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Research Papers

J. Comput. Nonlinear Dynam. 2008;3(3):031001-031001-6. doi:10.1115/1.2908160.

This paper is devoted to the theoretical and experimental investigation of a sample automotive belt-pulley system subjected to tension fluctuations. The equation of motion for transverse vibrations leads to a Duffing oscillator parametrically excited. The analysis is performed via the multiple scales approach for predicting the nonlinear response, considering longitudinal viscous damping. An experimental setup gives rise to nonlinear parametric instabilities and also exhibits more complex phenomena. The experimental investigation validates the assumptions made and the proposed model.

Topics: Pulleys , Belts , Damping
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(3):031002-031002-10. doi:10.1115/1.2908174.

The usefulness of the Lempel-Ziv complexity and the Lyapanov exponent as two metrics to characterize the dynamic patterns is studied. System output signal is mapped to a binary string and the complexity measure of the time-sequence is computed. Along with the complexity we use the Lyapunov exponent to evaluate the order and disorder in the nonlinear systems. Results from the Lempel-Ziv complexity are compared with the results from the Lyapunov exponent computation. Using these two metrics, we can distinguish the noise from chaos and order. In addition, using same metrics we study the complexity measure of the Fibonacci map as a quasiperiodic system. Our analytical and numerical results prove that for a system like Fibonacci map, complexity grows logarithmically with the evolutionary length of the data block. We conclude that the normalized Lempel-Ziv complexity measure can be used as a system classifier. This quantity turns out to be 1 for random sequences and non-zero value less than 1 for chaotic sequences. While for periodic and quasiperiodic responses as data string grows, their normalized complexity approaches zero. However, higher deceasing rate is observed for periodic responses.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(3):031003-031003-8. doi:10.1115/1.2908178.

This paper presents a methodology for trajectory planning and tracking control of a tractor with a steerable trailer based on the system’s dynamic model. The theory of differential flatness is used as the basic approach in these developments. Flat outputs are found that linearize the system’s dynamic model using dynamic feedback linearization, a subclass of differential flatness. It is demonstrated that this property considerably simplifies motion planning and the development of controller. Simulation results are presented in the paper, which show that the developed controller has the desirable performance with exponential stability.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(3):031004-031004-10. doi:10.1115/1.2908180.

The piano action is the mechanism that transforms the finger force applied to a key into a motion of a hammer that strikes a piano string. This paper presents a state-of-the-art model of a grand piano action, which is based on the five main components of the action mechanism (key, whippen, jack, repetition lever, and hammer). Even though some piano action researchers (e.g., Askenfelt and Jansson) detected some flexibility for the hammer shank in their experiments, all previous piano models have assumed the hammers to be rigid bodies. In this paper, we have accounted for the hammer shank flexibility using a Rayleigh beam model. It turns out that the flexibility of the hammer shank does not significantly affect the rotation of the other parts of the piano mechanism and the impact velocity of the hammer head, compared to the case that the hammer shank has been modeled as a rigid part. However, the flexibility of the hammer shank causes a greater scuffing motion for the hammer head during the contact with the string. To validate the theoretical results, experimental measurements were taken by two strain gauges mounted on the hammer shank, and by optical encoders at three of the joints.

Topics: Hammers , Mechanisms
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(3):031005-031005-12. doi:10.1115/1.2908258.

The present study formulates a model for a coupled oscillation of the convective flow and the solid membrane vibration, which occurs in a 2D domain of a fluid cell. The convection flow is induced by the transient thermal field of the membrane at the bottom of the fluid. The heat conduction in the solid material also causes the membrane to vibrate. This flow motion deviates from the conventional Rayleigh–Benard problem in that a transient thermal field causes the convection flow instead of a constant temperature gradient. A numerical computation reveals the synchronized motion behaviors between the Lorenz-type oscillator for the convection flow and the Duffing oscillator for the membrane motion. The bifurcation conditions from the stability analysis of the model justify the steady-state attractor behaviors and the difference in behavior from the oscillators without coupling.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(3):031006-031006-8. doi:10.1115/1.2908348.

An important problem that spans across many types of systems (e.g., mechanical and biological) is how to model the dynamics of joints or interfaces in built-up structures in such a way that the complex dynamic and energy-dissipative behavior that depends on microscale phenomena at the joint/interface is accurately captured, yet in a framework that is amenable to efficient computational analyses of the larger macroscale system of which the joint or interface is a (spatially) small part. Simulating joint behavior in finite element analysis by meshing the joint regions finely enough to capture relevant micromechanics is impractical for large-scale structural systems. A more practical approach is to devise constitutive models for the overall behavior of individual joints that accurately capture their nonlinear and energy-dissipative behavior and to locally incorporate the constitutive response into the otherwise often-linear structural model. Recent studies have successfully captured and simulated mechanical joint dynamics using computationally simple phenomenological models of combined elasticity and slip with associated friction and energy dissipation, known as Iwan models. In the present article, the author reviews the relationship, and in some cases equivalence, of one type of Iwan model to several other models of hysteretic behavior that have been used to simulate a wide range of physical phenomena. Specifically, it is shown that the “parallel-series” Iwan model has been referred to in other fields by different names, including “Maxwell resistive capacitor,” “Ishlinskii,” and “ordinary stop hysteron.” Given this, the author establishes the relationship of this Iwan model to several other hysteresis models, most significantly the classical Preisach model. Having established these relationships, it is then possible to extend analytical tools developed for a specific hysteresis model to all of the models with which it is related. Such analytical tools include experimental identification, inversion, and analysis of vibratory energy flow and dissipation. Numerical case studies of simple systems that include an Iwan-modeled joint illustrate these points.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(3):031007-031007-11. doi:10.1115/1.2908352.

The electromechanical integrated electrostatic harmonic drive is a new drive system invented by authors. The dynamic displacements of the flexible ring for the drive have important influence on operating performance of the drive system. In this paper, the three dimensional dynamic equations for the drive system are presented. The mode function equations and the frequency equation for the drive system are derived. The natural frequencies and dynamic displacements of the drive system are obtained. Using a finite element method analysis package, ANSYS , the natural frequencies and vibrating modes of the flexible ring for the drive system are simulated. The simulation results are compared to the analytical results above. The research is useful in design and manufacture of the drive system and can be used to design dynamic performance of the drive.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(3):031008-031008-6. doi:10.1115/1.2909083.

In this paper, an approach that enables one to investigate and to reduce the mechanical couplings between the manipulator links is proposed. The presented method is based on equations of motion expressed in terms of the normalized quasivelocities (NQV) introduced originally by Jain and Rodriguez (1995, “Diagonalized Lagrangian Robot Dynamics  ,” IEEE Trans. Rob. Autom., 11, pp. 571–584). As a result, some observations and an algorithm which can be used at the design step of manipulators are done. Moreover, it is shown that for a rigid manipulator a class of integrable desired trajectories of the NQV can be given. The presented strategy was tested by simulation on a 3 DOF spatial manipulator.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(3):031009-031009-6. doi:10.1115/1.2908272.

In this paper, we present an analytical method to investigate the behavior of a two-degree-of-freedom oscillator excited by dry friction. The system consists of two masses connected by linear springs. These two masses are in contact with a driving belt moving at a constant velocity. The contact forces between the masses and the belt are obtained assuming Coulomb’s friction law. Two families of periodic motions are found in closed form. The first one includes stick-slip oscillations with two switches per period, the second one is also composed of stick-slip motion, but includes three switches per period. In both cases, the initial conditions and the time duration of each kind of motions (stick or slip phases) are obtained in analytical form.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(3):031010-031010-8. doi:10.1115/1.2908317.

A complex nonlinear system under state feedback control with a time delay corresponding to two coupled nonlinear oscillators with a parametric excitation is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. In the system without control, phase-locked solutions with period equal to the parametric excitation period are possible only if the oscillator amplitudes are equal, but they depend on the system parameters and excitation amplitude. In many cases, the amplitudes of periodic solutions do not correspond to the technical requirements. It is demonstrated that, if the vibration control terms are added, stable periodic solutions with arbitrarily chosen amplitude and phase can be accomplished. Therefore, an effective vibration control is possible if appropriate time delay and feedback gains are chosen.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(3):031011-031011-8. doi:10.1115/1.2908347.

In this work, the basic problem of order reduction of nonlinear systems subjected to an external periodic excitation is considered. This problem deserves special attention because modes that interact (linearly or nonlinearly) with external excitation dominate the response. These dominant modes are identified and chosen as the “master” modes to be retained in the reduction process. The simplest idea could be to use a linear approach such as the Guyan reduction and choose those modes whose natural frequencies are close to that of external excitation as the master modes. However, this technique does not guarantee accurate results when nonlinear interactions are strong and a nonlinear approach must be adopted. Recently, the invariant manifold technique has been extended to forced problems by “augmenting” the state space, i.e., forcing is treated as an additional state and an invariant manifold is constructed. However, this process does not provide a clear picture of possible resonances and conditions under which an order reduction is possible. In a direct innovative approach suggested here, a nonlinear time-dependent relationship between the dominant and nondominant states is assumed and the dimension of the state space remains the same. This methodology not only yields accurate reduced order models but also explains the consequences of various primary and secondary resonances present in the system. One obtains various reducibility conditions in a closed form, which show interactions among eigenvalues, nonlinearities and the external excitation. One can also recover all “resonance conditions” obtained via perturbation or averaging techniques. The “linear” as well as the “extended invariant manifold” techniques are applied to some typical problems and results for large-scale and reduced order models are compared. It is anticipated that these techniques will provide a useful tool in the analysis and control of large-scale externally excited nonlinear systems.

Commentary by Dr. Valentin Fuster

Errata

J. Comput. Nonlinear Dynam. 2008;3(3):037001-037001-1. doi:10.1115/1.2909085.
FREE TO VIEW

Figures 8–13 in Ref. 1 contain results of the simulation of a very flexible pendulum. The results presented in this paper are obtained using 4 elements. While convergence is achieved in the case of stiff structures using this number of elements; in the extreme case of very flexible structure convergence is not achieved using the proposed number of finite elements (Figs. 8–13) regardless of the method used in formulating the elastic forces. This is mainly due to the extreme bending configurations that include loops. Some of these configurations are associated with singularities when linear constitutive models are used in the large deformation analysis (2-3). Therefore, the authors would like to bring to the attention of the reader that the results presented in Figs. 8–13 are currently being reexamined. However, all the conclusions made in the paper regarding the significance of the coupled deformation modes in the case of large deformation remain valid, as demonstrated by the results presented in Ref. 2. The results presented in Ref. 2 are obtained using a number of finite elements that ensure convergence. There are several ways that are currently being explored by the authors in order to improve the finite element solution in extreme bending problems (3).

Commentary by Dr. Valentin Fuster

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