0

IN THIS ISSUE

### Research Papers

J. Comput. Nonlinear Dynam. 2013;8(4):041001-041001-10. doi:10.1115/1.4023864.

A basic assumption on the data used for nonlinear dynamic model identification is that the data points are continuously collected in chronological order. However, there are situations in practice where this assumption does not hold and we end up with an identification problem from multiple data sets. The problem is addressed in this paper and a new cross-validation-based orthogonal search algorithm for NARMAX model identification from multiple data sets is proposed. The algorithm aims at identifying a single model from multiple data sets so as to extend the applicability of the standard method in the cases, such as the data sets for identification are obtained from multiple tests or a series of experiments, or the data set is discontinuous because of missing data points. The proposed method can also be viewed as a way to improve the performance of the standard orthogonal search method for model identification by making full use of all the available data segments in hand. Simulated and real data are used in this paper to illustrate the operation and to demonstrate the effectiveness of the proposed method.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041002-041002-6. doi:10.1115/1.4023914.

In this paper, the existence and uniqueness of the square-mean almost periodic solutions to a class of the semilinear stochastic equations is studied. In particular, the condition of the uniform exponential stability of the linear operator is essentially removed, only using the exponential dichotomy of the linear operator. Some new criteria ensuring the existence and uniqueness of the square-mean almost periodic solution for the system are presented. Finally, an example of a kind of the stochastic cellular neural networks is given. These obtained results are important in signal processing and the in design of networks.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041003-041003-11. doi:10.1115/1.4023915.

This contribution outlines a computational framework for the analysis of flexible multibody dynamics contact problems. The framework combines a flexible body formalism, specifically, the absolute nodal coordinate formulation (ANCF), with a discrete continuous contact force model to address many-body dynamics problems, i.e., problems with hundreds of thousands of rigid and deformable bodies. Since the computational effort associated with these problems is significant, the analytical framework is implemented to leverage the computational power available on today's commodity graphical processing unit (GPU) cards. The framework developed is validated against commercial and research finite element software. The robustness and efficiency of this approach is demonstrated through numerical simulations. The resulting simulation capability is shown to result in 2 orders of magnitude shorter simulation times for systems with a large number of flexible beams that might typically be encountered in hair or polymer simulations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041004-041004-8. doi:10.1115/1.4023589.

To improve the computational efficiency of nonlinear dynamic probabilistic analysis for aeroengine typical components, an extremum response surface method based on the support vector machine (SVM ERSM) was proposed in this paper. The basic principle was introduced and the mathematical model was established for the SVM ERSM. The probabilistic analysis of turbine casing radial deformation was taken as an example to validate the SVM ERSM considering the influences of nonlinear material property and dynamic heat loads. The results of probabilistic analysis imply that the distribution features of random parameters and the major factors are gained for more accurate the design of casing radial deformation. The SVM ERSM offers a feasible and valid method, which possesses high efficiency and high precision in the nonlinear dynamic probabilistic analysis. Moreover, the SVM ERSM is promising to provide an useful insight for casing dynamic optimal design and the blade-tip clearance control of aeroengine high pressure turbine.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041005-041005-7. doi:10.1115/1.4023739.

This paper addresses the problems of the robust stability and stabilization for fractional order systems based on the uncertain Takagi-Sugeno fuzzy model. A sufficient and necessary condition of asymptotical stability for fractional order uncertain T-S fuzzy model is given, and a parallel distributed compensate fuzzy controller is designed to asymptotically stabilize the model. The results are obtained in terms of linear matrix inequalities. Finally, a numerical example and fractional order Van der Pol system are given to show the effectiveness of our results.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041006-041006-8. doi:10.1115/1.4023916.

An original method for calculating the maximum vibration amplitude of the periodic solution of a nonlinear system is presented. The problem of determining the worst maximum vibration is transformed into a nonlinear optimization problem. The shooting method and the Floquet theory are selected to construct the general nonlinear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the effectiveness and ability of the proposed approach are illustrated through two numerical examples. Numerical examples show that the proposed method can give results with higher accuracy as compared with numerical results obtained by a parameter continuation method and the ability of the present method is also demonstrated.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041007-041007-10. doi:10.1115/1.4023934.

Passive dynamic walking is a manner of walking developed, partially or in whole, by the energy provided by gravity. Studying passive dynamic walking provides insight into human walking and is an invaluable tool for designing energy-efficient biped robots. The objective of this research was to develop a continuous mathematical model of passive dynamic walking, in which the Hunt–Crossley contact model, and the LuGre friction model were used to represent the normal and tangential ground reactions continuously. A physical passive walker was built to validate the proposed mathematical model. A traditional impact-based passive walking model was also used as a reference to demonstrate the advancement of the proposed passive dynamic walking model. The simulated gait of the proposed model matched the gait of the physical passive walker exceptionally well, both in trend and magnitude.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041008-041008-10. doi:10.1115/1.4023933.

Passive dynamic walking is an excellent tool for evaluating biped stability measures, due to its simplicity, but an understanding of the stability, in the classical definition, is required. The focus of this paper is on analyzing the stability of the passive dynamic gait. The stability of the passive walking model, validated in Part I, was analyzed with Lyapunov exponents, and the geometry of the basin of attraction was determined. A novel method was created to determine the 2D projection of the basin of attraction of the model. Using the insights gained from the stability analysis, the relation between the angular momentum and the stability of gait was examined. The angular momentum of the passive walker was not found to correlate to the stability of the gait.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041009-041009-12. doi:10.1115/1.4023866.

Intrinsic localized modes (ILMs) are investigated in an array with N Duffing oscillators that are weakly coupled with each other when each oscillator is subjected to sinusoidal excitation. The purpose of this study is to investigate the behavior of ILMs in nonlinear multi-degree-of-freedom (MDOF) systems. In the theoretical analysis, van der Pol's method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are shown for N = 2 and 3 and compared with the results of the numerical simulations. Basins of attraction are shown for a two-oscillator array with hard-type nonlinearities to examine the possibility of appearance of ILMs when an oscillator is disturbed. The influences of the connecting springs for both hard- and soft-type nonlinearities on the appearance of the ILMs are examined. Increasing the values of the connecting spring constants may cause Hopf bifurcation followed by amplitude modulated motion (AMM) including chaotic vibrations. The influence of the imperfection of an oscillator is also investigated. Bifurcation sets are calculated to show the influence of the system parameters on the excitation frequency range of ILMs. Furthermore, time histories are shown for the case of N = 10, and many patterns of ILMs may appear depending on the initial conditions.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041010-041010-8. doi:10.1115/1.4023548.

The complex orthogonal decomposition (COD), a process of extracting complex modes from complex ensemble data, is summarized, as is the use of complex modal coordinates. A brief assessment is made on how small levels of noise affect the decomposition. The decomposition is applied to the posturing of Caenorhabditis elegans, an intensively studied nematode. The decomposition indicates that the worm has a multimodal posturing behavior, involving a dominant forward locomotion mode, a secondary, steering mode, and likely a mode for reverse motion. The locomotion mode is closer to a pure traveling waveform than the steering mode. The characteristic wavelength of the primary mode is estimated in the complex plane. The frequency is obtained from the complex modal coordinate's complex whirl rate of the complex modal coordinate, and from its fast Fourier transform. Short-time decompositions indicate the variation of the wavelength and frequency through the time record.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041011-041011-18. doi:10.1115/1.4024025.

In this paper, global bifurcations and chaotic dynamics under bounded noise perturbation for the nonlinear normalized radial electric field near plasma are investigated using the Melnikov method. From this analysis, we get criteria that could be useful for designing the model parameters so that the appearance of chaos could be induced (when heating particles) or run out for quiescent H-mode appearance. For this purpose, we use a test of chaos to verify our prediction. We find that, chaos could be enhanced by noise amplitude growing. The results of numerical simulations also reveal that noise intensity modifies the attractor size through power spectra, correlation function, and Poincaré map. The criterion from the Melnikov method which is used to analytically predict the existence of chaotic behavior of the normalized radial electric field in plasma could be a valid tool for predicting harmful parameters values involved in experiment on Tokamak L–H transition.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041012-041012-6. doi:10.1115/1.4023966.

We present a method for detecting right half plane (RHP) roots of fractional order polynomials. It is based on a Nyquist-like criterion with a system-dependent contour which includes all RHP roots. We numerically count the number of origin encirclements of the mapped contour to determine the number of RHP roots. The method is implemented in Matlab, and a simple code is given. For validation, we use a Galerkin based strategy, which numerically computes system eigenvalues (Matlab code is given). We discuss how, unlike integer order polynomials, fractional order polynomials can sometimes have exponentially large roots. For computing such roots we suggest using asymptotics, which provide intuition but require human inputs (several examples are given).

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041013-041013-11. doi:10.1115/1.4024541.

The main contribution of this paper is to demonstrate the feasibility of using one computational environment for developing accurate geometry as well as performing the analysis of detailed biomechanics models. To this end, the finite element (FE) absolute nodal coordinate formulation (ANCF) and multibody system (MBS) algorithms are used in modeling both the contact geometry and ligaments deformations in biomechanics applications. Two ANCF approaches can be used to model the rigid contact surface geometry. In the first approach, fully parameterized ANCF volume elements are converted to surface geometry using parametric relationship that reduces the number of independent coordinate lines. This parametric relationship can be defined analytically or using a spline function representation. In the second approach, an ANCF surface that defines a gradient deficient thin plate element is used. This second approach does not require the use of parametric relations or spline function representations. These two geometric approaches shed light on the generality of and the flexibility offered by the ANCF geometry as compared to computational geometry (CG) methods such as B-splines and NURBS (Non-Uniform Rational B-Splines). Furthermore, because B-spline and NURBS representations employ a rigid recurrence structure, they are not suited as general analysis tools that capture different types of joint discontinuities. ANCF finite elements, on the other hand, lend themselves easily to geometric description and can additionally be used effectively in the analysis of ligaments, muscles, and soft tissues (LMST), as demonstrated in this paper using the knee joint as an example. In this study, ANCF finite elements are used to define the femur/tibia rigid body contact surface geometry. The same ANCF finite elements are also used to model the MCL and LCL ligament deformations. Two different contact formulations are used in this investigation to predict the femur/tibia contact forces; the elastic contact formulation which allows for penetrations and separations at the contact points, and the constraint contact formulation in which the nonconformal contact conditions are imposed as constraint equations, and as a consequence, no separations or penetrations at the contact points are allowed. For both formulations, the contact surfaces are described in a parametric form using surface parameters that enter into the ANCF finite element geometric description. A set of nonlinear algebraic equations that depend on the surface parameters is developed and used to determine the location of the contact points. These two contact formulations are implemented in a general MBS algorithm that allows for modeling rigid and flexible body dynamics.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041014-041014-7. doi:10.1115/1.4023865.

Proper initialization of fractional-order operators has been an ongoing problem, particularly in the application of Laplace transforms with correct initialization terms. In the last few years, a history-function-based initialization along with its corresponding Laplace transform has been presented. Alternatively, an infinite-dimensional state-space representation along with its corresponding Laplace transform has also been presented. The purpose of this paper is to demonstrate that these two approaches to the initialization problem for fractional-order operators are equivalent and that the associated Laplace transforms yield the correct initialization terms and can be used in the solution of fractional-order differential equations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041015-041015-7. doi:10.1115/1.4024542.

We consider a chain of N nonlinear resonators with natural frequency ratios of approximately 2:1 along the chain and weak nonlinear coupling that allows energy to flow between resonators. Specifically, the coupling is such that the response of one resonator parametrically excites the next resonator in the chain, and also creates a resonant back-action on the previous resonator in the chain. This class of systems, which is a generic model for passive frequency dividers, is shown to have rich dynamical behavior. Of particular interest in applications is the case when the high frequency end of the chain is resonantly excited, and coupling results in a cascade of subharmonic bifurcations down the chain. When the entire chain is activated, that is, when all N resonators have nonzero amplitudes, if the input frequency on the first resonator is Ω, the terminal resonator responds with frequency Ω/2N. The details of the activation depend on the strength and frequency of the input, the level of resonator dissipation, and the frequency mistuning in the chain. In this paper we present analytical results, based on perturbation methods, which provide useful predictions about these responses in terms of system and input parameters. Parameter conditions for activation of the entire chain are derived, along with results about other phenomena, such as the period doubling accumulation to full activation, and regions of multistability. We demonstrate the utility of the predictive results by direct comparison with simulations of the equations of motion, and we also present a sample mechanical system that embodies the desired properties. These results are useful for the design and operation of mechanical frequency dividers that are based on subharmonic resonances.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041016-041016-7. doi:10.1115/1.4024729.

In this work, a new three-dimensional fully parameterized triangular plate element based on the absolute nodal coordinate formulation (ANCF) is introduced. This plate element has 12 coordinates per node; therefore, it can be used in thick plate applications. The proposed 12 shape functions are obtained by adding three shape functions to the nine shape functions that were previously used with the ANCF thin triangular plate element. Unlike the existing ANCF thin triangular plate element, which allows only the use of classical Kirchoff's plate theory, the fully parameterized ANCF triangular plate element proposed in this work allows for the use of a general continuum mechanics approach and also allows for a straight forward implementation of general nonlinear constitutive equations. Moreover, all deformation modes including thickness deformation can be captured using the fully parameterized ANCF triangular plate element proposed in this paper. The numerical results obtained in this investigation show that in case of negligible deformation, the fully parameterized ANCF triangular plate element behaves like a rigid body. Moreover, it is found that there is a good agreement between the solutions obtained using the proposed fully parameterized ANCF triangular plate element and the theoretical model in the case of small deformations. Furthermore, it is shown that the results of the proposed element agree well with the results obtained using the existing fully parameterized ANCF rectangular plate element when large deformation conditions are applied. The twist behavior of the proposed element is verified by comparison with the results obtained using a conventional nonlinear rectangular plate element.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041017-041017-7. doi:10.1115/1.4024540.

This paper addresses the use of multidimensional scaling in the evaluation of controller performance. Several nonlinear systems are analyzed based on the closed loop time response under the action of a reference step input signal. Three alternative performance indices, based on the time response, Fourier analysis, and mutual information, are tested. The numerical experiments demonstrate the feasibility of the proposed methodology and motivate its extension for other performance measures and new classes of nonlinearities.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041018-041018-5. doi:10.1115/1.4024852.

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041019-041019-12. doi:10.1115/1.4024887.

The horizontal and vertical motions of a nonlinear spherical buoy, excited by synthetic ocean waves within a wave flume, is numerically and experimentally investigated. First, fluid motion in the wave tank is described using Airy's theory, and the forces on the buoy are determined using a modified form of Morison's equation. The system is then studied statically in order to determine the effects of varying system parameters. Numerical simulations then use the governing equations to compare predicted motions with experimentally observed behavior. Additionally, a commonly used linear formulation is shown to be insufficient in predicting buoy motion, while the nonlinear formulation presented is shown to be accurate.

Topics: Waves , Buoys , Fluids
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2013;8(4):041020-041020-7. doi:10.1115/1.4024970.

This paper presents the formulation of the time-fractional Camassa–Holm equation using the Euler–Lagrange variational technique in the Riemann–Liouville derivative sense and derives an approximate solitary wave solution. Our results witness that He's variational iteration method was a very efficient and powerful technique in finding the solution of the proposed equation.

Commentary by Dr. Valentin Fuster