Phase Modulation Equations Research Articles

Identification of nonlinear piezoelectric coefficients

An identification scheme that exploits the vibration response and generated voltage of an energy harvester is proposed to estimate parameters representing nonlinear piezoelectric coefficients. We develop the governing equations of a cantilever beam with tip mass and a piezoelectric layer using the generalized Hamilton's principle and by accounting for mechanical energy, virtual work, and electric enthalpy. We then use the method of multiple scales to determine the approximate solution of the response to a direct resonant excitation. We show that the nonlinear behavior captured by the method of multiple scales as approximate solution and amplitude and phase modulation equations can be used to estimate parameters of the nonlinear piezoelectric constitutive relations.

Vibration analysis of a beam on a nonlinear elastic foundation

Nonlinear vibrations of an Euler-Bernoulli beam resting on a nonlinear elastic foundation are discussed. In search of approximate analytical solutions, the classical multiple scales (MS) and the multiple scales Lindstedt Poincare (MSLP) methods are used. The case of primary resonance is investigated. Amplitude and phase modulation equations are obtained. Steady state solutions are considered. Frequency response curves obtained by both methods are contrasted with each other with respect to the effect of various physical parameters. For weakly nonlinear systems, MS and MSLP solutions are in good agreement. For strong hardening nonlinearities, MSLP solutions exhibit the usual jump phenomena whereas MS solutions are not reliable producing backward curves which are unphysical.

Nonlinear stability analysis of a composite laminated piezoelectric rectangular plate with multi-parametric and external excitations

The aim of this paper is to investigate the stability of a simply supported laminated composite piezoelectric rectangular plate under combined excitations. Analysis of the amplitude and phase modulation equations with the associated nonlinear interaction coefficients, as provided by the multiple scale analyses of various 1:1 internal resonance conditions and primary resonance case, where $$\upomega _2 \cong \upomega _1 $$ and $$\Omega _3 \cong \upomega _1 $$ is considered. The method of multiple time scale is applied to solve the non-linear differential equations describing the system up to the second-order approximation. All possible resonance cases at this approximation order are extracted. The stable/unstable periodic solutions are determined and are presented through frequency response plots. The analytical results are verified by comparing them with those of numerical integration of the modal equations. The influence of different parameters on the dynamic behavior of the composite laminated piezoelectric rectangular plate is studied. Variation of the some parameters leads to multi-valued amplitudes and hence to jump phenomena. A comparison with the available published work is reported.

Linear dynamical analysis of fractionally damped beams and rods

The aim of this study is to develop a general model for beams and rods with fractional derivatives. Fractional time derivatives can represent the damping term in dynamical models of continuous systems. Linear differential operators with spatial derivatives make it possible to generalize a wide range of problems. The method of multiple scales is directly applied to equations of motion. For the approximate solution, the amplitude and phase modulation equations are obtained in terms of the operators. Stability boundaries are derived from the solvability condition. It is shown that a fractional derivative influences the stability boundaries, natural frequencies, and amplitudes of vibrations. The solution procedure may be applied to many problems with linear vibrations of continuous systems.

Coherent solutions for the fundamental resonance of the Boussinesq equation

We consider the Boussinesq equation in an infinite wall under an external resonant weak excitation. Two slow flow equations are obtained and amplitude and phase modulation equations as well as external force-response and frequency–response curves are determined. Energy considerations are used in order to study the global behavior of the two slow flow equations and to demonstrate the existence of closed orbits around the equilibrium points with a second frequency in addition to the forcing frequency. Two period quasi-periodic motions are present with amplitudes depending on the initial conditions. Moreover, in certain cases librations can occur with a frequency depending on the amplitude of the external excitation. If the external excitation increases, the modulation period for the slow flow equations becomes infinite and an infinite-period bifurcation occurs, while the modulation amplitude remains finite and the libration transforms into a closed orbit. Finally, we demonstrate that the separatrix of the two slow flow equations corresponds to an asymptotically steady-state and phase-locked periodic solution.

Non-Linear Vibrations of a Microbeam Resting on an Elastic Foundation

In this study, response of a microbeam bonded to a non-linear elastic foundation is investigated. The model accounts for mid-plane stretching, an applied axial load, and an AC harmonic force. The microbeam is resting on a non-linear elastic foundation which introduces a cubic non-linear term to the equations of motion. Immovable end conditions introduce integral type nonlinearity. The integro-differential equations of motion are solved analytically by means of direct application of the method of multiple scales (a perturbation method). The amplitude and phase modulation equations are derived for the case of primary resonances. Frequency-response curves and non-linear frequencies are analyzed with respect to the effective physical parameters. Influence of elastic foundation coefficients, coefficients related to dielectric constants, and mid-plane stretching on the vibrations of the microbeam are investigated.

Global bifurcations and homoclinic trees in motion of a thin rectangular plate on a nonlinear elastic foundation

The global bifurcations and chaotic dynamics of a thin rectangular plate on a nonlinear elastic foundation subjected to a harmonic excitation are investigated. On the basis of the amplitude and phase modulation equations derived by the method of multiple scales, a near integrable two-degree-of-freedom Hamiltonian system is obtained by a transformation. The energy-phase method proposed by Haller and Wiggins is employed to analyze the global bifurcations for the thin rectangular plate. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case, which implies that chaotic motions may occur for this class of systems. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. To illustrate the theoretical predictions, we present visualizations of these complicated structures and numerical evidence of chaotic motions.

Principal parametric resonances of a general continuous system with cubic nonlinearities

A generalized vibrational model of cubic nonlinear continuous system with arbitrary parametric excitation is considered. The method of multiple scales (a perturbation method) is used to find an approximate analytical solution. The primary parametric resonance of the parametric excitation is considered. The amplitude and phase modulation equations are derived. Steady state solutions and their stability are discussed. The solution algorithm is applied to the problem of nonlinear vibrations of viscoelastic pipes conveying fluids. Natural frequencies of viscoelastic pipes are found. Frequency response curves and bifurcation points are drawn. Stable and unstable regions of trivial and nontrivial solutions are obtained.

Nonlinear dynamics and small damping signal control of chaos in a model of flow-induced oscillations of cylinders

Nonlinear dynamics of flow-induced oscillations of cylinders is investigated. The approach in our paper is made to introduce an harmonic forced vibration in the coupling term of the structural equation since this may be the consequence of approximating the potential force that could act as a periodic excitation. The method of multiple scales is used to determine the steady state responses. Amplitude and phase modulation equations as well as external force-response and frequency-response curves are obtained. We show that harmonic excitation can induce resonance phenomena in the oscillation of the structure for a range of frequencies of potential force, and also lock-in phenomena appear in the structure part. Also, we find that the structure can be damaged as the amplitude of the potential excitation increases. Numerical simulations confirm the existence of chaotic vibration in the system, a small damping signal control is used to suppress it since it may cause fatigue in the system. The model developed is expected to yield better results for structure in water.

A general solution procedure for the forced vibrations of a system with cubic nonlinearities: Three-to-one internal resonances with external excitation

A general vibrational model of a continuous system with arbitrary linear and cubic operators is considered. Approximate analytical solutions are found using the method of multiple scales. The primary resonances of the external excitation and three-to-one internal resonances between two arbitrary natural frequencies are treated. The amplitude and phase modulation equations are derived. The steady-state solutions and their stability are discussed. The solution algorithm is applied to two specific problems: (1) axially moving Euler–Bernoulli beam, and (2) axially moving viscoelastic beam.

A general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities: Primary resonance case

Nonlinear vibrations of a general model of continuous system is considered. The model consists of arbitrary linear and cubic operators. The equation of motion is solved by the method of multiple scales (a perturbation method). The primary resonances of external excitation is analysed. The amplitude and phase modulation equations are presented. Approximate analytical solution is derived. Steady-state solutions and their stability are discussed. Finally, the solution algorithm is applied to two different engineering problems. One of the application is the transverse vibration of an axially moving Euler–Bernoulli beam and the other is a viscoelastic beam.

Bifurcation control in the Burgers–KdV equation

We consider the bifurcation control for the forced Burgers–KdV equation by means of delay feedback linear terms. We use a perturbation method in order to find amplitude and phase modulation equations as well as external force-response and frequency-response curves. We observe in the resonance response a saddle-node bifurcation that leads to jump and hysteresis phenomena. We compare the uncontrolled and controlled systems and demonstrate that control terms can delay or remove the occurrence of the saddle-node bifurcation and reduce the amplitude peak of the resonant response.

Non linear vibrations of stepped beam system under different boundary conditions

In this study, the nonlinear vibrations of stepped beams having different boundary conditions were investigated. The equations of motions were obtained using Hamilton's principle and made non dimensional. The stretching effect induced non-linear terms to the equations. Forcing and damping terms were also included in the equations. The dimensionless equations were solved for six different set of boundary conditions. A perturbation method was applied to the equations of motions. The first terms of the perturbation series lead to the linear problem. Natural frequencies for the linear problem were calculated exactly for different boundary conditions. Second order non-linear terms of the perturbation series behave as corrections to the linear problem. Amplitude and phase modulation equations were obtained. Non-linear free and forced vibrations were investigated in detail. The effects of the position and magnitude of the step, as well as effects of different boundary conditions on the vibrations, were determined.

The Response of Nonlinear Single-Degree-of-Freedom Systems to Modulated High-Frequency Input

In this paper we study the response of single-degree-of-freedom with cubic, quartic and quintic nonlinearities to an amplitude-modulated excitation whose carrier frequency is much higher than the natural frequency of the system. The only restriction on the amplitude modulation is that it contains frequencies much lower than the carrier frequency of the excitation. The method of multiple scales is used to derive two coupled first-order ordinary differential equations that describe the evolution of the amplitude and phase with damping, nonlinearities and resonances. The evolution equations are used to determine the steady-state motions, while representative frequency-response curves are presented for each resonance. Stability analysis of the amplitude and phase modulation equations for both cases are performed. The bending of the response curves leads to multi-valued solutions and hence to jump phenomena.

Two-to-one internal resonances in a shallow curved beam resting on an elastic foundation

Vibrations of shallow curved beams are investigated. The rise function of the beam is assumed to be small. Sinusoidal and parabolic curvature functions are examined. The immovable end conditions result in mid-plane stretching of the beam which leads to nonlinearities. The beam is resting on an elastic foundation. The method of multiple scales, a perturbation technique, is used in search of approximate solutions of the problem. Two-to-one internal resonances between any two modes of vibration are studied. Amplitude and phase modulation equations are obtained. Steady state solutions and stability are discussed, and a bifurcation analysis of the amplitude and phase modulation equations are given. Conditions for internal resonance to occur are discussed, and it is found that internal resonance is possible for the case of parabolic curvature but not for that of sinusoidal curvature.

Nonlinear vibration of a post-buckled beam subjected to external and parametric excitations

An analytical investigation of the nonlinear vibration of a post-buckled beam subjected to harmonic excitations is presented. The system can be modelled as a nonlinear oscillator with parametric and external excitations having frequencies Ω 1 and Ω 2. The method of multiple scales is used to determine to second order the amplitude and phase-modulation equations. Attention is focused on subharmonic resonances (Ω 1 ≅ 2ω 0, Ω 2 ≅ 2Ω 0 ; Ω 1 ≅ 3ω 0, Ω 2 ≅ 3ω 0 ; Ω 1 ≅ 4ω 0, Ω 2 ≅ 4ω 0 ; Ω 1 ≅ 5ω 0 and Ω 1 ≅ 6ω 0) where ω 0 is the natural frequency. Steady-state amplitude for each case is plotted as a function of a detuning parameter showing the influences of the several parameters. Stability is performed on figures by solid and broken lines. There exist multivalued solutions which increase or decrease by the variation of some parameters. The response amplitude is not affected by increasing and decreasing the parameters F1 and α 3 for the cases of subharmonics of order one-half and one-third. The solution loses stability on increasing the parameters α 4, α 5 and F1 in the case of subharmonic resonance of order one-fifth and also for decreasing the parameters μ and α 4 in the case of subharmonic resonance of order one-sixth.

Single-mode control and chaos of cantilever beam under primary and principal parametric excitations

Abstract A non-linear control law is proposed to suppress the vibrations of the first mode of a cantilever beam when subjected to primary and principal parametric excitations. The dynamics of the beam are modeled with a second-order non-linear ordinary-differential equation. The model accounts for viscous damping air drag, and inertia and geometric non-linearities. A control law based on quantic velocity feedback is proposed. The method of multiple scales method is used to derive two-first ordinary differential equations that govern the evolution of the amplitude and phase of the response. These equations are used to determine the steady state responses and their stability. Amplitude and phase modulation equations as well as external force–response and frequency–response curves are obtained. Numerical simulations confirm this scenario and detect chaos and unbounded motions in the instability regions of the periodic solutions.

Non-linear vibrations of suspension bridges with external excitation

Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of non-linear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency–response equation is derived and the jump phenomenon in the frequency–response curves resulting from non-linearity is considered. Effects of initial amplitude and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.

Transient dynamics of a dispersive elastic wave guide weakly coupled to an essentially nonlinear end attachment

We consider the dynamics of a dispersive semi-infinite linear rod weakly connected to an essentially nonlinear end attachment. Using a Green’s function formulation we reduce the dynamics to an integro-differential equation, which we express in the form of an infinite set of ordinary differential equations using Neumann expansions. We examine resonant interactions of the attachment with incident traveling waves propagating in the pass band of the rod, as well as resonance capture phenomena where the attachment engages in transient 1:1 internal resonance with the in-phase mode of the rod, at the bounding frequency of its pass and stop bands. The studied resonant interactions of the nonlinear attachment with traveling waves are similar to resonance capture cascades occurring in finite-chain nonlinear attachment configurations; these are the transient dynamics of the attachment as it engages in a series of instantaneous resonances with a continuous spectrum of frequencies in the pass band of the medium. As the energy of the attachment decreases due to damping and energy radiation back to the rod, the attachment engages in 1:1 resonance capture with the in-phase mode of the rod; this resonance capture occurs due to the dispersion property in the linear medium. It is shown that the transient dynamics of resonance capture can be reduced to a set of two ‘slow flow’ amplitude and phase modulation equations. The analytical results are in agreement with direct numerical simulations.

Periodic behavior for the parametrically excited Boussinesq equation

Abstract The asymptotic perturbation (AP) method is applied to the study of the parametrically excited and weakly damped Boussinesq equation in an infinite wall. Amplitude and phase modulation equations as well as external force-response and frequency-response curves are obtained. The presence of jump phenomena and bifurcations is demonstrated.