Poincare Method Research Articles

Investigation for Synchronization of a Rotor-Pendulum System considering the Multi-DOF Vibration

This work is a continuation for our published literature for vibration synchronization. A new mechanism, two rotors coupled with a pendulum rod in a multi-DOF vibration system, is proposed to implement coupling synchronization, and the dynamics equation of mechanism is derived by Lagrange equation. In addition, the coupling relationship between the vibrobody and the pendulum rod is ascertained with the Laplace transformation method, based on the dimensionless equation of the dynamics system. The Poincare method is employed to study the synchronization state between the two unbalanced rotors, which is converted into that of existence and the stability of solutions for synchronization-balance equations. The obtained results are supported by computer simulations. It is demonstrated that the values of the spring stiffness coefficient, the length of the pendulum, and the angular installation of the pendulum are important parameters with respect to the synchronous behavior in the rotor-pendulum system.

Experimental Study of Thin Wall Milling Chatter Stability Nonlinear Criterion

The nonlinear dynamic behavior of milling process has been accompanied by the entire cutting process. In order to accurately determine and predict chatter stability of machining process, this article studied at both ends of the fixed thin part nonlinear criterion of milling chatter stability with experimental method. The experiment takes the vibration signal of thin part as the study object. And it analyses the vibration signal of different processing parameters based on the phase plane method, Poincare method and spectral analysis. Then, the relationship between the maximum Lyapunov exponent and the spindle speed and milling depth changes is discussed. Finally, taking the largest Lyapunov exponent as the criterion, the study determines the chatter stability domain of milling by using contour method. The comparative analysis is based on the milling chatter stability domain which obtained from the full discrete method. The experiments obtained the nonlinear stability criterion of aviation aluminium alloy 7075-T6 thin part.

Synchronization and Stability of Elasticity Coupling Two Homodromy Rotors in a Vibration System

The mechanical model of an elasticity coupling 1-DOF system is proposed to implement synchronization; the simplified model is composed of a rigid body, two induction motors, and a connecting spring. Based on the Lagrange equations, the dynamic equation of the system is established. Moreover, a typical analysis method, the Poincare method, is applied to study the synchronization characteristics, and the balanced equations and stability criterion of the system are obtained. Obviously, it can be seen that many parameters affect the synchronous state of the system, especially the stiffness of the support spring, the stiffness of the connecting spring, and the installation location of the motors. Meanwhile, choose a suitable stiffness of the connecting spring (k), which would play a significant role in engineering. Finally, computer simulations are used to verify the correctness of the theoretical analysis.

Single-Mode and Dual-Mode Nongomogeneous Dissipative Structures in the Nonlocal Model of Erosion

We consider a periodic boundary-value problem for a nonlinear equation with the deviating spatial argument in the case when the deviation is small. This equation is called a spatially nonlocal erosion equation. It describes the formation of undulating surface relief under the influence of ion bombardment and can be interpreted as a development of the well-known Bradley-Harper model. It is shown that the nonhomogeneous surface relief can occur when the stability of the homogeneous states of equilibrium changes. In this boundary value problem the loss of stability can occur at the higher modes and a number of such modes. The mode number depends on many factors. For example, it depends on the angle of incidence. It is also shown that the nonlinear boundary value problem can be included into the class of abstract parabolic equations. Solvability of this problem was studied in the works by P.E. Sobolevsky, and this method assumes to use the analytical theory of semigroups of bounded linear operators. In order to solve the occurring bifurcation problems there were used the investigation methods of dynamical systems with an infinite-dimensional phase space (a space of initial conditions) such as: the method of integral manifolds, the method of Poincare–Dulac normal forms and asymptotic methods of analysis. Both possible in the given situation problems were studied: in codimension one and in codimension two. In particular, asymptotic formulas were obtained for solutions which describe nonhomogeneous undulating surface relief. The question about the stability of these solutions was studied. And the analysis of normal form was given. Also the asymptotic formulas for the nonhomogeneous undulating solutions were obtained. In conclusion some possible interpretations of the obtained results are indicated.

Study of the stability of the perturbed solutions of the restricted three body problem

In this paper we have been examined the stability of the perturbed solutions of the restricted three body problem. We have been restricted ourselves only to the first order variational equations. Our variational equations depend on the periodic solutions. Here the applications of the method of Fuchs and Floquet Proves to be complicated and hence we have been preferred Poincare's Method of determination of the characteristic exponents. With the determination of the characteristic exponents we have been abled to conclude regarding the stability of the generating solution. We have obtained that the motions are unstable in all the cases. By Poincare's implicit function theorem we have concluded that the stability would remain the same for small value of the parameter m and in all types of motion of the restricted three-body problem.BIBECHANA 13 (2016) 18-22

A method to stochastic dynamical systems with strong nonlinearity and fractional damping

In this paper, a new technique is proposed to deal with strongly nonlinear stochastic systems with fractional derivative damping and random harmonic excitation. Combining the advantages of Linstedt–Poincare (L–P) method and multiple scales method, introducing a different frequency expansion form and a time transformation, a series of perturbation equations is obtained according to the powers of parameter. Then, we eliminate the secular producing terms to solve the perturbation equations to derive the second-order approximate solution. Furthermore, the steady-state frequency–amplitude function in deterministic case is analyzed, and the first-order and second-order steady-state moments of the amplitude are also discussed in the presence of random harmonic excitation. In order to explore the effectiveness of the proposed approximate method, two classical examples were proposed to verify the theoretical results through numerical simulations. Especially, the method can be used to investigate some types of extremely strong odd nonlinear terms via the discussions of each example.

Bifurcation analysis of high-speed railway wheel-set

In this paper, the lateral mathematical model of a railway wheel-set with two degrees of freedom has been built. To study the effect of yaw damper on the stability of high-speed wheel-set, the constrain between the wheel-set and the bogie is assumed as rigid. In this lateral model, only the nonlinear wheel/rail contact relationship has been taken into consideration in the lateral direction, due to both the nonlinear relationship between lateral displacement and contact angle of the wheel and the rail, and the nonlinear relationship between lateral displacement and the equivalent radius of the wheel at the contact point. And the nonlinear parameters are attained by using polynomial interpolation. By using Center Manifold Theorem, the method of Normal Form and Poincare method, the model is reduced to a planar dynamical system, and the symbolic expression of the first-order fine focus is given, which can be used to determine which kind of bifurcation will occur at the critical speed. The simulation is established by using the parameters of the wheel-set of Chinese high-speed railway vehicle CRH3, and the result is consistent with the determination of the first-order fine focus. At last, the influence of different parameters on the stability of CRH3 wheel-set is investigated.

Precession of a Planet with the Multiple Scales Lindstedt–Poincare Technique

Abstract The recently developed multiple scales Lindstedt–Poincare (MSLP) technique is successfully applied to a mathematical model of planet motion. The equation is originally developed to precisely understand the orbital motion of the planet Mercury around the Sun and the precession of the orbit due to the relativistic effects. The quadratic nonlinear equation is solved by the classical Lindstedt–Poincare method (LP) and then by the newly developed multiple scales Lindstedt–Poincare method (MSLP). Both approximate solutions are contrasted with the numerical simulations. When the relativistic effects are small, all three solutions coincide with each other. When the perturbation effects are increased, the MSLP solutions agree better with the numerical solutions than the LP solutions. The precession of the perihelion of the planet is calculated and compared for the approximate solutions.

Predicting Homoclinic and Heteroclinic Bifurcation of Generalized Duffing-Harmonic-van de Pol Oscillator

In this paper, a novel construction of solutions of nonlinear oscillators are proposed which can be called as the quadratic generalized harmonic function. Based on this novel solution, a modified generalized harmonic function Lindstedt–Poincare method is presented which call the quadratic generalized harmonic function perturbation method. Via this method, the homoclinic and heteroclinic bifurcations of Duffing-harmonic-van de Pol oscillator are investigated. The critical value of the homoclinic and heteroclinic bifurcation parameters are predicted. Meanwhile, the analytical solutions of homoclinic and heteroclinic orbits of this oscillator are also attained. To illustrate the accuracy of the present method, all the above-mentioned results are compared with those of Runge–Kutta method, which shows that the proposed method is effective and feasible. In addition, the present method can be utilized in study many other oscillators.

Study on Rub-Impact Dynamical Behavior of An Asymmetrical Rotor-Bearing with Variable Ratio of Stiffness

A rub-impact mathematic model of an Asymmetrical rotor-bearing with nonlinear oil film force is built in the paper. The bifurcation diagrams of the response were given following the changing of ratio of stiffness. We analyzed the bifurcation and the chaos character of an Asymmetrical rotor-bearing with operating rotor stiffness changing, then we get the dynamical character and the law when the rubbing happen. The nonlinear dynamic behaviors of the system were studied by using the numerical value integral and Poincare mapping methods. By analysis the process of rub-impact, it is so complex and many low frequencies with large amplitude are in the range of 0.3~0.6 X. The way of period-chaos-period and the phenomena of diverging backward whirl are discovered when the rubbing happen. These results provide important theoretical references for the safety operation of generating sets and the exact identification of the faults in rotating machinery.

Generalised perturbation techniques for strongly nonlinear oscillators with a positive, zero or negative linear stiffness term

This study is concerned with nonlinear oscillators whose equations of motion contain a linear stiffness term with a positive, zero or negative coefficient and a power-form nonlinear term whose coefficient is not necessarily small. The case of free conservative oscillations and forced non-conservative oscillations are treated separately. A new small parameter is introduced, enabling one to transform original oscillators with not necessarily small nonlinearity into weakly nonlinear systems. Two classical perturbation techniques (the Lindstedt–Poincare method and the method of multiple scales) are used subsequently to find an amplitude–frequency equation and approximations to motion.

Axisymmetric steady temperature field in FGM cylindrical shells with temperature-dependent heat conductivity and arbitrary linear boundary conditions

An approximate analytical solution to the axisymmetric heat conduction equation for a hollow cylinder made of functionally graded material with temperature-dependent heat conductivity is presented. General linear boundary conditions are considered. The Poincare method for regular perturbation problems is employed to obtain an analytical closed-form approximate solution for the temperature field. The hierarchical asymptotic problems are solved up to the second-order approximation. A numerical example is worked out, i.e., the one-dimensional heat conduction in the radial direction with prescribed temperatures at the boundaries. The approximate temperature profiles are compared with a numerical solution of the full nonlinear problem which provides the reference “exact” solution. A good agreement between the approximate and reference solutions is established. The convergence of the asymptotic series as well as the properties of the temperature field are studied.

Rotating waves in oscillators with Huygens’ coupling

This paper focuses on the onset of rotating waves in a network of pendulum-like oscillators that interact via Huygens’ coupling, i.e. in oscillators that are coupled by a flexible media. Conditions for the existence and stability of rotating wave solutions in the coupled system are derived by using the Poincare method of perturbation. Additionally, the amplitude and frequency of the rotating wave solution are also analytically determined. The theoretical results are further illustrated by means of computer simulations.

Perturbation Solutions of the Quintic Duffing Equation with Strong Nonlinearities

The quintic Duffing equation with strong nonlinearities is considered. Perturbation solutions are constructed using two different techniques: The classical multiple scales method (MS) and the newly developed multiple scales Lindstedt Poincare method (MSLP). The validity criteria for admissible solutions are derived. Both approximate solutions are contrasted with the numerical solutions. It is found that MSLP provides compatible solution with the numerical solution for strong nonlinearities whereas MS solution fail to produce physically acceptable solution for large perturbation parameters.

Edge-emitting semiconductor laser driven by a van der Pol oscillator: analytical and numerical analysis

A device constituted of an edge-emitting semiconductor laser driven by a van der Pol oscillator is studied analytically and numerically. For small value of the parameter controlling the dissipation in van der Pol oscillator, the Lindstedt–Poincare method is used to find analytically high accurate solutions of the van der Pol equation with a bias term. Then by using the harmonic balance method, we obtain the amplitude of the oscillatory states of the laser output. The analytical results are compared with those from numerical simulation and a good agreement is obtained. By varying the parameter controlling the dissipation in van der Pol oscillator and the normalized reverse bias saturation of the injection current in the laser, we numerically show that the laser output exhibits a period-doubling route to chaos. The coexistence between periodic and chaotic behaviors is found in edge-emitting semiconductor laser driven by a van der Pol oscillator for specific parameters values. It is demonstrated that the edge-emitting semiconductor laser driven by a van der Pol oscillator generates dynamical behaviors found in edge-emitting semiconductor laser subject to a sine modulated current demonstrating the advantage of using a simple and less bulky electronic device to modulate the semiconductor laser pump current.

Some remarks on Poincare's method of tangencial curves

Some remarks on Poincare's method of tangencial curves

Modeling and Simulation for the Feed System of CNC Machine

Study the linear and nonlinear stiffness dynamic characteristics of ball screw in feed system. According to the structure and the stiffness of ball screw, considering the influence of damping force, elastic force, friction force, driving force and load, establish the dynamic model of feed system. Use Linz Ted- Poincare (L-P) Method of singular perturbation to solve the model, obtain the quadratic approximate solution of the free vibration, analyze the multiple solution phenomenon of the model, and also conduct the numerical simulation analysis for the model.

Solutions of the Sitnikov's circular restricted three body problems when primaries are oblate spheroid

The solutions of sitnikov's circular restricted three body problem has been tried to obtain by using Lindstedt poincare method if the primaries are oblate spheroid. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9340 BIBECHANA 10 (2014) 92-99

A new perturbation technique in solution of nonlinear differential equations by using variable transformation

A perturbation algorithm using a new variable transformation is introduced. This transformation enables control of the independent variable of the problem. The problems are solved with new transformation: Classical Duffing equation with cubic nonlinear term and a singular perturbation problem. Results of multiple scales, Lindstedt Poincare method, new method and numerical solutions are contrasted.

High-order solutions of invariant manifolds associated with libration point orbits in the elliptic restricted three-body system

High-order analytical solutions of invariant manifolds, associated with Lissajous and halo orbits in the elliptic restricted three-body problem (ERTBP), are constructed in this paper. The equations of motion of ERTBP in the pulsating synodic coordinate system have five equilibrium points, and the three collinear libration points as well as the associated center manifolds are unstable. In our calculation, the general solutions of the invariant manifolds associated with Lissajous and halo orbits around collinear libration points are expressed as power series of five parameters: the orbital eccentricity, two amplitudes corresponding to the hyperbolic manifolds, and two amplitudes corresponding to the center manifolds. The analytical solutions up to arbitrary order are constructed by means of Lindstedt–Poincare method, and then the center and invariant manifolds, transit and non-transit trajectories in ERTBP are all parameterized. Since the circular restricted three-body problem (CRTBP) is a particular case of ERTBP when the eccentricity is zero, the general solutions constructed in this paper can be reduced to describe the dynamics around the collinear libration points in CRTBP naturally. In order to check the validity of the series expansions constructed, the practical convergence of the series expansions up to different orders is studied.