Research Papers

J. Comput. Nonlinear Dynam. 2017;13(2):021001-021001-17. doi:10.1115/1.4037927.

With the growing structural complexity and growing demands on structural reliability, nonlinear parameters identification is an efficient approach to provide better understanding of dynamic behaviors of the nonlinear system and contribute significantly to improve system performance. However, the dynamic response at nonlinear location, which cannot always be measured by the sensor, is the basis for most of these identification algorithms, and the clearance nonlinearity, which always exists to degrade the dynamic performance of mechanical structures, is rarely identified in previous studies. In this paper, based on the thought of output feedback which the nonlinear force is viewed as the internal feedback force of the nonlinear system acting on the underlying linear model, a frequency-domain nonlinear response reconstruction method is proposed to reconstruct the dynamic response at the nonlinear location from the arbitrary location where the sensor can be installed. For the clearance nonlinear system, the force graph method which is based on the reconstructed displacement response and nonlinear force is presented to identify the clearance value. The feasibility of the reconstruction method and identification method is verified by simulation data from a cantilever beam model with clearance nonlinearity. A clearance test-bed, which is a continuum structure with adjustable clearance nonlinearity, is designed to verify the effectiveness of proposed methods. The experimental results show that the reconstruction method can precisely reconstruct the displacement response at the clearance location from measured responses at reference locations, and based on the reconstructed response, the force graph method can also precisely identify the clearance parameter.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021002-021002-15. doi:10.1115/1.4037994.

A physics-based deformable tire–soil interaction simulation capability that can be fully integrated into the monolithic multibody dynamics computer algorithm is developed by extending a deformable tire model based on the flexible multibody dynamics approach to off-road mobility simulations with a moving soil patch technique and it is validated against test data. A locking-free nine-node brick element is developed for modeling large plastic soil deformation using the multiplicative finite strain plasticity theory along with the capped Drucker–Prager failure criterion. To identify soil parameters including cohesion and friction angle, the triaxial compression test is carried out, and the soil model developed is validated against the test data. In addition to the component level validation for the tire and soil models, the tire–soil interaction simulation capability developed in this study is validated against the soil bin mobility test results. The tire forces and rolling resistance coefficients predicted by the simulation model agree well with the test results. It is shown that effect of the wheel loads and tire inflation pressures is well captured in the simulation model. Furthermore, it is demonstrated that the moving soil patch technique, with which soil behavior only in the vicinity of the rolling tire is solved to reduce the soil model dimensionality, leads to a significant reduction in computational time, thereby enabling use of the high-fidelity physics-based tire–soil interaction model in the large-scale off-road mobility simulation.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021003-021003-8. doi:10.1115/1.4037929.

This paper is mainly concerned with asymptotic stability for a class of fractional-order (FO) nonlinear system with application to stabilization of a fractional permanent magnet synchronous motor (PMSM). First of all, we discuss the stability problem of a class of fractional time-varying systems with nonlinear dynamics. By employing Gronwall–Bellman's inequality, Laplace transform and its inverse transform, and estimate forms of Mittag–Leffler (ML) functions, when the FO belongs to the interval (0, 2), several stability criterions for fractional time-varying system described by Riemann–Liouville's definition is presented. Then, it is generalized to stabilize a FO nonlinear PMSM system. Furthermore, it should be emphasized here that the asymptotic stability and stabilization of Riemann–Liouville type FO linear time invariant system with nonlinear dynamics is proposed for the first time. Besides, some problems about the stability of fractional time-varying systems in existing literatures are pointed out. Finally, numerical simulations are given to show the validness and feasibleness of our obtained stability criterions.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021004-021004-10. doi:10.1115/1.4038203.

The problem of robust exponential stability for a class of switched nonlinear dynamical systems with uncertainties and time-varying delays is investigated. On the assumption that each isolated subsystem of the interconnected system can be exponentially stabilized and the corresponding Lyapunov functions are available, using M-matrix property, the differential inequalities with time-varying delays are constructed. By the stability analysis of the differential inequalities, the sufficient conditions to ensure the robust exponential stability of the switched interconnected systems (SIS) under arbitrary switching are obtained. The proposed method, which neither requires the individual subsystems to share a common Lyapunov function (CLF), nor needs to know the values of individual Lyapunov functions at each switching time, would provide a new mentality for studying stability of arbitrary switching. In addition, by resorting to average dwell time approach, conditions for guaranteeing the robust exponential stability of SIS under constrained switching are derived. The proposed criteria are explicit, and they are convenient for practical applications. Finally, two numerical examples are given to illustrate the validity and correctness of the proposed theories.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021005-021005-8. doi:10.1115/1.4037418.

This paper comprises of a finite difference method with implicit scheme for the Riesz fractional reaction–diffusion equation (RFRDE) by utilizing the fractional-centered difference for approximating the Riesz derivative, and consequently, we obtain an implicit scheme which is proved to be convergent and unconditionally stable. Also a novel analytical approximate method has been dealt with namely optimal homotopy asymptotic method (OHAM) to investigate the solution of RFRDE. The numerical solutions of RFRDE obtained by proposed implicit finite difference method have been compared with the solutions of OHAM and also with the exact solutions. The comparative study of the results establishes the accuracy and efficiency of the techniques in solving RFRDE. The proposed OHAM renders a simple and robust way for the controllability and adjustment of the convergence region and is applicable to solve RFRDE.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021006-021006-12. doi:10.1115/1.4037596.

This paper, for the first time, investigates the nonlinear forced dynamics of a three-layered microplate taking into account all the in-plane and out-of-plane motions. The Kirchhoff's plate theory, along with von Kármán nonlinear strains, is employed to derive the nonlinear size-dependent transverse and in-plane equations of motion in the modified couple stress theory (MCST) framework, based on Hamilton's energy principle. A nonconservative damping force of viscous type as well as an external excitation load consisting of a harmonic term is considered in the model. All the transverse and in-plane displacements and inertia are accounted for in both the theoretical modeling and numerical simulations; this leads to further complexities in the nonlinear model and simulations. These complexities arising in the theoretical model are overcome through the use of a well-optimized numerical scheme. The effects of different layer arrangements and different layer material percentages on the force–amplitude and frequency–amplitude curves of the microsystem are investigated. The results of this study shed light in the nonlinear resonant behavior of multilayered microplates and could be helpful in design and analysis of multilayered microplates in microelectromechanical systems (MEMS) applications.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021007-021007-9. doi:10.1115/1.4037931.

The fractional differential equations of the single-degree-of-freedom (DOF) quarter vehicle with a magnetorheological (MR) suspension system under the excitation of sine are established, and the numerical solution is acquired based on the predictor–corrector method. The analysis of phase trajectory, time domain response, and Poincaré section shows that the nonlinear dynamic characteristics between fractional and integer-order suspension systems are quite different, which proves the superiority of using fractional order to describe the physical properties. By discussing the influence of each parameter on the vibration, the range of parameters to avoid the chaotic vibration is obtained. The variable feedback control is used to control the chaotic vibration effectively.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021008-021008-18. doi:10.1115/1.4037797.

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases, the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices (STMs) are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021009-021009-9. doi:10.1115/1.4038148.

In this paper, a special type of beam element is developed with three nodes and with only translational degrees-of-freedom (DOFs) at each node. This element can be used effectively to build low degree-of-freedom models of rotors. The initial model from the Bernoulli theory is fitted to experimental results by nonlinear optimization. This way, we can avoid the complex modeling of contact problems between the parts of squirrel cage rotors. The procedure is demonstrated on the modeling of a machine tool spindle.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021010-021010-11. doi:10.1115/1.4038142.

In this work, an efficient topology optimization approach is proposed for a three-dimensional (3D) flexible multibody system (FMBS) undergoing both large overall motion and large deformation. The FMBS of concern is accurately modeled first via the solid element of the absolute nodal coordinate formulation (ANCF), which utilizes both nodal positions and nodal slopes as the generalized coordinates. Furthermore, the analytical formulae of the elastic force vector and the corresponding Jacobian are derived for efficient computation. To deal with the dynamics in the optimization process, the equivalent static load (ESL) method is employed to transform the topology optimization problem of dynamic response into a static one. Besides, the newly developed topology optimization method by moving morphable components (MMC) is used and reevaluated to optimize the 3D FMBS. In the MMC-based framework, a set of morphable structural components serves as the building blocks of optimization and hence greatly reduces the number of design variables. Therefore, the topology optimization approach has a potential to efficiently optimize an FMBS of large scale, especially in 3D cases. Two numerical examples are presented to validate the accuracy of the solid element of ANCF and the efficiency of the proposed optimization methodology, respectively.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021011-021011-8. doi:10.1115/1.4037765.

In this work, Lie symmetry analysis for the time fractional third-order evolution (TOE) equation with Riemann–Liouville (RL) derivative is analyzed. We transform the time fractional TOE equation to nonlinear ordinary differential equation (ODE) of fractional order using its Lie point symmetries with a new dependent variable. In the reduced equation, the derivative is in Erdelyi–Kober (EK) sense. We obtain a kind of an explicit power series solution for the governing equation based on the power series theory. Using Ibragimov's nonlocal conservation method to time fractional partial differential equations (FPDEs), we compute conservation laws (CLs) for the TOE equation. Two dimensional (2D), three-dimensional (3D), and contour plots for the explicit power series solution are presented.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021012-021012-9. doi:10.1115/1.4031194.

The generalized polynomial chaos (gPC) mathematical technique, when integrated with the extended Kalman filter (EKF) method, provides a parameter estimation and state tracking method. The truncation of the series expansions degrades the link between parameter convergence and parameter uncertainty which the filter uses to perform the estimations. An empirically derived correction for this problem is implemented, which maintains the original parameter distributions. A comparison is performed to illustrate the improvements of the proposed approach. The method is demonstrated for parameter estimation on a regression system, where it is compared to the recursive least squares (RLS) method.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021013-021013-9. doi:10.1115/1.4038444.

In this paper, we formulate a new nonstandard finite difference (NSFD) scheme to study the dynamic treatments of a class of fractional chaotic systems. To design the new proposed scheme, an appropriate nonlocal framework is applied for the discretization of nonlinear terms. This method is easy to implement and preserves some important physical properties of the considered model, e.g., fixed points and their stability. Additionally, this scheme is explicit and inexpensive to solve fractional differential equations (FDEs). From a practical point of view, the stability analysis and chaotic behavior of three novel fractional systems are provided by the proposed approach. Numerical simulations and comparative results confirm that this scheme is also successful for the fractional chaotic systems with delay arguments.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;13(2):021014-021014-9. doi:10.1115/1.4038447.

This paper deals with the nonlinear fluid structure interaction (FSI) dynamics of a Dipteran flight motor inspired flapping system in an inviscid fluid. In the present study, the FSI effects are incorporated to an existing forced Duffing oscillator model to gain a clear understanding of the nonlinear dynamical behavior of the system in the presence of aerodynamic loads. The present FSI framework employs a potential flow solver to determine the aerodynamic loads and an explicit fourth-order Runge–Kutta scheme to solve the structural governing equations. A bifurcation analysis has been carried out considering the amplitude of the wing actuation force as the control parameter to investigate different complex states of the system. Interesting dynamical behavior including period doubling, chaotic transients, periodic windows, and finally an intermittent transition to stable chaotic attractor have been observed in the response with an increase in the bifurcation parameter. Similar dynamics is also reflected in the aerodynamic loads as well as in the trailing edge wake patterns.

Commentary by Dr. Valentin Fuster

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