Lindstedt-Poincare Perturbation Method Research Articles

Analytical Approximate Solution of a Coupled Two Frequency Hill\'s Equation

Abstract: A coupled two frequency Hill's equation is solved. Analytically approximate solution correct up-to first order is derived using modified Lindstedt-Poincare perturbation method. For a wide range of controlling parameters, we compare the numerical and analytical solutions. The solution is the first step towards developing a comprehensive understanding of the electrodynamics of charged particles in a combinational ion trap utilizing both electrostatic DC and RF fields along with a constant static magnetic field with prospects of confining antimatter such as anti-hydrogen for a reasonably long durations of time.

Review of Wind-Induced Effects Estimation through Nonlinear Analysis of Tall Buildings, High-Rise Structures, Flexible Bridges and Transmission Lines

The nonlinear effects exhibited by structures under the action of wind loads have gradually stepped into the vision of wind-resistant researchers. By summarizing the prominent wind-induced nonlinear problems of four types of wind-sensitive structures, namely tall buildings, high-rise structures, flexible bridges, and transmission lines, the occurrence mechanism of their nonlinear effects is revealed, providing cutting-edge research progress in theoretical studies, experimental methods and vibration control. Aerodynamic admittance provides insights into the aerodynamic nonlinearity (AN) between the wind pressure spectrum and wind speed spectrum of tall building surfaces. The equivalent nonlinear equation method is used to solve nonlinear vibration equations with generalized van-der-Pol-type aerodynamic damping terms. The elastic–plastic finite element method and multiscale modeling method are widely employed to analyze the effects of geometric nonlinearity (GN) and material nonlinearity (MN) at local nodes on the wind-induced response of latticed tall structures. The AN in blunt sections of bridges arises from the amplitude dependence of the aerodynamic derivative and the higher-order term of the self-excited force. Volterra series aerodynamic models are more suitable for the nonlinear aerodynamic modeling of bridges than the polynomial models studied more in the past. The improved Lindstedt–Poincare perturbation method, which considers the strong GN in the response of ice-covered transmission lines, offers high accuracy. The complex numerical calculations and nonlinear analyses involved in wind-induced nonlinear effects continue to consume significant computational resources and time, especially for complex wind field conditions and flexible and variable structural forms. It is necessary to further develop analytical, modeling and identification tools to facilitate the modeling of nonlinear features in the future.

Active control of wave propagation in nonlinear planar networks using piezoelectric actuation

Due to the fascinating effects of nonlinearity on the wave propagation and dynamic properties of periodic structures, nonlinear phononic crystals have been an important research topic in recent years. However, while a lot of efforts were devoted to the wave propagation analysis of nonlinear chains, 2D planar lattices with weak nonlinearities have been left relatively unexplored. Hence, this work presents an active control technique of wave propagation in nonlinear planar lattices using piezoelectric actuation. To formulate the problem, the governing equations are derived for the transverse motion of weakly nonlinear lattices with square topology considering the effects of auxiliary piezoelectric springs. The Lindstedt–Poincare perturbation method is applied to obtain semi-analytical solutions of the governing equations. Furthermore, the analytical nonlinear dispersion relations are derived for the discrete nonlinear phononic lattices with triangular topology, for the first time. The results reveal that in addition to the effect of piezoelectric springs with both positive and negative gains on the dispersion curves, the wave directionality of nonlinear planar lattices is also affected by such modifications. Hence, a tuning approach is proposed to control the directionality and isotropy of wave propagation in nonlinear planar lattices, as well as their wave attenuation performance. According to the results, piezoelectric springs with negative control gain can open new tunable ultra-low frequency stop-bands in the dispersion curves of planar nonlinear lattices.

Bifurcation analysis and approximate analytical periodic solution of ER3BP with radiation and albedo effects

The nonlinear dynamics of an elliptic restricted three-body system with perturbations such as radiation and albedo are studied in this paper. The effects of different perturbation parameters on the nonlinear dynamic behavior of the third-body are analyzed through the system’s bifurcation diagram. The second- and third-order approximate periodic solutions near the collinear libration points in the plane are obtained using the Lindstedt-Poincare perturbation method.

Comparing a Three-Term Perturbation Solution of the Nonlinear ODE of the Jacobi Elliptic SN Function to Its Approximation into Circular Functions

In this paper, the nonlinear differential equation of the elliptic sn function is solved analytically using the Lindstedt-Poincare perturbation method. This differential equation has a cubic nonlinearity and a con- stant known as the modulus of elliptic integral. This constant takes any value from zero to one and the square of its value is used as a small parameter to expand the dependent variable in series and start analyti- cal iterations. Fortunately, there is an exact solution to this differential equation known as the Jacobi sn elliptic function. When the modulus approaches zero the elliptic differential equation becomes linear with the circular sine function as exact solution. Thus, the sine function is con- sidered as the unperturbed solution and is used as the basis to add more correction terms through analytical iterations. The Lindstedt-Poincare technique is used to render the perturbation solution uniformly valid at larger values of the independent variable. A three-term perturbation so- lution is obtained and shows good convergence and boundedness. This solution is compared with the exact, numerically calculated, sn elliptic function. It is also compared analytically with the approximate expan- sion of the elliptic function into circular functions in case of a small modulus. The relative percentage error between the perturbation solution and the exact one is calculated at certain values of the modulus and for all values of the independent variable. The relative error is reasonably small but increases at larger values of the modulus. In addition, the ap- proximation of the exact solution gives smaller relative error than that of the perturbation solution including the same order of the modulus.

Lagrangian motion of fluid particles in gravity–capillary standing waves

A third-order analytical solution for the gravity–capillary standing wave is derived in Lagrangian coordinates through the Lindstedt–Poincare perturbation method. By numerical computation, the dynamical properties of nonlinear standing waves with surface tension in finite water depth, including particle trajectory and surface profile are investigated. We find that the presence of surface tension leads to a change of the crest form. Moreover, we also find that the particle trajectories near the surface oscillate back and forth along the arcs which will change from concave to convex as the inverse Bond number increases. There is no mass transport of the particles in a wave period.

A Generalization of Lindstedt–Poincaré Perturbation Method to Strongly Mixed-Parity Nonlinear Oscillators

In this paper, generalization for the Lindstedt–Poincare perturbation method for nonlinear oscillators to a class of strongly mixed-parity oscillating system is established. In this extended and enhanced approach, two new odd nonlinear oscillators are introduced in terms of the mixed-parity oscillator. By combining the analytical approximate solutions corresponding to the two new systems, the accurate approximate solutions of the original mixed-parity oscillator are obtained. Comparing with the existing methods such as the perturbation method, the new solution methodology for the singular nonlinear system introduced is not only simple, but the combinatory solution is straight forward and it yields very accurate and physically insightful solutions. Using two typical examples, we demonstrated that this new proposed approach is capable of establishing highly accurate approximate analytical frequency and periodic solutions for small as well as large amplitude of oscillation. The new analytical methodology established will potentially shed new insights to the physical interpretations of strongly nonlinear oscillators including optoelectronic oscillators, pendulums and spring-masses.

Nonlinear vibrations of elastic beam with piezoelectric actuators

This paper presents theoretical and experimental investigations into nonlinear flexural vibrations of a structure composed of a host beam with piezoelectric ceramic actuators symmetrically bonded to its top and bottom surfaces. The composite beam is supported at its ends to completely restrain axial displacements or to impose the displacement of one or both ends. Applying voltage to piezoelectric actuators one creates prestress which can stabilize the structure when the external compressive force appears. The analytical model for describing flexural vibration of a beam under both the external load and piezoelectric actuation is based on the Euler–Bernoulli beam. The piezo material exhibits linear piezoelectricity with constitutive equations including electromechanical coupling. Due to geometrical nonlinearity, the solution to the problem has been obtained by using the Lindstedt-Poincare perturbation method. The main results concern the effect of the residual piezo force on the non-linear vibration frequency of the structure. In the experimental part of the study two laboratory stands have been designed and built for three and five segmented beams to find out and prove the effect of the electric field on the residual force and the natural frequency of both systems. Very good agreement between theoretical and experimental results has been observed.

Analytical and numerical studies on the nonlinear dynamic response of orthotropic membranes under impact load

Orthotropic membrane components and structures are widely used in building structures, instruments and meters, electronic engineering, space and aeronautics, etc., because of their light weights. However, the same lightweight combined with low stiffness make membranes prone to vibration under dynamic loads, and in some cases the vibration may lead to structural failure. Herein, the undamped nonlinear vibration response of pretension rectangular orthotropic membrane structures subjected to impact loading is studied by analytical and numerical methods. The analytical solution is obtained by solving the governing equations by the Bubnov-Galerkin method and the Lindstedt-Poincare perturbation method. Numerical analysis has also been carried out based on the same theoretical model. The analytical and numerical results have been compared and analyzed, and the influence of various model parameters on membrane vibration discussed. The results obtained herein provide some theoretical basis for the vibration control and dynamic design of orthotropic membrane components and structures.

Solution of natural characteristics of a hard-coating plate based on Lindstedt–Poincaré perturbation method and its valedictions by FEM and measurement

The solution of natural characteristics of a hard-coating cantilever plate, which is made of a layer of anisotropic hard-coating material and the substrate of isotropic metal, is investigated based on the theory of nonlinear elasticity of plate. The hard-coating plate is regarded as a nonlinear elastic plate, which is made of equivalent nonlinear material. Nonlinear governing differential equations of the hard-coating plate are built based on the theory of nonlinear elasticity of plate and transformed from nonlinear partial differential equations into nonlinear ordinary differential ones by Galerkin method. These equations are solved by Lindstedt–Poincare perturbation method (small parameter method), and the approximate analytic solutions of natural characteristics are obtained. To test the validity and computational accuracy of approximate analytic method, comparison and validation of these solutions are proceeded by FE method and experimental test. The results indicate that, the natural frequencies obtained by approximate analytical method of hard-coating plate natural characteristics, which is based on Lindstedt–Poincare perturbation method, and by FE method are extremely close to the experimental test results, the relative errors are less than 6 %. However, the calculated natural frequencies obtained by linear analytical method have a comparatively big difference after the third order.

Linear and nonlinear free vibration of a multilayered magneto-electro-elastic doubly-curved shell on elastic foundation

Nonlinear and linear free vibration of symmetrically laminated magneto-electro-elastic doubly-curved thin shell resting on an elastic foundation is studied analytically. The shell is considered to be simply-supported on all edges and the magneto-electro-elastic body is poled along the z direction and subjected to electric and magnetic potentials between the upper and lower surfaces. To obtain the equations of motion, the Donnell shell theory in the presence of rotary inertia effect is used. Moreover, Gauss' laws for electrostatics and magnetostatics are used to model the electric and magnetic behavior. The nonlinear partial differential equations of motion are reduced to a single nonlinear ordinary differential equation by introducing a force function and using the single-term Galerkin method. The resulting equation is solved analytically by Lindstedt-Poincare perturbation method. After validation of the present study, several numerical studies are done to investigate the effects of foundation parameters, geometrical properties of the shell, and electric and magnetic potentials on the linear and nonlinear behavior of these smart shells.

Sitnikov restricted four-body problem with radiation pressure

An analytical study of the elliptic Sitnikov restricted four-body problem when all the primaries as source of same radiation pressure is presented. We find a solution, which is valid for small bounded oscillations in case of moderate eccentricity of the primary. We have linearized the equation of motion to obtain the Hill’s type equation. Using the Courant and Snyder transformation, Hill’s equation transformed into harmonic oscillator type equation. We have used the Lindstedt-Poincare perturbation method and again we have applied the Courant and Snyder transformation to obtain the final result.

Description of the suppression of the soliton self-frequency shift by bandwidth-limited amplification

A perturbation study of the suppression of the soliton self-frequency shift by the bandwidth-limited optical amplification is proposed. The stability of the equilibrium point for the soliton amplitude and velocity identified by the adiabatic approximation of the soliton perturbation theory (SPT) is analyzed by a numerical solution of a linearized system in the neighborhood of the equilibrium point. The obtained analytical expressions for the eigenvalues of the linearized system allow the determination of the values of pulse and material parameters for which the equilibrium point is stable. A perturbation approach that leads to the research of the equation of strongly nonlinear Duffing-Van der Pol oscillator is suggested. The last equation is explored by two different methods. First, the recently obtained results for this equation by the hyperbolic perturbation method are used. Next, the hyperbolic Lindstedt-Poincare perturbation method is applied to the exploration of this equation. The equilibrium velocity of the perturbed stationary solution was calculated as a critical value of the control parameter in both methods. It turned out that the coupling of the equilibrium velocity and the amplitude of the perturbed stationary solution in both methods is similar to the relation between the soliton amplitude and velocity derived by the adiabatic approximation of SPT. The change in the form of the perturbed stationary solution has also been identified by means of the hyperbolic Lindstedt-Poincare perturbation method.

Solution of the Sitnikov Problem

A new analytic expression for the position of the infinitesimal body in the elliptic Sitnikov problem is presented. This solution is valid for small bounded oscillations in cases of moderate primary eccentricities. We first linearize the problem and obtain solution to this Hill's type equation. After that the lowest order nonlinear force is added to the problem. The final solution to the equation with nonlinear force included is obtained through first the use of a Courant and Snyder transformation followed by the Lindstedt–Poincare perturbation method and again an application of Courant and Snyder transformation. The solution thus obtained is compared with existing solutions, and satisfactory agreement is found.

Application of vector-valued rational approximations to a class of non-linear oscillations

By combining a perturbation technique with a rational approximation of vector-valued function, we propose a new approach to non-linear oscillations of conservative single-degree-of-freedom systems with odd non-linearity. The equation of motion does not require to contain a small parameter. First, the Lindstedt–Poincare perturbation method is used to obtain an asymptotic analytical solution. Then the range of validity of the analytical representation is extended by using the vector-valued rational approximation of functions. For constructing the rational approximations, all that is needed is the coefficients of the perturbation expansion being considered. General approximate formulas for period and the corresponding periodic solution of a non-linear system are established. Two examples are used to illustrate the effectiveness of the proposed method.

Nonlinear free vibration of orthotropic shallow shells of revolution under the static loads

In this paper, the axisymmetric nonlinear free vibration problems of cylindrically orthotropic shallow thin spherical and conical shells under uniformly distributed static loads are studied by using MWR and Lindstedt-Poincare perturbation method, from which, the characteristic relation between frequency ratio and amplitude is obtained. The effects of static loads, geometric and material parameters on vibrational behavior of shells are also discussed.

Transient Dynamic Large-Deflection Analysis of Panel Structure under Blast Loading

The transient dynamic response of a panel structure subjected to blast loading was investigated, as simulated by the step function and the exponentially decaying function. Based on the Bauer methodology, and applying Galerkin's method, the von Karman nonlinear equations of plates could be transformed into Duffing-type nonlinear ordinary differential equations and then solved by adopting the Lindstedt-Poincare perturbation method. The complete analysis of square plates was carried out for all-clamped and all-simply supported edge conditions. The results of deflection history and stress resultants using one perturbation term were sufficiently correlated with results both from related literature and obtained by the finite-element method.

THE NONLINEAR MATHIEU EQUATION

The purpose of this paper is to classify the different sequences of bifurcation that can occur for small amplitude solutions to the nonlinear Mathieu equation near to the Mathieu regions of instability. We do this by using the Lindstedt-Poincare perturbation method to construct a vector field which interpolates the successive iterations of the Poincare map. These vector fields are then analysed to determine the sequence of bifurcations.

Vibration problems of flexible circular plates with initial deflection

In this paper, the differential equations of flexible circular plates with initial deflection are derived. The stability of motion is investigated in phase plane. The periodical solutions of nonlinear vibration for circular plates with initial deflection are obtained by use of Galerkin method and Lindstedt-Poincare perturbation method. The effect of initial deflection on the dynamic behavior of the flexible plates are also discussed.