Research Papers

J. Comput. Nonlinear Dynam. 2018;14(1):011001-011001-15. doi:10.1115/1.4041771.

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;14(1):011002-011002-16. doi:10.1115/1.4041775.

A systematic theoretical approach is presented, revealing dynamics of a class of multibody systems. Specifically, the motion is restricted by a set of bilateral constraints, acting simultaneously with a unilateral constraint, representing a frictional impact. The analysis is carried out within the framework of Analytical Dynamics and uses some concepts of differential geometry, which provides a foundation for applying Newton's second law. This permits a successful and illuminating description of the dynamics. Starting from the unilateral constraint, a boundary is defined, providing a subspace of allowable motions within the original configuration manifold. Then, the emphasis is focused on a thin boundary layer. In addition to the usual restrictions imposed on the tangent space, the bilateral constraints cause a correction of the direction where the main impulse occurs. When friction effects are negligible, the dominant action occurs along this direction and is described by a single nonlinear ordinary differential equation (ODE), independent of the number of the original generalized coordinates. The presence of friction increases this to a system of three ODEs, capturing the essential dynamics in an appropriate subspace, arising by bringing the image of the friction cone from the physical to the configuration space. Moreover, it is shown that the classical Darboux–Keller approach corresponds to a special case of the new method. Finally, the theoretical results are complemented by a selected set of numerical results for three examples.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;14(1):011003-011003-19. doi:10.1115/1.4041967.

A hybrid uncertain parameter model (HUPM) is introduced to predict the luffing angular response (LAR) field of the dual automobile cranes system (DACS) with random and interval parameters. In the model, all random parameters with specified probabilistic distributions comprise a random vector, while all interval parameters with determined bounds comprise an interval vector. A hybrid uncertain LAR equilibrium equation is established, and a novel approach named as hybrid perturbation compound function-based moment method is proposed based on the HUPM. In the hybrid perturbation compound function-based moment method, the expression of LAR is developed according to the random interval perturbation compound function-based method. More, by using the random interval compound function-based moment method and the monotonic technique, the expectations and variances of the bounds for LAR are calculated. Compared with the hybrid Monte Carlo method (HMCM) and interval perturbation method (IPM), numerical results on different uncertain cases of the DACS demonstrate the feasibility and efficiency of the proposed algorithm. The proposed method is proved to be an effective engineering method to quantify the effects of hybrid uncertain parameters on the LAR of DACS.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;14(1):011004-011004-9. doi:10.1115/1.4041911.

This paper considers a design problem of dissipative and observer-based finite-time nonfragile control for a class of uncertain discrete-time system with time-varying delay, nonlinearities, external disturbances, and actuator saturation. In particular, in this work, it is assumed that the nonlinearities satisfy Lipschitz condition for obtaining the required results. By choosing a suitable Lyapunov–Krasovskii functional, a new set of sufficient conditions is obtained in terms of linear matrix inequalities, which ensures the finite-time boundedness and dissipativeness of the resulting closed-loop system. Meanwhile, the solvability condition for the observer-based finite-time nonfragile control is also established, in which the control gain can be computed by solving a set of matrix inequalities. Finally, a numerical example based on the electric-hydraulic system is provided to illustrate the applicability of the developed control design technique.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;14(1):011005-011005-6. doi:10.1115/1.4041912.

We consider an extension of the well-known Hamilton–Jacobi–Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;14(1):011006-011006-10. doi:10.1115/1.4042016.

Identification of aircraft flight dynamic modes has been implemented by adopting highly nonlinear flight test data. This paper presents a new algorithm for identification of the flight dynamic modes based on Hilbert–Huang transform (HHT) due to its superior potential capabilities in nonlinear and nonstationary signal analysis. Empirical mode decomposition and ensemble empirical mode decomposition (EEMD) are the two common methods that apply the HHT transform for decomposition of the complex signals into instantaneous mode frequencies; however, experimentally, the EMD faces the problem of “mode mixing,” and EEMD faces with the signal precise reconstruction, which leads to imprecise results in the estimation of flight dynamic modes. In order to overcome (handle) this deficiency, an improved EEMD (IEEMD) algorithm for processing of the complex signals that originate from flight data record was introduced. This algorithm disturbing the original signal using white Gaussian noise, IEEMD, is capable of making a precise reconstruction of the original signal. The second improvement is that IEEMD performs signal decomposition with fewer number of iterations and less complexity order rather than EEMD. This algorithm has been applied to aircraft spin maneuvers flight test data. The results show that implication of IEEMD algorithm on the test data obtained more precise signal extractions with fewer iterations in comparison to EEMD method. The signal is reconstructed by summing the flight modes with more accuracy respect to the EEMD. The IEEMD requires a smaller ensemble size, which results in saving of a significant computational cost.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;14(1):011007-011007-5. doi:10.1115/1.4041968.

This paper has dedicated to study the control of chaos when the system dynamics is unknown and there are some limitations on measuring states. There are many chaotic systems with these features occurring in many biological, economical and mechanical systems. The usual chaos control methods do not have the ability to present a systematic control method for these kinds of systems. To fulfill these strict conditions, we have employed Takens embedding theorem which guarantees the preservation of topological characteristics of the chaotic attractor under an embedding named “Takens transformation.” Takens transformation just needs time series of one of the measurable states. This transformation reconstructs a new chaotic attractor which is topologically similar to the unknown original attractor. After reconstructing a new attractor its governing dynamics has been identified. The measurable state of the original system which is one of the states of the reconstructed system has been controlled by delayed feedback method. Then the controlled measurable state induced a stable response to all of the states of the original system.

Commentary by Dr. Valentin Fuster

Technical Brief

J. Comput. Nonlinear Dynam. 2018;14(1):014501-014501-11. doi:10.1115/1.4041828.

In this work, vibration characteristic diagnosis of misalignment rotor in loosely fitted bearing is investigated using dimensional analysis (DA) approach. A comprehensive empirical model (EM) using nondimensional parameters is developed to diagnose the rotor-bearing system, and EM model has been validated through an experimental setup developed in-house. Experiments are performed for various defects such as misalignment and bearing looseness. The EM results can be used to monitor the real-time conditions of the rotor-bearing system. This work also presents the effect of misalignment and bearing looseness under various load and speed conditions. Further, work has been extended to predict the combined effect of bearing looseness and misalignment. It has been found that EM model predictions of the vibration amplitude are better when compared to experimental results.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;14(1):014502-014502-6. doi:10.1115/1.4041891.

The present work deals with the solutions of the Gerdjikov–Ivanov(G–I) equation with the Riesz fractional derivative by means of the time-splitting spectral approach. In this approach, the G–I equation is split into two equations and the proposed technique viz. time-splitting spectral method is employed for discretizing the equation in space and then subsequently integrating in time exactly. Furthermore, an implicit finite difference method (IMFD) is utilized here to compare the results with the above-mentioned seminumerical method viz. time-splitting spectral technique. Moreover, it has been established that the proposed method is unconditionally stable. In addition to these, the error norms have been also presented here.

Commentary by Dr. Valentin Fuster

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