Showing papers in "International Journal of Non-linear Mechanics in 1996"
TL;DR: In this paper, explicit traveling wave solutions to a Kolmogorov-Petrov-skii-Piskunov equation are presented through two ansatze transformations.
Abstract: Some explicit traveling wave solutions to a Kolmogorov-Petrovskii-Piskunov equation are presented through two ansatze. By a Cole-Hopf transformation, this Kolmogorov-Petrov-skii-Piskunov equation is also written as a bilinear equation and two solutions to describe nonlinear interaction of traveling waves are further generated. Backlund transformations of the linear form and some special cases are considered.
446 citations
TL;DR: In this paper, a neural network system identification model is employed for simulation of output when measured system input is available, and also demonstrates the ability to match higher order spectral characteristics, such as power spectral density and central moments through fourth order.
Abstract: This study addresses the simulation of a class of non-normal processes based on measured samples and sample characteristics of the system input and output. The class of non-normal processes considered here concerns environmental loads, such as wind and wave loads, and associated structural responses. First, static transformation techniques are used to perform simulations of the underlying Gaussian time or autocorrelation sample. An optimization procedure is employed to overcome errors associated with a truncated Hermite polynomial transformation. This method is able to produce simulations which closely match the sample process histogram, power spectral density, and central moments through fourth order. However, it does not retain the specific structure of the phase relationship between frequency components, demonstrated by the inability to match higher order spectra. A Volterra series up to second order with analytical kernels is employed to demonstrate the bispectral matching made possible with memory models. A neural network system identification model is employed for simulation of output when measured system input is available, and also demonstrates the ability to match higher order spectral characteristics.
99 citations
TL;DR: In this paper, the curvature ratio of a curved annular pipe is used to study steady, fully developed flow of Oldroyd-B fluids through curved pipes for both pipes of circular and annular cross-section.
Abstract: Perturbation methods are used to study steady, fully developed flow of Oldroyd-B fluids through curved pipes for both pipes of circular and annular cross-section. The perturbation parameter is the curvature ratio, given by the cross-sectional radius of the pipe divided by the constant radius of the pipe centerline. We compare results for creeping and non-creeping flows for both viscoelastic and Newtonian fluids. In pipes of circular cross-section, the velocity field for creeping flows of Oldroyd-B fluids is qualitatively similar to that found for non-creeping flows of Newtonian fluids. Namely, in addition to the primary flow, there is a secondary motion consisting of counter-rotating vortices. In curved annular pipes, two pairs of counter-rotating vortices are generated by either inertial or elastic effects. In this geometry, the differences between creeping flow of viscoelastic fluids and non-creeping flow of Newtonian fluids are dramatically accentuated at small values of inner to outer pipe radius, riro. For Newtonian fluids, as riro approaches zero, the magnitude and size of the vortices adjacent to the inner cylinder shrink to zero. However, for creeping flow of Oldroyd-B fluids, the inner vortex pair is comparable to the outer vortex pair in both size and strength, even for values of riro as small as 0.01. For pipes of circular cross-section, the effect of elasticity on the drag is considered and earlier predictions by Bowen et al. for the upper convected Maxwell fluid are extended to the Oldroyd-B fluid for non-zero Reynolds number.
62 citations
TL;DR: In this article, a perturbation-incremental method is presented for the analysis of strongly non-linear oscillators of the form x + g(x) = λf(x,.x).x, where g ( x ) and f ( x,. x ) are arbitrary nonlinear functions of their arguments.
Abstract: A perturbation-incremental method is presented for the analysis of strongly non-linear oscillators of the form x + g(x) = λf(x, .x).x , where g ( x ) and f ( x , . x ) are arbitrary non-linear functions of their arguments. The method is an extension of the perturbation-iterative method to the case where λ is not necessarily small. It incorporates salient features from both the perturbation method and the incremental method. Limit cycles of the oscillators can be calculated to any desired degree of accuracy. The stability of the limit cycle is also discussed.
56 citations
TL;DR: In this paper, an algorithm was developed and presented for the efficient and automated generation of the moment equations for dynamical systems with an arbitrary number of states and closed at an arbitrary level (limited only by available computational resources).
Abstract: This paper examines the efficacy of cumulant-neglect closure methods for complex dynamical systems. Previous efforts to employ this method were limited by the complicated and lengthy nature of the governing moment equations. To address this difficulty, an algorithm was developed and is presented herein for the efficient and automated generation of the moment equations for dynamical systems with an arbitrary number of states and closed at an arbitrary level (limited only by available computational resources). Both stationary and non-stationary moment responses were obtained through solution of the closed set of moment equations. The Duffing oscillator subjected to additive Gaussian white noise and the linear oscillator subjected to the square of a Gaussian process were examined in order to demonstrate the effectiveness and accuracy of the algorithm. A number of additional systems were examined to determine the range of applicability of the cumulant neglect closure method. Some general observations regarding stability and performance of cumulant-neglect closure methods are given.
52 citations
TL;DR: In this paper, a second-order averaging of the autoparametric system with a pendulum attached to a primary spring-mass is performed to answer questions about the range of small parameter e (a function of the forcing amplitude) for which the solutions are valid, and about the persistence of some unstable dynamical behavior, like saturation.
Abstract: The autoparametric system consisting of a pendulum attached to a primary spring-mass is known to exhibit 1:2 internal resonance, and amplitude-modulated chaos under harmonic forcing conditions. First-order averaging studies and an analysis of the amplitude dynamics predicts that the response curves of the system exhibit saturation. The period-doubling route to chaos is observed following a Hopf bifurcation to limit cycles. However, to answer questions about the range of the small parameter e (a function of the forcing amplitude) for which the solutions are valid, and about the persistence of some unstable dynamical behavior, like saturation, higher-order non-linear effects need to be taken into account. Second-order averaging of the system is undertaken to address these issues. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of e appearing in the averaged equations. For larger e, second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response. For the response between the two Hopf bifurcation points from the coupled-mode solution branch, the period-doubling as well as the Silnikov mechanism for chaos are observed. The predictions of the averaged equations are verified qualitatively for the original equations.
49 citations
TL;DR: The role and implications of microstructure heterogeneity and non-linearity in interpreting a variety of experimental observations during plastic flow and fracture are outlined in this paper, in connection with the phenomena of dislocation and strain patterning, as well as with existing observations on oscillatory fracture.
Abstract: The role and implications of microstructure heterogeneity and non-linearity in interpreting a variety of experimental observations during plastic flow and fracture are outlined. Traditionally, these observations were not related to theoretical modelling efforts as recent techniques in non-linear dynamics were not available and usual applications were limited to standard materials and processes where micro- and macroscales do not interact and can be treated independently. Advanced technology has imposed the need for the development of models accounting for scale coupling effects. One class of such models, the so-called gradient models, will be described in the paper in connection with the phenomena of dislocation and strain patterning, as well as with existing observations on oscillatory fracture.
49 citations
TL;DR: In this article, a solution for non-linear free vibrations of a symmetrically laminated, cross-ply, geometrically perfect, thick, conical shell with its two ends both clamped and both simply supported is expressed in the form of generalized double Fourier series with time-dependent coefficients.
Abstract: Non-linear equations of transverse motion for a generally laminated, truncated, conical shell are derived by use of the dynamic virtual work principle. The effects of transverse shear deformation, rotatory inertia and geometric initial imperfection are taken into account. A solution for non-linear free vibrations of a symmetrically laminated, cross-ply, geometrically perfect, thick, conical shell with its two ends both clamped and both simply supported is expressed in the form of generalized double Fourier series with time-dependent coefficients. The Galerkin procedure furnishes a system of equations for time functions which are solved by the method of harmonic balance. Non-linear vibrations of circular cylindrical shells and annular plates are treated as special cases. Numerical results in non-linear vibrations of symmetric cross-ply conical laminates are presented graphically for different parameters. Present results are also compared with available data for special cases.
48 citations
TL;DR: In this article, a unified framework from which emerge the Lagrange equations, the Gibbs-Appell Equations and the Generalized Inverse Equations for describing the motion of constrained mechanical systems is presented.
Abstract: This paper presents a unified framework from which emerge the Lagrange equations, the Gibbs-Appell Equations and the Generalized Inverse Equations for describing the motion of constrained mechanical systems. The unified approach extends the applicability of the first two approaches to systems where the constraints are non-linear functions of the generalized velocities and are not necessarily independent. Furthermore, the approach leads to the Explicit Gibbs-Appell Equations.
46 citations
TL;DR: In this article, the energy-release rates of elastic paramagnets and soft ferromagnets were established on the basis of the rotationally invariant (finite-strain) quasi-magnetostatic theory for which neither magnetic ordering nor spin effects need to be introduced.
Abstract: Expressions of the energy-release rates are established on the basis of the rotationally invariant (finite-strain) quasi-magnetostatic theory of elastic paramagnets and soft ferromagnets for which neither magnetic ordering nor spin effects need to be introduced. The corresponding path-independent integrals useful in fracture studies are constructed and this is shown to yield essentially the same results as the canonical field-theoretic approach using the notions of Eshelby srress and material forces . A simple application to the antiplane crack problem of a magnetostrictive body in an axial bias magnetic field illustrates the use of the obtained expressions. The same notions could be used to study the so-called magneto-plastic effect .
44 citations
TL;DR: In this paper, an exact solution technique is developed to obtain stationary probability densities for a class of MDOF nonlinear systems under Gaussian white-noise excitations, without the restriction of equipartition of kinetic energies as with the case of previous exact solutions.
Abstract: An exact solution technique is developed to obtain stationary probability densities for a class of multi-degree-of-freedom (MDOF) non-linear systems under Gaussian white-noise excitations, without the restriction of equipartition of kinetic energies as with the case of previous exact solutions. The conditions under which exact solutions are obtainable are given. When these conditions are not satisfied, approximate solutions are obtained on the basis of minimum weighted residuals. Examples are given for illustration.
TL;DR: In this paper, the uniqueness of the solution of the title problem has been examined through a simple mathematical procedure, and the solution has been shown to be unique in a number of applications.
Abstract: The uniqueness of the solution of the title problem has been examined through a simple mathematical procedure.
TL;DR: In this article, a non-linear contact problem between the stator and the rotor of an ultrasonic traveling were motor is discussed. But the stators are modelled as a Bernoulli-Euler beam and the slider (rotor) is assumed to be rigid.
Abstract: In this paper we discuss the non-linear contact problem between the stator and the rotor of an ultrasonic travelling were motor. For a first simplified mathematical model the problem is formulated for a linear motor in which the stator is modelled as a Bernoulli-Euler beam and the slider (rotor) is assumed to be rigid. A thin layer of visco-elastic material is assumed to exist between stator and slider. Expressions are obtained for the contact pressure between the two parts. The frictional forces both in the sticking and in the sliding zone can then be easily obtained assuming Coulomb friction.
TL;DR: In this article, the thermodynamics and stability of a viscoelastic second grade solid whose action is characterized by two microstructural coefficients α 1 and α 2 in addition to the Newtonian viscosity was studied.
Abstract: We study the thermodynamics and stability of a viscoelastic second grade solid whose action is characterized by two microstructural coefficients α 1 and α 2 in addition to the Newtonian viscosity μ. We show that it is both necessary and sufficient that μ ⩾ 0, α 1 ⩾ 0 and α 1 + α 2 = 0 if the material model is to be compatible with thermodynamics and its free energy is to be at a local minimum in equilibrium. Then, we construct a stability theorem for second grade solids which undergo mechanically isolated motions wherein it is shown that the motion of the body relative to its center of mass will dissipate away in time. The stability theorem is exemplified by investigating the free oscillation of cylindrical and spherical shells where the equilibrium state is globally stable. When μ = 0, but α 1 ≠ 0, the shells exhibit a larger period than if they were purely elastic in the classical sense.
TL;DR: In this article, the combined effect of viscous and ohmic dissipations on magnetoconvection in a vertical enclosure heated at the vertical side walls in the presence of applied electric field parallel to gravity and magnetic field normal to gravity is investigated.
Abstract: The combined effect of viscous and ohmic dissipations on magnetoconvection in a vertical enclosure heated at the vertical side walls in the presence of applied electric field parallel to gravity and magnetic field normal to gravity is investigated The coupled non-linear equations governing the motion are solved both analytically valid for small buoyancy parameter N and numerically valid for large N Solutions for large N reveal a marked change in velocity profile, mass flow rate, skin friction and rate of heat transfer These results are presented for various Hartmann number M , electric field loading parameter E and buoyancy parameter N It is shown in the case of open circuit (ie E ≠ 0) that the effect of magnetic field is to increase both the velocity and temperature in contrast with the short circuit case (ie E = 0) The results for the case when the walls are maintained at the same temperatures (ie T 1 = T 2 ) are obtained as a particular case
TL;DR: In this article, the path integral solution (PIS) technique has been applied to calculate the response statistics of a white noise excited oscillator with bilinear hysteresis.
Abstract: The path integral solution (PIS) technique has been applied to calculate the response statistics of a white noise excited oscillator with bilinear hysteresis. It is shown that the PIS method provides an interesting alternative for investigating the random vibration of hysteretic oscillators. Computational aspects of the application of the PIS technique for estimating the statistics of the response of such oscillators are discussed, and the obtained numerical results are compared with the results produced by stochastic averaging and Monte Carlo simulation.
TL;DR: In this paper, a procedure for obtaining the dynamic response of non-linear systems with parameter uncertainties is presented, where the uncertain parameters are modeled as time-independent random variables, and a set of deterministic nonlinear differential equations is derived using the weighted residual method.
Abstract: This paper presents a procedure for obtaining the dynamic response of non-linear systems with parameter uncertainties. Consideration is given to systems with polynomial non-linearity subjected to deterministic excitation. The uncertain parameters are modeled as time-independent random variables. The set of orthogonal polynomials associated with the probability density function is used as the solution basis, and the response variables are expanded in terms of a finite sum of these polynomials. A set of deterministic non-linear differential equations is derived using the weighted residual method. The discrete-time solutions to the equation set are evaluated numerically using a step-by-step time-integration scheme and the response statistics are determined. Application of the proposed method is illustrated through the analysis of non-linear single-degree-of-freedom structural systems exhibiting uncertain stiffnesses. Both hardening and softening stiffness characteristics are examined. The accuracy of the results is validated by direct numerical integration.
TL;DR: In this article, the stochastic Hopf bifurcation behavior of the noisy Duffing-van der Pol oscillator x = (α + σW)x + β x − β x 2 x − x 3, (A) is studied numerically.
Abstract: The stochastic Hopf bifurcation behavior of the noisy Duffing-van der Pol oscillator x = (α + σW)x + β x − x 2 x − x 3 , (A) is studied numerically. (α, β are bifurcation parameters, W is white noise, and σ is an intensity parameter.) When the qualitative change of the stationary solution of the Fokker-Planck equation is considered, the stochastic Hopf bifurcation appears as a change from a Dirac measure to a crater-like density. Unfortunately, this behavior is not related to the sample stability of the trivial solution zero. To capture all the stochastic dynamics of the equation, it is necessary to investigate the change of stability of invariant measures and the occurrence of new invariant measures for the generated random dynamical system.
TL;DR: In this paper, a non-linear axisymmetric free vibration analysis of the bottom plate of a cylindrical tank coupled with liquid contained is presented by means of the Ritz averaging method.
Abstract: Theoretical analyses and experimental studies have been carried out on the non-linear hydroelastic vibration of a cylindrical tank with an elastic bottom. In this paper, non-linear axisymmetric free vibration analysis of the bottom plate of the tank, coupled with liquid contained, is presented by means of the Ritz averaging method. In the analysis, the effect of an in-plane force in the plate due to the static liquid pressure is taken into account. The effect of liquid contained on the non-linearity of the backbone curve of the principal resonance of both sloshing- and bulging-type responses was clarified, and it was found that with increase of the liquid height, the non-linearity with a hard-spring type of bottom plate decreased in degree, and in fact became close to linear characteristics. The effect of elastic stiffness of the foundation on the bottom plate response was also clarified.
TL;DR: In this article, the authors revisited the initial postbuckling behavior of a finite beam on an elastic foundation using a double-scale perturbation method and showed that the initial beam may have periodic, modulated and localized modes.
Abstract: The initial post-buckling behavior of a finite beam on an elastic foundation is revisited using a double-scale perturbation method. The results show that the initial post-buckling behavior may have periodic, modulated and localized modes. All these modes are obtained as solutions to the Duffing's equation that arise in the solution process. It is found that the modulated initial post-buckling mode converges to a local mode when the length of the beam increases to infinity.
TL;DR: In this article, the non-linear interaction of the in-plane and out-of-plane motions of a suspended cable in the neighbourhood of 2:1 internal resonance under random loading is studied.
Abstract: The non-linear interaction of the in-plane and out-of-plane motions of a suspended cable in the neighbourhood of 2:1 internal resonance under random loading is studied. The random loading acts externally on the in-plane mode, while the out-of-plane mode is non-linearly coupled with the in-plane mode. Any non-trivial motion of the out-of-plane mode is mainly due to this non-linear coupling, which becomes significant in the neighbourhood of internal resonance. The response statistics are estimated by employing the Fokker-Planck equation together with Gaussian and non-Gaussian closures. Monte-Carlo simulation is also used for numerical verification. Away from the internal resonance condition, the response is governed by the inplane motion, and the non-Gaussian closure solution is found to be in good agreement with numerical simulation results. The stochastic bifurcation of the out-of-plane mode is predicted by Gaussian and non-Gaussian closures, and by Monte-Carlo simulation. The non-Gaussian closure can only predict bounded solutions within a limited region. The on-off intermittency of the second mode is observed in the Monte-Carlo simulation over a small range of excitation level. The influence on response statistics of excitation level and cable parameters, such as internal detuning, damping ratios, and sag-to-span ratio, is reported.
TL;DR: In this paper, the authors investigated the Lie symmetries and similarity reductions of the new completely integrable dispersive shallow water equation, discussed recently by Camassa and Holm, through classical and non-classical methods.
Abstract: In this paper we have investigated the Lie symmetries and similarity reductions of the new completely integrable dispersive shallow water equation, discussed recently by Camassa and Holm, through classical and non-classical methods. We compare the similarity reduction thus obtained by the direct method of Clarkson and Kruskal. The resultant ordinary differential equation satisfies the Painleve property and the ARS conjecture. We also point out how the similarity reduction of the present integrable system differs from that of the non-integrable, but closely related, BBM equation.
TL;DR: In this article, the necessary conditions for chaotic motion in the system of coupled Duffing equations of the buckled beam have been analyzed, and it is shown that the higher components of the modified Melnikov vector do not change the critical condition obtained from consideration of only the first component.
Abstract: Part II of this work completes the analysis of the necessary conditions for the chaos in the system of modal equations of motion of the buckled beam, i.e. a system of coupled Duffing equations. The direct calculation of the stable and unstable perturbed manifolds is used to establish the necessary condition for chaotic motion. It is shown that the higher components of the modified Melnikov vector do not change the critical condition obtained from consideration of only the first component. Analysis of the Lyapunov exponents of the system demonstrates the difference between necessary and sufficient conditions for steady-state chaos. Finally, the additional consideration of the non-hyperbolic modes shows that they can be neglected while developing the condition for the intersection of the stable and unstable manifolds.
TL;DR: In this paper, a theory for non-linear bending of symmetrically laminated, cylindrically orthotropic, shallow, conical shells subjected to an axisymmetrically distributed load including transverse shear effects is established.
Abstract: In this paper, a theory for non-linear bending of symmetrically laminated, cylindrically orthotropic, shallow, conical shells subjected to an axisymmetrically distributed load including transverse shear effects is established. By means of this theory and the modified iteration method, the analytical solution of the critical buckling pressure for a symmetrically laminated, cylindrically orthotropic, shallow, conical shell with a rigidly clamped edge under the action of uniform lateral pressure is obtained.
TL;DR: In this article, a general qualitative approach for dynamic buckling and stability of autonomous dissipative structural systems is comprehensively presented, focusing on systems which under the same statically applied loading exhibit a limit point instability or an unstable branching point instability with a non-linear fundamental path.
Abstract: A general qualitative approach for dynamic buckling and stability of autonomous dissipative structural systems is comprehensively presented. Attention is focused on systems which under the same statically applied loading exhibit a limit point instability or an unstable branching point instability with a non-linear fundamental path. Using the total energy equation, the theory of point and periodic attractors of the basin of attraction of a stable equilibrium point, of local and global bifurcations, of the inset and outset manifolds of a saddle and of the geometry of the channel of motion, the stability of the fundamental equilibrium path and the mechanism of dynamic buckling are thoroughly discussed. This allows us to establish useful qualitative criteria leading to exact, approximate and upper/lower bound buckling estimates without integrating the highly non-linear initial-value problem. The individual and coupling effect of geometric and material non-linearities of damping and mass distribution on the dynamic buckling load are also examined. A comparison of the results of the above qualitative analysis with those obtained via numerical simulation is performed on several two- and three-degree-of-freedom models of engineering importance.
TL;DR: In this article, it was shown that the difference between ideal white noise and physical white noise is only related to different forms of integration of the Dirac delta occurrence, and that the two integrals coincide.
Abstract: The concept of ideal white noise and the subsequent Ito and Stratonovich integral concepts seem to be useful only for mathematical manipulation in deriving the moment equations for non-linear systems under parametric white noise. For physical white noise, that is for a band limited white noise, the two integrals coincide. This paper is devoted to a further insight into this problem starting from a single deterministic parametric impulse showing that a difference between physical and ideal white noise is only related to different forms of integration of the Dirac delta occurrence.
TL;DR: The analysis of stochastic exponential p-stability and some types of stability in probability of dynamic systems, which is described by a normal homogeneous system of differential equations of the first order with parametric perturbations, is analyzed as a logical extension of the exponential, asymptotic and weak stability of deterministic systems as discussed by the authors.
Abstract: The stochastic exponential p-stability and some types of stability in probability of dynamic systems, which is described by a normal homogeneous system of differential equations of the first order with parametric perturbations, is analysed as a logical extension of the exponential, asymptotic and weak stability of deterministic systems. The analysis of the stochastic stability has been based on the second Lyapunov method. The Lyapunov function possessing certain properties, corresponding to the type of stochastic stability in question, is constructed. This function must always be positive definite, and the result of the application of the adjoined Fokker-Planck-Kolgomorov operator must be the function of negative or zero value, with the exception of the origin. The paper deduces the structure of other necessary and sufficient properties which the Lyapunov function has to satisfy, if the system is to be stable. The differences between analogous definitions in deterministic and stochastic domains have been shown. The case of a non-linear system is compared with the linear and linearised systems, in order to decide, whether, and under which conditions the system can be linearised from the point of view of the analysis of the stability, and whether the analysis can be performed using, for example, the Rous-Hurwitz determinants. The fundamental problem is of course the level of stability which the simplified fictitious system has to satisfy, in order to imply the required type of stability of the original system. This alignment is, of course, limited on such a type of non-linearities, which do not cause energetically unstable branches, multiple equilibrium states and the polymodal character of the response. Some comments about the physical interpretation of the Lyapunov function have been added, as well as about the possibilities of its construction in individual cases.
TL;DR: In this article, a non-Newtonian fluid model in which the stress is an arbitrary function of the symmetric part of the velocity gradient is considered, and symmetry groups of the two-dimensional boundary layer equations of the model are found by using exterior calculus.
Abstract: A non-Newtonian fluid model in which the stress is an arbitrary function of the symmetric part of the velocity gradient is considered. Symmetry groups of the two-dimensional boundary layer equations of the model are found by using exterior calculus. The complete isovector field corresponding to some cases, such as arbitrary shear function, Newtonian fluids, and powerlaw fluids, are found. Similarly, solutions for some special transformations are presented. Results obtained in a previous paper [M. Pakdemirli, Int. J. Non-Linear Mech. 29, 187 (1994)] using special groups of transformations (scaling, spiral) are verified in this study using a general approach.
TL;DR: In this paper, the random properties of a random linear system are described by a Karhunen-Loeve-expansion, and for the response of a linear system with random properties subjected to a stochastic loading a spectral approach introducing Hermitian polynomials for the random response quantities is used according to a proposal of Ghanem and Spanos.
Abstract: The goal of the paper is to include into statistical linearization random system properties, which may be important in the case of a narrow-band excitation. Random properties of the system are described by a Karhunen-Loeve-expansion, and for the response of a linear system with random properties subjected to a stochastic loading a spectral approach introducing Hermitian polynomials for the random response quantities is used according to a proposal of Ghanem and Spanos. Whereas for a deterministic non-linear contribution to a random linear system the procedure is rather similar to the well-established linearization for deterministic systems, some additional operations are necessary to take into account the randomness of the non-linear contributions.
TL;DR: In this paper, a study of the dynamics of an inverted pendulum under the regulation of a linear feedback control system is conducted, and the authors show that the boundaries of the stable state-space hyper-volume exhibit a jagged, possibly fractal, morphology.
Abstract: A study is conducted into the dynamics of an inverted pendulum under the regulation of a linear-feedback control system. Analysis of the resulting dynamic equations indicates the presence of stable regions within the four-dimensional state space of the system, which are confirmed through simulation. System trajectories are examined in this state space, and complex, apparently chaotic behaviour is observed outside the stable regions of the state space. An algorithm is developed to map the boundaries of the stable state-space hyper-volume; this algorithm is broadly applicable to the study of other systems. Results are presented, generated by executing the algorithm on the inverted pendulum dynamics. These comprise a series of three-dimensional plots, outlining the four-dimen-sional stability volume. The boundaries of this volume are seen to exhibit a jagged, possibly fractal, morphology.