Research Papers

J. Comput. Nonlinear Dynam. 2018;13(10):101001-101001-15. doi:10.1115/1.4040780.

In this paper, we present efficient algorithms for computation of the residual of the constrained discrete Euler–Lagrange (DEL) equations of motion for tree structured, rigid multibody systems. In particular, we present new recursive formulas for computing partial derivatives of the kinetic energy. This enables us to solve the inverse dynamics problem of the discrete system with linear computational complexity. The resulting algorithms are easy to implement and can naturally be applied to a very broad class of multibody systems by imposing constraints on the coordinates by means of Lagrange multipliers. A comparison is made with an existing software package, which shows a drastic improvement in computational efficiency. Our interest in inverse dynamics is primarily to apply direct transcription optimal control methods to multibody systems. As an example application, we present a digital human motion planning problem, which we solve using the proposed method. Furthermore, we present detailed descriptions of several common joints, in particular singularity-free models of the spherical joint and the rigid body joint, using the Lie groups of unit quaternions and unit dual quaternions, respectively.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(10):101002-101002-12. doi:10.1115/1.4040709.

Taking the flow field features of labyrinth seal into consideration, the fluid force generated from the seal cavity, which is spatially separated into two regions, is modeled with the perturbation method. The rotor orbit defined in the perturbation analysis is spatio-temporal varied, which is quite different from the usually preconditioned elliptical track. Meanwhile, the nonlinear fluid force originating from the seal clearance is delineated by the Muszynska's model. Based on the short bearings assumption, a nonlinear oil-film force model is employed. The rotating shaft is simulated by Timoshenko beam finite element with the consideration of geometric asymmetry. Applying the Lagrange's equations, the motion equations of the rotor-bearing-labyrinth seal system are derived. By means of spectrum cascades, bifurcation diagrams, Poincaré maps, etc., the numerical analysis of the system dynamic characteristics is conducted. The results show that abundant nonlinear behaviors can be triggered in the speed-up. The instability threshold and the vibration amplitude of the rotor system are, respectively, enhanced and reduced by the increasing inlet pressure. With shorter seal length, the sealing effect is decreased, whereas the system stability is improved. The fluid-whip phenomenon can be eliminated by increasing the mass unbalance eccentricity at a certain rotational speed.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(10):101003-101003-7. doi:10.1115/1.4040708.

A family of Runge–Kutta (RK) methods designed for better stability is proposed. Authors have optimized the stability of RK method by increasing the stability region by trading some of the higher order terms in the Taylor series. For flow involving shocks, compromising a few higher order terms will not affect convergence rate that is justified with an example. Though this kind of analysis began about three decades ago, most of the papers dealt with classical optimization and ended up in relatively nonoptimal values. Here, authors have overcome that by using evolutionary algorithm (EA), the result is refined using multisection method (MSM). The schemes designed based on this procedure have better stability than the classical RK methods, strong stability RK methods (SSPRK), and low dispersive and dissipative RK methods (LDDRK) of the same number of stages. Authors have tested the schemes on a variety of test cases and found some significant improvement.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(10):101004-101004-10. doi:10.1115/1.4040871.

This paper focuses on the implementation of a Hamiltonian model of multi-unit hydropower systems (MUHSs). First, a nonlinear mathematical model of the MUHS is established considering the occurrence of water hammer during the transient process. From the point of view of the energy transmission and dissipation of the system, a novel Hamiltonian model of the MUHS is proposed. Moreover, numerical simulations are carried out to further investigate the effectiveness and consistency of the implemented model. Finally, a comparative analysis is performed to validate the proposed approach against existing methods. The results demonstrate that the proposed Hamiltonian function not only reflects the energy change but also describes the complex dynamic evolution of MUHSs in transient processes. It is also found that the transient dynamic behavior of the system is influenced by the coupled effect of common penstock and the interaction of basic system variables. This study provides theoretical basis for the safe and stable operation of hydropower stations during transient processes.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(10):101005-101005-13. doi:10.1115/1.4040870.

Model-based design facilitates quick development of vehicle controllers early in the development cycle. The goal is to develop simple, accurate, and computationally efficient physics based models that are capable of real-time simulation. We present models that serve the purpose of both plant and anti-jerk control design of electric vehicles (EVs). In this research, we propose a procedure for quick identification of longitudinal dynamic parameters for a high-fidelity plant and control-oriented model of an EV through road tests. Experimental data were gathered on our test vehicle, a Toyota Rav4EV, using an integrated measurement system to collect data from multiple sensors. A matlab/simulink nonlinear least square parameter estimator with a trust-reflective algorithm was used to identify the vehicle parameters. The models have been validated against experimental data.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(10):101006-101006-8. doi:10.1115/1.4041028.

Computational rod models have emerged as efficient tools to simulate the bending and twisting deformations of a variety of slender structures in engineering and biological applications. The dynamics of such deformations, however, strongly depends on the constitutive law in bending and torsion that, in general, may be nonlinear, and vary from material to material. Jacobian-based computational rod models require users to change the Jacobian if the functional form of the constitutive law is changed, and hence are not user-friendly. This paper presents a scheme that automatically modifies the Jacobian based on any user-defined constitutive law without requiring symbolic differentiation. The scheme is then used to simulate force-extension behavior of a coiled spring with a softening constitutive law.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(10):101007-101007-6. doi:10.1115/1.4041030.

In this paper, we construct and analyze a Legendre spectral-collocation method for the numerical solution of distributed-order fractional initial value problems. We first introduce three-term recurrence relations for the fractional integrals of the Legendre polynomial. We then use the properties of the Caputo fractional derivative to reduce the problem into a distributed-order fractional integral equation. We apply the Legendre–Gauss quadrature formula to compute the distributed-order fractional integral and construct the collocation scheme. The convergence of the proposed method is discussed. Numerical results are provided to give insights into the convergence behavior of our method.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(10):101008-101008-11. doi:10.1115/1.4041032.

In this work, small-world outer synchronization of coupled small-world networks is presented. In particular, we use Newman and Watts model to achieve small-world outer synchronization of small-world chaotic networks with Chua's oscillators like chaotic nodes. By means of extensive numerical simulations, we show that the new outer connections between existing networks decrease the necessary coupling strength to achieve outer synchronization. Two scenarios of interest are studied, (i) small-world outer synchronization with unidirectional outer connections (with chaotic master network), and (ii) small-world outer synchronization with bidirectional outer connections (without chaotic master network). In both scenarios, the isolated networks are bidirectionally coupled using Chua's oscillators like chaotic nodes.

Topics: Synchronization
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(10):101009-101009-13. doi:10.1115/1.4041031.

The dissipative contact force model plays a key role in predicting the response of multibody mechanical systems. Contact-impact event can frequently take place in multibody systems and the impact pair is often affected by supporting forces which are treated as external spring forces. However, the external spring forces are ignored during the derivation process of existing dissipative contact force models. Considering the influences of external spring forces, the fact is discussed that the crucial issues, including relative velocity and energy loss, in modeling dissipative contact force are different compared to the same issues analyzed in existing literatures. These differences can result in obvious errors in describing the collision response in multibody systems. Thus, a comparative study is carried out for examining the performances of several popular dissipative contact force models in multibody dynamics. For this comparison, a method associated with Newton's method is proposed to calculate the contact force that meets the Strong's law of energy loss and this force is used as reference. The comparative results show that the models suitable for both hard and soft contact exhibit good accuracy when contact equivalent stiffness is far larger than external spring stiffness by two orders of magnitude. Conversely, these models can cause varying degree and obvious errors in contact force, number of collisions, etc., especially when the difference in stiffness is close to or less than one order of magnitude.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(10):101010-101010-7. doi:10.1115/1.4041085.

In this paper, we introduce a class of time-fractional diffusion model with singular source term. The derivative employed in this model is defined in the Caputo sense to fit the conventional initial condition. With assistance of corresponding linear fractional differential equation, we verify that the solution of such model may not be globally well-defined, and the dynamics of this model depends on the order of fractional derivative and the volume of spatial domain. In simulation, a finite difference scheme is implemented and interesting numerical solutions of model are illustrated graphically. Meanwhile, the positivity, monotonicity, and stability of the proposed scheme are proved. Numerical analysis and simulation coincide the theoretical studies of this new model.

Commentary by Dr. Valentin Fuster

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