Bíngen Yang

Abstract The determination of the eigenvalues of damped vibrating continua plays an important role in analysis of this type of distributed systems in structural dynamics. Unlike a proportionally damped system, for a nonproportionally damped vibrating continuum, its eigenvalues in closed form are usually difficult to obtain, and consequently, a discretized model is often used for determining an approximate estimation of the system eigenvalues. In this paper, a new analytical method is proposed for systematic estimation of the eigenvalues of one-dimensional vibrating continua with nonproportional damping. The proposed method, which combines the Distributed Transfer Function Method (DTFM) and the root locus method, delivers eigenvalues of such systems in closed-form without the need for performing model discretization. The solution procedure of the proposed method is detailed and illustrated on two typical nonproportionally damped continua: a single-body system composed of an elastic bar with end mass and viscous damper, and a stepped system composed of a simply-supported Euler-Bernoulli beam with partially distributed viscous damping. Numerical examples and comparison with traditional assumed modes method indicate that this DTFM-based root locus method maintains the accuracy and efficiency in eigenvalue computation, while provided with a systematic establishment of the transcendental characteristic equation of a given distributed system.

The sandwich plates in consideration are structures composed of a number of Lévy plate components laminated with viscoelastic layers, and they are seen in broad engineering applications. In vibration analysis of a sandwich plate, conventional analytical methods are limited due to the complexity of the geometric and material properties of the structure, and consequently, numerical methods are commonly used. In this paper, an innovative analytical method is proposed for vibration analysis of sandwich Lévy plates having different configurations of viscoelastic layers and using various models of viscoelastic materials. The focus of the investigation is on the determination of closed-form frequency response at any given frequencies. In the development, an s-domain state-space formulation is established by the Distributed Transfer Function Method (DTFM). With this formulation, closed-form analytical solutions of the frequency response problem of sandwich plates are obtained, without the need for spatial discretization. As one unique feature, the DTFM-based approach has consistent formulas and unified solution procedures by which analytical solutions at both low and high frequencies are obtained. The accuracy, efficiency, and versatility of the proposed analytical method are demonstrated in three numerical examples, where the DTFM-based analysis is compared with the finite element method and certain existing analytical solutions.

Multiconductor transmission lines (MCTLs) have various applications in electrical engineering. In modeling and analysis of MCTLs, which are described by a set of coupled partial differential equations, most existing methods rely on approximation or discretization. For high-speed high-frequency applications, accurate analytical methods are always desirable. Such a method, however, is not currently available for complex MCTLs. This paper presents an innovative analytical method for modeling and analysis of MCTLs with various configurations. The proposed method, which is called the Distributed Transfer Function Method, is capable of delivering closed-form analytical solutions for complex MCTLs, in both the frequency domain and the time domain. One highlight of the proposed method is that it gives exact and closed-form solutions for branched transmission lines for the first time. The accuracy, efficiency, and high-frequency utility of the method is demonstrated in numerical examples.

Abstract The Inductrack system, a magnetic levitation strategy based on permanent magnets, has been utilized in many engineering applications since it was envisioned in 1990s. To further implement such Maglev system in high-speed ground transportation, its dynamic behaviors and control methods should be comprehensively studied. Most previous investigations on this topic adopted the steady-state Inductrack model, which is notable but has limited consideration from a dynamics perspective. Recently, a transient model for a 2D, 3DOF Inductrack system has been available, which characterizes the nonlinear, time-varying, and motion-dependent couplings in transient scenarios. Dynamic analysis has shown that both vertical and pitch vibrations of the Maglev vehicle cannot be naturally damped out in most cases, necessitating a feedback control design in the Inductrack system. Presented in this paper is a control design approach for the rigid-body Inductrack system. Firstly, a state feedback linear controller is placed based on a linear reference model and tuned using linear quadratic integral optimal control theory. The linear controller computes the actuation force for each on-board active Halbach array. Secondly, a nonlinear mapping function is identified that outputs the active current density of each array as the control effort. As verified in the numerical examples, the vertical and pitch motions of the Inductrack vehicle can be effectively stabilized under various of traveling speed. Although only a 2D setup is considered, the modeling and control technique developed in this work are fully extendible to address more complicated scenarios.

Abstract Combined systems, which are flexible structures carrying moving subsystems, are seen in various applications. Due to structure–subsystem interactions, the structure in a combined system encounters jump discontinuities in its internal forces (such as the bending moment and shear force of a beam). Accurate estimation of such jump discontinuities is important to the performance, safety, and longevity of a combined system. Because of the time-varying nature and complexity of structure–subsystem interactions, conventional series solution methods experience slow convergence, and the Gibbs phenomenon in computation and the improved series expansion methods are limited to certain proportionally damped continua under moving forces and moving oscillators. In this paper, a novel modified series expansion method (MSEM) is proposed to resolve the aforementioned issues with the existing series solution methods. Through the introduction of a jump influence function, the proposed method produces fast-convergent series solutions and accurately predicts the jump discontinuities without the Gibbs phenomenon. The MSEM is applicable to structures with nonproportional damping and subject to arbitrary boundary conditions, and it can easily manage general M-DOF moving subsystems having multiple contact points with a supporting structure. As an important result of this investigation, a mathematical proof of the convergence of the MSEM-based solutions is given for the first time. Additionally, two numerical examples are presented to demonstrate the accuracy, efficiency, and versatility of the proposed MSEM in modeling and analysis of combined systems.