Research Papers

J. Comput. Nonlinear Dynam. 2018;13(7):071001-071001-11. doi:10.1115/1.4039899.

Nonlinear vibration characteristics of three-blade wind turbines are theoretically investigated. The wind turbine is modeled as a coupled system, consisting of a flexible tower with two degrees-of-freedom (2DOF), and three blades, each with a single degree of freedom (SDOF). The blades are subjected to steady winds. The wind velocity increases proportionally with height due to vertical wind shear. The natural frequency diagram is calculated with respect to the rotational speed of the wind turbine. The corresponding linear system with parametric excitation terms is analyzed to determine the rotational speeds where unstable vibrations appear and to predict at what rotational speeds the blades may vibrate at high amplitudes in a real wind turbine. The frequency response curves are then obtained by applying the swept-sine test to the equations of motion for the nonlinear system. They exhibit softening behavior due to the nonlinear restoring moments acting on the blades. Stationary time histories and their fast Fourier transform (FFT) results are also calculated. In the numerical simulations, localization phenomena are observed, where the three blades vibrate at different amplitudes. Basins of attraction (BOAs) are also calculated to examine the influence of a disturbance on the appearance of localization phenomena.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(7):071002-071002-8. doi:10.1115/1.4040081.

A method to calculate static solutions for mechanical systems containing rigid and flexible bodies modeled by finite elements is described. The formulation of the equations makes use of generalized strains, which leads to an extended set of equations for both these generalized strains and nodal coordinates, together with constraint equations imposing the relations between these two groups of coordinates. The associated Lagrangian multipliers are the generalized stresses. The resulting iteration scheme appears to be quite robust in comparison with more traditional methods, especially if some displacements are prescribed. Once a static solution has been found, the linearized equations of motion about this solution can be obtained in terms of a set of minimal coordinates, that is, in the degrees-of-freedom (DOFs). In addition, a continuation method is described for tracing a branch of static solutions if some parameters are varied. The method is of the familiar predictor–corrector type with a linear or cubic predictor and a corrector with a step size constraint. Applications to a large-deflection problem of a curved cantilever beam, large deflections of a fluid-conveying tube and its resulting instability, and the buckling of an overconstrained parallel leaf-spring mechanism due to misalignment are given.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(7):071003-071003-12. doi:10.1115/1.4040021.

We present an efficient method to significantly reduce the offline cost associated with the construction of training sets for hyper-reduction of geometrically nonlinear, finite element (FE)-discretized structural dynamics problems. The reduced-order model is obtained by projecting the governing equation onto a basis formed by vibration modes (VMs) and corresponding modal derivatives (MDs), thus avoiding cumbersome manual selection of high-frequency modes to represent nonlinear coupling effects. Cost-effective hyper-reduction is then achieved by lifting inexpensive linear modal transient analysis to a quadratic manifold (QM), constructed with dominant modes and related MDs. The training forces are then computed from the thus-obtained representative displacement sets. In this manner, the need of full simulations required by traditional, proper orthogonal decomposition (POD)-based projection and training is completely avoided. In addition to significantly reducing the offline cost, this technique selects a smaller hyper-reduced mesh as compared to POD-based training and therefore leads to larger online speedups, as well. The proposed method constitutes a solid alternative to direct methods for the construction of the reduced-order model, which suffer from either high intrusiveness into the FE code or expensive offline nonlinear evaluations for the determination of the nonlinear coefficients.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(7):071004-071004-9. doi:10.1115/1.4040128.

Active magnetic bearing (AMB) is the device to support and control rotating shaft. Feedback linearization is one of the methods to compensate the system nonlinearity, and it is often used in the control of AMB. Some parameters in the electromagnetic force model have often been ignored or their parametric uncertainty from the nominal values has been calibrated; however, their influence on the stability has not been investigated. In this paper, the influence of the parametric uncertainty in the electromagnetic force model on the stability of AMB is investigated. The equilibrium positions and their stability are investigated and clarified analytically. Furthermore, the choice of the parameter value for improving the stability of AMB with feedback linearization is proposed, and its effectiveness is explained analytically. It is shown that the proposed choice of the parameter value also reduces the remained nonlinearity significantly. The validity of theoretical results and proposed choice of the parameter value are confirmed by experiment.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(7):071005-071005-12. doi:10.1115/1.4040253.

Parameter estimation and model order reduction (MOR) are important system identification techniques used in the development of models for mechanical systems. A variety of classical parameter estimation and MOR methods are available for nonlinear systems but performance generally suffers when little is known about the system model a priori. Recent advancements in information theory have yielded a quantity called causation entropy (CSE), which is a measure of influence between elements in a multivariate time series. In parameter estimation problems involving dynamic systems, CSE can be used to identify which state transition functions in a discrete-time model are important in driving the system dynamics, leading to reductions in the dimensionality of the parameter space. This method can likewise be used in black box system identification problems to reduce model order and limit issues with overfitting. Building on the previous work, this paper illustrates the use of CSE-enabled parameter estimation for nonlinear mechanical systems of varying complexity. Furthermore, an extension to black-box system identification is proposed wherein CSE is used to identify the proper model order of parameterized black-box models. This technique is illustrated using nonlinear differential equation (NDE) models of physical devices, including a nonlinear spring–mass–damper, a pendulum, and a nonlinear model of a car suspension. Overall, the results show that CSE is a promising new tool for both gray-box and black-box system identification that can speed convergence toward a parameter solution and mitigate problems with model overfitting.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(7):071006-071006-10. doi:10.1115/1.4040254.

In light of two wheel–rail contact relations, i.e., displacement compatibility and force equilibrium, a newly developed three-dimensional (3D) model for vehicle–track interactions is presented in this paper. This model is founded on the basis of an assumption: wheel–rail rigid contact. Unlike most of the dynamic models, where the interconnections between the vehicle and the track entirely depend on the wheel–rail contact forces, the subsystems of the vehicle and the tracks in the present study are effectively united as an entire system with interactive matrices of stiffness, damping and mass by the energy-variational principle and wheel–rail contact geometry. With wheel–rail nonlinear creepage/equivalent stiffness, this proposed model can derive dynamic results approaching to those of vehicle-track coupled dynamics. However, it is possible to apply a relatively large time integral step with numerical stability in computations. By simplifying into a linearized model, pseudo-excitation method (PEM) can be theoretically implemented to characterize the dominant vibration frequencies of vehicle-track systems due to random excitations. Finally, a trail method is designed to achieve the wheel climbing derailment process and a full derailment case where the bottom of the wheel flange has completely reached the rail top to form a complete derailment is presented.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(7):071007-071007-8. doi:10.1115/1.4040129.

The robust control for a class of disturbed fractional-order systems is presented in this paper. The proposed controller considers a dynamic observer to exactly compensate for matched disturbances in finite time, and a procedure to compensate for unmatched disturbances is then derived. The proposed disturbance observer is built upon continuous fractional sliding modes, producing a fractional-order reaching phase, leading to a continuous control signal, yet able to reject for some continuous but not necessarily differentiable disturbances. Numerical simulations and comparisons are presented to highlight the reliability of the proposed scheme.

Commentary by Dr. Valentin Fuster

Technical Brief

J. Comput. Nonlinear Dynam. 2018;13(7):074501-074501-6. doi:10.1115/1.4040022.

This paper deals with the issue of robustness in control of robots using the proportional plus derivative (PD) controller and the augmented PD controller. In the literature, a variety of PD and model-based controllers for multilink serial manipulator have been claimed to be asymptotically stable for trajectory tracking, in the sense of Lyapunov, as long as the controller gains are positive. In this paper, we first establish that for simple PD controllers, the criteria of positive controller gains are insufficient to establish asymptotic stability, and second that for the augmented PD controller the criteria of positive controller gains are valid only when there is no uncertainty in the model parameters. We show both these results for a simple planar two-degrees-of-freedom (2DOFs) robot with two rotary (R) joints, following a desired periodic trajectory, using the Floquet theory. We provide numerical simulation results which conclusively demonstrate the same.

Commentary by Dr. Valentin Fuster

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