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IN THIS ISSUE

### Research Papers

J. Comput. Nonlinear Dynam. 2017;13(1):011001-011001-7. doi:10.1115/1.4037922.

The multiterm fractional variable-order differential equation has a massive application in physics and engineering problems. Therefore, a numerical method is presented to solve a class of variable order fractional differential equations (FDEs) based on an operational matrix of shifted Chebyshev polynomials of the fourth kind. Utilizing the constructed operational matrix, the fundamental problem is reduced to an algebraic system of equations which can be solved numerically. The error estimate of the proposed method is studied. Finally, the accuracy, applicability, and validity of the suggested method are illustrated through several examples.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011002-011002-11. doi:10.1115/1.4037593.

Catastrophic chaotic and hyperchaotic dynamical behaviors have been experimentally observed in the so-called “smart valves” network, given certain critical parameters and initial conditions. The centralized network-based control of these coupled systems may effectively mitigate the harmful dynamics of the valve-actuator configuration which can be potentially caused by a remote set and would gradually affect the whole network. In this work, we address the centralized control of two bi-directional solenoid actuated butterfly valves dynamically coupled in series subject to the chaotic and hyperchaotic dynamics. An interconnected adaptive scheme is developed and examined to vanish both the chaotic and hyperchaotic dynamics and return the coupled network to its safe domain of operation.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011003-011003-8. doi:10.1115/1.4037923.

In this paper, we investigate the linear and nonlinear response of shape memory alloy (SMA)-based Duffing and quadratic oscillator under large deflection conditions. In this study, we first present thermomechanical constitutive modeling of SMA with a single degree-of-freedom system. Subsequently, we solve equation to obtain linear frequency and nonlinear frequency response using the method of harmonic balance and validate it with numerical solution as well as averaging method under the isothermal condition. However, for nonisothermal condition, we analyze the influence of cubic and quadratic nonlinearity on nonlinear response based on method of harmonic balance. Analysis of results leads to various ways of controlling the nature and extent of nonlinear response of SMA-based oscillators. Such findings can be effectively used to control external vibration of different systems.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011004-011004-6. doi:10.1115/1.4037414.

In this study, we describe the fractional convection operator for the first time and present its discrete form with second-order convergence. A numerical scheme for the fractional-convection–diffusion equation is also constructed in order to get insight into the fractional convection behavior visually. Then, we study the fractional-convection-dominated diffusion equation which has never been considered, where the diffusion is normal and is characterized by the Laplacian. The interesting fractional convection phenomena are observed through numerical simulation. Moreover, we investigate the fractional-convection-dominated-diffusion equation which is studied for the first time either, where the convection and the diffusion are both in the fractional sense. The corresponding fractional convection phenomena are displayed via computer graphics as well.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011005-011005-7. doi:10.1115/1.4037672.

This paper considers the robust synchronization of chaotic systems in the presence of nonsymmetric input saturation constraints. The synchronization happens between two nonlinear master and slave systems in the face of model uncertainties and external disturbances. A new adaptive sliding mode controller is designed in a way that the robust synchronization occurs. In this regard, a theorem is proposed, and according to the Lyapunov approach the adaptation laws are derived, and it is proved that the synchronization error converges to zero despite of the uncertain terms in master and slave systems and nonsymmetric input saturation constraints. Finally, the proposed method is applied on chaotic gyro systems to show its applicability. Computer simulations verify the theoretical results and also show the effective performance of the proposed controller.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011006-011006-11. doi:10.1115/1.4037595.

This paper investigates the possibility of energy generation via pendulum rotations when the source of vertical excitation is chaotic in nature. The investigations are conducted using an additional height-adjustable mechanism housing a secondary spring to optimize a configuration of experimental pendulum setup. Chaotic oscillations of the pendulum pivot are made possible at certain excitation conditions due to a piecewise-linear stiffness characteristic introduced by the modification. A velocity control method is applied to maintain the rotational motion of the pendulum as it interacts with the vertical oscillator. The control input is affected by a motor, and a generator is used to quantify the energy extraction. The experimental results imply the feasibility of employing a pendulum device in a chaotic vibratory environment for energy harvesting purpose.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011007-011007-9. doi:10.1115/1.4037225.

The Kidder equation, $y″(x)+2xy′(x)/1−βy(x)=0, x∈[0,∞), β∈[0,1]$ with $y(0)=1$, and $y(∞)=0$, is a second-order nonlinear two-point boundary value ordinary differential equation (ODE) on the semi-infinite domain, with a boundary condition in the infinite that describes the unsteady isothermal flow of a gas through a semi-infinite micro–nano porous medium and has widely used in the chemical industries. In this paper, a hybrid numerical method is introduced for solving this equation. First, by using the method of quasi-linearization, the equation is converted to a sequence of linear ODEs. Then these linear ODEs are solved by using the rational Legendre functions (RLFs) collocation method. By using 200 collocation points, we obtain a very good approximation solution and the value of the initial slope $y′(0)=−1.19179064971942173412282860380015936403$ for $β=0.50$, highly accurate to 38 decimal places. The convergence of numerical results is shown by decreasing the residual errors when the number of collocation points increases.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011008-011008-11. doi:10.1115/1.4037416.

A new three-dimensional (3D) chaotic system is proposed with four nonlinear terms which include two quadratic terms. To analyze the dynamical properties of the new system, mathematical tools such as Lyapunov exponents (LEs), Kaplan–York dimensions, observability constants, and bifurcation diagram have been exploited. The results of these calculations verify the specific features of the new system and further determine the effect of different system parameters on its dynamics. The proposed system has been experimentally implemented as an analog circuit which practically confirms its predicted chaotic behavior. Moreover, the problem of master–slave synchronization of the proposed chaotic system is considered. To solve this problem, we propose a new method for designing a nonfragile Takagi–Sugeno (T–S) fuzzy static output feedback synchronizing controller for a general chaotic T–S system and applied the method to the proposed system. Some practical advantages are achieved employing the new nonlinear controller as well as using system output data instead of the full-state data and considering gain variations because of the uncertainty in values of practical components used in implementation the controller. Then, the designed controller has been realized using analog devices to synchronize two circuits with the proposed chaotic dynamics. Experimental results show that the proposed nonfragile controller successfully synchronizes the chaotic circuits even with inexact analog devices.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011009-011009-7. doi:10.1115/1.4037417.

A number of strategies can be followed for the real-time simulation of multibody systems. The main contributing factor to computational efficiency is usually the algorithm itself (the number of equations and their structure, the number of coordinates, the time integration scheme, etc.). Additional (but equally important) aspects have to do with implementation (linear solvers, sparse matrices, parallel computing, etc.). In this paper, an iterative refinement technique is introduced into a semirecursive multibody formulation. First, the formulation is summarized and its basic features are highlighted. Then, the basic goal is to iteratively solve the fundamental system of equations to obtain the accelerations. The iterative process is applied to compute corrections of the solution in an economic way, terminating as soon as a given precision is reached. We show that, upon implementation of this method, the computation time can be reduced at a very low implementation and accuracy costs. Two vehicles are simulated to prove the numerical benefits, namely a 16-degrees-of-freedom (DOF) sedan vehicle and a 40-degrees-of-freedom semitrailer truck. In short, a simple method to iteratively solve for the accelerations of vehicle systems in an efficient way is presented.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011010-011010-8. doi:10.1115/1.4037597.

This paper presents an efficient approximation schemes for the numerical solution of a fractional variational problem (FVP) and fractional optimal control problem (FOCP). As basis function for the trial solution, we employ the shifted Jacobi orthonormal polynomial. We state and derive a new operational matrix of right-sided Caputo fractional derivative of such polynomial. The new methodology of the present schemes is based on the derived operational matrix with the help of the Gauss–Lobatto quadrature formula and the Lagrange multiplier technique. Accordingly, the aforementioned problems are reduced into systems of algebraic equations. The error bound for the operational matrix of right-sided Caputo derivative is analyzed. In addition, the convergence of the proposed approaches is also included. The results obtained through numerical procedures and comparing our method with other methods demonstrate the high accuracy and powerful of the present approach.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011011-011011-10. doi:10.1115/1.4037796.

The complex architecture of aircraft engines requires demanding computational efforts when the dynamic coupling of their components has to be predicted. For this reason, numerically efficient reduced-order models (ROM) have been developed with the aim of performing modal analyses and forced response computations on complex multistage assemblies being computationally fast. In this paper, the flange joint connecting two turbine disks of a multistage assembly is studied as a source of nonlinearities due to friction damping occurring at the joint contact interface. An analytic contact model is proposed to calculate the local microslip based on the different deformations that the two flanges in contact take during vibration. The model is first introduced using a simple geometry representing two flanges in contact, and then, it is applied to a reduced finite element model in order to calculate the nonlinear forced response.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011012-011012-9. doi:10.1115/1.4037924.

Based on the special dynamical property of continuous transition at certain degenerate grazing points in the single-degree-of-freedom impact oscillator, the control problem of the grazing-induced chaos is investigated in this paper. To design degenerate grazing bifurcations, we show how to obtain the degenerate grazing points of the 1/n impact periodic motions by the existence and stability analysis first. Then, a discrete-in-time feedback control strategy is used to suppress the grazing-induced chaos into the 1/n impact periodic motions precisely by the desired degenerate grazing bifurcation. The feasibility of the control strategy is verified by numerical simulations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011013-011013-12. doi:10.1115/1.4037226.

While several curvature expressions have been used in the literature, some of these expressions differ from basic geometry definitions and lead to kinematic coupling between bending and shear deformations. This paper uses three different elastic force formulations in order to examine the effect of the curvature definition in the large displacement analysis of beams. In the first elastic force formulation, a general continuum mechanics approach (method 1) based on the nonlinear strain–displacement relationship is used. The second approach (method 2) is based on a classical nonlinear beam theory, in which a curvature expression consistent with differential geometry and independent of the shear deformation is used. The third elastic force formulation (method 3) employs a curvature expression that depends on the shear angle. In order to examine numerically the effect of using different curvature definitions, three different planar beam elements are used. The first element (element I) is the fully parameterized absolute nodal coordinate formulation (ANCF) shear deformable beam element. The second element (element II) is an ANCF consistent rotation-based formulation (CRBF) shear deformable beam element obtained from element I by consistently replacing the position gradient vectors by rotation parameters. The third element (element III) is a low-order bilinear ANCF/CRBF finite element in which nonzero differential geometry-based curvature definition cannot be obtained because of the low order of interpolation. Numerical results are obtained using the three elastic force formulations and the three finite elements in order to shed light on the definition of bending and shear in the large displacement analysis of beams. The results obtained in this investigation show that the use of method 2, with a penalty formulation that restricts the excessive cross section deformation, can improve significantly the convergence of the ANCF finite element.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011014-011014-13. doi:10.1115/1.4037764.

A complete dynamic model of a timing belt drive system with an oval cogged pulley and an auto-tensioner is established in this work. Periodic torsional vibrations of all accessory pulleys and the tensioner arm are calculated using a modified incremental harmonic balance (MIHB) method based on the complete dynamic model. Calculated results from the MIHB method are verified by comparing them with those obtained from Runge–Kutta method. Influences of tensioner parameters and oval pulley parameters on torsional vibrations of camshafts and other accessory pulleys are investigated. A sequence quadratic programing (SQP) method with oval pulley parameters selected as design variables is applied to minimize the overall torsional vibration amplitude of all the accessory pulleys and the tensioner arm in the timing belt drive system at different operational speeds. It is demonstrated that torsional vibrations of the timing belt drive system are significantly reduced by matching belt stretch with speed variations of the crankshaft and fluctuating torque loads on camshafts. The timing belt drive system with optimal oval parameters given in this work has better performance in the overall torsional vibration of the system than that with oval parameters provided by the kinematic model and the simplified dynamic model in previous research.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):011015-011015-7. doi:10.1115/1.4038290.

This research introduces a model of a delayed reaction–diffusion fractional neural network with time-varying delays. The Mittag–Leffler-type stability of the solutions is investigated, and new sufficient conditions are established by the use of the fractional Lyapunov method. Mittag–Leffler-type synchronization criteria are also derived. Three illustrative examples are established to exhibit the proposed sufficient conditions.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Comput. Nonlinear Dynam. 2017;13(1):014501-014501-7. doi:10.1115/1.4037413.

A topological analysis of the attractor associated with the Moore–Spiegel nonlinear system is performed, following the basic idea laid down by Gilmore and Lefranc (2002, The Topology of Chaos, Wiley, Hoboken, NJ). Starting with the usual fixed point analysis and their stability, we proceed to study in detail the process of chaotic orbit extraction with the help of close return map. This is then used to construct the symbolic dynamics associated with it, which is helpful in understanding the sequential change taking place inside the attractor. In the next part, we show how to characterize the evolution of the attractor from its birth to the crisis by finding out the homoclinic orbit and the corresponding unstable manifold. In the concluding part of the paper, we show how all the pertinent information of the attractor can be encoded in the template, leading to the explicit realization of linking numbers and the relative rotation rates. In the concluding section, we have touched upon a new approach to chaotic dynamics, using the flow curvature manifold to display the relative positioning of the attractor in relation to the fixed points and the null lines.

Topics: Attractors
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):014502-014502-5. doi:10.1115/1.4037594.

The forced vibration of gyroscopic continua is investigated by taking the pipes conveying fluid as an example. The nonlinear normal modes and a numerical iterative approach are used to perform numerical response analysis. The nonlinear nonautonomous governing equations are transformed into a set of pseudo-autonomous ones by using the harmonic balance method. Based on the pseudo-autonomous system, the nonlinear normal modes are constructed by the invariant manifold method on the state space and substituted back into the original discrete equations. By repeating the above mentioned steps, the dynamic responses can be numerically obtained asymptotically using such iterative approach. Quadrature phase difference between the general coordinates is verified for the gyroscopic system and traveling waves instead of standing waves are found in the time-domain complex modal analysis.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(1):014503-014503-6. doi:10.1115/1.4037415.

This technical brief revisits the method outlined in Tasora and Anitescu 2011 [“A Matrix-Free Cone Complementarity Approach for Solving Large-Scale, Nonsmooth, Rigid Body Dynamics,” Comput. Methods Appl. Mech. Eng., 200(5–8), pp. 439–453], which was introduced to solve the rigid multibody dynamics problem in the presence of friction and contact. The discretized equations of motion obtained here are identical to the ones in Tasora and Anitescu 2011 [“A Matrix-Free Cone Complementarity Approach for Solving Large-Scale, Nonsmooth, Rigid Body Dynamics,” Comput. Methods Appl. Mech. Eng., 200(5–8), pp. 439–453]; what is different is the process of obtaining these equations. Instead of using maximum dissipation conditions as the basis for the Coulomb friction model, the approach detailed uses complementarity conditions that combine with contact unilateral constraints to augment the classical index-3 differential algebraic equations of multibody dynamics. The resulting set of differential, algebraic, and complementarity equations is relaxed after time discretization to a cone complementarity problem (CCP) whose solution represents the first-order optimality condition of a quadratic program with conic constraints. The method discussed herein has proven reliable in handling large frictional contact problems. Recently, it has been used with promising results in fluid–solid interaction applications. Alas, this solution is not perfect, and it is hoped that the detailed account provided herein will serve as a starting point for future improvements.

Commentary by Dr. Valentin Fuster