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### Research Papers

J. Comput. Nonlinear Dynam. 2017;12(4):041001-041001-15. doi:10.1115/1.4035408.

This study deals with the determination of Lagrangians, first integrals, and integrating factors of the modified Emden equation by using Jacobi and Prelle–Singer methods based on the Lie symmetries and λ-symmetries. It is shown that the Jacobi method enables us to obtain Jacobi last multipliers by means of the Lie symmetries of the equation. Additionally, via the Lie symmetries of modified Emden equation, we analyze some mathematical connections between λ-symmetries and Prelle–Singer method. New and nontrivial Lagrangian forms, conservation laws, and exact solutions of the equation are presented and discussed.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041002-041002-9. doi:10.1115/1.4034996.

This paper investigates the influences of nonzero mean Poisson impulse amplitudes on the response statistics of dynamical systems. New correction terms of the extended Itô calculus, as a generalization of the Wong–Zakai correction terms in the case of normal excitations, are adopted to consider the non-normal property in the case of Poisson process. Due to these new correction terms, the corresponding drift and diffusion coefficients of Fokker–Planck–Kolmogorov (FPK) equation have to be modified and they become more complicated. Herein, the exponential–polynomial closure (EPC) method is employed to solve such a complex FPK equation. Since there are no exact solutions, the efficiency of the EPC method is numerically evaluated by the simulation results. Three examples of different excitation patterns are considered. Numerical results indicate that the influence of nonzero mean impulse amplitudes on system responses depends on the excitation patterns. It is negligible in the case of parametric excitation on displacement. On the contrary, the influence becomes significant in the cases of external excitation and parametric excitation on velocity.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041003-041003-8. doi:10.1115/1.4035412.

A simplest fractional-order delayed memristive chaotic system is investigated in order to analyze the nonlinear dynamics of the system. The stability and bifurcation behaviors of this system are initially investigated, where time delay is selected as the bifurcation parameter. Some explicit conditions for describing the stability interval and the transversality condition of the emergence for Hopf bifurcation are derived. The period doubling route to chaos behaviors of such a system is discussed by using a bifurcation diagram, a phase diagram, a time-domain diagram, and the largest Lyapunov exponents (LLEs) diagram. Specifically, we study the influence of time delay on the chaotic behavior, and find that when time delay increases, the transitions from one cycle to two cycles, two cycles to four cycles, and four cycles to chaos are observed in this system model. Corresponding critical values of time delay are determined, showing the lowest orders for chaos in the fractional-order delayed memristive system. Finally, numerical simulations are provided to verify the correctness of theoretical analysis using the modified Adams–Bashforth–Moulton method.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041004-041004-10. doi:10.1115/1.4035266.

In this paper, an analytical technique is proposed to determine the exact solution of fractional order modified Fornberg–Whitham equation. Since exact solution of fractional Fornberg–Whitham equation is unknown, first integral method has been applied to determine exact solutions. The solitary wave solution of fractional modified Fornberg–Whitham equation has been attained by using first integral method. The approximate solutions of fractional modified Fornberg–Whitham equation, obtained by optimal homotopy asymptotic method (OHAM), are compared with the exact solutions obtained by the first integral method. The obtained results are presented in tables to demonstrate the efficiency of these proposed methods. The proposed schemes are quite simple, effective, and expedient for obtaining solution of fractional modified Fornberg–Whitham equation.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041005-041005-6. doi:10.1115/1.4034997.

This paper describes the stabilization of a fractional-order nonlinear brushless DC motor (BLDCM) with the Caputo derivative. Based on the Laplace transform, a Mittag-Leffler function, Jordan decomposition, and Grönwall's inequality, sufficient conditions are proposed that ensure the local stabilization of a BLDCM as fractional-order $α$: $0<α≤1$ is proposed. Then, numerical simulations are presented to show the feasibility and validity of the designed method. The proposed scheme is simpler and easier to implement than previous schemes.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041006-041006-9. doi:10.1115/1.4035409.

Impact may excite intense vibration responses of the flexible robotic arm and thus deteriorate its working performance. A vibration absorption method is put forward to alleviate impact influence of the flexible robotic arm. To dissipate the impact vibration energy, a slider mass–spring–dashpot mechanism is used as a vibration absorber and attached to the flexible robotic arm. Internal resonance is sufficiently utilized to provide a bridge for the transfer of impact vibration energy between the flexible link and the absorber via nonlinear coupling. In the presence of damping of the absorber, the impact vibration energy of the flexible link can be effectively migrated to and dissipated by the absorber. Numerical simulations and virtual prototype simulations verify its effectiveness and feasibility in alleviating impact vibration of the flexible robotic arm undergoing a collision.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041007-041007-8. doi:10.1115/1.4035413.

This paper introduces a new planar gradient deficient beam element based on the absolute nodal coordinate formulation. In the proposed formulation, the centerline position is interpolated using cubic polynomials while shear deformation is taken into account via independently interpolated linear terms. The orientation of the cross section, which is defined by the axial slope of the element's centerline position combined with the independent shear terms, is coupled with the displacement field. A structural mechanics based formulation is used to describe the strain energy via generalized strains derived using a local element coordinate frame. The accuracy and the convergence properties of the proposed formulation are verified using numerical tests in both static and dynamics cases. The numerical results show good agreement with reference formulations in terms of accuracy and convergence.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041008-041008-6. doi:10.1115/1.4035078.

In this work, a class of kicked systems perturbed with an irregular kicks pattern is studied and formation of a chaos in the senses of Devaney and Li–Yorke in the corresponding discretized system is investigated. Beside a discussion on chaotic stability, an example is presented. Then, the existence of a period three orbit of a 2D map which governs a class of dynamic problems on time scales is studied. As an application, a chaotic encryption scheme for a time-dependent plain text with the help of chaos induction in the sense of Li–Yorke is presented.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041009-041009-7. doi:10.1115/1.4035410.

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041010-041010-10. doi:10.1115/1.4035667.

Miniature wireless inertial measurement units (IMUs) hold great promise for measuring and analyzing multibody system dynamics. This relatively inexpensive technology enables noninvasive motion tracking in broad applications, including human motion analysis. This paper advances the use of an array of IMUs to estimate the joint reactions (forces and moments) in multibody systems via inverse dynamic modeling. In particular, this paper reports a benchmark experiment on a double-pendulum that reveals the accuracy of IMU-informed estimates of joint reactions. The estimated reactions are compared to those measured by high-precision miniature (6 degrees-of-freedom) load cells. Results from ten trials demonstrate that IMU-informed estimates of the three-dimensional reaction forces remain within 5.0% RMS of the load cell measurements and with correlation coefficients greater than 0.95 on average. Similarly, the IMU-informed estimates of the three-dimensional reaction moments remain within 5.9% RMS of the load cell measurements and with correlation coefficients greater than 0.88 on average. The sensitivity of these estimates to mass center location is discussed. Looking ahead, this benchmarking study supports the promising and broad use of this technology for estimating joint reactions in human motion applications.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041011-041011-10. doi:10.1115/1.4035192.

System identification of the sinusoidal steady-state response of the Phantom Omni using a local linear model revealed that friction has a non-negligible effect on the accuracy of a global linear model, particularly at low frequencies. Some of the obvious errors observed with the global linear model at low frequencies were (i) the response amplitude was lower; (ii) local linear model coefficients became physically impossible (e.g., negative) at low frequencies; and (iii) low frequency inputs resulted in a greater nonlinearity in the response compared to higher frequency inputs. While standard friction models such as Coulomb friction could be used to model the nonlinearity, there is a desire to create a friction model that is not only accurate for sinusoidal steady-state responses, but can also be generalized to any input response. One measure that is universally present in dynamical systems is energy, and in this paper the relationship between a generalized measure of energy and damping for modeling the effect of friction is developed. This paper introduces the “α-invariant” as a means of generalizing the friction behavior observed with sinusoidal steady-state responses to other waveforms. For periodic waveforms, the α-invariant is shown to be equivalent to the energy dissipated in each cycle, which demonstrates the physical significance of this quantity. The α-invariant nonlinear model formulation significantly outperforms the linear model for both sinusoidal steady state and step responses, demonstrating that this method accurately represents the physical mechanisms in the Phantom Omni. Overall, the α-invariant provides an efficient way of capturing nonlinear dynamics with a small number of parameters and experiments.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041012-041012-16. doi:10.1115/1.4035411.

A formal impulse-based analysis is presented for the collision of two rigid bodies at single contact point under Coulomb's friction in three dimensions (3D). The tangential impulse at the contact is known to be linear in the sliding velocity whose trajectory, parametrized with the normal impulse and referred to as the hodograph, is governed by a generally nonintegrable ordinary differential equation (ODE). Evolution of the hodograph is bounded by rays in several invariant directions of sliding in the contact plane. Exact lower and upper bounds are derived for the number of such invariant directions, utilizing the established positive definiteness of the matrix defining the governing ODE. If the hodograph reaches the origin, it either terminates (i.e., the contact sticks) or continues in a new direction (i.e., the contact resumes sliding) whose existence and uniqueness, only assumed in the literature, are proven. Closed-form integration of the ODE becomes possible as soon as the sliding velocity turns zero or takes on an invariant direction. Assuming Stronge's energy-based restitution, a complete algorithm is described to combine fast numerical integration (NI) with a case-by-case closed-form analysis. A number of solved collision instances are presented. It remains open whether the modeled impact process will always terminate under Coulomb's friction and Stronge's (or Poisson's) restitution hypothesis.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041013-041013-8. doi:10.1115/1.4035190.

This paper studies inducing robust stable oscillations in nonlinear systems of any order. This goal is achieved through creating stable limit cycles in the closed-loop system. For this purpose, the Lyapunov stability theorem which is suitable for stability analysis of the limit cycles is used. In this approach, the Lyapunov function candidate should have zero value for all the points of the limit cycle and be positive in the other points in the vicinity of it. The proposed robust controller consists of a nominal control law with an additional term that guarantees the robust performance. It is proved that the designed controller results in creating the desirable stable limit cycle in the phase trajectories of the uncertain closed-loop system and leads to induce stable oscillations in the system's output. Additionally, in order to show the applicability of the proposed method, it is applied on two practical systems: a time-periodic microelectromechanical system (MEMS) with parametric errors and a single-link flexible joint robot in the presence of external disturbances. Computer simulations show the effective robust performance of the proposed controllers in generating the robust output oscillations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041014-041014-13. doi:10.1115/1.4035670.

Correlation and calibration using test data are natural ingredients in the process of validating computational models. Model calibration for the important subclass of nonlinear systems which consists of structures dominated by linear behavior with the presence of local nonlinear effects is studied in this work. The experimental validation of a nonlinear model calibration method is conducted using a replica of the École Centrale de Lyon (ECL) nonlinear benchmark test setup. The calibration method is based on the selection of uncertain model parameters and the data that form the calibration metric together with an efficient optimization routine. The parameterization is chosen so that the expected covariances of the parameter estimates are made small. To obtain informative data, the excitation force is designed to be multisinusoidal and the resulting steady-state multiharmonic frequency response data are measured. To shorten the optimization time, plausible starting seed candidates are selected using the Latin hypercube sampling method. The candidate parameter set giving the smallest deviation to the test data is used as a starting point for an iterative search for a calibration solution. The model calibration is conducted by minimizing the deviations between the measured steady-state multiharmonic frequency response data and the analytical counterparts that are calculated using the multiharmonic balance method. The resulting calibrated model's output corresponds well with the measured responses.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041015-041015-11. doi:10.1115/1.4035470.

Multibody dynamics approaches have nowadays been an essential part in examining train crashworthiness. A typical passenger train structure has been investigated on its crashworthiness using three-dimensional (3D) models of a single passenger car and multiple cars formulated using multibody dynamics approaches. The simulation results indicate that the crush length or crush force or both of the crush mechanisms in the high and low energy (HE and LE) crush zones of a passenger train have to be increased for the higher crash speeds. The results on multiple cars (up to ten cars) show that the design of HE and LE crush zones is significantly influenced by the number of cars. The energy absorbed by the HE zone is reasonably consistent for train models with more than four cars at the crash speed of 35 km/h. The comparison of simulations can identify the contribution of the number of cars to the head-on crash forces. The influence of train mass on the design of both HE and LE crush zones, and the influence of design of the crush zones on the wheel-rail contacts are examined.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041016-041016-11. doi:10.1115/1.4035823.

Irregularities in the geometry and flexibility of railway crossings cause large impact forces, leading to rapid degradation of crossings. Precise stress and strain analysis is essential for understanding the behavior of dynamic frictional contact and the related failures at crossings. In this research, the wear and plastic deformation because of wheel–rail impact at railway crossings was investigated using the finite-element (FE) method. The simulated dynamic response was verified through comparisons with in situ axle box acceleration (ABA) measurements. Our focus was on the contact solution, taking account not only of the dynamic contact force but also the adhesion–slip regions, shear traction, and microslip. The contact solution was then used to calculate the plastic deformation and frictional work. The results suggest that the normal and tangential contact forces on the wing rail and crossing nose are out-of-sync during the impact, and that the maximum values of both the plastic deformation and frictional work at the crossing nose occur during two-point contact stage rather than, as widely believed, at the moment of maximum normal contact force. These findings could contribute to the analysis of nonproportional loading in the materials and lead to a deeper understanding of the damage mechanisms. The model provides a tool for both damage analysis and structure optimization of crossings.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041017-041017-10. doi:10.1115/1.4035669.

The chaotic vibration analysis of a heavy articulated vehicle (HAV) under consecutive speed control humps (SCHs) excitation is studied. The vehicle is modeled as a nonlinear half-truck oscillatory system with three axles. The suspension system between the truck bodies and axles is equipped with passive viscous damper and magnetorheological (MR) damper. The consecutive SCHs-speed coupling excitation function is presented by a half-sine wave with constant amplitude and variable frequency. The nonlinear dynamic behavior of the system is investigated by special respective techniques. Also, the ride comfort is assessed by the RMS value of truck bodies' accelerations. The results reveal that the quasi-periodic motion is observed at lower speeds when the truck moves on SCHs without load; while in the presence of the load, the dynamic characteristics of the system confirm the chaotic vibration possibility in a widespread range at higher speeds. Further studies indicate that the chaotic behaviors can directly affect on driving comfort and lead to the ride comfort becoming lower. The obtained results can be helpful in designing the oscillatory system for the heavy vehicles to preserve the comfort of drivers and the protection of load.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):041018-041018-8. doi:10.1115/1.4035668.

The size-dependent model is studied based on the modified couple stress theory for the geometrically nonlinear curvilinear Timoshenko beam made from a functionally graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the curvilinear beams is investigated with the use of the setup method. The second-order accuracy finite difference method is used to solve the problem of nonlinear partial differential equations (PDEs) by reducing it to the Cauchy problem. The obtained set of nonlinear ordinary differential equations (ODEs) is then solved by the fourth-order Runge–Kutta method. The relaxation method is employed to solve numerous static problems based on the dynamic approach. Eight different combinations of size-dependent coefficients and the functionally graded material coefficient are used to study the stress-strain responses of Timoshenko beams. Stability loss of the curvilinear Timoshenko beams is investigated using the Lyapunov criterion based on the estimation of the Lyapunov exponents. Beams with/without the size-dependent behavior, homogeneous beams, and functionally graded beams having the same stiffness are investigated. It is shown that in straight-line beams, the size-dependent effect decreases the beam deflection. The same is observed if the most rigid layer is located on the top of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid layer essentially improves the beam strength against stability loss. The observed transition of the largest Lyapunov exponent from a negative to positive value corresponds to the transition from a precritical to postcritical beam state.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Comput. Nonlinear Dynam. 2017;12(4):044501-044501-8. doi:10.1115/1.4034734.

Linear spring mass systems placed on a moving belt have been subjected to numerous investigations. Dynamical characteristics like amplitude and frequency of oscillations and bifurcations have been well studied along with different control mechanisms for this model. But the corresponding nonlinear system has not received comparable attention. This paper presents an analytical investigation of the behavior of a Duffing oscillator placed on a belt moving with constant velocity and excited by dry friction. A negative gradient friction model is considered to account for the initial decrease and the subsequent increase in the frictional forces with increasing relative velocity. Approximate analytical expressions are obtained for the amplitudes and base frequencies of friction-induced stick–slip and pure-slip phases of oscillations. For the pure-slip phase, an expression for the equilibrium point is obtained, and averaging procedure is used to arrive at approximate analytical expressions of the periodic amplitude of oscillations around this fixed-point. For stick–slip oscillations, analytical expressions for amplitude are arrived at by using perturbation analysis for the finite time interval of the stick phase, which is linked to the subsequent slip phase through conditions of continuity and periodicity. These analytical results are validated by numerical studies and are shown to be in good agreement with them. It is shown that the pure-slip oscillation phase and the critical velocity of the belt remain unaffected by the nonlinear term. It is also shown that the amplitude of the stick–slip phase varies inversely with nonlinearity. The effect of different system parameters on the vibration amplitude is also studied.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):044502-044502-7. doi:10.1115/1.4035484.

Simulations of three-dimensional train system dynamics for long freight railway trains with consideration being given to all degrees-of-freedom of all essential components of all vehicles have not been reported due to the challenge of long computing time. This paper developed a parallel computing scheme for three-dimensional train system dynamics. Key modeling techniques were discussed, which include modeling of longitudinal train dynamics, single vehicle system dynamics and multibody coupler systems. Assume that there are n vehicles in the train, then, n + 2 cores are needed. The first core (core 0) is used as the master core; the last core (core n + 1) is used for air brake simulation; the rest of the cores (core 1 to core n) are used for the computing of single vehicle system dynamics for all n vehicles in parallel. During the simulation, the master core collects the results from core n + 1 and then sends the air brake pressures and knuckle forces to core 1 to core n. core 1 to core n execute vehicle system dynamics simulations and then send the coupler kinematics to the master core. The details of the parallel computing scheme were presented in this paper. The feasibility of the computing scheme has been demonstrated by a simulation of a long heavy haul train that has 214 vehicles. A 3 h train trip was simulated; 216 cores were used. The accumulated computing time of all cores was about 253 days, while the wall-clock time was about 29 h. Such computing speed has made the simulations of three-dimensional train system dynamics practical.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):044503-044503-5. doi:10.1115/1.4035672.

This paper analyses forest fires (FF) in the U.S. during 1984–2013, based on data collected by the monitoring trends in burn severity (MTBS) project. The study adopts the tools of dynamical systems to tackle information about space, time, and size. Computational visualization methods are used for reducing the information dimensionality and to unveil the relationships embedded in the data.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):044504-044504-3. doi:10.1115/1.4035786.

We investigate the dynamics of a two degrees-of-freedom oscillator excited by dry friction. The system consists of two masses connected by linear springs and in contact with a belt moving at a constant velocity. The contact forces between the masses and the belt are given by Coulomb's laws. Several periodic orbits including slip and stick phases are obtained. In particular, the existence of periodic orbits involving a part where one of the masses moves at a higher speed than the belt is proved.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(4):044505-044505-7. doi:10.1115/1.4035825.

A rotating flexible beam undergoing large deformation is known to exhibit chaotic motion for certain parameter values. This work deals with an approach for control of chaos known as chaos synchronization. A nonlinear controller based on the Lyapunov stability theory is developed, and it is shown that such a controller can avoid the sensitive dependence of initial conditions seen in all chaotic systems. The proposed controller ensures that the error between the controlled and the original system, for different initial conditions, asymptotically goes to zero. A numerical example using the parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.

Commentary by Dr. Valentin Fuster