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Research Papers

J. Comput. Nonlinear Dynam. 2018;13(8):081001-081001-7. doi:10.1115/1.4040459.

To synchronize quadratic chaotic systems, a synchronization scheme based on simultaneous estimation of nonlinear dynamics (SEND) is presented in this paper. To estimate quadratic terms, a compensator including Jacobian matrices in the proposed master–slave schematic is considered. According to the proposed control law and Lyapunov theorem, the asymptotic convergence of synchronization error to zero is proved. To identify unknown parameters, an adaptive mechanism is also used. Finally, a number of numerical simulations are provided for the Lorenz system and a memristor-based chaotic system to verify the proposed method.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(8):081002-081002-11. doi:10.1115/1.4040343.

This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

Topics: Wavelets
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(8):081003-081003-9. doi:10.1115/1.4040344.

We study the stability of a pre-tensioned, heavy cable traveling vertically against gravity at a constant speed. The cable is modeled as a slender beam incorporating rotary inertia. Gravity modifies the tension along the traveling cable and introduces spatially varying coefficients in the equation of motion, thereby precluding an analytical solution. The onset of instability is determined by employing both the Galerkin method with sine modes and finite element (FE) analysis to compute the eigenvalues associated with the governing equation of motion. A spectral stability analysis is necessary for traveling cables where an energy stability analysis is not comprehensive, because of the presence of gyroscopic terms in the governing equation. Consistency of the solution is checked by direct time integration of the governing equation of motion with specified initial conditions. In the stable regime of operations, the rate of change of total energy of the system is found to oscillate with bounded amplitude indicating that the system, although stable, is nonconservative. A comprehensive stability analysis is carried out in the parameter space of traveling speed, pre-tension, bending rigidity, external damping, and the slenderness ratio of the cable. We conclude that pre-tension, bending rigidity, external damping, and slenderness ratio enhance the stability of the traveling cable while gravity destabilizes the cable.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(8):081004-081004-7. doi:10.1115/1.4040411.

Complex and real valued exact solutions to some reaction-diffusion equations are suggested by using homogeneous balance and Sine-Gordon equation expansion method. The predicted solution of finite series of some hyperbolic functions is determined by using some relations between the hyperbolic functions and the trigonometric functions based on Sine-Gordon equation and traveling wave transform. The Newel–Whitehead–Segel (NWSE) and Zeldovich equations (ZE) are solved explicitly. Some complex valued solutions are depicted in real and imaginary components for some particular choice of parameters.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(8):081005-081005-10. doi:10.1115/1.4040540.

This paper deals with the forced response analysis of chains of thin elastic beams that are subject to periodic external loading and frictionless intermittent contact between the beams. Our study shows that the beams show nonlinear resonances whose frequencies are the same as the linear resonant frequencies if all the beams have the same stiffness. Furthermore, it is also shown that small gaps between the beams and small deviation or mistuning in the stiffness of each beam can cause drastic changes in the nonlinear resonant frequencies of the system dynamics. The system is modeled as a semidiscrete system of piecewise-linear oscillators with multiple degrees-of-freedom (DOF) that are subject to unilateral constraints, which is derived from a finite element discretization of the beams. The resulting equations of motions are solved by a second-order numerical integration scheme, and steady-state solutions are sought for various driving frequencies. Results of parametric studies with respect to the gaps between the beams and the number of beams are presented to discuss how these parameters affect the resonant behavior of the system.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(8):081006-081006-7. doi:10.1115/1.4039838.

Dynamic relaxation (DR) is the most widely used approach for static equilibrium analyses. Specifically, DR compels dynamic systems to converge to a static equilibrium through the addition of fictitious damping. DR methods are classified by the method in which fictitious damping is applied. Conventional DR methods use a fictitious mass matrix to increase the fictitious damping while maintaining numerical stability. There are many calculation methods for the fictitious mass matrix; however, it is difficult to select the appropriate method. In addition, these methods require a stiffness matrix of a model, which makes it difficult to apply nonlinear models. To resolve these problems, a new DR method that uses continuous kinetic damping (CKDR) is proposed in this study. The proposed method does not require the fictitious mass matrix and any tuning coefficients, and it possesses a second-order convergence rate. The aforementioned advantages are unique and significant when compared to those of conventional methods. The stability and convergence rate were analyzed by using an eigenvalue analysis and demonstrated by simulating nonlinear models of a pendulum and cable. Simple but representative models were used to clearly demonstrate the features of the proposed DR method and to enable the reproducibility of the verification results.

Commentary by Dr. Valentin Fuster

Technical Brief

J. Comput. Nonlinear Dynam. 2018;13(8):084501-084501-5. doi:10.1115/1.4040342.

This paper presents an accurate and efficient hybrid solution method, based on Newmark-β algorithm, for solving nonlinear oscillators containing fractional derivatives (FDs) of arbitrary order. Basically, this method employs a quadrature method and the Newmark-β algorithm to handle FDs and integer derivatives, respectively. To reduce the computational burden, the proposed approach provides a strategy to avoid nonlinear algebraic equations arising routinely in the Newmark-β algorithm. Numerical results show that the presented method has second-order accuracy. Importantly, it can be applied to both linear and nonlinear oscillators with FDs of arbitrary order, without losing any precision and efficiency.

Topics: Algorithms , Algebra
Commentary by Dr. Valentin Fuster

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