Modified Lindstedt Poincare Method Research Articles

A new modified Lindstedt–Poincare method for nonlinear damped forced oscillations

Nonlinear damped forced oscillators are very importance in the fields of mechanical, electrical and other dynamical systems. Small oscillations are being investigated by classical perturbations. Earlier Cheung et al. presented a modified Lindstedt–Poincare method for large amplitude of oscillation; but its mathematical manipulation is tremendously difficult. Moreover, it fails to determine the solutions when linear restoring force vanishes. In this article, a modification of the same method is made which greatly reduces mathematical calculations. On the other hand, it provides more accurate results and is used whether linear restoring force is present or not.

Combination of modified Lindstedt-Poincare and homotopy perturbation methods

Some strongly nonlinear oscillators have been investigated with the help of combined modified Lindstedt–Poincare and homotopy perturbation methods. The solutions are more accurate than those obtained by the modified Lindstedt–Poincare method. Moreover, the determination of solutions is simpler than usual homotopy perturbation methods. Some similarities and dissimilarities between the present method and other existing methods have also been discussed.

The modified Lindstedt–Poincare method for solving quadratic nonlinear oscillators

Recently, an analytical solution of a quadratic nonlinear oscillator has been presented based on the harmonic balance method. By introducing a small parameter, a set of nonlinear algebraic equations have been solved which usually appear among unknown coefficients of several harmonic terms. But the method is not suitable for all quadratic oscillators. Earlier, introducing a small parameter to the frequency series, Cheung et al. modified the Lindstedt–Poincare method and used it to solve strong nonlinear oscillators including a quadratic oscillator. But due to some limitations of both parameters, a changed form of frequency-related parameter (introduced by Cheung et al.) has been presented for solving various quadratic oscillators.

On the length scale dependence of DNA conformational change under local perturbation

Conformational change of a DNA molecule is frequently observed in multiple biological processes and has been modeled using a chain of strongly coupled oscillators with a nonlinear bistable potential. While the mechanism and properties of conformational change in the model have been investigated and several reduced order models developed, the conformational dynamics as a function of the length of the oscillator chain is relatively less clear. To address this, we use a modified Lindstedt–Poincare method and numerical computations. We calculate a perturbation expansion of the frequency of the model’s nonzero modes, finding that approximating these modes with their unperturbed dynamics, as in a previous reduced order model, may not hold when the length of the DNA model increases. We investigate the conformational change to the local perturbation in models of varying lengths, finding that for the chosen input and parameters, there are two regions of DNA length in the model — first, where the minimum energy required to undergo the conformational change increases with the DNA length; and second, where it is almost independent of the length of the DNA model. We analyze the conformational change in these models by adding randomness to the local perturbation, finding that the tendency of the system to remain in a stable conformation against random perturbation decreases with increase in DNA length. These results should help to understand the role of the length of a DNA molecule in influencing its conformational dynamics.

An Analytical Solution for Nonlinear Vibrations Analysis of Functionally Graded Plate Using Modified Lindstedt–Poincare Method

In this research, the nonlinear free vibrations analysis of functionally graded (FG) rectangular plate which simply supported all edges are investigated analytically using modified Lindstedt–Poincare (MLP) method for the first time. For this purpose, with the aid of von Karman nonlinearity strain-displacement relations, the partial differential equations of motion are developed based on first-order shear deformation theory (FSDT). Afterward, by applying Galerkin method, the nonlinear partial differential equations are transformed into the time-dependent nonlinear ordinary differential equations. The nonlinear equation of motion is then solved analytically by MLP method to determine the nonlinear frequencies of the FG rectangular plate. The material properties are assumed to be graded through the direction of plate thickness according to power law distribution. The effects of some system parameters such as vibration amplitude, volume fraction index and aspect ratio on the nonlinear to linear frequency ratio are discussed in detail. To validate the analysis, the results of this paper are compared with both the published data and numerical method, and good agreements are found.

Nonlinear vibration analysis of a beam subjected to a random axial force

In this paper, nonlinear vibration of an Euler–Bernoulli beam excited by a harmonic random axial force is studied. The equation of motion contains a term with time-varying coefficient which is solved by modified Lindstedt–Poincare method. Then the effect of the random axial force on the response is investigated using the obtained approximate solution. Two cases are considered in random analysis, namely random amplitude and random phase of the axial force. For both cases, the ensemble average, mean square value and the autocorrelation function are obtained. The results have indicated that for each case the mean and the mean square value are a function of time which means that the lateral displacement of the beam is a nonstationary process. A numerical study is also conducted based on the iteration of numerical solution in order to verify the derived analytical formulae. It is shown that a good agreement is seen between the analytical and numerical solution statistical properties.

Analysis of vibration characteristics and adaptive continuous perturbation control of some torsional vibration system with backlash

The dynamic equation of some nonlinear torsional vibration system with two masses is established, which contains backlash. In the case of primary resonance, the frequency response equation of the system is deduced with the modified Lindstedt–Poincare method combined with the multiple scales method. The influence of the backlash change on the amplitude of the torsional vibration system is analyzed by the amplitude frequency response map, and by using the method of numerical analysis, the influence of the backlash change on the system entering the chaotic motion is analyzed by bifurcation diagram, the maximum Lyapunov exponent map, phase diagram and Poincare map. After adding the method of the adaptive continuous perturbation control, the amplitude of the system decreases, and there is a transformation from chaotic motion to periodic motion.

Closed-form criterion for convergence and stability of pseudo-force method for nonlinear dynamic analysis

The pseudo-force (PF) method has been successfully applied to numerous problems with positive secant stiffness or positive tangential stiffness; however, it is widely believed to diverge for problems with the ratio of actual stiffness to pseudo-tangential stiffness being negative or larger than two. In the present paper, this point of view is investigated by applying PF method to single- and multi-degree of freedom systems with elasto-plastic nonlinearities which involve repeated cycles of loading and unloading, where the elastic behavior is either very high positive tangential stiffness or very high negative tangential stiffness (e.g., in Coulomb friction or velocity-dependent friction) or geometrically nonlinear problems. The objective of this study is to in essence seek the reasons for convergence of PF method in the presence of stiff nonlinearities. Closed-form expressions for convergence and stability criteria are derived. Approximate solution by modified Lindstedt–Poincare method (L–P method) is formulated. The PF method is also applied to a base-isolated structure with nonlinear sliding or velocity-dependent friction bearings. The results of the study indicate the wide applicability of PF method for highly nonlinear problems, including those with high positive or negative tangential stiffness.

Modified Lindstedt-Poincaré method for obtaining resonance periodic solutions of nonlinear non-autonomous oscillators

A modified Lindstedt-Poincare (LP) method for obtaining the resonance periodic solutions of nonlinear non-autonomous vibration systems is proposed in this paper. In the modified method, nonlinear non-autonomous equations are converted into a group of linear ordinary differential equations by introducing a set of simple transformations. An approximate resonance solution for the original equation can then be obtained. The periodic solutions of primary, super-harmonic, sub-harmonic, zero-frequency and combination resonances can be solved effectively using the modified method. Some examples, such as damped cubic nonlinear systems under single and double frequency excitation, and damped quadratic nonlinear systems under double frequency excitation, are given to illustrate its convenience and effectiveness. Using the modified LP method, the first-order approximate solutions for each equation are obtained. By comparison, the modified method proposed in this paper produces the same results as the method of multiple scales.

Calculation of secular axial frequencies in a nonlinear ion trap with hexapole, octopole and decapole superpositions by a modified Lindstedt–Poincare method

In this paper the modified Lindstedt–Poincare method is used for calculation of axial secular frequencies of a nonlinear ion trap with hexapole, octopole and decapole superpositions. The motion of the ion in a rapidly oscillating field is transformed to the motion in an effective potential. The equations of ion motion in the effective potential are in the form of a Duffing-like equation. With only octopole superposition the resulted nonlinear equations are symmetric, however, in the presence of hexapole and decapole superpositions, they are asymmetric. For asymmetric oscillators, it has been pointed out that the angular frequency for positive amplitudes is different from the angular frequency for negative amplitudes. Considering this problem, the modified Lindstedt–Poincare method is used for solving the resulted nonlinear equations. As a result, the ion secular frequencies as a function of nonlinear field parameters are obtained. The calculated secular frequencies are compared with the results of some other methods and the exact results. There is an excellent agreement between the results of this paper and the exact results.

A new perturbation procedure for limit cycle analysis in three-dimensional nonlinear autonomous dynamical systems

By introducing a new parametric transformation and a suitable nonlinear frequency expansion, the modified Lindstedt–Poincare method is extended to derive analytical approximations for limit cycles in three-dimensional nonlinear autonomous dynamical systems. By considering two typical examples, it can be seen that the results of the present method are in good agreement with those obtained numerically even if the control parameter is moderately large. Moreover, the present prediction is considerably more accurate than some published results obtained by the multiple time scales method and the normal form method.

Approximate period calculation for some strongly nonlinear oscillation by He’s parameter-expanding methods

In this paper, He’s modified Lindstedt–Poincare method and bookkeeping parameter method, also known as He’s parameter-expanding method, is applied to various kinds of strongly nonlinear oscillators. We obtained sufficiently accurate solutions with one iteration which is valid for whole domain to the contrary of classical perturbation techniques.

Reliable analysis for obtaining exact soliton solutions of nonlinear Schrödinger (NLS) equation

In this paper, we predict a new type of dispersion managed exact soliton solutions which can be expressed in trigonometric and hyperbolic functions of nonlinear Schrödinger equation by applying He’s modified Lindstedt–Poincare method. Particular cases are discussed.

He's parameter-expanding methods for strongly nonlinear oscillators

Ji-Huan He's work on asymptotic techniques is briefly reviewed, and his parameter-expanding methods including modified Lindstedt–Poincare method and bookkeeping parameter method are discussed in detail. Some remarkable virtues of the methods are exploited, and their applications are illustrated.

Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt–Poincare method

Abstract In the paper, J.H. He's modified Lindstedt–Poincare method is applied to the nonlinear oscillators with discontinuities. Only one iteration leads to high accuracy of the solution.

On a problem of modified Lindstedt–Poincare for certain strongly non-linear oscillators

In the present paper concerns the modified Lindstedt–Poincare method is presented for extending the rang of the validity of perturbation expansion to strongly non-linear oscillation. It is an effective method for dealing with strongly non-linear vibration of single degree-of-freedom system which cannot be treated by the standard L–P method. Therefore a strongly non-linear system with large parameter μ is transformed into a small parameter system with respect to α. This method is suitable for the system with even non-linearities as well as with odd non-linearities. All numerical results computed are in pretty good and displayed graphysically. This method is suitable for the system with even non-linearities as well as with odd non-linearities. All numerical results computed are in pretty good and displayed graphysically.

Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant

In this paper, a modified Lindstedt–Poincare method is proposed. In this technique, a constant, rather than the non-linear frequency, is expanded in powers of the expanding parameter to avoid the occurrence of secular terms in the perturbation series solution. Some examples are given here to illustrate its effectiveness and convenience. The results show that the obtained approximate solutions are uniformly valid on the whole solution domain, and they are suitable not only for weakly non-linear systems, but also for strongly non-linear systems.

Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part II: a new transformation

In this paper, a modified Lindstedt–Poincare method is proposed. In this technique, we introduce a new transformation of the independent variable. This transformation will also allow us to avoid the occurrence of secular terms in the perturbation series solution. Some examples are given here to illustrate its effectiveness and convenience. The results show that the obtained approximate solutions are uniformly valid on the whole solution domain, and they are suitable not only for weakly non-linear systems, but also for strongly non-linear systems.