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IN THIS ISSUE

### Research Papers

J. Comput. Nonlinear Dynam. 2015;11(4):041001-041001-10. doi:10.1115/1.4031288.

This paper aims at analyzing the coupled nonlinear dynamical behavior of geometrically imperfect shear deformable extensible microbeams based on the third-order shear deformation and modified couple stress theories. Using Hamilton's principle and taking into account extensibility, the three nonlinear coupled continuous expressions are obtained for an initially slightly curved (i.e., a geometrically imperfect) microbeam, describing the longitudinal, transverse, and rotational motions. A high-dimensional Galerkin scheme is employed, together with an assumed-mode technique, in order to truncate the continuous system with an infinite number of degrees of freedom into a discretized model with sufficient degrees of freedom. This high-dimensional discretized model is solved by means of the pseudo-arclength continuation technique for the system at the primary resonance, and also by direct time-integration to characterize the dynamic response at a fixed forcing amplitude and frequency; stability analysis is conducted via the Floquet theory. Apart from analyzing the nonlinear resonant response, the linear natural frequencies are obtained via an eigenvalue analysis. Results are shown through frequency–response curves, force–response curves, time traces, phase-plane portraits, and fast Fourier transforms (FFTs). The effect of taking into account the length-scale parameter on the coupled nonlinear dynamic response of the system is also highlighted.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041002-041002-11. doi:10.1115/1.4031290.

This paper investigates the internal energy transfer and modal interactions in the dynamical behavior of slightly curved microplates. Employing the third-order shear deformation theory, the microplate model is developed taking into account geometric nonlinearities as well as the modified couple stress theory; the initial curvature is modeled by an initial imperfection in the out-of-plane direction. The in-plane displacements and inertia are retained, and the coupled out-of-plane, rotational, and in-plane motion characteristics are analyzed. Specifically, continuous models are developed for kinetic and potential energies as well as damping and external works; these are balanced and reduced via Lagrange's equations along with an assumed-mode technique. The reduced-order model is then solved numerically by means of a continuation technique; stability analysis is performed by means of the Floquet theory. The possibility of the occurrence of modal interactions and internal energy transfers is verified via a linear analysis on different natural frequencies of the system. The nonlinear resonant response of the system is obtained for the cases with internal energy transfer, and energy transfer mechanisms are analyzed; as we shall see, the presence of an initial curvature affects the system dynamics substantially. The importance of taking into account small-size effects is also shown by discovering this fact that both the linear and nonlinear internal energy transfer mechanisms are shifted substantially if this effect is ignored.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041003-041003-9. doi:10.1115/1.4031675.

This paper presents two approaches to the stability analysis of flexible dynamical systems in the time domain. The first is based on the partial Floquet theory and proceeds in three steps. A preprocessing step evaluates optimized signals based on the proper orthogonal decomposition (POD) method. Next, the system stability characteristics are obtained from partial Floquet theory through singular value decomposition (SVD). Finally, a postprocessing step assesses the accuracy of the identified stability characteristics. The Lyapunov characteristic exponent (LCE) theory provides the theoretical background for the second approach. It is shown that the system stability characteristics are related to the LCE closely, for both constant and periodic coefficient systems. For the latter systems, an exponential approximation is proposed to evaluate the transition matrix. Numerical simulations show that the proposed approaches are robust enough to deal with the stability analysis of flexible dynamical systems and the predictions of the two approaches are found to be in close agreement.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041004-041004-7. doi:10.1115/1.4031839.

In the study, a novel fuzzy generalized predictive control (GPC) scheme is proposed for the stability control of nonlinear brushless DC motor (BLDCM). First, the fuzzy predictive model of BLDCM is presented via Takagi–Sugeno fuzzy model. Then, based on the controlled autoregressive moving average (CARMA) model transformed by Takagi–Sugeno fuzzy model of BLDCM, a new fuzzy GPC scheme for the nonlinear BLDCM is designed by combining fuzzy techniques and GPC theory, and the rigorous mathematical derivation is given. Finally, numerical simulations are implemented to verify the effectiveness and superiority of the proposed scheme. It also provides reference for other nonlinear even chaos control in actual project.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041005-041005-6. doi:10.1115/1.4031840.

This paper presents a technique to obtain the transition curves of fractional periodic time-delayed (FPTD) systems based on a proposed explicit harmonic balance (EHB) method. This method gives the analytical Hill matrix of FPTD systems explicitly with a symbolic computation-free algorithm. Furthermore, all linear operations on Fourier basis vectors including fractional order derivative operators and time-delayed operators for a linear FPTD system are obtained. This technique is illustrated with parametrically excited simple and double pendulum systems, with both time-delayed states and fractional damping.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041006-041006-11. doi:10.1115/1.4031286.

An iterative method is proposed for finding periodic orbits of strongly nonlinear oscillators. The method combines the strength of analytical approaches, where the candidate solution is assumed in the form of a Fourier series, and the convenience of numerical methods that can be applied to larger systems with strong nonlinearity. The proposed method does not require integration of the vector field over any period of time and examples presented here illustrate that it is faster than traditional collocation algorithms, has a large radius of convergence, and is capable of finding several periodic orbits in each solution.

Topics: Algorithms
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041007-041007-9. doi:10.1115/1.4031841.

Lyapunov stability of linear noncommensurate order fractional systems is treated in this paper. The proposed methodology is based on the concept of fractional energy stored in inductor and capacitor components, where natural decrease of the stored energy is caused by internal Joule losses. The Lyapunov function is expressed as the sum of the different reversible fractional energies, whereas its derivative is interpreted in terms of internal and external Joule losses. Stability conditions are derived from the energy balance principle, adapted to the fractional case. Examples are taken from electrical systems, but this methodology applies also directly to mechanical and electromechanical systems.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041008-041008-9. doi:10.1115/1.4031980.

This paper investigates the robust reliable β-dissipative control for uncertain dynamical systems with mixed actuator faults via sampled-data approach. In particular, a more general reliable controller containing both linear and nonlinear parts is constructed for the considered system. Then, by applying the input delay approach, the sampling measurement of the digital control signal is transformed into time-varying delayed one. The aim of this paper is to design state feedback sampled-data controller to guarantee that the resulting closed-loop system to be strictly (Q, S, R)-β-dissipative. By constructing appropriate Lyapunov function and employing a delay decomposition approach, a new set of delay-dependent sufficient stabilization criteria is obtained in terms of linear matrix inequalities (LMIs). Moreover, the obtained LMIs are dependent, not only upon upper bound of time delay but also depend on the dissipative margin β and the actuator fault matrix. As special cases, $H∞$ and passivity control performances can be deduced from the proposed dissipative control result. Finally, numerical simulation is provided based on a flight control model to verify the effectiveness and applicability of the proposed control scheme.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041009-041009-9. doi:10.1115/1.4031979.

Falling is the leading cause of both fatal and nonfatal injury in the elderly, often requiring expensive hospitalization and rehabilitation. We study the stability of human balance during stance using inverted single- and double-pendulum models, accounting for physiological reflex delays in the controller. The governing second-order neutral delay differential equation (NDDE) is transformed into an equivalent partial differential equation (PDE) constrained by a boundary condition and then into a system of ordinary differential equations (ODEs) using the Galerkin method. The stability of the ODE system approximates that of the original NDDE system; convergence is achieved by increasing the number of terms used in the Galerkin approximation. We validate our formulation by deriving analytical expressions for the stability margins of the double-pendulum human stance model. Numerical examples demonstrate that proportional–derivative–acceleration (PDA) feedback generally, but not always, results in larger stability margins than proportional–derivative (PD) feedback in the presence of reflex delays.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041010-041010-8. doi:10.1115/1.4031956.

Compared with chaotic systems over the real number field, complex chaotic dynamics have some unique properties. In this paper, a kind of novel hybrid synchronizations of complex chaotic systems is discussed analytically and numerically. Between two nonidentical complex chaotic systems, modified projective synchronization (MPS) in the modulus space and complete synchronization in the phase space are simultaneously achieved by means of active control. Based on the Lyapunov stability theory, a controller is developed, in which time delay as an important consideration is involved. Furthermore, a switch-modulated digital secure communication system based on the proposed synchronization scheme is carried out. Different from the previous works, only one set of drive-response chaotic systems can implement switch-modulated secure communication, which could simplify the complexity of design. Furthermore, the latency of a signal transmitted between transmitter and receiver is simulated by channel delay. The corresponding numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041011-041011-9. doi:10.1115/1.4032074.

A novel input–output linearization minimum sliding mode error feedback control (I/OMSMEFC) is proposed for the synchronization between two uncoupled FitzHugh–Nagumo (FHN) neurons with different ionic currents and external electrical stimulations. To estimate and offset the system uncertainties and external disturbances, the concept of equivalent control error is introduced, which is the key to utilization of I/OMSMEFC. A cost function is formulated on the basis of the principle of minimum sliding mode covariance constraint; then the equivalent control error is estimated and fed back. It is shown that the proposed I/OMSMEFC can compensate various kinds of system uncertainties and external disturbances. Meanwhile, it can reduce the steady-state error more than the conventional sliding mode control (SMC). In addition, the sliding mode after the I/OMSMEFC will tend to be the ideal SMC, resulting in improved control performance and quantity. Sufficient conditions are given based on the Lyapunov stability theorem and numerical simulations are performed to verify the effectiveness of presented I/OMSMEFC for the chaotic synchronization accurately.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):041012-041012-9. doi:10.1115/1.4031978.

Modeling and analysis of a system of two self-balancing pendulums is presented in this paper. Such systems are commonly used as elements of automotive door latch mechanisms that can be subjected to oscillatory excitation or vibratory inertia forces occurring during crash events. In order to avoid an unwanted behavior such as opening of the door, the considered mechanism should be properly designed and its dynamical response well understood and predictable. One pendulum of the double-pendulum system, playing the role of a counterweight (CW), is used to reduce the second (or main) pendulum motion under inertia loading. The interaction force between the pendulums is defined as the reaction of a holonomic constraint linking the rotations of both pendulums. Another reaction force acts between one of the pendulums and the support, reinforced by the action of a preloaded spring. An important aspect of the model is its discontinuous nature due to the presence of a gap in the interface area. This may result in impacts between both pendulums and between one of the pendulums and the support. High-frequency/high-acceleration amplitude vibratory motion of the base part provides inertia input to the system. Classical multibody dynamics approach is adopted first to solve the equations of motion. It is shown that the considered system under certain conditions responds with a high-amplitude irregular motion. A special methodology is used in order to study the regions of chaotic motion, with the goal to gain more understanding of the considered system dynamics. Bifurcation diagrams are presented together with quantitative and qualitative analysis of the motion. The sensitivity of solutions to variation of system parameters and input characteristics is also analyzed in the paper.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041013-041013-13. doi:10.1115/1.4031769.

In this investigation, a new singularity-free formulation of a three-dimensional Euler–Bernoulli beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope, which can be easily expressed by Euler parameters. The hyperspherical interpolation function is used to guarantee that the normalization constraint equation of Euler parameters is always satisfied. Each node of a beam element has only four nodal coordinates, which are significantly fewer than those in an absolute node coordinate formulation (ANCF) and the finite element method (FEM). Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by a differential-algebraic equation (DAE) solver. The current formulation can be used to calculate the static equilibrium and linear and nonlinear dynamics of an Euler–Bernoulli beam under arbitrary, concentrated, and distributed forces. While the mass matrix is more complex than that in the ANCF, the stiffness matrix and generalized forces are simpler, which is amenable for calculating the equilibrium of the beam. Several numerical examples are presented to demonstrate the performance of the current formulation. It is shown that the current formulation can achieve the same accuracy as the ANCF and FEM with a fewer number of coordinates.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041014-041014-7. doi:10.1115/1.4032695.

In this paper, the initialization of fractional order systems is analyzed. The objective is to prove that the usual pseudostate variable $x(t)$ is unable to predict the future behavior of the system, whereas the infinite dimensional variable $z(ω, t)$ fulfills the requirements of a true state variable. Two fractional systems, a fractional integrator and a one-derivative fractional system, are analyzed with the help of elementary tests and numerical simulations. It is proved that the dynamic behaviors of these two fractional systems differ completely from that of their integer order counterparts. More specifically, initialization of these systems requires knowledge of $z(ω,t0)$ initial condition.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041015-041015-5. doi:10.1115/1.4032767.

In this paper, we apply two decomposition methods, the Adomian decomposition method (ADM) and a well-established iterative method, to solve time-fractional Klein–Gordon type equation. We compare these methods and discuss the convergence of them. The obtained results reveal that these methods are very accurate and effective.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041016-041016-15. doi:10.1115/1.4033003.

Reduced order models (ROMs) of turbine bladed disks (blisks) are essential to quickly yet accurately characterize vibration characteristics and effectively design for high cycle fatigue. Modeling blisks with contacting shrouds at adjacent blades is especially challenging due to friction damping and localized nonlinearities at the contact interfaces which can lead to complex stick–slip behavior. While well-known techniques such as the harmonic balance method (HBM) and Craig–Bampton component mode synthesis (CB-CMS) have generally been employed to generate ROMs in the past, they do not reduce degrees-of-freedom (DoFs) at the interfaces themselves. In this paper, we propose a novel method to obtain a set of reduction basis functions for the contact interface DoFs as well as the remaining DoFs called “adaptive microslip projection” (AMP). The method is based on analyzing a set of linear systems with specifically chosen boundary conditions on the contact interface. Simulated responses of full order baseline models and the novel ROMs under various conditions are studied. Results obtained from the ROMs compare very favorably with the baseline model. This study addresses the case of a shrouded blisk in microslip close to stick. The AMP procedure may be possibly applied to other systems with Coulomb friction contacts, but its accuracy and effectiveness will need to be evaluated separately.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041017-041017-12. doi:10.1115/1.4033440.

Mesh reflectors with large apertures have been used in many communication satellites. The performance of antenna reflectors crucially depends on the faceting error of the reflective surface, which is approximated by using meshes. The force density method (FDM) has been widely used for the form-finding analysis of mesh reflectors. However, after performing form-finding of some meshes, the effective reflective area will decrease. In addition, the form-finding of the auxiliary mesh has received little attention, and it cannot be achieved by using the FDM. Thus, in this study, an effective form-finding methodology that combines the iterative FDM and the minimum norm method (MNM) is proposed. To consider the flexibility of the reflector ring truss, a static analysis of the ring truss under the tension force actions is also performed in the form-finding processes. The reflector flexible parts are described by the absolute nodal coordinate formulation (ANCF). Finally, the form-finding analysis of the reflector with the standard configuration, the central hub configuration, and the circular configuration is performed to validate the proposed methodology. The influence of the mesh tension force on the reflector natural frequencies is also studied. After performing the form-finding analysis, the initial configuration of the reflector with tensioned meshes for the deployment dynamics study can be determined. Based on this paper, the deployment dynamics of a complex AstroMesh reflector will be studied in a successive paper “Dynamics of a Deployable Mesh Reflector of Satellite Antenna: Parallel Computation and Deployment Simulation.”

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041018-041018-10. doi:10.1115/1.4032160.

In industrial practice, the floating frame of reference formulation (FFRF)—often combined with the component mode synthesis (CMS) in order to reduce the number of flexible degrees-of-freedom—is the common approach to describe arbitrarily shaped bodies in flexible multibody systems. Owed to the relative formulation of the flexible deformation with respect to the reference frame, the equations of motion show state-dependent nonconstant inertia terms. Such relative description, however, comes along with considerable numerical costs, since both the mass matrix and gyroscopic forces, i.e., the quadratic velocity vector, need to be evaluated in every integration step. The state dependency of the inertia terms can be avoided by employing an alternative formulation based on the mode shapes as in the classical CMS approach. In this approach, which is referred to as generalized component mode synthesis (GCMS), the total (absolute) displacements are approximated directly. Consequently, the mass matrix is constant, no quadratic velocity vector appears, and the stiffness matrix is a corotated but otherwise constant matrix. In order to represent the same flexible deformation as in the classical FFRF-based CMS, however, a comparatively large number of degrees-of-freedom is required. The approach described in the present paper makes use of the fact that a majority of components in technical systems are constrained to motions showing large rotations only about a single spatially fixed axis. For this reason, the GCMS is adapted for multibody systems that are subjected to small flexible deformations and undergo a rigid body motion showing large translations, large rotations about one axis, but small rotations otherwise. Thereby, the number of shape functions representing the flexible deformation is reduced, which further increases numerical efficiency compared to the original GCMS formulation for arbitrary rotations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041019-041019-18. doi:10.1115/1.4033005.

Clearance is unavoidable in many engineering structures due to the manufacturing and installation errors. These clearances can cause intense impact and wear of the contacting pairs, which may change the dynamic response and eventually reduce the movement precision and the service life of the transmission system. Parameters identification of the clearance would provide better understanding of dynamic behaviors of the clearance and contribute significantly for the control of the induced disturbance and deviation. In this paper, based on dynamic characteristics of the clearance nonlinearity, the piecewise fitting method is first proposed to identify the clearance value of the continuum structure. During the proposed method, first, the rough scope of the clearance value extracted from the displacement response is divided into subintervals. And then, the nonlinear force is fitted by the piecewise linear function in the subintervals. Once the equivalent stiffness is obtained, the clearance value can be calculated by the sorting nonlinear force–displacement curve. The feasibility of the piecewise fitting method was verified by a cantilever beam system with clearances in simulation. Besides, some influence factors of this identification method, including the clearance value, exciting force level and measurement noise, are fully discussed to illustrate the robustness of this method. Moreover, an experiment system of a cantilever beam with adjustable clearances was designed to experimentally validate the effectiveness of the proposed method, and the results show that the piecewise fitting method can precisely identify the clearance value of continuous systems.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041020-041020-11. doi:10.1115/1.4033254.

When a rigid body negotiates a curve, the centrifugal force takes a simple form which is function of the body mass, forward velocity, and the radius of curvature of the curve. In this simple case of rigid body dynamics, curve negotiation does not lead to Coriolis forces. In the case of a flexible body negotiating a curve, on the other hand, the inertia of the body becomes function of the deformation, curve negotiations lead to Coriolis forces, and the expression for the deformation-dependent centrifugal forces becomes more complex. In this paper, the nonlinear constrained dynamic equations of motion of a flexible body negotiating a circular curve are used to develop a systematic procedure for the calculation of the centrifugal forces during curve negotiations. The floating frame of reference (FFR) formulation is used to describe the body deformation and define the nonlinear centrifugal and Coriolis forces. The algebraic constraint equations which define the motion trajectory along the curve are formulated in terms of the body reference and elastic coordinates. It is shown in this paper how these algebraic motion trajectory constraint equations can be used to define the constraint forces that lead to a systematic definition of the resultant centrifugal force in the case of curve negotiations. The embedding technique is used to eliminate the dependent variables and define the equations of motion in terms of the system degrees of freedom. As demonstrated in this paper, the motion trajectory constraints lead to constant generalized forces associated with the elastic coordinates, and as a consequence, the elastic velocities and accelerations approach zero in the steady state. It is also shown that if the motion trajectory constraints are imposed on the coordinates of a flexible body reference that satisfies the mean-axis conditions, the centrifugal forces take the same form as in the case of rigid body dynamics. The resulting flexible body dynamic equations can be solved numerically in order to obtain the body coordinates and evaluate numerically the constraint and centrifugal forces. The results obtained for a flexible body negotiating a circular curve are compared with the results obtained for the rigid body in order to have a better understanding of the effect of the deformation on the centrifugal forces and the overall dynamics of the body.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041021-041021-7. doi:10.1115/1.4033252.

This paper investigates the voltage–amplitude response of soft alternating current (AC) electrostatically actuated micro-electro-mechanical system (MEMS) clamped circular plates for sensing applications. The case of soft AC voltage of frequency near half natural frequency of the plate is considered. Soft AC produces small to very small amplitudes away from resonance zones. Nearness to half natural frequency results in primary resonance of the system, which is investigated using the method of multiple scales (MMS) and numerical simulations using reduced order model (ROM) of seven terms (modes of vibration). The system is assumed to be weakly nonlinear. Pull-in instability of the voltage–amplitude response and the effects of detuning frequency and damping on the response are reported.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041022-041022-10. doi:10.1115/1.4033382.

Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix (STM) $Φ(t,α)$, associated with the linear part of the equation, can be expressed in terms of the periodic Lyapunov–Floquét (L-F) transformation matrix $Q(t,α)$ and a time-invariant matrix $R(α)$ containing a set of symbolic system parameters $α.$ Computation of $Q(t,α)$ and $R(α)$ in symbolic form as a function of $α$ is of paramount importance in stability, bifurcation analysis, and control system design. In earlier studies, since $Q(t,α)$ and $R(α)$ were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In 2009, an attempt was made by Butcher et al. (2009, “Magnus' Expansion for Time-Periodic Systems: Parameter Dependent Approximations,” Commun. Nonlinear Sci. Numer. Simul., 14(12), pp. 4226–4245) to compute the $Q(t,α)$ matrix in a symbolic form using the Magnus expansions with some success. In this work, an efficient technique for symbolic computation of $Q(t,α)$ and $R(α)$ matrices is presented. First, $Φ(t,α)$ is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method as suggested in the literature. Then, $R(α)$ is computed using a Gaussian quadrature integral formula. Finally, $Q(t,α)$ is computed using the matrix exponential summation method. Using mathematica, this approach has successfully been applied to the well-known Mathieu equation and a four-dimensional time-periodic system in order to demonstrate the applications of the proposed method to linear as well as nonlinear problems.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041023-041023-10. doi:10.1115/1.4033385.

In this paper, a robust adaptive sliding mode controller is proposed. Under the existence of external disturbances, modified hybrid projective synchronization (MHPS) between two identical and two nonidentical fractional-order complex chaotic systems is achieved. It is shown that the response system could be synchronized with the drive system up to a nondiagonal scaling matrix. An adaptive controller and parameter update laws are investigated based on the Lyapunov stability theorem. The closed-loop stability conditions are derived based on the fractional-order Lyapunov function and Mittag-Leffler function. Finally, numerical simulations are given to verify the theoretical analysis.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041024-041024-6. doi:10.1115/1.4033383.

This paper presents an adaptive robust controller for a class of uncertain chaotic Rossler system with time-varying mismatched parameters. The proposed controller is designed based on Lyapunov stability theory, and it is shown that using this controller all signals of the closed-loop system are uniformly ultimately bounded (UUB). In addition, the proposed scheme is such that it does not require a priori information about the bound of uncertainties. Furthermore, since all the signals are UUB, the control signal is smooth and feasible to implement. Simulation results on a third-order Rossler system with time-varying parameters confirm the effectiveness of the proposed controller.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041025-041025-12. doi:10.1115/1.4033307.

In this paper, the dynamic oscillation of a rectangular thin plate under parametric and external excitations is investigated and controlled. The motion of a rectangular thin plate is modeled by coupled second-order nonlinear ordinary differential equations. The formulas of the thin plate are derived from the von Kármán equation and Galerkin's method. A control law based on negative acceleration feedback is proposed for the system. The multiple time scale perturbation technique is applied to solve the nonlinear differential equations and obtain approximate solutions up to the second-order approximations. One of the worst resonance case of the system is the simultaneous primary resonances, where $Ω1≅ω1 and Ω2≅ω2$. From the frequency response equations, the stability of the system is investigated according to the Routh–Hurwitz criterion. The effects of the different parameters are studied numerically. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. The simulation results are achieved using matlab 7.0 software. A comparison is made with the available published work.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041026-041026-8. doi:10.1115/1.4033554.
OPEN ACCESS

Bacteriophage T4 is one of the most common and complex of the tailed viruses that infect host bacteria using an intriguing contractile tail assembly. Despite extensive progress in resolving the structure of T4, the dynamics of the injection machinery remains largely unknown. This paper contributes a first model of the injection machinery that is driven by elastic energy stored in a structure known as the sheath. The sheath is composed of helical strands of protein that suddenly collapse from an energetic, extended conformation prior to infection to a relaxed, contracted conformation during infection. We employ Kirchhoff rod theory to simulate the nonlinear dynamics of a single protein strand coupled to a model for the remainder of the virus, including the coupled translation and rotation of the head (capsid), neck, and tail tube. Doing so provides an important building block toward the future goal of modeling the entire sheath structure which is composed of six interacting helical protein strands. The resulting numerical model exposes fundamental features of the injection machinery including the time scale and energetics of the infection process, the nonlinear conformational change experienced by the sheath, and the contribution of hydrodynamic drag on the head (capsid).

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041027-041027-8. doi:10.1115/1.4033439.

In this paper, a Jacobi spectral Galerkin method is developed for nonlinear Volterra integral equations (VIEs) of the second kind. The spectral rate of convergence for the proposed method is established in the $L∞$-norm and the weighted L2-norm. Global superconvergence properties are discussed by iterated Galerkin methods. Numerical results are presented to demonstrate the effectiveness of the proposed method.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041028-041028-6. doi:10.1115/1.4033606.

This paper considers the problem of finite-time output tracking for a class of nonautonomous nonlinear fractional-order (FO) systems in the presence of model uncertainties and external disturbances. The finite-time control methods indicate better properties in terms of robustness, disturbance rejection, and settling time. Thus, design of a robust nonsingular controller for finite-time output tracking of a time-varying reference signal is considered in this paper, and a novel FO nonsingular terminal sliding mode controller (TSMC) is designed, which can conquer the uncertainties and guarantees the finite-time convergence of the system output toward the desired time-varying reference signal. For this purpose, an appropriate nonsingular terminal sliding manifold is designed, where maintaining the system's states on this manifold leads to finite-time vanishing of error signal (i.e., ensures the finite-time occurrence of both reaching and sliding phases). Moreover, by tacking the fractional derivative of the sliding manifold, the convergence of system's trajectories into the terminal sliding manifold in a finite time is proven, and the convergence time is estimated. Finally, in order to verify the theoretical results, the proposed method is applied to an FO model of a horizontal platform system (FO-HPS), and the computer simulations show the efficiency of the proposed method in finite-time output tracking.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041029-041029-8. doi:10.1115/1.4033442.

In this paper, the size-dependent resonant behavior of a microcantilever immersed in an incompressible fluid cavity is investigated. The nonclassical modified couple stress theory (MCST) is employed to capture the effects of length scale. The microbeam is deflected by applying a bias direct current (DC) voltage and then driven to vibrate around its deflected position by a harmonic alternating (AC) voltage. Regarding the nonlinear electrostatic force and the fluid pressure exerted upon the microbeam, the governing equations of the system are derived based on the MCST. Multiple scales method is used to obtain an approximate analytical solution for nonlinear resonance curves. Initially, the effect of length scale parameter on the dynamic response of system is studied, and then, a parametric study is conducted to evaluate the effects of MCST as well as the fluidic confinement on the resonance curves. The obtained results reveal that the frequency response along with the softening behavior of the system decreases when MCST is used. It is shown that the resonance amplitude obtained by the MCST is considerably smaller than those obtained by the classical theory (CT). Finally, it is found that the dynamic stability margins of the system could be extended by the size effect perspective.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):041030-041030-5. doi:10.1115/1.4033555.

This study aims to investigate the harmonic resonance of third-order forced van der Pol oscillator with fractional-order derivative using the asymptotic method. The approximately analytical solution for the system is first determined, and the amplitude–frequency equation of the oscillator is established. The stability condition of the harmonic solution is then obtained by means of Lyapunov theory. A comparison between the traditional integer-order of forced van der Pol oscillator and the considered fractional-order one follows the numerical simulation. Finally, the numerical results are analyzed to show the influences of the parameters in the fractional-order derivative on the steady-state amplitude, the amplitude–frequency curves, and the system stability.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Comput. Nonlinear Dynam. 2015;11(4):044501-044501-13. doi:10.1115/1.4031287.

Regarding constrained mechanical systems, we are faced with index-3 differential-algebraic equation (DAE) systems. Direct discretization of the index-3 DAE systems only enforces the position constraints to be fulfilled at the integration-time points, but not the hidden constraints. In addition, order reduction effects are observed in the velocity variables and the Lagrange multipliers. In literature, different numerical techniques have been suggested to reduce the index of the system and to handle the numerical integration of constrained mechanical systems. This paper deals with an alternative concept, called collocated constraints approach. We present index-2 and index-1 formulations in combination with implicit Runge–Kutta methods. Compared with the direct discretization of the index-3 DAE system, the proposed method enforces also the constraints on velocity and—in case of the index-1 formulation—the constraints on acceleration level. The proposed method may very easily be implemented in standard Runge–Kutta solvers. Here, we only discuss mechanical systems. The presented approach can, however, also be applied for solving nonmechanical higher-index DAE systems.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):044502-044502-5. doi:10.1115/1.4031197.

Two approaches are commonly used for handling frictional contact within the framework of the discrete element method (DEM). One relies on the complementarity method (CM) to enforce a nonpenetration condition and the Coulomb dry-friction model at the interface between two bodies in mutual contact. The second approach, called the penalty method (PM), invokes an elasticity argument to produce a frictional contact force that factors in the local deformation and relative motion of the bodies in contact. We give a brief presentation of a DEM-PM contact model that includes multi-time-step tangential contact displacement history. We show that its implementation in an open-source simulation capability called Chrono is capable of accurately reproducing results from physical tests typical of the field of geomechanics, i.e., direct shear tests on a monodisperse material. Keeping track of the tangential contact displacement history emerges as a key element of the model. We show that identical simulations using contact models that include either no tangential contact displacement history or only single-time-step tangential contact displacement history are unable to accurately model the direct shear test.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2015;11(4):044503-044503-4. doi:10.1115/1.4032075.

Due to the high computing demand of whole-trip train dynamics simulations and the iterative nature of optimizations, whole-trip train dynamics optimizations using sequential computing schemes are practically impossible. This paper reports advancements in whole-trip train dynamics optimizations enabled by using the parallel computing technique. A parallel computing scheme for whole-trip train dynamics optimizations is presented and discussed. Two case studies using parallel multiobjective particle swarm optimization (pMOPSO) and parallel multiobjective genetic algorithm (pMOGA), respectively, were performed to optimize a friction draft gear design. Linear speed-up was achieved by using parallel computing to cut down the computing time from 18 months to just 11 days. Optimized results using pMOPSO and pMOGA were in agreement with each other; Pareto fronts were identified to provide technical evidence for railway manufacturers and operators.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):044504-044504-5. doi:10.1115/1.4032076.

This technical brief describes the procedure and demonstrates the feasibility of integrating soil/tire models using the absolute nodal coordinate formulation (ANCF). The effects of both the soil plasticity and the tire elasticity are captured using ANCF finite elements (FEs). Capturing the tire/soil dynamic interaction is necessary for the construction of higher fidelity off-road vehicle models. ANCF finite elements, as will be demonstrated in this paper, can be effectively used for the modeling of tire and soil mechanics. In this investigation, the soil model is developed using ANCF hexahedral finite elements, while the tire model can be developed using different ANCF finite elements including beam, plate, or solid elements; ANCF plate elements are used in this investigation for demonstration purposes. The Drucker–Prager plastic material, which is used to model the behavior of the soil, is appropriate for the simulation of a number of types of soils and offers a good starting point for computational plasticity in terramechanics applications. Such higher fidelity simulations can be fruitfully applied toward the investigation of complex dynamic phenomena in terramechanics. The proposed ANCF/Drucker–Prager soil model is implemented in a multibody system (MBS) algorithm which allows for using the ANCF reference node (ANCF-RN) to apply linear connectivity conditions between ANCF finite elements and the rigid components of the vehicle. This new implementation is demonstrated using a tire of an off-road wheeled vehicle. The generality of the approach allows for the simulation of general vehicle maneuvers over unprepared terrain. Unlike other approaches that implement force or superelement models into an MBS simulation environment, in the approach proposed in this paper both the soil material and vehicle parameters can be altered independently. This allows for a greater degree of flexibility in the development of computational models for the evaluation of the off-road wheeled vehicle performance.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):044505-044505-5. doi:10.1115/1.4032245.

This paper designs a preload adjustable rotary nut ball screw dual-driven micro feed system, due to the elastic property of the feed system has great influence on its own frequency–response characteristics, which can be identified by analyzing the amplitude relationship between the torque input signal and the acceleration output signal. In order to get the structural dynamic of the dual-driven servomechanism, which is first modeled through lumped mass method, the frequency–response characteristics are calculated using the Lagrange equation and the state-space method. Finally, the frequency–response characteristics of a macro–macro dual-driven and single-driven systems are compared via numerical analysis, and the influence of changes in the preload, torsional rigidity, and table's total mass on the frequency–response characteristics are studied.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;11(4):044506-044506-6. doi:10.1115/1.4033438.

Gyroelastic body refers to a flexible structure with a distribution of stored angular momentum (called gyricity). In previous studies, it was assumed that each volume element of the structure possesses an infinitesimal spinning rotor so that the distribution of the gyricity is continuous. However, the momentum devices must be discretely distributed in engineering applications; therefore, this paper studies the gyroelastic body formed by directly mounting a set of variable speed control moment gyroscopes (CMGs) on the flexible structure. The detailed dynamics of the CMGs is incorporated to capture the interactions between the CMGs and the structure. The gyroelastic modes and pseudorigid modes are discussed based on the linearized mathematical model. The examples of a gyroelastic beam and a gyroelastic parabolic structure demonstrate several involved concepts and properties.

Commentary by Dr. Valentin Fuster