Determination of limit cycles for a modified van der Pol oscillator

The determination of amplitude and period of limit cycles is a crucial question in non-linear mechanics. In this paper, a van der Pol oscillator containing a periodic potential is considered. Amplitude and period of limit cycles are calculated by He’s variational method and Krylov–Bogoliubov–Mitropolsky (KBM) method.     

Stability of limit cycles in autonomous nonlinear systems

Periodical solutions or limit cycles (LC) comprise a significant family among the response types of nonlinear autonomous systems. Their identification and stability assessment is of a great importance during the analysis of an unknown system. A new analytical/iterative method of LC identification and portrait investigation was presented recently. The current study proposes a novel technique for their stability assessment. This strategy facilitates the distinction of stable and unstable LCs, thereby allowing the definition of attractive and repulsive response fields. A narrow toroidal domain is constructed around the LC, which is arithmetized by an orthogonal system that is positioned by tangential and normal vectors to the LC. The stability of the LC is investigated using the transformed differential system of the normal components of the response, which are functions of the coordinate along the LC trajectory. Exponential LC stability criteria are also proposed, which are based on the first degree of the perturbation procedure. Theoretical considerations are illustrated using single and two degree of freedom systems including demonstrations with specific systems. The strengths, future steps, and shortcomings of this method are evaluated.     

Continuation of limit cycles near saddle homoclinic points using splines in phase space

We present a new algorithm for continuation of limit cycles of autonomous systems as a system parameter is varied. The algorithm works in phase space with an ordered set of points on the limit cycle, along with spline interpolation. Currently popular algorithms in bifurcation analysis packages compute time-domain approximations of limit cycles using either shooting or collocation. The present approach seems useful for continuation near saddle homoclinic points, where it encounters a corner while time-domain methods essentially encounter a discontinuity (a relatively short period of rapid variation). Other phase space-based algorithms use rescaled arclength in place of time, but subsequently resemble the time-domain methods. Compared to these, we introduce additional freedom through a variable stretching of arclength based on local curvature, through the use of an auxiliary index-based variable. Several numerical examples are presented. Comparisons with results from the popular package, MATCONT, are favorable close to saddle homoclinic points.     

Switching-induced stable limit cycles

Physical limits place bounds on the divergent behaviour of dynamical systems. The paper explores this situation, providing an example where generator field-voltage limits capture behaviour, giving rise to a stable, though non-smooth, limit cycle. It is shown that shooting methods can be adapted to solve for such non-smooth switching-induced limit cycles. By continuing branches of switching-induced and smooth limit cycles, the paper established the co-existence of equilibria, smooth and non-smooth limit cycles. Furthermore, it is shown that when branches of switching-induced and smooth limit cycles merge, the limit cycles are annihilated at a grazing bifurcation.     

Identification of limit cycles for piecewise nonlinear aeroelastic systems

This paper describes two methods for the analysis of aeroelastic systems with complex piecewise nonlinear structural stiffness. These methods are tested and compared for low speed incompressible and transonic flows. The first technique employed in this paper uses a new application of the analytical solution of linear algebraic systems, the second technique utilises logarithmic and tanh functions to both represent discrete nonlinearities and to act as a switch between different nonlinear areas. The transonic aerodynamic models used are generated using an eigenvalue realisation algorithm (ERA) which produces reduced order models (ROMs) from the pulse responses of time linearised Euler simulations. It is shown that such aerodynamic models are well suited to use with continuation methods. Flutter boundaries and limit cycle oscillations can then be rapidly identified with good accuracy.     

Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method

The perturbation-incremental method is applied to determine the separatrices and limit cycles of strongly nonlinear oscillators. Conditions are derived under which a limit cycle is created or destroyed. The latter case may give rise to a homoclinic orbit or a pair of heteroclinic orbits. The limit cycles and the separatrices can be calculated to any desired degree of accuracy. Stability and bifurcations of limit cycles will also be discussed.     

Upper bound limit analysis of active earth pressure with different fracture surface and nonlinear yield criterion

Conventional calculations of static and seismic active earth pressures of soils on a retaining wall are formulated assuming the soils obeying a linear Mohr–Coulomb yield criterion. However, experimental evidences show that the strength envelopes of almost all geomaterials are nonlinear in nature over a wide range of normal stresses. In this paper, the strength envelope of the backfill behind a retaining wall is considered to follow a nonlinear yield criterion. A simple method is proposed for calculating the static and seismic active earth pressures acting against a retaining wall using a nonlinear yield criterion. This method is based on the upper bound theorem of limit analysis. Both translational and rotational fracture surfaces are employed in the formulation for calculating active earth pressures. Quasi-static representation of earthquake effects using a seismic coefficient concept is adopted for seismic active earth pressure calculations. Instead of using directly the actual nonlinear yield criterion, a linear Mohr–Coulomb yield criterion, which is tangential to the nonlinear yield criterion, is used to formulate the active earth pressure problem as a classical nonlinear programming problem. A nonlinear sequential quadratic programming algorithm is used to search for the maximum solution. In order to assess the validity of the proposed method, values of active earth pressures for different values of seismic coefficients and nonlinear parameters in the yield criterion are calculated and compared with solutions obtained using an extended Rankine’s active earth pressure theory. For the case of static active earth pressure, the upper bound solutions using the present method with a translational fracture surface are equal to the extended Rankine’s theoretical solutions and are slightly smaller than those obtained using the present method with a rotational fracture surface. For the case of seismic active earth pressure, numerical results obtained using the present method with a rotational fracture surface is very close to the extended Rankine’s theoretical solutions. A study is conducted to investigate the effects of the parameters in the nonlinear yield criterion on the active earth pressures.     

Output frequency properties of nonlinear systems

Nonlinear systems usually have complicated output frequencies. For the class of Volterra systems, some interesting properties of the output frequencies are studied in this paper. These properties show theoretically the periodicity of the output super-harmonic and inter-modulation frequencies and clearly demonstrate the mechanism of the interaction between different output harmonics incurred by different input nonlinearities in system output spectrum. These new results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities. Examples and discussions are given to illustrate these new results.     

Bifurcation of limit cycles in fluid film bearings

Rotors supported by journal bearings may become unstable due to self-excited vibrations when a critical rotor speed is exceeded. Linearised analysis is usually used to determine the stability boundaries. Non-linear bifurcation theory or numerical integration is required to predict stable or unstable periodic oscillations close to the critical speed. In this paper, a dynamic model of a short journal bearing is used to analyse the bifurcation of the steady state equilibrium point of the journal centre. Numerical continuation is applied to determine stable or unstable limit cycles bifurcating from the equilibrium point at the critical speed. Under certain working conditions, limit cycles themselves are shown to disappear beyond a certain rotor speed and to exhibit a fold bifurcation giving birth to unstable limit cycles surrounding the stable supercritical limit cycles. Numerical integration of the system of equations is used to support the results obtained by numerical continuation. Numerical simulation permitted a partial validation of the analytical investigation.     

Analysis of time-periodic nonlinear dynamical systems undergoing bifurcations

In this study a new procedure for analysis of nonlinear dynamical systems with periodically varying parameters under critical conditions is presented through an application of the Liapunov-Floquet (L-F) transformation. The L-F transformation is obtained by computing the state transition matrix associated with the linear part of the problem. The elements of the state transition matrix are expressed in terms of Chebyshev polynomials in timet which is suitable for algebraic manipulations. Application of Floquet theory and the eigen-analysis of the state transition matrix at the end of one principal period provides the L-F transformation matrix in terms of the Chebyshev polynomials. Since this is a periodic matrix, the L-F transformation matrix has a Fourier representation. It is well known that such a transformation converts a linear periodic system into a linear time-invariant one. When applied to quasi-linear equations with periodic coefficients, a dynamically similar system is obtained whose linear part is time-invariant and the nonlinear part consists of coefficients which are periodic. Due to this property of the L-F transformation, a periodic orbit in original coordinates will have a fixed point representation in the transformed coordinates. In this study, the bifurcation analysis of the transformed equations, obtained after the application of the L-F transformation, is conducted by employingtime-dependent center manifold reduction andtime-dependent normal form theory. The above procedures are analogous to existing methods that are employed in the study of bifurcations of autonomous systems. For the two physical examples considered, the three generic codimension one bifurcations namely, Hopf, flip and fold bifurcations are analyzed. In the first example, the primary bifurcations of a parametrically excited single degree of freedom pendulum is studied. As a second example, a double inverted pendulum subjected to a periodic loading which undergoes Hopf or flip bifurcation is analyzed. The methodology is semi-analytic in nature and provides quantitative measure of stability when compared to point mappings method. Furthermore, the technique is applicable also to those systems where the periodic term of the linear part does not contain a small parameter which is certainly not the case with perturbation or averaging methods. The conclusions of the study are substantiated by numerical simulations. It is believed that analysis of this nature has been reported for the first time for this class of systems.     

Interaction of free and forced nonlinear normal modes in two-DOF dissipative systems under resonance conditions

The resonance dynamics of a dissipative spring-mass and of a dissipative spring-pendulum system is studied. Internal resonance case is considered for the first system; both external resonances and simultaneous external and internal resonance are studied for the second one. Analysis of the systems resonance behavior is made on the base of the concept of nonlinear normal vibration modes (NNMs) by Kauderer and Rosenberg, which is generalized for dissipative systems. The multiple time scales method under resonance conditions is applied. The resulting equations are reduced to a system with respect to the system energy, arctangent of the amplitudes ratio and the difference of phases of required solution in the resonance vicinity. Equilibrium positions of the reduced system correspond to nonlinear normal modes; in energy dissipation case they are quasi-equilibriums. Analysis of the equilibrium states of the reduced system permits to investigate stability of nonlinear normal modes in the resonance vicinity and to describe transfer from unstable vibration mode to stable one. New vibration regimes, which are called transient nonlinear normal modes (TNNMs) are obtained. These regimes take place only for some particular levels of the system energy. In the vicinity of values of time, corresponding to these energy levels, the TTNM attract other system motions. Then, when the energy decreases, the transient modes vanish, and the system motions tend to another nonlinear normal mode, which is stable in the resonance vicinity. The reliability of the obtained analytical results is confirmed by numerical and numerical-analytical simulations.     

Residue harmonic balance approach to limit cycles of non-linear jerk equations

A residue harmonic balance is established for accurately determining limit cycles to parity- and time-reversal invariant general non-linear jerk equations with cubic non-linearities. The new technique incorporates the salient features of both methods of harmonic balance and parameter bookkeeping to minimize the total residue. The residue is separated into two parts in each step; one conforms to the present order of approximation and the remaining part for use in the next order. The corrections are governed by a set of linear ordinary differential equations that can be solved easily. Three specific cases of non-linear jerk equations are given to illustrate the validity and efficiency. The approximations to the angular frequency and the limit cycle are obtained and compared. The results show that the approximations obtained are in excellent agreement with the exact solutions for a wide range of initial velocities. The new technique is simple in principle and can be applied to other non-linear oscillating systems.     

An exact nonlinear hybrid-coordinate formulation for flexible multibody systems

The previous low-order approximate nonlinear formulations succeeded in capturing the stiffening terms, but failed in simulation of mechanical systems with large deformation due to the neglect of the high-order deformation terms. In this paper, a new hybrid-coordinate formulation is proposed, which is suitable for flexible multibody systems with large deformation. On the basis of exact strain–displacement relation, equations of motion for flexible multibody system are derived by using virtual work principle. A matrix separation method is put forward to improve the efficiency of the calculation. Agreement of the present results with those obtained by absolute nodal coordinate formulation (ANCF) verifies the correctness of the proposed formulation. Furthermore, the present results are compared with those obtained by use of the linear model and the low-order approximate nonlinear model to show the suitability of the proposed models. The project supported by the National Natural Science Foundation of China (10472066, 50475021).     

Existence, number, and stability of limit cycles in weakly dissipative, strongly nonlinear oscillators

Oscillators control many functions of electronic devices, but are subject to uncontrollable perturbations induced by the environment. As a consequence, the influence of perturbations on oscillators is a question of both theoretical and practical importance. In this paper, a method based on Abelian integrals is applied to determine the emergence of limit cycles from centers, in strongly nonlinear oscillators subject to weak dissipative perturbations. It is shown how Abelian integrals can be used to determine which terms of the perturbation are influent. An upper bound to the number of limit cycles is given as a function of the degree of a polynomial perturbation, and the stability of the emerging limit cycles is discussed. Formulas to determine numerically the exact number of limit cycles, their stability, shape and position are given.     

On the use of neural networks for identification of linear and nonlinear systems

A procedure based on neural networks for the classification of linear and nonlinear systems is presented, using excitation and response data under swept sine excitation. Special attention is paid to the classification and identification of linear and bilinear systems, the latter being considered since they exhibit typical characteristics of cracked systems. The computer simulations show that: (1) using the procedure presented in this paper the trained classification network can reliably classify a linear system and different nonlinear systems; (2) the output of the trained identification neural network for a linear system and a bilinear system can be used as a quantitative indicator of characteristics of bilinear systems having different stiffness ratios (k _(x>0)/k _(x<0)) with respect to the bilinear system used in the training stage; (3) for two-degree-of-freedom systems, the trained network can not only determine the existence of a bilinear stiffness and the magnitude of its stiffness ratio, but also specify which stiffness is bilinear, i.e. indicate its position. These results provide a possibility of using the trained neural networks to detect and locate structural cracks which have the characteristics of bilinear systems.Visiting scholar, from People's Republic of China.     

Connection between Volterra series and perturbation method in nonlinear systems analyses

This paper is concerned with the connection between the Volterra series and the regular perturbation method in nonlinear systems analyses. It is revealed for the first time that, for a forced polynomial nonlinear system, if its derived linear system is a damped dissipative system, the steady response obtained through the regular perturbation method is exactly identical to the response given by the Volterra series. On the other hand, if the derived linear system is an undamped conservative system, then the Volterra series is incapable of modeling the forced polynomial nonlinear system. Numerical examples are further presented to illustrate these points. The results provide a new criterion for quickly judging whether the Volterra series is applicable for modeling a given polynomial nonlinear system.     

Numerical calculation of nonlinear normal modes in structural systems

This paper presents two methods for numerical calculation of nonlinear normal modes (NNMs) in multi-degree-of-freedom, conservative, nonlinear structural dynamics models. The approaches used are briefly described as follows. Method 1: Starting with small amplitude initial conditions determined by a selected mode of the associated linear system, a small amount of negative damping is added in order to “artificially destabilize” the system; numerical integration of the system equations of motion then produces a simulated response in which orbits spiral outward essentially in the nonlinear modal manifold of interest, approximately generating this manifold for moderate to strong nonlinearity. Method 2: Starting with moderate to large amplitude initial conditions proportional to a selected linear mode shape, perform numerical integration with the coefficient ε of the nonlinearity contrived to vary slowly from an initial value of zero; this simulation methodology gradually transforms the initially flat eigenspace for ε = 0 into the manifold existing quasi-statically for instantaneous values of ε. The two methods are efficient and reasonably accurate and are intended for use in finding NNMs, as well as interesting behavior associated with them, for moderately and strongly nonlinear systems with relatively many degrees of freedom (DOFs).     

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

In this paper a general technique for the analysis of nonlinear dynamical systems with periodic-quasiperiodic coefficients is developed. For such systems the coefficients of the linear terms are periodic with frequency ω while the coefficients of the nonlinear terms contain frequencies that are incommensurate with ω. No restrictions are placed on the size of the periodic terms appearing in the linear part of system equation. Application of Lyapunov-Floquet transformation produces a dynamically equivalent system in which the linear part is time-invariant and the time varying coefficients of the nonlinear terms are quasiperiodic. Then a series of quasiperiodic near-identity transformations are applied to reduce the system equation to a normal form. In the process a quasiperiodic homological equation and the corresponding ‘solvability condition’ are obtained. Various resonance conditions are discussed and examples are included to show practical significance of the method. Results obtained from the quasiperiodic time-dependent normal form theory are compared with the numerical solutions. A close agreement is found.     

Dynamics modeling and stability of robotic systems with discrete-time force control

This paper investigates the dynamic behavior of robotic echanical systems with discrete-time force control. Force control is associated with the constrained motion of a mechanical system. A novel approach is presented to analyze the stability and performance based on the separation of constrained and admissible motions. This results in a model representing the dynamics of the constrained motion of the system. The analysis connects the complex nonlinear model of a mechanical system to a set of abstract delayed oscillators. These oscillator models make it possible to perform a detailed closed-form mathematical analysis of the stability behavior. A planar two-degree-of-freedom (DoF) mechanism is presented as an example to illustrate the material. Results are illustrated by stability charts in the parameter space of mechanical parameters, control gains and the sampling rate.     

Globally convergent homotopy algorithms for nonlinear systems of equations

Probability-one homotopy methods are a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely. These new globally convergent homotopy techniques have been successfully applied to solve Brouwer fixed point problems, polynomial systems of equations, constrained and unconstrained optimization problems, discretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements, and finite difference, collocation, and Galerkin approximations to nonlinear partial differential equations. This paper introduces, in a tutorial fashion, the theory of globally convergent homotopy algorithms, deseribes some computer algorithms and mathematical software, and presents several nontrivial engineering applications.This work was supported in part by DOE Grant DE-FG05-88ER25068, NASA Grant NAG-1-1079, and AFOSR Grant 89-0497.