Perturbation Method Of Multiple Scales Research Articles

Nonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectric functionally graded laminated composite shells: Part II: Numerical results

Nonlinear vibration and dynamic response of the simply supported piezoelectric functionally graded material (FGM) shells under combined electrical, thermal, mechanical and aerodynamic loading are studied in this paper. The material properties of the shell are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of the constituents. The third-order piston theory is employed to evaluate the aerodynamic pressure. The governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity and the physical neutral surface concept are taken into consideration. The Galerkin method, Volmir’s assumption and the multiple time scales perturbation methods are used for the nonlinear dynamical analysis of shells to give the expression of natural frequencies, the nonlinear dynamic responses and the primary resonance phenomena. The influences of the shell geometry and piezoelectric thickness, temperature change, external constant electric voltage and aerodynamic loads on the nonlinear dynamic behavior of the piezoelectric functionally graded shells through a comprehensive parametric study are discussed in details.

Nonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectric functionally graded laminated composite shells, Part I: Theory and analytical solutions

Nonlinear vibration and dynamic response of simply supported piezoelectric functionally graded material (FGM) shells under combined electrical, thermal, mechanical and aerodynamic loading are studied in this paper. The material properties of the shell are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of the constituents. The third-order piston theory is employed to evaluate the aerodynamic pressure. The governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity and the physical neutral surface concept are taken into consideration. The Galerkin method, Volmir’s assumption and the multiple time scales perturbation methods are used for the nonlinear dynamical analysis of shells to give the expression of natural frequencies, the nonlinear dynamic responses and the primary resonance phenomena. The influences of the shell geometry and piezoelectric thickness, temperature change, external constant electric voltage and aerodynamic loads on the nonlinear dynamic behavior of the piezoelectric functionally graded shells through a comprehensive parametric study are discussed in details.

Advective Diffusion of Contaminants in the Surf Zone

AbstractA theory on the horizontal spreading of contaminants in the surf zone is described in this paper. The surf zone is viewed as a narrow strip comparable to the characteristic wavelength, in which turbulence caused by wave breaking is strong. Flows inside the surf zone consist of both oscillatory motions and a steady current induced by breaking waves. The wave field in the shoaling zone is modeled by the linear theory for monochromatic waves, and it is assumed that the breaking wave height is proportional to the local depth in the surf zone. The simplest scenario of longshore current on a straight beach of constant slope is considered, for which the longshore current velocity is predicted by a slightly modified version of Longuet-Higgins' original theory developed in the 1970s. On the basis of the estimation that the time scale of horizontal diffusion is much longer than the wave period, the perturbation method of multiple scales is applied to derive the transport equation for the advective diffusion...

Vibration, Stability, and Resonance of Angle-Ply Composite Laminated Rectangular Thin Plate under Multiexcitations

An analytical investigation of the nonlinear vibration of a symmetric cross-ply composite laminated piezoelectric rectangular plate under parametric and external excitations is presented. The method of multiple time scale perturbation is applied to solve the nonlinear differential equations describing the system up to and including the second-order approximation. All possible resonance cases are extracted at this approximation order. The case of 1 : 1 : 3 primary and internal resonance, whereΩ3≅ω1,ω2≅ω1, andω3≅3ω1, is considered. The stability of the system is investigated using both phase-plane method and frequency response curves. The influences of the cubic terms on nonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate are studied. The analytical results given by the method of multiple time scale is verified by comparison with results from numerical integration of the modal equations. Reliability of the obtained results is verified by comparison between the finite difference method (FDM) and Runge-Kutta method (RKM). It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system. Such cases should be avoided as working conditions for the system. Variation of the parametersμ1,μ2,α7,β8,ω1,ω2,f1,f2leads to multivalued amplitudes and hence to jump phenomena. Some recommendations regarding the different parameters of the system are reported. Comparison with the available published work is reported.

Numerical Study of the Influence of Waving Bottom on the Fluid Surface Wave

The effect of waving bottom on the fluid surface wave was investigated. Starting from the basic equations of potential flow theory and boundary conditions, we used the multiple scales perturbation method to deduce fluid surface waves satisfy the first-order approximate equation and second-order approximate equation. Under the second-order approximation, the fluid surface waveform was simulated with the Matlab in the presence of different waving bottom form. The results show that there are three solitary waves on the surface of the fluid. With the development of time, the amplitude of each solitary wave has not changed. It seems that they are not affected each other and propagate independently. So it suggests that the waving bottom is effective for maintaining surface wave energy balance income and expenditure in spreading process.

Vibration suppression in ultrasonic machining described by non-linear differential equations via passive controller

In the present work, the coupling of two non-linear oscillators of the main system and absorber representing ultrasonic cutting process is investigated. This leads to a two-degree-of-freedom system subjected to multi-external excitation force. The aim of this work is to control the main system behavior at simultaneous primary and internal resonance condition, where the system damage is probable. Multiple scale perturbation method is applied to obtain the solution up to the second order approximations. The different resonance cases are reported and studied numerically. The stability of the system is investigated applying both phase-plane and frequency response techniques. The effects of the different parameters of the absorber on the system behavior are studied numerically. Comparison with the available published work is reported.

Shear-deformable thin-walled composite Beams in internal and external resonance

The non-linear dynamic response of thin-walled composite beams is analyzed considering the effect of shear deformation. The model is based on a small strain and large rotation and displacements theory, which is formulated through the adoption of a higher-order displacement field and takes into account shear flexibility (bending and warping shear). The beam is assumed to be in internal resonance conditions of the kind 2:3:1, so that quadratic, cubic and combination resonances occur. In the analysis of a weakly nonlinear continuous system, the Galerkin’s method is employed to express the problem in terms of generalized coordinates. Then, the perturbation method of multiple scales is applied to the reduced system in order to obtain the equations of amplitude and modulation. The equilibrium solution is governed by the modal coupling and experience a complex behavior composed by saddle–noddle and Hopf bifurcations. The results of the analysis show that the equilibrium solutions are influenced by the shear effect, when this effect is ignored the amplitude of vibration is reduced significantly, thus altering the dynamic response of the beam. This alteration can infer in an incorrect stability prediction of the periodic solutions.

Viscous Potential Flow Analysis of Nonlinear Rayleigh–Taylor Instability with Heat and Mass Transfer

The nonlinear analysis of Rayleigh–Taylor instability of two viscous fluids is studied when there is heat and mass transfer across the interface, using viscous potential flow theory. The fluids are considered to be viscous and incompressible with different kinematic viscosities. The analysis is based on the method of multiple scale perturbation and the nonlinear stability is governed by first-order nonlinear partial differential equation. The stability conditions are obtained and stability is discussed theoretically as well as numerically. Regions of stability and instability have been shown graphically indicating the effect of various parameters. It has been observed that the heat and mass transfer has stabilizing effect on the stability of the system in the nonlinear analysis.

Secondary resonances of a microresonator under AC-DC electrostatic and DC piezoelectric actuations

This article studies the secondary resonances of a clamped-clamped microresonator under combined electrostatic and piezoelectric actuations. The electrostatic actuation is induced by applying the AC-DC voltage between the microbeam and the electrode plate that lies at the opposite side of the microbeam. The piezoelectric actuation is induced by applying the DC voltage between upper and lower sides of piezoelectric layer. It is assumed that the neutral axis of bending is stretched when the microbeam is deflected. The drift effect of piezoelectric layer (the phenomenon where there is a slow increase of the free strain after the application of a DC field) is neglected. The equations of motion are solved by using the multiple scale perturbation method. The system possesses a subharmonic resonance of order one-half and a superharmonic resonance of order two. It is shown that using the DC piezoelectric actuation, the sensitivity of AC-DC electrostatically actuated microresonator under subharmonic and superharmonic resonances may be tuned. In addition, it is shown that the tuning domain of the microbeam under combined electrostatic and piezoelectric actuations at subharmonic and superharmonic conditions is larger than the tuning domain of microbeam under only the electrostatic actuation.

Three-dimensional modulation of electron-acoustic waves: 3 + 1 Davey–Stewartson system

In this paper we consider the three-dimensional modulation of an electron-acoustic wave by means of a multiple-scale perturbation method. We find that the governing equation for the amplitude that describes the asymptotic properties of the wave is a Davey–Stewartson system in three-space variables. We use the system to study the linear stability of the modulation wave and we compare our results with preceding studies by Kourakis and Shukla (2004) [8] on the oblique modulation problem based on a non-linear Schrödinger equation.

Second-order approximation of angle-ply composite laminated thin plate under combined excitations

The influence of the quadratic and cubic terms on non-linear dynamic characteristics of the angle-ply composite laminated rectangular plate with parametric and external excitations is investigated. The method of multiple time scale perturbation is applied to solve the non-linear differential equations describing the system up to and including the second-order approximation. All possible resonance cases are extracted and investigated at this approximation order. Two cases of the sub-harmonic resonances cases (Ω2≅2ω1 and Ω2≅2ω2) in the presence of 1:2 internal resonance ω2≅2ω1 are considered. The stability of the system is investigated using both frequency response equations and phase-plane method. It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system. Such cases should be avoided as working conditions for the system. Some recommendations regarding the different parameters of the system are reported. Comparison with the available published work is reported.

An investigation into the vibration analysis of a plate with a surface crack of variable angular orientation

An analytical approach is presented for the forced vibration analysis of a plate containing a surface crack of variable angular orientation, based on three different boundary conditions. The method is based on classical plate theory. Firstly, the equation of motion is derived for the plate containing the angled surface crack with respect to one side of the plate and subjected to transverse harmonic excitation. The crack formulation representing the angled surface crack is based on a simplified line-spring model. Then, by employing the Berger formulation, the derived governing equation of motion of the cracked plate model is transformed into a cubic nonlinear system. The nonlinear behaviour of the cracked plate model is thus investigated from the amplitude–frequency equation by use of the multiple scales perturbation method. For both cracked square and rectangular plate models, the influence of the boundary conditions, the crack orientation angle, crack length, and location of the point load is demonstrated. It is found that the vibration characteristics and nonlinear characteristics of the plate structure can be greatly affected by the orientation of the crack in the plate. Finally the validity of the developed model is shown through comparison of the results with experimental work.

Solving systems of nonlinear difference equations by the multiple scales perturbation method

In this paper, we apply an improved version of the multiple scales perturbation method to a system of weakly nonlinear, regularly perturbed ordinary difference equations. Such systems arise as a result of the discretization of a system of nonlinear differential equations, or as a result in the stability analysis of nonlinear oscillations. In our procedure, asymptotic approximations of the solutions of the difference equations will be constructed which are valid on long iteration scales.

Study of Stability and Vibration Reduction in Multi-Tool Ultrasonic Machining under Simultaneous Primary and Internal Resonance

The main object of this paper is the mathematical study of the vibration behavior in ultrasonic machining (USM) described by non-linear differential equations. The ultrasonic machining (USM) consists of the tool holder and the absor-bers representing the tools. This leads to four-degree-of-freedom system subject to multi-external excitation forces. The aim of this project is the reduction of the vibrations in the tool holder and have reasonable amplitudes for the tools represented by the multi-absorbers. Multiple scale perturbation method is applied to obtain the solution up to the second order approximation and to study the stability of the steady state solution near different simultaneous resonance cases. The resulting different resonance cases are reported and studied numerically. The stability of the steady state solution near the selected resonance cases is studied applying both frequency response equations and phase-plane technique. The effects of the different parameters of the system and the absorbers on the system behavior are studied numerically. Optimum working conditions for the tools were obtained. Comparison with the available published work is reported.

Non-linear vibration response of laminated fibre-reinforced rectangular plates

This paper investigates the non-linear free and forced vibration response of immovable simply-supported composite rectangular plates. The plates are symmetric and are subjected to a transverse pressure which is harmonic in time. A novel procedure is implemented to obtain the non-linear ordinary differential equation by using first order shear deformation theory and Galerkin method. The multiple time scales perturbation method is utilised to obtain an analytical closed-form expression for the non-linear frequency and frequency response equation of the studied plates. After validation of present study with available literature, the non-linear free and forced vibrations of a fibre-metal laminated plate, named Glare 3, have been investigated. Also, the effects of some system parameters on the non-linear frequency have been studied.

Nonlinear analysis of capillary instability with heat and mass transfer

The nonlinear capillary instability of the cylindrical interface between the vapor and liquid phases of a fluid is studied when there is heat and mass transfer across the interface, using viscous potential flow theory. The fluids are considered to be viscous and incompressible with different kinematic viscosities. Both asymmetric and axisymmetric disturbances are considered. The analysis is based on the method of multiple scale perturbation and the nonlinear stability is governed by first-order nonlinear partial differential equation. The stability conditions are obtained and discussed theoretically as well as numerically. Regions of stability and instability have been shown graphically indicating the effect of various parameters. It has been observed that the heat and mass transfer has stabilizing effect on the stability of the system in the nonlinear analysis for both axisymmetric as well as asymmetric disturbances.

Dynamic instability of inclined cables under combined wind flow and support motion

In this paper an inclined nearly taut stay, belonging to a cable-stayed bridge, is considered. It is subject to a prescribed motion at one end, caused by traveling vehicles, and embedded in a wind flow blowing simultaneously with rain. The cable is modeled as a non-planar, nonlinear, one-dimensional continuum, possessing torsional and flexural stiffness. The lower end of the cable is assumed to undergo a vertical sinusoidal motion of given amplitude and frequency. The wind flow is assumed uniform in space and constant in time, acting on the cable along which flows a rain rivulet. The imposed motion is responsible for both external and parametric excitations, while the wind flow produces aeroelastic instability. The relevant equations of motion are discretized via the Galerkin method, by taking one in-plane and one out-of-plane symmetric modes as trial functions. The two resulting second-order, non-homogeneous, time-periodic, ordinary differential equations are coupled and contain quadratic and cubic nonlinearities, both in the displacements and velocities. They are tackled by the Multiple Scale perturbation method, which leads to first-order amplitude-phase modulation equations, governing the slow dynamics of the cable. The wind speed, the amplitude of the support motion and the internal and external frequency detunings are set as control parameters. Numerical path-following techniques provide bifurcation diagrams as functions of the control parameters, able to highlight the interactions between in-plane and out-of-plane motions, as well as the effects of the simultaneous presence of the three sources of excitation.

Vibration and stability of an aircraft tail under simultaneous primary-combined and internal resonance

This paper adds a negative velocity feedback to the dynamical system of twin-tail aircraft to suppress the vibration. The system is represented by two coupled second-order nonlinear differential equations having both quadratic and cubic nonlinearities. The system describes the vibration of an aircraft tail subjected to both multi-harmonic and multi-tuned excitations. The method of multiple time scale perturbation is adopted to solve the nonlinear differential equations and obtain approximate solutions up to the third order approximations. The stability of the proposed analytic solution near the simultaneous primary, combined and internal resonance is studied and its conditions are determined. The effect of different parameters on the steady state response of the vibrating system is studied and discussed by using frequency response equations. Some different resonance cases are investigated numerically

Nonlinear response of a resonant viscoelastic microbeam under an electrical actuation

In this paper, using perturbation and Galerkin method, the response of a resonant viscoelastic microbeam to an electric actuation is obtained. The microbeam is under axial load and electrical load. It is assumed that midplane is stretched, when the beam is deflected. The equation of motion is derived using the Newton's second law. The viscoelastic model is taken to be the Kelvin-Voigt model. In the first section, the static deflection is obtained using the Galerkin method. Exact linear symmetric mode shape of a straight beam and its deflection function under constant transverse load are used as admissible functions. So, an analytical expression that describes the static deflection at all points is obtained. Comparing the result with previous research show that using deflection function as admissible function decreases the computation errors and previous calculations volume. In the second section, the response of a microbeam resonator system under primary and secondary resonance excitation has been obtained by analytical multiple scale perturbation method combined with the Galerkin method. It is shown, that a small amount of viscoelastic damping has an important effect and causes to decrease the maximum amplitude of response, and to shift the resonance frequency. Also, it shown, that an increase of the DC voltage, ratio of the air gap to the microbeam thickness, tensile axial load, would increase the effect of viscoelastic damping, and an increase of the compressive axial load would decrease the effect of viscoelastic damping.

Slow-light solitons in three-level atomic systems modified by a microwave field

We study the modified effect of slow-light soliton in a resonant three-level atomic system via electromagnetically induced transparency (EIT) by utilizing a microwave field. We derive a high-order nonlinear Schrodinger equation by using a perturbation method of multiple-scales, and calculate the modification of soliton velocity and frequency shift. We find that in the presence of the microwave field an obvious decrease of propagating velocity of soliton can be realized, which provides an effective method to slow down optical solitons in EIT systems. We also find that the down shift of oscillating frequency of soliton in such system can be largely suppressed by the microwave field.