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IN THIS ISSUE

### Guest Editorial

J. Comput. Nonlinear Dynam. 2016;11(6):060301-060301-2. doi:10.1115/1.4034296.
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My kinematics and dynamics activities began in 1963 when I was offered a job as assistant (not to be confused with an assistant professor) to teach seminars and laboratory for theory of machinery and mechanisms (TMM) at the Polytechnic Institute of CLUJ Romania.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):060302-060302-3. doi:10.1115/1.4034308.
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The following is the story of the growth of the technology now called multibody dynamics. However, more than that, it is the story of the times and some of the people who allowed that technology to develop.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):060303-060303-3. doi:10.1115/1.4034295.
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This note is dedicated to Nicolae Orlandea, John Uicker, and Roger Wehage in recognition of their significant contributions to the field of multibody system dynamics and in appreciation of the value of their innovative work which influenced generations of students, researchers, and practicing engineers

Commentary by Dr. Valentin Fuster

### Research Papers

J. Comput. Nonlinear Dynam. 2016;11(6):061001-061001-7. doi:10.1115/1.4032574.

This article presents a numerical method based on the Adams–Bashforth–Moulton scheme to solve variable-order fractional delay differential equations (VFDDEs). In these equations, the variable-order (VO) fractional derivatives are described in the Caputo sense. The existence and uniqueness of the solutions are proved under Lipschitz condition. Numerical examples are presented showing the applicability and efficiency of the novel method.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061002-061002-8. doi:10.1115/1.4033723.

This paper derives a new operational matrix of the variable-order (VO) time fractional partial derivative involved in anomalous diffusion for shifted Chebyshev polynomials. We then develop an accurate numerical algorithm to solve the 1 + 1 and 2 + 1 VO and constant-order fractional diffusion equation with Dirichlet conditions. The contraction of the present method is based on shifted Chebyshev collocation procedure in combination with the derived shifted Chebyshev operational matrix. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations, and it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, we analyze the convergence of the present method graphically. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061003-061003-7. doi:10.1115/1.4033308.

When deoxyribonucleic (DNA), held at a fixed tension, is subjected to torsional deformations, it responds by forming plectonemic supercoils accompanied by a reduction in its end-to-end extension. This transition from the extended state to the supercoiled state is marked by an abrupt buckling of the DNA accompanied by a rapid “hopping” of the DNA between the extended and supercoiled states. This transition is studied by means of Brownian dynamics simulations using a discrete wormlike-chain (dWLC) model of DNA. The simulations reveal, among other things, the distinct regimes that occur during DNA supercoiling and the probabilities of states within the buckling transition regime.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061004-061004-6. doi:10.1115/1.4033899.

In the present article, we apply a numerical scheme, namely, homotopy analysis Sumudu transform algorithm, to derive the analytical and numerical solutions of a nonlinear fractional differential-difference problem occurring in nanohydrodynamics, heat conduction in nanoscale, and electronic current that flows through carbon nanotubes. The homotopy analysis Sumudu transform method (HASTM) is an inventive coupling of Sumudu transform algorithm and homotopy analysis technique that makes the calculation very easy. The fractional model is also handled with the aid of Adomian decomposition method (ADM). The numerical results derived with the help of HASTM and ADM are approximately same, so this scheme may be considered an alternative and well-organized technique for attaining analytical and numerical solutions of fractional model of discontinued problems. The analytical and numerical results derived by the application of the proposed technique reveal that the scheme is very effective, accurate, flexible, easy to apply, and computationally very appropriate for such type of fractional problems arising in physics, chemistry, biology, engineering, finance, etc.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061005-061005-16. doi:10.1115/1.4033657.

The finite-element approach of absolute nodal coordinate formulation (ANCF) is a possible way to simulate the deployment dynamics of a large-scale mesh reflector of satellite antenna. However, the large number of finite elements of ANCF significantly increases the dimension of the dynamic equations for the deployable mesh reflector and leads to a great challenge for the efficient dynamic simulation. A new parallel computation methodology is proposed to solve the differential algebraic equations for the mesh reflector multibody system. The mesh reflector system is first decomposed into several independent subsystems by cutting its joints or finite-element grids. Then, the Schur complement method is used to eliminate the internal generalized coordinates of each subsystem and the Lagrange multipliers for joint constraint equations associated with the internal variables. With an increase of the number of subsystems, the dimension of simultaneous linear equations generated in the numerical solution process will inevitably increase. By using the multilevel decomposition approach, the dimension of the simultaneous linear equations is further reduced. Two numerical examples are used to validate the efficiency and accuracy of the proposed parallel computation methodology. Finally, the dynamic simulation for a 500 s deployment process of a complex AstroMesh reflector with over 190,000 generalized coordinates is efficiently completed within 78 hrs.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061006-061006-9. doi:10.1115/1.4034047.

In this paper, the problem of reliable sampled-data control design with strict dissipativity for a class of linear continuous-time-delay systems against nonlinear actuator faults is studied. The main objective of this paper is to design a reliable sampled-data controller to ensure a strictly dissipative performance for the closed-loop system. Based on the linear matrix inequality (LMI) optimization approach and Wirtinger-based integral inequality, a new set of sufficient conditions is established for reliable dissipativity analysis of the considered system by assuming the mixed actuator fault matrix to be known. Then, the proposed result is extended to unknown fault matrix case. Also, the reliable sampled-data controller with strict dissipativity is designed by solving a convex optimization problem which can be easily solved by using standard numerical algorithms. Finally, a numerical example based on liquid propellant rocket motor with a pressure feeding system model is presented to illustrate the effectiveness of the developed control design technique.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061007-061007-12. doi:10.1115/1.4034049.

To protect the aircraft flight safety across the envelope of angle of attack, bifurcation analysis and backstepping method are investigated to predict and suppress the unstable nonlinear flight phenomena. By applying bifurcation analysis and continuation method to the flight motion, the onsets of both the chaotic phenomenon called falling leaf motion and the catastrophe phenomenon named coupled jump behavior are derived. To stabilize these unstable motions, a backstepping and disturbance observer based flight controller is designed. According to the main function of the control surface, we divide the flight controller into the airspeed subsystem and the flight path subsystem, where the airspeed subsystem is regulated by an adaptive dynamic inversion controller while the flight path subsystem is stabilized by a third-order compound controller. Considering the parametric uncertainties of aerodynamics, three sliding mode disturbance observers are presented as compensators to approximate the compound uncertainties. Simulations demonstrate that the proposed controller can recover the aircraft from falling leaf motion or coupled jump behavior to straight level flying.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061008-061008-13. doi:10.1115/1.4033755.

In this paper, two approaches are used to solve a class of the distributed optimal control problems defined on rectangular domains. In the first approach, a meshless method for solving the distributed optimal control problems is proposed; this method is based on separable representation of state and control functions. The approximation process is done in two fundamental stages. First, the partial differential equation (PDE) constraint is transformed to an algebraic system by weighted residual method, and then, Bezier curves are used to approximate the action of control and state. In the second approach, the Bernstein polynomials together with Galerkin method are utilized to solve partial differential equation coupled system, which is a necessary and sufficient condition for the main problem. The proposed techniques are easy to implement, efficient, and yield accurate results. Numerical examples are provided to illustrate the flexibility and efficiency of the proposed method.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061009-061009-8. doi:10.1115/1.4033920.
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Forecasting bifurcations is a significant challenge and an important need in several applications. Most of the existing forecasting approaches focus on bifurcations in nonoscillating systems. However, subcritical and supercritical flutter (Hopf) bifurcations are very common in a variety of systems, especially fluid–structural systems. This paper presents a unique approach to forecast (nonlinear) flutter based on observations of the system only in the prebifurcation regime. The proposed method is based on exploiting the phenomenon of critical slowing down (CSD) in oscillating systems near certain bifurcations. Techniques are introduced to enhance the prediction accuracy for cases of low-frequency oscillations and large-dimensional dynamical systems. The method is applied to an aeroelastic system responding to gust loads. Numerical results are provided to demonstrate the performance of the method in predicting the postbifurcation regime accurately in both supercritical and subcritical cases.

Topics: Bifurcation
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061010-061010-7. doi:10.1115/1.4034048.

In this paper, we first propose a fractional-order energy demand–supply system, with the background of the energy resources demand in the eastern regions of China and the energy resources supply in the western regions of China. Then, we confirm the energy resource attractor with a necessary condition about the existence of chaotic behaviors. By employing an improved version of Adams Bashforth Moulton algorithm, we use three cases with different fractional values to verify the necessary condition. Finally, chaos control of fractional-order energy demand–supply system is investigated by two different control strategies: a linear feedback control and an adaptive switching control strategy via a single control input. Numerical simulations show that the energy demand and import in Eastern China and energy supply in Western China are self-feedback controlled around the system’s equilibrium point.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061011-061011-12. doi:10.1115/1.4033895.

In this paper, we establish a mathematical framework that allows us to optimize the speed profile and select the optimal gears for heavy-duty vehicles (HDVs) traveling on highways while varying parameters. The key idea is to solve the analogous boundary value problem (BVP) analytically for a simple scenario (linear damped system with quadratic elevation profile) and use this result to initialize a numerical continuation algorithm. Then, the numerical algorithm is used to investigate how the optimal solution changes with parameters. In particular, we gradually introduce nonlinearities (air resistance and engine saturation), implement different elevation profiles, and incorporate external perturbations (headwind and traffic). This approach enables real-time optimization in dynamic traffic conditions, therefore may be implemented on-board.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061012-061012-10. doi:10.1115/1.4034123.

This paper investigates the nonlinear responses of a typical two-dimensional airfoil with control surface freeplay and cubic pitch stiffness in an incompressible flow. The differential transform (DT) method is applied to the aeroelastic system. Due to the nature of this method, it is capable of providing analytical solutions in forms of Taylor series expansions in each subdomain between two adjacent sampling points. The results demonstrate that the DT method can successfully detect nonlinear aeroelastic responses such as limit cycle oscillations (LCOs), chaos, bifurcation, and flutter phenomenon. The accuracy and efficiency of this method are verified by comparing it with the RK (Henon) method. In addition to ordinary differential equations (ODEs), the DT method is also a powerful tool for directly solving integrodifferential equations. In this paper, the original aeroelastic system of integrodifferential equations is handled directly by the DT method. With no approximation or simplification imposed on the integral terms of aerodynamic function, the resulted solutions are closer to representing the real dynamical behavior.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061013-061013-7. doi:10.1115/1.4033685.

The objective of this paper is to identify the suitable advance controller among optimized proportional–integral–derivative (O-PID), improved self-tuning fuzzy-PID (ISTF-PID), advanced fuzzy nonadaptive PID (AF-NA-PID), and AF-adaptive PID (AF-A-PID) controllers for speed control of nonlinear hybrid electric vehicle (HEV) system. The conventional PID (C-PID) controller cannot tackle the nonlinear systems effectively and gives a poor tracking and disturbance rejection performance. The performances of HEV with the proposed advance controllers are compared with existing C-PID, STF-PID, and conventional fuzzy PID (C-F-PID) controllers. The proposed controllers are designed to achieve the desired vehicle speed and rejection of disturbance due to road grade with reduced pollution and fuel economy.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061014-061014-7. doi:10.1115/1.4034391.

The time fractional subdiffusion equation (FSDE) as a class of anomalous diffusive systems has obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order $0<α<1$. Since analytically solving this problem is often impossible, proposing numerical methods for its solution has practical importance. In this paper, an efficient and accurate Galerkin method based on the Legendre wavelets (LWs) is proposed for solving this equation. The time fractional derivatives are described in the Riemann–Liouville sense. To do this, we first transform the original subdiffusion problem into an equivalent problem with fractional derivatives in the Caputo sense. The LWs and their fractional operational matrix (FOM) of integration together with the Galerkin method are used to transform the problem under consideration into the corresponding linear system of algebraic equations, which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account, automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061015-061015-10. doi:10.1115/1.4034393.

The determination of the contact points between two bodies with analytically described boundaries can be viewed as the limiting case of the extremal point problem, where the distance between the bodies is vanishing. The advantage of this approach is that the solutions can be computed efficiently along with the generalized state during time integration of a multibody system by augmenting the equations of motion with the corresponding extremal point conditions. Unfortunately, these solutions can degenerate when one boundary is concave or both boundaries are nonconvex. We present a novel method to derive degeneracy and nondegeneracy conditions that enable the determination of the type and codimension of all the degenerate solutions that can occur in plane contact problems involving two bodies with smooth boundaries. It is shown that only divergence bifurcations are relevant, and thus, we can simplify the analysis of the degeneracy by restricting the system to its one-dimensional center manifold. The resulting expressions are then decomposed by applying the multinomial theorem resulting in a computationally efficient method to compute explicit expressions for the Lyapunov coefficients and transversality conditions. Furthermore, a procedure to analyze the bifurcation behavior qualitatively at such solution points based on the Tschirnhaus transformation is given and demonstrated by examples. The application of these results enables in principle the continuation of all the solutions simultaneously beyond the degeneracy as long as their number is finite.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061016-061016-16. doi:10.1115/1.4034394.

Most of the nonlinear system identification techniques described in the existing literature required force and response information at all excitation degrees-of-freedom (DOFs). For cases, where the excitation comes from base motion, those methods cannot be applied as it is not feasible to obtain the measurements of motion at all DOFs from an experiment. The objective of this research is to develop the methodology for the nonlinear system identification of continuous, multimode, and lightly damped systems, where the excitation comes from the moving base. For this purpose, the closed-form expression for the equivalent force also known as the pseudo force from the measured data for the base-excited structure is developed. A hybrid model space is developed to find out the nonlinear restoring force at the nonlinear DOFs. Once the nonlinear restoring force is obtained, the nonlinear parameters are extracted using “multilinear least square regression” in a modal space. A modal space is chosen to express the direct and cross-coupling nonlinearities. Using a cantilever beam as an example, the proposed methodology is demonstrated, where the experimental setup, testing procedure, data acquisition, and data processing are presented. The example shows that the method proposed here is systematic and constructive for nonlinear parameter identification for base-excited structure.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061017-061017-11. doi:10.1115/1.4034432.

In this paper, we present a new type of fractional operator, the Caputo–Katugampola derivative. The Caputo and the Caputo–Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy-type problem, with dependence on the Caputo–Katugampola derivative, is proved. A decomposition formula for the Caputo–Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation (FDE).

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061018-061018-9. doi:10.1115/1.4034491.

This paper focuses on multiscale dynamics occurring in steam supply systems. The dynamics of interest are originally described by a distributed-parameter model for fast steam flows over a pipe network coupled with a lumped-parameter model for slow internal dynamics of boilers. We derive a lumped-parameter model for the dynamics through physically relevant approximations. The derived model is then analyzed theoretically and numerically in terms of existence of normally hyperbolic invariant manifold in the phase space of the model. The existence of the manifold is a dynamical evidence that the derived model preserves the slow–fast dynamics, and suggests a separation principle of short-term and long-term operations of steam supply systems, which is analog to electric power systems. We also quantitatively verify the correctness of the derived model by comparison with brute-force simulation of the original model.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(6):061019-061019-10. doi:10.1115/1.4034566.

This work presents an experimental and numerical study on the dynamic properties of viscoelastic (VE) microvibration damper under microvibration conditions at different frequencies and temperatures. The experimental results show that the storage modulus and the loss factor of VE microvibration damper both increase with increasing frequency but decrease with increasing temperature. To explicitly and accurately represent the temperature and frequency effects on the dynamic properties of VE microvibration damper, a modified standard solid model based on a phenomenological model and chain network model is proposed. A Gaussian chain spring and a temperature-dependent dashpot are employed to reflect the temperature effect in the model, and the frequency effect is considered with the nature of the standard solid model. Then, the proposed model is verified by comparing the numerical results with the experimental data. The results show that the proposed model can accurately describe the dynamic properties of VE microvibration damper at different temperatures and frequencies.

Commentary by Dr. Valentin Fuster

### Book Review

J. Comput. Nonlinear Dynam. 2016;11(6):066501-066501-1. doi:10.1115/1.4034731.
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Commentary by Dr. Valentin Fuster