Multiple Time Scales Perturbation Technique Research Articles

Non-reciprocity and selective wave amplification in nonlinear inertia-induced autoparametric periodic structures

Inspired by the spatiotemporally modulated metamaterials, we propose a one-dimensional autoparametric periodic structure induced by nonlinear inertia that is realized simply by constructing an array of pendula coupled to local oscillators. The mechanism of autoparametric modulation is revealed using a unit cell. The evolution equations for a superlattice describing the global dynamics of the wave-vibration coupling are obtained analytically by combining the Floquet-Bloch theorem and the multiple-time scales perturbation technique. The band structure exists corresponding to the fixed point in the phase space, wherein large amplitude local vibrations with distributed traveling phases modulate the inertial properties of the host pendula propagating with a small amplitude signal wave by playing the role of a pump wave. Non-reciprocal dispersion relations are demonstrated both by numerical results and perturbation results with two-order accuracy. Selective wave amplification with two-mode directional emission is observed in a two-port system which is composed of a sandwich composite medium. The present research based on the proposed autoparametric structure paves a way towards building tunable wave manipulators, vibration energy migrators and mechanical amplifiers.

Dynamical analysis of an inverted pendulum with positive position feedback controller approximate uniform solution

The inverted pendulum is controlled in this article by using the nonlinear control theory. From classical analytical mechanics, its substructure equation of motion is derived. Because of the inclusion of the restoring forces, the Taylor expansion is employed to facilitate the analysis. An estimated satisfactory periodic solution is obtained with the aid of the modified Homotopy perturbation method. A numerical technique based on the fourth-order Runge–Kutta method is employed to justify the previous solution. On the other hand, a positive position feedback control is developed to dampen the vibrations of an IP system subjected to multi-excitation forces. The multiple time scale perturbation technique of the second order is introduced as a mathematical method to solve a two-degree-of-freedom system that simulates the IP with the PPF at primary and 1:1 internal resonance. The stability of these solutions is checked with the aid of the Routh–Hurwitz criterion. A set of graphs, based on the frequency response equations resulting from the MSPT method, is incorporated. Additionally, a numerical simulation is set up with RK-4 to confirm the overall controlled performance of the studied model. The quality of the solution is confirmed by the match between the approximate solution and the numerical simulation. Numerous other nonlinear systems can be controlled using the provided control method. Illustrations are offered that pertain to implications in design and pedagogy. The linearized stability of IP near the fixed points as well as the phase portraits is depicted for the autonomous and non-autonomous cases. Because of the static stability of the IP, it is found that its instability can be suppressed by the increase of both the generalized force as well as the torsional constant stiffness of the spring. Additionally, the presence of the magnetic field enhances the stability of IP.

A nonlinear time delay control influence in simulation aircraft tail contain a vertical cantilever beam

Nonlinear time delay saturation controller (NTDSC) is used to minimize the vibrations of a vertical cantilever beam simulated as an airplane tail within harmonic excitation force. In the case of primary and 1:2 internal resonance as a simultaneous resonance case, an analytic solution is obtained by employing the multiple time scale perturbation technique to solve the nonlinear differential equations and modeling the system with NTDSC. Time histories and Poincare portraits were created to demonstrate the system without and with controller. The time-history response, as well as the impacts of the parameters on the system and controller, were simulated numerically using the MATLAB program. A good match is found when the approximate solutions are compared to the numerical simulations via the Runge–Kutta method fourth-order (RK4). The stability investigation of the steady state solution in the studied simultaneous resonance case is reviewed and analyzed via the Routh-Huriwitz approach. The impact of controller parameters and time delays on system response curves are examined. Also, the operation’s safe zone is established from the outcomes of the effect via various values of time delays on the system after adding the control scheme. The numerical and analytical solutions at time-history were compared using the MATLAB program to confirm their comparability. The outcomes of this study were used to develop a theoretical foundation for the system’s design and construction. A comparison is made with recently released papers to appear the difference between this study and the others.

Nonlinear Vibration Analysis of Viscoelastic Smart Sandwich Plates Through the use of Fractional Derivative Zener Model

In this paper, the nonlinear vibrational behavior of a sandwich plate with embedded viscoelastic material is studied through the use of constitutive equations with fractional derivatives. The studied sandwich structure is consisted of a viscoelastic core that is located between the faces of functionally graded magneto-electro-elastic (FG-MEE). In order to determine the frequency-dependent feature of the viscoelastic layer, four-parameter fractional derivative model is utilized. The material properties of FG-MEE face sheets have been distributed considering the power law scheme along the thickness. In addition, for derivation of the governing equations on the sandwich plate, first-order shear deformation plate theory along with von Karman-type of kinematic nonlinearity are implemented. The derived partial differential equations (PDEs) have been transformed to the ordinary differential equations (ODEs) through the Galerkin method. After that, the nonlinear vibration equations for the sandwich plate have been solved by multiple time scale perturbation technique. Moreover, for evaluating the effect of different parameters such as electric and magnetic fields, fractional order, the ratio of the core-to-face thickness and the power low index on the nonlinear vibration characteristics of sandwich plates with FG-MEE face sheets, the parametric analysis has been performed. The obtained results revealed the enhanced nonlinear natural frequency through an increment in the fractional order. Furthermore, the prominent influence of fractional order on the nonlinear frequency of sandwich plate was declared at the negative electric potential and positive magnetic potential.

Nonlinear extensional-flexural vibrations in variable cross section beams with eccentric intermediate mass

Beams involving intermediate axially and transversely eccentric masses have many industrial applications such as plane wings with an intermediately positioned heavy gas turbine. In most of these applications, the beams undergo large deflections which give rise to geometric nonlinearities. So, the objective of the current research is to provide a nonlinear dynamic model for extensional-flexural vibration of such beams with variable cross sections. Hamilton's principle is employed for finding the normalized coupled extensional-flexural equations of motion and the corresponded boundary conditions. Then a new finite difference scheme is developed for finding the natural frequencies of the system and the corresponded mode shapes. The solution of the Eigen value-Eigen function problem is then verified by comparing the numerical findings with the exact solutions which are available for the simple case of constant cross section beams. A parametric study is also carried out to characterize the effect of the intermediate mass as well as its eccentricity parameters on the natural frequencies of the system. It is observed, that except for fundamental frequency, other natural frequencies are noticeably reduced with increasing the weight of the mass. Moreover, it is observed that with increasing the axial and transverse eccentric parameters, the third natural frequency is decreased, while the first, second and fourth ones are not appreciably changed. The derived modes are utilized in Lagrange equations along with a single mode approximation to derive the nonlinear coupled temporal equations of motion in time domain. These nonlinear equations are then solved analytically using the multiple time scales perturbation technique, and closed form expressions are suggested for the extensional and flexural responses of the system. The analytical findings are closely verified via numerical simulations. Finally, the frequency response of the system in the primary resonance case is derived analytically and the dependence of the amplitude of the response on the system and excitation parameters is studied in detail. The results reveal that the system exhibit hardening behavior and with increasing the excitation frequency or amplitude of the excitation, a discontinuous increase in the vibration amplitude may occur. The modeling approach suggested in this paper can be effectively used for studying the dynamic behavior of more complex systems involving single or multiple eccentric intermediate masses.

Size-dependent nonlinear vibration analysis of shear deformable microarches using strain gradient theory

Non-classical continuum theories are able to capture the size-dependent behavior of microscale structures. Among them, strain gradient modeling is one of the most accurate ones. The objective of this paper is to develop for the first time a nonlinear strain gradient formulation for the vibrational analysis of electrostatically actuated shear deformable microarches. The normalized equations of motion and the corresponding boundary conditions are derived based on Hamilton’s principle. Novel closed form expressions are derived for the mode shapes of the system using the perturbation technique. These mode shapes are then utilized in a Galerkin projection method to reduce the partial differential equations of motion to time domain nonlinear ordinary differential equations. Multiple time scales perturbation technique is then employed to derive analytical solutions for the nonlinear vibration of the system resulting from a DC step voltage excitation. The analytical results are compared with numerical simulations, and excellent agreement is observed. The formulations and the results presented in this paper can be utilized for design, analysis and synthesis of microscale bistable structures with pre-deformed thick microbeams as their main operating element.

Nonlinear analysis of functionally graded piezoelectric energy harvesters

This study deals with nonlinear vibrational analysis of functionally graded piezoelectric energy harvesters (FGPEH) carrying concentrated mass at free end, while taking into account the geometrical nonlinear terms as well as electromechanical couplings. Hamilton’s principle and Rayleigh’s beam theory are utilized to obtain the coupled electromechanical governing equations. The first mode shape of the system is derived analytically which is also validated by comparing it with numerical findings of a three-dimensional finite element simulation performed in COMSOL Multiphysics commercial software. The Galerkin projection method is employed along with a single mode approximation to derive the nonlinear coupled electromechanical governing equations which are then solved analytically using the method of multiple time scales perturbation technique for both cases of free and forced vibrations. Time domain responses of free vibration are compared with those of numerical simulations carried out in MATLAB and excellent agreement is observed. Moreover, the nonlinear frequency responses of the forced vibration are obtained for the case of near resonance excitation. A parametric study is also accomplished to reveal the effects of power low index of the FG material as well as the scaled amplitude of the input displacements of the base on the shape of the frequency response.

Breather structures in degenerate relativistic non-extensive plasma

We examine the excitation of breather structures in a degenerate relativistic plasma consisting of non-extensive electrons and cold ions. For this purpose, the multiple time scale perturbation technique is used to obtain a nonlinear Schrödinger equation (NLSE). We then consider different localized solutions regarding analytical breather solutions of the NLSE, and examine their properties in the frame of the present plasma system, i.e. a degenerate relativistic non-extensive plasma. The results of the present investigation may be useful for the understanding of the basic features of the nonlinear excitations that may occur in dense astrophysical plasmas.

Harmonic Oscillations and Resonances in 3-D Nonlinear Dynamical System

This paper is concerned with the three dimensional motion of a nonlinear dynamical system. The motion is described by nonlinear partial differential equation, which is converted by Galerkin method to three dimensional ordinary differential equations. The three dimensional differential equations, under the influence of external forces, are solved analytically and numerically by the multiple time scales perturbation technique and the Runge-Kutta fourth order method. Phase plane technique and frequency response equations are used to investigate the stability of the system and the effects of the parameters of the system, respectively.

Effects of time delay on an active vibration control of a forced and Self-excited nonlinear beam

Loop delays are taken into our consideration when positive position feedback controller is used to control the vibrations of forced and self-excited nonlinear beam. External excitation is a harmonic excitation caused by support motion of the cantilever beam. Self-excitation is caused by fluid flow and modeled by a nonlinear damping with a negative linear part (Rayleigh’s function). The multiple timescales perturbation technique is applied to obtain a first-order approximate solution. Effects of time delay on the system are extensively studied, and optimal conditions for the system operation are deduced. The equilibrium solution curves are plotted for several values of controller parameters. The stability of the steady-state solution is investigated using frequency-response equations. The analytical results are validated by numerical integration of the original closed-loop system model equations, time histories and Poincare map for the system. We realized that all predictions from analytical solutions are in good agreement with the numerical simulation.

Active Control of a Rectangular Thin Plate Via Negative Acceleration Feedback

In this paper, the dynamic oscillation of a rectangular thin plate under parametric and external excitations is investigated and controlled. The motion of a rectangular thin plate is modeled by coupled second-order nonlinear ordinary differential equations. The formulas of the thin plate are derived from the von Kármán equation and Galerkin's method. A control law based on negative acceleration feedback is proposed for the system. The multiple time scale perturbation technique is applied to solve the nonlinear differential equations and obtain approximate solutions up to the second-order approximations. One of the worst resonance case of the system is the simultaneous primary resonances, where Ω1≅ω1 and Ω2≅ω2. From the frequency response equations, the stability of the system is investigated according to the Routh–Hurwitz criterion. The effects of the different parameters are studied numerically. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. The simulation results are achieved using matlab 7.0 software. A comparison is made with the available published work.

Analytical modeling of large amplitude free vibration of non-uniform beams carrying a both transversely and axially eccentric tip mass

The objective of this paper is to study large amplitude flexural–extensional free vibration of non-uniform cantilever beams carrying a both transversely and axially eccentric tip mass. The effects of variable axial force is also taken into account. Hamilton׳s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. A numerical finite difference scheme is proposed to find the natural frequencies and mode shapes of the system which is validated specifically for a beam with linearly varying cross section. Using a single mode approximation in conjunction with the Lagrange method, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. These temporal coupled equations are then solved analytically using the multiple time scales perturbation technique. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. The qualitative and quantitative knowledge resulting from this research is expected to enable the study of the effects of eccentric tip mass and non-uniformity on the large amplitude flexural–extensional vibration of beams for improved dynamic performance.

Vibration suppression of a time-varying stiffness AMB bearing to multi-parametric excitations via time delay controller

The applications of active magnetic bearings are growing in industry due to its amazing advantages in reducing friction losses. In this research, the vibration of a two-degree-of-freedom rotor, active magnetic bearings system is suppressed via a nonlinear time delay controller at the confirmed worst resonance case. The selected resonance case is the simultaneous primary and sub-harmonic resonance case. The main aim of this paper was to study the effects of the nonlinear, time delay controller on the behavior of the vibrating system. The multiple time scale perturbation technique is applied to obtain an approximate solution to the second-order approximation. The steady-state solution is obtained around the worst resonance case. The stability of the system is studied applying both frequency response equations and phase-plane method. The worst resonance case is confirmed applying numerical technique. The effects of the different parameters on the steady-state response of the vibrating system are investigated. The obtained approximate solution is validated numerically. Some recommendations are given regarding the design of such system. At the end of the work, a comparison is made with the available published work.

Modeling Geometric Nonlinearities in the Free Vibration of a Planar Beam Flexure With a Tip Mass

The objective of this work is to analytically study the nonlinear dynamics of beam flexures with a tip mass undergoing large deflections. Hamilton's principle is utilized to derive the equations governing the nonlinear vibrations of the cantilever beam and the associated boundary conditions. Then, using a single mode approximation, these nonlinear partial differential equations are reduced to two coupled nonlinear ordinary differential equations. These equations are solved analytically using the multiple time scales perturbation technique. Parametric analytical expressions are presented for the time domain response of the beam around and far from its internal resonance state. These analytical results are compared with numerical ones to validate the accuracy of the proposed analytical model. Compared with numerical solution methods, the proposed analytical technique shortens the computational time, offers design insights, and provides a broader framework for modeling more complex flexure mechanisms. The qualitative and quantitative knowledge resulting from this effort is expected to enable the analysis, optimization, and synthesis of flexure mechanisms for improved dynamic performance.

A Comparison between Active and Passive Vibration Control of Non-Linear Simple Pendulum. Part II: Longitudinal Tuned Absorber and Negative Gφ and Gφn Feedback

In part I [1] we dealt with a tuned absorber, which can move in the transversally direction, where it is added to an externally excited pendulum. Active control is applied to the system via negative velocity feedback or its square or cubic value. The multiple time scale perturbation technique is applied throughout. An approximate solution is derived up to second order approximation. The stability of the system is investigated applying both frequency response equations and phase plane methods. The effects of the absorber on system behavior are studied numerically. Optimum working conditions of the system are obtained applying passive and active control methods. Both control methods are demonstrated numerically. In this paper, a tuned absorber, in the longitudinal direction, is added to an externally excited pendulum. Active control is applied to the system via negative acceleration feedback or via negative angular displacement or its square or cubic value. An approximate solution is derived up to the second order approximation for the system with absorber. The stability of the system is investigated applying both frequency response equations and phase plane methods. The effects of the absorber on system behavior are studied numerically. Optimum working conditions of the system are extracted when applying both passive and active control methods.

A Comparison between Active and Passive Vibration Control of Non-Linear Simple Pendulum. Part I: Transversally Tuned Absorber and Negative Gφn Feedback

Vibrations and dynamic chaos should be controlled in structures and machines. The most famous method of vibration control is using absorbers or tuned dampers or neutralizers. The main aim is to improve the system behavior at different resonance conditions. Two main strategies are used. They are passive and active control methods. In this paper, a tuned absorber, in the transversally direction, is added to an externally excited pendulum, which is simulated by a second order non-linear differential equation having both quadratic and cubic non-linearties, subjected to harmonic excitation. The absorber is usually designed to control one frequency at primary resonance where system damage is probable. The quenching efficiency of the system is studied. Active control is applied to the system via negative velocity feedback or its square or cubic value. The multiple time scale perturbation technique is applied throughout. An approximate solution is derived up to the second order approximation. The stability of the system is investigated applying both frequency response equations and phase plane methods. The effects of the absorber on system behavior are studied numerically. Optimum working conditions of the system are extracted when applying both passive and active control methods.

Vibration reduction of a three DOF non-linear spring pendulum

The dynamic response of mechanical and civil structures subject to high-amplitude vibration is often dangerous and undesirable. Vibrations and dynamic chaos should be controlled or eliminated in both structures and machines. This can be employed via passive and active control methods. In this paper, a tuned absorber, in the transversally direction, is connected to an externally excited spring–pendulum system (three degree of freedom), subjected to harmonic excitation. The tuned absorber is usually designed to control one frequency at primary resonance where system damage is probable. Active control is also applied to the considered system via negative displacement feedback to change the linear frequency of the system and to shift it away from the resonating one. Also active control is applied to improve the behavior of the spring–pendulum at the primary resonance via negative velocity feedback or its square or cubic value. The multiple time scale perturbation technique is applied throughout. The stability of the system is investigated applying both frequency response function and phase-plane method. The effects of the absorber and different parameters on system behavior are studied numerically. Optimum working conditions of the system are extracted applying both passive and active control methods, to be used in the design of such systems.

Response of Parametrically Excited One Degree of Freedom System With Non-linear Damping and Stiffness

The behavior of one-degree-of-freedom system with different quadratic damping and cubic stiffness non-linearities simulating the axial vibration of a cantilever beam under multi-parametric excitation forces is studied. The approximate solution of the given system up to and including the third order approximation is determined applying the multiple time scale perturbation technique. The steady state solution and its stability are investigated. The theoretical resonance conditions are confirmed applying numerical techniques. The stability of the numerical solution is studied using the phase-plane method. Effects of the different parameters on the system behavior and jump phenomenon on the frequency response curves are also investigated. A comparison with other similar published works is reported.

On the Solution of One-Degree-of-Freedom Nonlinear System to Modulated High-Frequency Input

In this paper a single-degree of freedom system with quadratic and cubic nonlinearities to an amplitude-modulated excitation whose carrier frequency is much higher than the natural frequency of the system is investigated by the multiple time scales perturbation technique [MTSPT]. The perturbation analysis is used to obtain a second order nonlinear equation governing the low-frequency response of the system. The response of the system to (constant) unmodulated and (harmonically) modulated excitations is investigated. Numerical investigations are given to study the effects of various parameters on the stability of the system response.

An MSPT of a parametrically excited cantilever beam

A system of nonlinear, coupled, second-order differential equations describing the oscillations of a cantilever beam has been considered and solved. The multiple-time scale perturbation technique (MSPT) has been applied to solve those differential equations up to the fourth approximation. The obtained solutions are presented as the response of the system's different modes. The theoretical resonance conditions of the system (three different modes) have been obtained from the third and fourth approximations. Some of these resonance conditions have been confirmed applying well-known numerical techniques.