0

IN THIS ISSUE

### Research Papers

J. Comput. Nonlinear Dynam. 2017;13(3):031001-031001-14. doi:10.1115/1.4038445.

Nonlinear dynamics, control, and stability analysis of dry friction damped system under state feedback control with time delay are investigated. The dry friction damped system is harmonically excited, and the nonlinearities in the equation of motion arise due to nonlinear damping and spring force. In this paper, a frequency domain-based method, viz., incremental harmonic balance method along with arc-length continuation technique (IHBC) is first employed to identify the primary responses which may be present in such system. The IHBC is then reformulated in a manner to treat the dry friction damped system under state feedback control with time delay and is applied to obtain control of responses in an efficient and systematic way. The stability of uncontrolled responses is obtained by Floquet's theory using Hsu' scheme, and the stability of the controlled responses is obtained by applying a semidiscretization method for delay differential equation (DDE).

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(3):031002-031002-12. doi:10.1115/1.4038508.

In this paper, the dynamic response of a spatial four-bar mechanism with a spherical clearance joint with flexible socket is investigated. Previous research treats the socket as a whole rigid part and neglects the flexibility of the socket. In order to better describe the influence of the spherical clearance joint, a rigid-flexible coupling model of a four-bar mechanism is established, in which the socket of the spherical clearance joint is treated as flexible body. The dynamic responses of this spatial mechanism are discussed for the mechanism with a flexible socket and the case with traditional rigid socket. Furthermore, the effects of clearance size and driving speed are also separately discussed. The results demonstrated that the dynamic response of mechanism is affected by the clearance joint. The socket flexibility can relieve the undesired effects of the clearance on the responses of the mechanism with clearance. The flexible socket acts as a suspension for the mechanism with clearance joint.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(3):031003-031003-8. doi:10.1115/1.4038443.

This study is interested in the stability and stabilization of a class of fractional-order nonlinear systems with Caputo derivatives. Based on the properties of the Laplace transform, Mittag-Leffler function, Jordan decomposition, and Grönwall's inequality, some sufficient conditions that ensure local stability and stabilization of a class of fractional-order nonlinear systems under the Caputo derivative with $1<α<2$ are presented. Finally, typical instances, including the fractional-order three-dimensional (3D) nonlinear system and the fractional-order four-dimensional (4D) nonlinear hyperchaos, are implemented to demonstrate the feasibility and validity of the proposed method.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(3):031004-031004-8. doi:10.1115/1.4038642.

This paper investigates a novel adaptive fuzzy fractional-order nonsingular terminal sliding mode controller (AFFO-NTSMC) for second-order nonlinear dynamic systems. The technique of fractional calculus and nonsingular terminal sliding mode control (NTSMC) are combined to establish fractional-order NTSMC (FO-NTSMC), in which a new fractional-order (FO) nonsingular terminal sliding mode (NTSM) surface is proposed. Then, a corresponding controller is designed to provide robustness, high performance control, finite time convergence in the presence of uncertainties and external disturbances. Furthermore, a fuzzy system with online adaptive learning algorithm is derived to eliminate the chattering phenomenon in conventional sliding mode control (SMC). The stability of the closed-loop system is rigorously proven. Numerical simulation results are presented to demonstrate the effectiveness of the proposed control method.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(3):031005-031005-11. doi:10.1115/1.4038776.

The effect of wind disturbances on the stability of six-rotor unmanned aerial vehicles (UAVs) was investigated, exploring the various disturbances in different directions. The simulation model-based Euler–Poincare equation was established to investigate the spectra of Lyapunov exponents. Next, the value of the Lyapunov exponents was used to evaluate the stability of the systems. The results obtained show that the various speeds of rotors are optimized to keep up the stability after disturbances. In addition, the flight experiment with the hitting gust has been carried out to verify the validity and accuracy of the simulation results.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(3):031006-031006-14. doi:10.1115/1.4038446.

In-plane vibration of cyclically symmetric ring structures is examined with emphasis on the comparison of instabilities estimated by complete and simplified models. The aim of this paper is to understand under what conditions and to what degree the simplified models can approach the complete model. Previous studies develop time-variant models and employ perturbation method by assuming weak support. This work casts the rotating-load problem into a nonrotating load problem. A complete model with time-invariant coefficients is developed in rotating-support-fixed frame, where the bending and extensional deformations are incorporated. It is then reduced into two simplified ones based on different deformation restrictions. Due to the time-invariant effect observed in the rotating-support-fixed frame, the eigenvalues are formulated directly by using classical vibration theory and compared based on a sample structure. The comparisons verify that the two types of models are comparable only for weak support. Furthermore, the simplified models cannot accurately predict all unstable behaviors in particular for strong support. The eigenvalues are different even in comparable regions. For verification purpose, the time-invariant models are transformed into time-variant ones in the inertial frame, based on which instabilities are estimated by using Floquét theory. Consistence between the time-invariant and -variant models verifies the comparisons.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(3):031007-031007-12. doi:10.1115/1.4038820.

The dynamic vehicle–track interactions are complex processes due to the highly nonlinear terms and spatially varying excitations in vehicle design, track maintenance, dynamic prediction, etc. Therefore, it is of importance to clarify the key factors affecting the dynamic behaviors of system components. In this paper, a comprehensive model is presented, which is capable of analyzing the global sensitivity of vehicle–track interactions. In this model, the vehicle–track interactions considering the nonlinear wheel–rail contact geometries are depicted in three-dimensional (3D) space, and then the approaches for global sensitivity analysis (GSA) and time–frequency analysis are combined with the dynamic model. In comparison to the local sensitivity analysis, the proposed model has accounted for the coupling contributions of various factors. Thus, it is far more accurate and reliable to evaluate the critical factors dominating the vehicle–track interactions. Based on the methods developed in the present study, numerical examples have been conducted to draw the following marks: track irregularities possess the dominant role in guiding the dynamic performance of vehicle–track systems, besides, the vertical stiffness of primary suspension and rail pads also shows significant influence on vertical acceleration of the car body and the wheel–rail vertical force, respectively. Finally, a method is developed to precisely extract the characteristic wavelengths and amplitude limits of track irregularities.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(3):031008-031008-14. doi:10.1115/1.4038641.

This paper considers a global sliding mode control (GSMC) approach for the stabilization of uncertain chaotic systems with multiple delays and input nonlinearities. By designing the global sliding mode surface, the offered scheme eliminates reaching phase problem. The offered control law is formulated based on state estimation, Lyapunov–Krasovskii stability theory, and linear matrix inequality (LMI) technique which present the asymptotic stability conditions. Moreover, the proposed design approach guarantees the robustness against multiple delays, nonlinear inputs, nonlinear functions, external disturbances, and parametric uncertainties. Simulation results for the presented controller demonstrate the efficiency and feasibility of the suggested procedure.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Comput. Nonlinear Dynam. 2018;13(3):034501-034501-4. doi:10.1115/1.4038777.

Fractional calculus is viewed as a novel and powerful tool to describe the stress and strain relations in viscoelastic materials. Consequently, the motions of engineering structures incorporated with viscoelastic dampers can be described by fractional-order differential equations. To deal with the fractional differential equations, initialization for fractional derivatives and integrals is considered to be a fundamental and unavoidable problem. However, this issue has been an open problem for a long time and controversy persists. The initialization function approach and the infinite state approach are two effective ways in initialization for fractional derivatives and integrals. By comparing the above two methods, this technical brief presents equivalence and unification of the Riemann–Liouville fractional integrals and the diffusive representation. First, the equivalence is proved in zero initialization case where both of the initialization function and the distributed initial condition are zero. Then, by means of initialized fractional integration, equivalence and unification in the case of arbitrary initialization are addressed. Connections between the initialization function and the distributed initial condition are derived. Besides, the infinite dimensional distributed initial condition is determined by means of input function during historic period.

Commentary by Dr. Valentin Fuster

### Announcement

J. Comput. Nonlinear Dynam. 2018;13(3):038001-038001-1. doi:10.1115/1.4038750.

The development and acceptance of modeling and simulation tools in the design of large and complex dynamical systems require reliable simulation mechanisms under model and data uncertainty. Sensitivity analysis and uncertainty quantification (UQ) are recognized as critical elements in improving credibility and robustness in prediction of phenomena and behaviors associated with dynamical systems. Sensitivities to model parameters, model forms, as well as initial and boundary conditions, are essential in simulation result interpretation, model calibration, simulation-based optimization under uncertainty, model reduction, and reliability assessment.

Commentary by Dr. Valentin Fuster