Showing papers in "Journal of Computational and Nonlinear Dynamics in 2006"

Computational Dynamics of Multibody Systems: History, Formalisms, and Applications

Abstract: Multibody dynamics is based on analytical mechanics and is applied to engineering systems such as a wide variety of machines and all kind of vehicles. Multibody dynamics depends on computational dynamics and is closely related to control design and vibration theory. Recent developments in multibody dynamics focus on elastic or flexible systems, respectively, contact and impact problems, and actively controlled systems. Some fundamentals in multibody dynamics, recursive algorithms and methods for dynamical analysis are presented. In particular, methods from linear system analysis and nonlinear dynamics approaches are discussed. Also, applications from vehicles, manufacturing science and molecular dynamics are shown.

Optimal Inputs for Phase Models of Spiking Neurons

Abstract: Variational methods are used to determine the optimal currents that elicit spikes in various phase reductions of neural oscillator models. We show that, for a given reduced neuron model and target spike time, there is a unique current that minimizes a squareintegral measure of its amplitude. For intrinsically oscillatory models, we further demonstrate that the form and scaling of this current is determined by the model’s phase response curve. These results reflect the role of intrinsic neural dynamics in determining the time course of synaptic inputs to which a neuron is optimally tuned to respond, and are illustrated using phase reductions of neural models valid near typical bifurcations to periodic firing, as well as the Hodgkin-Huxley equations. DOI: 10.1115/1.2338654 Phase-reduced models of neurons have traditionally been used to investigate either the patterns of synchrony that result from the type and architecture of coupling 1–8 or the response of large groups of oscillators to external stimuli 9–11. In all of these cases, the inputs to the model cells were fixed by definition of the model at the outset and the dynamics of phase models of networks or populations were analyzed in detail. The present paper takes a complementary, control-theoretic approach that is related to probabilistic “spike-triggered” methods 12: we fix at the outset a feature of the dynamical trajectories of interest—spiking at a precise time t1—and study the neural inputs that lead to this outcome. By computing the optimal such input, according to a measure of the input strength required to elicit the spike, we identify the signal to which the neuron is optimally “tuned” to respond. We view the present work as part of the first attempts 13,14 to understand the dynamical response of neurons using control theory, and, as we expect that insights from this general perspective will be combined with the “forward” dynamics results that Phil Holmes and many others have derived to ultimately enhance our understanding of neural processing, we hope that it will serve as a fitting tribute to his work.

Normal Force-Displacement Relationship of Spherical Joints With Clearances

Abstract: The spherical joint with clearances can be modeled as an axi-symmetric quasi-static contact of a sphere in a cavity The Hertz theory based on the assumption of nonconformal contact is often used to represent the normal force-displacement relationship for the spherical joint with clearances This assumption limits the application of the theory, especially in the case of occurring large deformation in contact area The Steuermann theory is effective to solve the conformal contact problem in some cases, but it is strictly dependent on the selecting index n of the polynomial, which is used to represent the contact profiles In this paper, an approximate contact model of the spherical joints with clearances is developed that is based on using the distributed elastic forces to model the compliant of the surfaces in contact The new formulation is simple and straightforward, and its validity has been tested by comparison to the finite element results DOI: 101115/12162872

Model Reduction of Systems With Localized Nonlinearities

Abstract: An LDRD funded approach to development of reduced order models for systems with local nonlinearities is presented. This method is particularly useful for problems of structural dynamics, but has potential application in other fields. The key elements of this approach are (1) employment of eigen modes of a reference linear system, (2) incorporation of basis functions with an appropriate discontinuity at the location of the nonlinearity. Galerkin solution using the above combination of basis functions appears to capture the dynamics of the system with a small basis set. For problems involving small amplitude dynamics, the addition of discontinuous (joint) modes appears to capture the nonlinear mechanics correctly while preserving the modal form of the predictions. For problems involving large amplitude dynamics of realistic joint models (macro-slip), the use of appropriate joint modes along with sufficient basis eigen modes to capture the frequencies of the system greatly enhances convergence, though the modal nature the result is lost. Also observed is that when joint modes are used in conjunction with a small number of elastic eigen modes in problems of macro-slip of realistic joint models, the resulting predictions are very similar to those of the full solution when seen through a lowmore » pass filter. This has significance both in terms of greatly reducing the number of degrees of freedom of the problem and in terms of facilitating the use of much larger time steps.« less

A computational approach to conley's decomposition theorem

Abstract: Background. The discrete dynamics generated by a continuous map can be represented combinatorially by an appropriate multivalued map on a discretization of the phase space such as a cubical grid or triangulation. Method of approach. We describe explicit algorithms for computing dynamical structures for the combinatorial multivalued maps. Results. We provide computational complexity bounds and numerical examples. Specifically we focus on the computation attractor-repeller pairs and Lyapunov functions for Morse decompositions. Conclusions. The computed discrete Lyapunov functions are weak Lyapunov functions and well-approximate a continuous Lyapunov function for the underlying map.

Theoretical and experimental analysis of a base-excited rotor

Abstract: In this study, the dynamic behavior of a flexible rotor system subjected to support excitation (imposed displacements of its base) is analyzed. The effect of an excitation on lateral displacements is investigated from theoretical and experimental points of view. The study focuses on behavior in bending. A mathematical model with two gyroscopic and parametrical coupled equations is derived using the Rayleigh-Ritz method. The theoretical study is based on both the multiple scales method and the normal form approach. An experimental setup is then developed to observe the dynamic behavior permitting the measurement of lateral displacements when the system's support is subjected to a sinusoidal rotation. The experimental results are favorably compared with the analytical and numerical results.

Development of elastic forces for a large deformation plate element based on the absolute nodal coordinate formulation

Abstract: Dynamic analysis of large rotation and deformation can be carried out using the absolute nodal coordinate formulation. This formulation, which utilizes global displacements and slope coordinates as nodal variables, make it possible to avoid the difficulties that arise when a rotation is interpolated in three-dimensional applications. In the absolute nodal coordinate formulation, a continuum mechanics approach has become the dominating procedure when elastic forces are defined. It has recently been perceived, however, that the continuum mechanics based absolute nodal coordinate elements suffer from serious shortcomings, including Poisson's locking and poor convergence rate. These problems can be circumvented by modifying the displacement field of a finite element in the definition of elastic forces. This allows the use of the mixed type interpolation technique, leading to accurate and efficient finite element formulations. This approach has been previously applied to two- and three-dimensional absolute nodal coordinate based finite elements. In this study, the improved approach for elastic forces is extended to the absolute nodal coordinate plate element. The introduced plate element is compared in static examples to the continuum mechanics based absolute nodal coordinate plate element, as well as to commercial finite element software. A simple dynamic analysis is performed using the introduced element in order to demonstrate the capability of the element to conserve energy.

Co-dimension-Two Grazing Bifurcations in Single-Degree-of-Freedom Impact Oscillators

Abstract: Grazing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and nonimpacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper, the transition between two characteristically different co-dimension-one grazing bifurcation scenarios is found to be associated with the presence of certain co-dimension-two grazing bifurcation points and their unfolding in parameter space. The analysis investigates the distribution of such degenerate bifurcation points along the grazing bifurcation manifold in examples of single-degree-of-freedom oscillators. Unfoldings obtained with the discontinuity-mapping technique are used to explore the possible influence on the global dynamics of the smooth co-dimension-one bifurcations of the impacting dynamics that emanate from such co-dimension-two points. It is shown that attracting impacting motion may result from parameter variations through a co-dimension-two grazing bifurcation of an initially unstable limit cycle in a nonlinear micro-electro-mechanical systems (MEMS) oscillator.

Lobes and Lenses in the Stability Chart of Interrupted Turning

Abstract: In this paper, a new method for the stability analysis of interrupted turning processes is introduced. The approach is based on the construction of a characteristic function whose complex roots determine the stability of the system. By using the argument principle, the number of roots causing instability can be counted, and thus, an exact stability chart can be drawn. In the special case of period doubling bifurcation, the corresponding multiplier - 1 is substituted into the characteristic function, leading to an implicit formula for the stability boundaries. Further investigations show that all the period doubling boundaries are closed curves, except the first lobe at the highest cutting speeds. Together with the stability boundaries of Neimark-Sacker (or secondary Hopf) bifurcations, the unstable parameter domains are formed from the union of lobes and lenses.

Tautochronic Vibration Absorbers for Rotating Systems

Abstract: This paper describes an analytical and experimental investigation of the dynamic response and performance of a special type of centrifugal pendulum vibration absorber used for reducing torsional vibrations in rotating systems. This absorber has the property that it behaves linearly out to large amplitudes, and thus experiences no frequency de-tuning. Previous analytical work on such tautochronic absorbers has considered the response, dynamic stability, and performance of single- and multi-absorber systems. In particular, it is known that these absorbers, when perfectly tuned to the order of the applied torque, do not exhibit hysteretic jumps in the response, but multi-absorber systems can experience instabilities that destroy the symmetry of their synchronous response. In this work we extend the theory to include linear de-tuning of the absorbers, which can be used as a design parameter to influence absorber performance, both in terms of rotor vibration reduction and operating range. This paper reviews the basic analysis, which employs scaling and averaging, and extends it to include the detuning. In addition, systematic experiments of systems with one and two absorbers are carried out. The experimental results are unique in that the test facility is capable of varying the excitation order, thereby allowing one to obtain order-response curves that are useful for design purposes. The experimental results are found to be in excellent agreement with the analytical predictions, and these clearly demonstrate the tradeoffs faced when selecting absorber tuning.

Nonlinear Energy Sink With Uncertain Parameters

Abstract: The effects of a nonlinear energy sink during the instationary regime are analyzed by introducing uncertain parameters to verify the robustness of the transient spatial energy transfer when parameters are not well known. It was shown that it is possible to passively absorb energy from a linear nonconservative system (damped) structure to a nonlinear attachment weakly coupled to the linear one. This rapid and irreversible transfer of energy, named energy pumping, is studied by taking into account uncertainties on parameters, especially damping (since damping plays a great role and there is a lack of knowledge about it). In essence, the nonlinear subsystem acts as a passive nonlinear energy sink for impulsively applied external vibrational disturbances. The aim is to be able to apply energy pumping in practice where the nonlinear attachment realization will never perfectly reflect the design. Since strong nonlinearities are involved, polynomial chaos expansions are used to obtain information about random displacements. Not only are numerical investigations done, but nonlinear normal modes and the role of damping are also analytically studied, which confirms the numerical studies and shows the supplementary information obtained compared to a parametrical study.

A Practical Approach for the Linearization of the Constrained Multibody Dynamics Equations

Abstract: The paper presents an approach to linearize the set of index 3 nonlinear Differential Algebraic Equations (DAE) that govern the dynamics of constrained mechanical systems. The proposed method handles heterogeneous systems that might contain flexible bodies, friction, control elements (user-defined differential equations), and non-holonomic constraints. Analytically equivalent to a state-space formulation of the system dynamics in Lagrangian coordinates, the proposed method augments the governing equations and then computes a set of sensitivities that provide the linearization of interest. The attributes associated with the method are the ability to handle large heterogeneous systems, ability to linearize the system in terms of arbitrary user-defined coordinates, and straightforward implementation. The proposed approach has been released in the 2005 version of the MSC.ADAMS/Solver(C++) and compares favorably with a reference method previously available. The approach was also validated against MSC.NASTRAN and experimental results.

Dynamics and Stability of a Two Degree of Freedom Oscillator With an Elastic Stop

Abstract: A two degree of freedom oscillator with a colliding component is considered. The aim of the study is to investigate the dynamic behavior of the system when the stiffness obstacle changes to a finite value to an infinite one. Several cases are considered. First, in the case of rigid impact and without external excitation, a family of periodic solutions are found in analytical form. In the case of soft impact, with a finite time duration of the shock, and no external excitation, the existence of periodic solutions, with an arbitrary value of the period, is proved. Periodic motions are also obtained when the system is submitted to harmonic excitation, in both cases of rigid or soft impact. The stability of these periodic motions is investigated for these four cases.

Viscous Shocks in the Destabilized Kuramoto-Sivashinsky Equation

Abstract: We study stationary periodic solutions of the Kuranwto-Sivashinsky (KS) model for complex spatio-temporal dynamics in the presence of an additional linear destabilizing term. In particular, we show the phase space origins of the previously observed stationary "viscous shocks" and related solutions. These arise in a reversible four-dimensional dynamical system as perturbed heteroclinic connections whose tails are joined through a reinjection mechanism due to the linear term. We present numerical evidence that the transition to the KS limit contains a rich bifurcation structure even within the class of stationary reversible solutions.

A Simple Stabilizing Control for Sagittal Plane Locomotion

Abstract: The spring loaded inverted pendulum model has been shown to accurately model sagittal plane locomotion for a variety of legged animals and has been used as a target for control for higher dimensional robotic implementations. Tuned appropriately, the model exhibits passively stable periodic gaits using either fixed leg touch-down angle or swing-leg retraction leg touch-down protocols. In this work, we examine the performance of the model when model parameters are set to values characteristic of an insect, in particular the cockroach Blaberus discoidalis. While body motions and forces exhibited during a stride are shown to compare well with those observed experimentally, almost all of the resulting periodic gaits are unstable. We therefore develop and analyze a simple adaptive control scheme that improves periodic gait stability properties. Since it is unlikely that neural reflexes can act quickly enough during a stride to effect control, control is applied once per stance phase through appropriate choice of the leg touch-down angle. The control law developed is novel since it achieves gait stabilization solely through a judicious combination of leg lift-off and touch-down angles, instead of utilizing all of the system positions and velocities in full-state feedback control. Implementing the control law improves the stability properties of a large number of periodic gaits and enables movement between stable periodic gaits by changing a single parameter.

The dynamic response of tuned impact absorbers for rotating flexible structures

Abstract: This paper describes an analytical investigation of the dynamic response and performance of impact vibration absorbers fitted to flexible structures that are attached to a rotating hub. This work was motivated by experimental studies at NASA, which demonstrated the effectiveness of these types of absorbers for reducing resonant transverse vibrations in periodically-excited rotating plates. Here we show how an idealized model can be used to describe the essential dynamics of these systems, and used to predict absorber performance. The absorbers use centrifugally induced restoring forces so that their non-impacting dynamics are tuned to a given order of rotation, whereas their large amplitude dynamics involve impacts with the primary flexible system. The linearized, non-impacting dynamics are first explored in detail, and it is shown that the response of the system has some rather unique features as the hub rotor speed is varied. A class of symmetric impacting motions is also analyzed and used to predict the effectiveness of the absorber when operating in its impacting mode. It is observed that two different types of grazing bifurcations take place as the rotor speed is varied through resonance, and their influence on absorber performance is described. The analytical results for the symmetric impacting motions are also used to generate curves that show how important absorber design parameters—including mass, coefficient of restitution, and tuning—affect the system response. These results provide a method for quickly evaluating and comparing proposed absorber designs

Periodic Motions in a Periodically Forced Oscillator Moving on an Oscillating Belt With Dry Friction

Abstract: In this paper, periodic motion in an oscillator moving on a periodically oscillating belt with dry friction is investigated. The conditions of stick and nonstick motions for such an oscillator are obtained in the relative motion frame, and the grazing and stick (or sliding) bifurcations are presented as well. The periodic motions are predicted analytically and numerically, and the analytical prediction is based on the appropriate mapping structures. The eigenvalue analysis of such periodic motions is carried out. The periodic motions are illustrated through the displacement, velocity, and force responses in the absolute and relative frames. This investigation provides an efficient method to predict periodic motions of such an oscillator involving dry friction. The significance of this investigation lies in controlling motion of such a friction-induced oscillator in industry.

New Periodic Orbits for the n-Body Problem

Abstract: Since the discovery of the figure-8 orbit for the three-body problem [Moore 1993] a large number of periodic orbits of the n-body problem with equal masses and beautiful symmetries have been discovered. However, most of those that have appeared in the literature are either planar or are obtained from perturbations of planar orbits. Here we exhibit a number of new three-dimensional periodic n-body orbits with equal masses and cubic symmetry, including some whose moment of inertia tensor is a scalar. We found these orbits numerically by minimizing the action as a function of the trajectories’ Fourier coefficients. We also give numerical evidence that a planar 3-body orbit first found �

Control of impact microactuators for precise positioning

Abstract: Electrically driven impact microactuators generate nanoscale displacements without large driving distances and high voltages These systems exhibit complex dynamics because of inherent nonlinearities due to impacts, friction, and electric forces As a result, dramatic changes in system behavior, associated with so-called grazing bifurcations, may occur during the transition between impacting and nonimpacting dynamics, including the presence of robust chaos For successful open-loop operating conditions, the system design is limited to certain parameter regions, where desired system responses reside The objective of this paper is to overcome this limitation to allow for a more precise displacement manipulation using impact microactuators This is achieved through a closed-loop feedback scheme that successfully controls the system dynamics in the near-grazing region

Verification of Absolute Nodal Coordinate Formulation in Flexible Multibody Dynamics via Physical Experiments of Large Deformation Problems

Abstract: A review of the current state of the absolute nodal coordinate formulation (ANCF) is proposed for large-displacement and large-deformation problems in flexible multibody dynamics. The review covers most of the known implementations of different kinds of finite elements including thin and thick planar and spatial beams and plates, their geometrical description inherited from FEM, and formulations of the most important elements of equations of motion. Much attention is also paid to simulation examples that show reasonableness and accuracy of the formulations applied to real physical problems and that are compared with experiments having significant geometrical nonlinearity. Current and further development directions of the ANCF are also briefly outlined.

A Compliant Contact Model Including Interference Geometry for Polyhedral Objects

Abstract: Modeling of contact with the environment is an essential capability for the simulation of space robotics system, which includes tasks such as berthing and docking The effect of interbody contact on the robotic system has to be determined to predict potential problems in the design cycle A compliant contact dynamics model is proposed here that considers most possible contact situations for a wide diversity of possible object shapes and using interference geometry information A uniform formula is provided to determine the contact force as a function of geometric parameters and material properties A corresponding geometric algorithm is provided in order to obtain the necessary geometric parameters Some simulation results are presented based on the implementation of the geometric algorithm

New Model of a CVT Rocker-Pin Chain With Exact Joint Kinematics

Abstract: The improvement of continuously variable transmissions (CVTs) is a challenging task. Detailed dynamic models of the system are needed for the optimization process. This paper introduces a new model of a rocker-pin chain, which is the central part of a chain CVT. During the derivation of the equations of motion, special attention is turned on the exact description of the joint kinematics. An interesting method to speed up the numerical simulation of the chain drive is introduced, which takes into account the special structure of the mass matrix. Simulation results show the strong influence of the geometry of the rocker-pin joints on the dynamics of the whole gear. Further reasonable gear efficiencies can be estimated for arbitrary joint geometries. This model is an excellent basis for further work on optimization of the CVT chain.

Nonlinear Modeling of Flexible Manipulators Using Nondimensional Variables

Abstract: This paper deals with nonlinear modeling of planar one- and two-link, flexible manipulators with rotary joints using finite element method (FEM) based approaches. The equations of motion are derived taking into account the nonlinear strain-displacement relationship and two characteristic velocities, $U_a$ and $U_g$, representing material and geometric properties (also axial and flexural stiffness) respectively, are used to nondimensionalize the equations of motion. The effect of variation of $U_a$ and $U_g$ on the dynamics of a planar flexible manipulator is brought out using numerical simulations. It is shown that above a certain $U_g$ value (approximately \geq 45 m/s), a linear model (using a linear strain-displacement relationship) and the nonlinear model give approximately the same tip deflection. Likewise, it was found that the effect of $U_a$ is prominent only if $U_g$ is small. The natural frequencies are seen to be varying in a nonlinear manner with $U_a$ and in a linear manner with $U_g$.

Response Scenario and Nonsmooth Features in the Nonlinear Dynamics of an Impacting Inverted Pendulum

Abstract: In this work, we perform a systematic numerical investigation of the nonlinear dynamics of an inverted pendulum between lateral rebounding barriers. Three different families of considerably variable attractors-periodic, chaotic, and rest positions with subsequent chattering-are found. All of them are investigated, in detail, and the response scenario is determined by both bifurcation diagrams and behavior charts of single attractors, and overall maps. Attention is focused on local and global bifurcations that lead to the attractor-basin metamorphoses. Numerical results show the extreme richness of the dynamical response of the system, which is deemed to be of interest also in view of prospective mechanical applications.

Nonlinear response and suppression of chaos by weak harmonic perturbation inside a triple well Φ6-rayleigh oscillator combined to parametric excitations

Abstract: The nonlinear response and suppression of chaos by weak harmonic perturbation inside a triple well Φ 6 -Rayleigh oscillator combined to parametric excitations is studied in this paper. The main attention is focused on the dynamical properties of local bifurcations as well as global bifurcations including homoclinic and heteroclinic bifurcations. The original oscillator is transformed to averaged equations using the method of harmonic balance to obtain periodic solutions. The response curves show the saddle-node bifurcation and the multi-stability phenomena. Based on the Melnikov's method, horseshoe chaos is found and its control is made by introducing an external periodic perturbation.

Rail Geometry and Euler Angles

Abstract: In railroad vehicle dynamics, Euler angles are often used to describe the track geometry (track centerline and rail space curves). The tangent and curvature vectors as well as local geometric properties such as the curvature and torsion can be expressed in terms of Euler angles. Some of the local geometric properties and Euler angles can be related to measured parameters that are often used to define the track geometry. The Euler angles employed, however, define a coordinate system that may differ from the Frenet frame used in the classical differential geometry. The relationship between the track frame used in railroad vehicle dynamics and the Frenet frame used in the theory of curves is developed in this paper and is used to shed light on some of the formulas and identities used in the geometric description in railroad vehicle dynamics. The conditions under which the two frames (track and Frenet) become equivalent are presented and used to obtain expressions for the curvature and torsion in terms of Euler angles and their derivatives with respect to the arc length.

Analysis of the chatter instability in a nonlinear model for drilling

Abstract: In this paper we present stability analysis of a non-linear model for chatter vibration in a drilling operation. The results build our previous work [Stone, E., and Askari, A., 2002, "Nonlinear Models of Chatter in Drilling Processes," Dyn. Syst., 17(1), pp. 65-85 and Stone, E., and Campbell, S. A., 2004, "Stability and Bifurcation Analysis of a Nonlinear DDE Model for Drilling," J. Nonlinear Sci., 14(1), pp. 27-57], where the model was developed and the nonlinear stability of the vibration modes as cutting width is varied was presented. Here we analyze the effect of varying cutting depth. We show that qualitatively different stability lobes are produced in this case. We analyze the criticality of the Hopf bifurcation associated with loss of stability and show that changes in criticality can occur along the stability boundary, resulting in extra periodic solutions.

Polynomial interpolated taylor series method for parameter identification of nonlinear dynamic system

Abstract: This research work is in the area of structural health monitoring and structural damage mitigation. It addresses and advances the technique in parameter identification of structures with significant nonlinear response dynamics. The method integrates a nonlinear hybrid parameter multibody dynamic system (HPMBS) modeling technique with a parameter identification scheme based on a polynomial interpolated Taylor series methodology. This work advances the model based structural health monitoring technique, by providing a tool to accurately estimate damaged structure parameters through significant nonlinear damage. The significant nonlinear damage implied includes effects from loose bolted joints, dry frictional damping, large articulated motions, etc. Note that currently most damage detection algorithms in structures are based on finding changed stiffness parameters and generally do not address other parameters such as mass, length, damping, and joint gaps. This work is the extension of damage detection practice from linear structure to nonlinear structures in civil and aerospace applications. To experimentally validate the developed methodology, we have built a nonlinear HPMBS structure. This structure is used as a test bed to fine-tune the modeling and parameter identification algorithms. It can be used to simulate bolted joints in aircraft wings, expansion joints of bridges, or the interlocking structures in a space frame also. The developed technique has the ability to identify unique damages, such as systematic isolated and noise-induced damage in group members and isolated elements. Using this approach, not just the damage parameters, such as Young's modulus, are identified, but other structural parameters, such as distributed mass, damping, and friction coefficients, can also be identified.

Global Dynamics of an Autoparametric System With Multiple Pendulums

Abstract: The Hamiltonian dynamics of a resonantly excited linear spring-mass-damper system coupled to an array of pendulums is investigated in this study under 1:1:1:...:2 internal resonance between the pendulums and the linear oscillator. To study the small-amplitude global dynamics, a Hamiltonian formulation is introduced using generalized coordinates and momenta, and action-angle coordinates. The Hamilton's equations are averaged to obtain equations for the first-order approximations to free and forced response of the system. Equilibrium solutions of the averaged Hamilton's equations in action-angle or comoving variables are determined and studied for their stability characteristics. The system with one pendulum is known to be integrable in the absence of damping and external excitation. Exciting the system with even a small harmonic forcing near a saddle point leads to stochastic response, as clearly demonstrated by the Poincare sections of motion. Poincare sections are also computed for motions started with initial conditions near center-center, center-saddle and saddle-saddle-type equilibria for systems with two, three and four pendulums. In case of the system with more than one pendulum, even the free undamped dynamics exhibits irregular exchange of energy between the pendulums and the block. The increase in complexity is also demonstrated as the number of pendulums is increased, and when external excitation is present.

Subharmonic-Response Computation and Stability Analysis for a Nonlinear Oscillator Using a Split-Frequency Harmonic Balance Method

Abstract: A split-frequency harmonic balance method (SF-HBM) is developed to obtain subharmonic responses of a nonlinear single-degree-of-freedom oscillator driven by periodic excitation. This method is capable of generating highly accurate periodic solutions involving a large number of solution harmonics. Responses at the excitation period, or corresponding multiples (such as period 2 and period 3), can be readily obtained with this method, either in isolation or as combinations. To achieve this, the oscillator equation error is first expressed in terms of two Mickens functions, where the assumed Fourier series solution is split into two groups, nominally associated with low-frequency or high-frequency harmonics. The number of low-frequency harmonics remains small compared to the number of high-frequency harmonics. By exploiting a convergence property of the equation-error functions, accurate low-frequency harmonics can be obtained in a new iterative scheme using a conventional harmonic balance method, in a separate step from obtaining the high-frequency harmonics. The algebraic equations (needed in the HBM part of the method) are generated wholly numerically via a fast Fourier transform, using a discrete-time formulation to include inexpansible nonlinearities. A nonlinear forced-response stability analysis is adapted for use with solutions obtained with this SF-HBM. Period-3 subharmonic responses are obtained for an oscillator with power-law nonlinear stiffness. The paper shows that for this type of oscillator, two qualitatively different period-3 subharmonic response branches can be obtained across a broad frequency range. Stability analysis reveals, however, that for an increasingly stiff model, neither of these subharmonic branches are stable.