analysis published in 1999"

Multiple resonances in suspended cables: direct versus reduced-order models

Abstract: We apply two analytical approaches to construct asymptotic models for the non-linear three-dimensional responses of an elastic suspended shallow cable to a harmonic excitation. We investigate the case of primary resonance of the first in-plane symmetric mode when it is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and a two-to-one internal resonance with the first symmetric out-of-plane mode. First, we apply the method of multiple scales directly to the governing two integral-partial-differential equations and associated boundary conditions. Reconstitution of the solvability conditions at second and third orders leads to a system of four coupled non-linear complex-valued equations describing the modulation of the amplitudes and phases of the interacting modes. The homogeneous solutions associated with the first in-plane and out-of-plane modes in the second-order problem are needed to make the reconstituted modulation equations derivable from a Lagrangian. However, this procedure leads to an indeterminacy, indicating a likely inconsistency with this specific application of the method of multiple scales. Then, we apply the method to a four-degree-of-freedom Galerkin discretized model obtained using the pertinent excited eigenmodes. Again, the homogeneous solutions associated with the first in-plane and out-of-plane modes in the second-order problems are required to make the reconstituted modulation equations derivable from a Lagrangian. Frequency–response curves obtained using the two generated asymptotic models, for a specific choice of the arbitrary constant appearing in both models, show different qualitative as well as quantitative predictions for some classes of motions. The effects of an inconsistent reconstitution in the direct approach are also investigated.

Nonlinear Vibration of Parametrically Excited Moving Belts, Part I: Dynamic Response

Abstract: The dynamic response and stability of parametrically excited viscoelastic belts are investigated in these two consecutive papers. In the first paper, the generalized equation of motion is obtained for a viscoelastic moving belt with geometric nonlinearity. The linear viscoelastic differential constitutive law is employed to characterize the material property of belts. The method of multiple scales is applied directly to the governing equation which is in the form of continuous gyroscopic systems. No assumptions regarding the spatial dependence of the motion are made. Closed-form solutions for the amplitude and the existence conditions of nontrivial limit cycles of the summation resonance are obtained. It is shown that there exists an upper boundary for the existence condition of the summation parametric resonance due to the existence of viscoelasticity. The effects of viscoelastic parameters, excitation frequencies, excitation amplitudes, and axial moving speeds on dynamic responses and existence boundaries are investigated.

Dynamics of a Cubic Nonlinear Vibration Absorber

Abstract: We study the dynamics of a nonlinear active vibration absorber. We consider a plant model possessing curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is approximately equal to the plant natural frequency, we show the existence of a saturation phenomenon. As the forcing amplitude is increased beyond a certain threshold, the response amplitude of the directly excited mode (plant) remains constant, while the response amplitude of the indirectly excited mode (absorber) increases. We obtain an approximate solution to the governing equations using the method of multiple scales and show that the system possesses two possible saturation values. Using numerical techniques, we perform stability analyses and demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic responses.

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

Abstract: The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in 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 three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. 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 principal parametric resonance of the first mode or combination parametric resonance of the additive type, 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 trivial and 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. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.

Nonlinear Vibration of Parametrically Excited Viscoelastic Moving Belts, Part II: Stability Analysis

Abstract: The amplitude and existence conditions of nontrivial limit cycles are derived in the companion paper by the use of the method of multiple scales. In this paper, the stability for parametrically excited viscoelastic moving belts is studied. Stability boundaries of the trivial limit cycle for general summation parametric resonance are obtained. The Routh-Hurwitz criterion is used to investigate the stability of nontrivial limit cycles. Closed-form expressions are found for the stability of nontrivial limit cycles of general summation parametric resonance. It is shown that the first limit cycle is always stable while the second limit cycle is always unstable for the viscoelastic moving belts. The effects of viscoelastic parameters, excitation frequencies, excitation amplitudes, and axial moving speeds on stability boundaries are discussed.

Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances

Abstract: The non-linear behaviour of a slender beam carrying a lumped mass subjected to principal parametric base excitation is investigated. The dimension of the beam–mass system and the position of the attached mass are so adjusted that the system exhibits 3 : 1 internal resonance. Multi-mode discretization of the governing equation which retains the cubic non-linearities of geometrical and inertial type is carried out using Galerkin’s method. The method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and the stability of the system. The linear first-order perturbation results show new zones of instability due to the presence of internal resonance. For low amplitude of excitation and damping Hopf bifurcations are observed in the trivial steady-state response. The multi-branched non-trivial response curves show turning point, pitch-fork and Hopf bifurcations. Cascade of period and torus doubling, crises as well as the Shilnikov mechanism for chaos are observed. This is the first natural physical system exhibiting a countable infinity of horseshoes in a neighbourhood of the homoclinic orbit.

Experimental Nonlinear Identification of a Single Mode of a Transversely Excited Beam

Abstract: A procedure is presented for using a primary resonance excitation in experimentally identifying the nonlinear parameters of a model approximating the response of a cantilevered beam by a single mode. The model accounts for cubic inertia and stiffness nonlinearities and quadratic damping. The method of multiple scales is used to determine the frequency-response function for the system. Experimental frequency- and amplitude-sweep data is compared with the prediction of the frequency-response function in a least-squares curve-fitting algorithm. The algorithm is improved by making use of experimentally known information about the location of the bifurcation points. The method is validated by using the extracted parameters to predict the force-response curves at other nearby frequencies.

Bifurcation in a parametrically excited two-degree-of-freedom nonlinear oscillating system with 1:2 internal resonance

Abstract: The nonlinear response of a two-degree-of-freedom nonlinear oscillating system to parametric excitation is examined for the case of 1∶2 internal resonance and, principal parametric resonance with respect to the lower mode. The method of multiple scales is used to derive four first-order autonomous ordinary differential equations for the modulation of the amplitudes and phases. The steadystate solutions of the modulated equations and their stability are investigated. The trivial solutions lose their stability through pitchfork bifurcation giving rise to coupled mode solutions. The Melnikov method is used to study the global bifurcation behavior, the critical parameter is determined at which the dynamical system possesses a Smale horseshoe type of chaos.

Nonlinear nearly Wilton ripples between two uniform stream flows

Abstract: A study is made of the small-amplitude disturbances at the interface of two semi-infinite ideal fluids and which arise when two harmonics are in the ratio 1:2. By employing the method of multiple scales, a system of coupled nonlinear partial differential equations is derived. These equations model the propagation of the wave packet, correct up to third order and remain valid when there are imperfections in the resonance. Solutions to the equations are determined and the corresponding Stokes-type profiles exhibited. They are shown to be stable against plane-wave perturbations

Two-Mode Interaction of a Beam with a Nonlinear Boundary Condition

Abstract: In order to investigate modal interactions in a subharmonic resonance of a beam with a nonlinear boundary condition, we consider a beam constrained by a nonlinear spring to a harmonic excitation. The resonance conditions considered are ω n 3ω m and Ω 3ω n , where ω m and ω n are the natural frequencies and Ω is the excitation frequency. This nonlinear problem is governed by a linear partial differential equation, initial conditions and a nonlinear and inhomogeneous boundary condition. The method of multiple scales is used to transform the problem into a system of autonomous ordinary differential equations for amplitude and phase variables. The steady-state responses and their stability are determined by use of this system. In order to check the validity of the analytical solution we solve the initial and boundary value problem by means of a finite difference analysis.

Nonlinear bending-torsion oscillations of cantilever beams to combination parametric excitations

Abstract: The nonlinear nonplanar oscillations of cantilever beams excited by a combination parametric resonance are investigated. The effect of geometric and inertia nonlinearities in the governing equations of motion and boundary conditions is considered. The method of multiple scales is directly applied to the partial-differential equations and boundary conditions to derive the equations governing the modulations of the amplitudes and phases of the interacting modes. Then, the influence of the forcing amplitude and frequency on the response is analyzed. The results show that, through this mechanism, a small-amplitude high-frequency excitation can produce a large-amplitude low-frequency response, which cannot be predicted by linear theory.

Multiple-Scale Analysis

Abstract: Multiple-scale analysis is a very general collection of perturbation techniques that embodies the ideas of both boundary-layer theory and WKB theory. Multiple-scale analysis is particularly useful for constructing uniformly valid approximations to solutions of perturbation problems.

Effect of Two-to-One Internal Resonances in a Nonlinearly Restrained Fluid Valve

Abstract: The effect of two-to-one internal resonances on the nonlinear response of a pressure relief valve is studied. The fluid valve is modeled as a distributed parameter system at one end and nonlinearly restrained at the other. The method of multiple scales is used to solve the system of partial differential equation and boundary conditions. Frequency-response curves are presented for the primary resonance of either mode in the presence of a two-to-one internal resonance. Stability of the steady-state solutions is investigated. Parameters of the system leading to two-to-one internal resonances are tabulated.

Nonlinear Dynamics and Stability of a Two D.O.F. Elastic/Elasto-Plastic Model System

Abstract: The local and global nonlinear dynamics of a two-degree-of-freedom model system is studied. The undeflected model consists of an inverted T formed by three rigid bars, with the tips of the two horizontal bars supported on springs. The springs exhibit an elasto-plastic response, including the Bauschinger effect. The vertical rigid bar is subjected to a conservative (dead) or non-conservative (follower) force having static and periodic components. First, the method of multiple scales is used for the analysis of the local dynamics of the system with elastic springs. The attention is focused at modal interaction phenomena in weak excitation at primary resonance and in hard sub-harmonic excitation. Three different asymptotic expansions are utilised to get a structural response for typical ranges of excitation parameters. Numerical integration of the governing equations is then performed to validate results of asymptotic analysis in each case. A full global nonlinear dynamics analysis of the elasto-plastic system is performed to reveal the role of plastic deformations in the stability of this system. Static 'force-displacement' curves are plotted and the role of plastic deformations in the destabilisation of the system is discussed. Large-amplitude non-linear oscillations of the elasto-plastic system are studied, including the influence of material hardening and of static and sinusoidal components of the applied force. A practical method is proposed for the study of a non-conservative elasto-plastic system as a non-conservative elastic system with an 'equivalent' viscous damping.

Parametric Third‐Subharmonic Resonance in Nonlinear Electrohydrodynamic Rayleigh‐Taylor Instability with Mass and Heat Transfer

Abstract: The nonlinear electrohydrodynamic Rayleigh-Taylor instability with mass and heat transfer is investigated. The fluids are stressed by a periodic acceleration and a normal electric field. Based on the method of multiple scales, a parametric nonlinear Schrodinger equation with complex coefficients is derived in the third-subharmonic resonance case. A standard nonlinear Schrodinger equation with complex coefficients is obtained in the non-resonance case. A temporal solution is carried out for the parametric nonlinear Schrodinger equation. Necessary and sufficient conditions for stability are obtained. Numerical calculations show that the thickness of the fluid, the normal electric field, and the coefficient of mass and heat transfer have destabilizing effect. It is found that the external frequency plays a dual role in the stability criteria.

Nonlinear Capillary Waves on the Surface of an Electrohydrodynamic Planar Jet

Abstract: The method of multiple scales is used to analyse the nonlinear breakup of a planar jet in the presence of electric field taking into account surface tension. The evolution of the amplitude is governed by a nonlinear Schrodinger equation which gives the criterion for modulational instability. Numerical result is given in the graphical form.

A new perturbation technique

Abstract: In this new perturbation technique method, the unperturbed equation is the linearized equation of the original one, so the solution of the unperturbed solution is actually the approximate solution of the exact solution. The present theory is as simple as the straightforward expansion, while can eliminate the secular expansions completely.

Simultaneous Parametric and Internal Resonances in Systems Simulating Liquid Sloshing Impact

Abstract: The nonlinear dynamic interaction between liquid sloshing impact and elastic structural dynamics is examined in the neighborhood of simultaneous parametric and internal resonance conditions. The analytical modeling of the liquid impact forces is represented by a simple pendulum. The impact forces with the tank walls are phenomenologically described by a power function. In the absence of internal resonance, Pilipchuk and Ibrahim (1997) analyzed the system to describe the in-phase and out-of-phase strongly non-linear periodic regimes. The present work considers both weakly and strongly nonlinear forces of interaction. The method of multiple scales is used to determine the system response in the neighborhood of combination parametric resonance and internal resonance conditions. Under combination parametric resonance, and in the absence of impact forces, the response is found to sensitive to initial conditions. Depending on the initial conditions and internal detuning parameter, the response can be quasi-periodic or chaotic with irregular jumps between two unstable equilibria. In the presence of impact forces, the system preserves fixed response amplitude response within a small range of internal detuning parameter. Beyond that range, the response exhibits quasi-periodic and snap-through regimes. Each regime is governed by the initial conditions, internal detuning parameter, damping ratios and excitation level. The paper briefly describes the response characteristics under fist and second mode parametric excitations.