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As a new approach to magnetic levitation by permanent magnet envisioned in 1990s, the concept of Inductrack has been applied to designs of various Maglev systems, including rocket launchers, Maglev trains, and the Hyperloop system as a conceptual future transportation tool. In such a system, the interaction forces between the source and induced magnetic fields are motion-coupled, and inevitably complicated with transient responses. Most previous investigations have mainly relied on the assumptions of ideal magnetic field, steady-state response and averaged magnetic force. Although these analyses provide some useful information for system design, due to limited understanding of the magnetic field-motion coupling and transient responses, they cannot predict the dynamic behaviors of an Inductrack system with fidelity and accuracy. Presented in this paper is a new transient model of Inductrack dynamic systems that is derived from fundamental physics laws, with the interactions of magnetic forces and motion in consideration. In the development, a state space representation of the transient model is established for numerical simulations. As verified in numerical examples, a benchmark 2-DOF transient model not only is able to reproduce the “steady-state” results reported in the literature, but also can exhibit the transient behaviors of Inductrack systems in longitudinal and vertical motion, which otherwise may not be possible by the existing models. The modeling approach presented in this paper is extendible to multi-degree-of-freedom (M-DOF) cases and it provides a foundation for stability analyses and feedback control of Inductrack systems.

This paper presents a theoretical study on the dynamic shape control problem of deployable mesh reflectors (DMRs) via feedback approaches. The reflector structure is simplified from a nonlinear model to be quasi-static with respect to temperature variations but dynamic with respect to mechanical vibrations. The orbital cycle is segmented into multiple temperature zones, and an H∞ robust state feedback controller is designed for each zone to guarantee the local stability of the system under the model uncertainty caused by thermal effects and to reject external force disturbances. At the same time, gain scheduling control method is adopted to compensate thermal distortions and to ensure smooth transition response when switching among the local robust controllers. A DMR model is considered in the case study to show the effectiveness of the control approach. The structural vibrations caused by external force disturbances can be sufficiently suppressed in a much shorter time. The closed loop response of the DMR structure shows that much higher surface accuracy is obtained during the orbiting mission compared to the open-loop configuration, and transient focal length and transient de-focus of the reflector are well controlled within the satisfactory bounds, demonstrating the numerical feasibility of the proposed method to solve the dynamic shape control problem of DMRs.

Medium-frequency (mid-frequency) vibration analysis of complex structures plays an important role in automotive, aerospace, mechanical, and civil engineering. Flexible beam structures modeled by the classical Euler–Bernoulli beam theory have been widely used in various engineering problems. A kinematic hypothesis made in the Euler–Bernoulli beam theory is that the plane sections of a beam normal to its neutral axis remain planes after the beam experiences bending deformation, which neglects shear deformation. However, previous investigations found out that the shear deformation of a beam (even with a large slenderness ratio) becomes noticeable in high-frequency vibrations. The Timoshenko beam theory, which describes both bending deformation and shear deformation, would naturally be more suitable for medium-frequency vibration analysis. Nevertheless, vibrations of Timoshenko beam structures in a medium frequency region have not been well studied in the literature. This paper presents a new method for mid-frequency vibration analysis of two-dimensional Timoshenko beam structures. The proposed method, which is called the augmented Distributed Transfer Function Method (DTFM), models a Timoshenko beam structure by a spatial state-space formulation in the [Formula: see text]-domain. The augmented DTFM determines the frequency response of a beam structure in an exact and analytical form, in any frequency region covering low, middle, or high frequencies. Meanwhile, the proposed method provides the local information of a beam structure, such as displacement, shear deformation, bending moment and shear force at any location, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated in numerical examples, where the efficiency and accuracy of the proposed method is demonstrated. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are examined through comparison of the Timoshenko beam and Euler–Bernoulli beam theories.

A coupled dynamic system consisting of a supporting beam structure and multiple passing rigid bodies is seen in various engineering applications. The dynamic response of such a coupled system is quite different from that of the beam structure subject to moving loads or moving oscillators. The dynamic interactions between the beam and moving rigid bodies are complicated, mainly because of the time-varying number and locations of contact points between the beam and bodies. Due to lack of an efficient modeling and solution technique, previous studies on these coupled systems have been limited to a beam carrying one or a few moving rigid bodies. As such, dynamic interactions between a supporting structure and arbitrarily many moving rigid bodies have not been well investigated, and parametric resonance induced by a sequence of moving rigid bodies, which has important engineering implications, is missed. In this paper, a new semi-analytical method for modeling and analysis of the above-mentioned coupled systems is developed. The method is based on an extended solution domain, by which the number of degrees of freedom of a coupled system is fixed regardless of the number of contact points between the beam and moving rigid bodies at any given time. This feature allows simple and concise description of flexible–rigid body interactions in modeling, and easy and effective implementation of numerical algorithms in solution. The proposed method provides a useful platform for thorough study of flexible–rigid body interactions and parametric resonance for coupled beam–moving rigid body systems. The accuracy and efficiency of the proposed method in computation is demonstrated in several examples.

A magnet can affect the vibration of the metal strings in a musical instrument. Here it is shown that a magnetic field can cause splitting of a metal guitar string's normally degenerate transverse vibration modes, leading to a beat note in the resulting sound. A distributed-spring model accurately predicts the natural frequencies of a vibrating guitar string in the non-uniform magnetic field.

Dynamic analysis of a multi-span beam structure carrying moving rigid bodies is essentially important in various engineering applications. With many rigid bodies having different speeds and varying inter-distances, number of degrees of freedom of the coupled beam-moving rigid body system is time-varying and the beam-rigid body interaction is thus complicated. Developed in this paper is a method of extended solution domain (ESD) that resolves the issue of time-varying number of degrees and delivers a consistent mathematical model for the coupled system. The governing equation of the coupled system is derived with generalized assumed mode method through use of exact eigenfunctions and solved via numerical integration. Numerical simulation shows the accuracy and efficiency of the proposed method. Moreover, a preliminary study on parametric resonance on a beam structure with 10 rigid bodies provides guidance for future development of conditions on parametric resonance induced by moving rigid bodies, which can be useful for operation of certain coupled structure systems.

A multi-span beam structure carrying multiple moving oscillators is seen in a variety of engineering applications, including highway bridges, elevated guideways and railways with moving vehicles, and tubes conveying fast-moving pods. With the oscillators having different speeds and varying inter-distances, the dynamic interactions between the supporting structure and the moving oscillators are usually complicated. Indeed, the number of moving oscillators on the structure is time-varying, and as such, a conventional solution method must frequently check the number of oscillators on the structure and adjust the numerical algorithm accordingly. Because of this, most investigations have been limited to just one or a few moving oscillators. Proposed in this paper is a new semi-analytical method that can systematically handle a beam structure with an arbitrary number of moving oscillators, without tedious number checking and algorithm adjustment. In the development, an extended solution domain (ESD) is firstly defined and a generalized assumed-mode method is then developed based on the ESD, which eventually yields a set of time-varying state equations. Solution of the state equations by a standard numerical integration algorithm gives the dynamic response of the coupled beam-oscillator system. Because the proposed method makes use of the exact eigenfunctions of the multi-span structure that are obtained by a distributed transfer function method, it is highly accurate and efficient in computation, as shown in a numerical study.

Flexible multistage rotating systems that are supported or guided by long water-lubricated rubber bearings (WLRBs) have a variety of engineering applications. Vibration analysis of this type of machinery for performance and duality requires accurate modeling of WLRBs and related rotor–bearing assemblies. This work presents a new model of WLRBs, with attention given to the determination of bearing dynamic coefficients. Due to its large length-to-diameter ratio, a WLRB cannot be described by conventional pointwise bearing models with good fidelity. The bearing model proposed in this paper considers spatially distributed bearing forces. For the first time in the literature, the current study addresses the issue of mixed lubrication in the operation of WLRBs, which involves interactions of shaft vibration, elastic deformation of rubber material and fluid film pressure, and validates the WLRB model in experiments. Additionally, with the new bearing model, vibration analysis of WLRB-supported flexible multistage rotating systems is performed through use of a distributed transfer function method, which delivers accurate and closed-form analytical solutions of steady-state responses without discretization.

Summary Structures made of shape memory polymer composite (SMPC), due to their ability to be formed into a desired compact loading shape and then transformed back to their original aperture by means of an applied stimulus, are an ideal solution to deployment problems of large and lightweight space structures. In the literature, there is a wide array of work on constitutive laws and qualitative analyses of SMP materials; dynamic equations and numerical solution methods for SMPC structures have rarely been addressed. In this work, a macroscopic model for the shape fixation and shape recovery processes of SMPC structures and a finite element formulation for relevant numerical solutions are developed. To demonstrate basic concepts, a cantilever SMPC beam is used in the presentation. In the development, a quasi-static beam model that combines geometric nonlinearity in beam deflection with a temperature-dependent constitutive law of SMP material is obtained, which is followed by derivation of the dynamic equations of the SMPC beam. Furthermore, several finite element models are devised for numerical solutions, which include both beam and shell elements. Finally, in numerical simulation, the quasi-static SMPC beam model is used to show the physical behaviors of the SMPC beam in shape fixation and shape recovery. Copyright © 2015 John Wiley & Sons, Ltd.

This paper presents a new approach to optimal bearing placement that minimizes the vibration amplitude of a flexible rotor system with a minimum number of bearings. The thrust of the effort is the introduction of a virtual bearing method (VBM), by which a minimum number of bearings can be automatically determined in a rotor design without trial and error. This unique method is useful in dealing with the issue of undetermined number of bearings. In the development, the VBM and a distributed transfer function method (DTFM) for closed-form analytical solutions are integrated to formulate an optimization problem of mixed continuous-and-integer type, in which bearing locations and bearing index numbers (BINs) (specially defined integer variables representing the sizes and properties of all available bearings) are selected as design variables. Solution of the optimization problem by a real-coded genetic algorithm yields an optimal design that satisfies all the rotor design requirements with a minimum number of bearings. Filling a technical gap in the literature, the proposed optimal bearing placement approach is applicable to either redesign of an existing rotor system for improvement of system performance or preliminary design of a new rotor system with the number of bearings to be installed being unforeknown.

Flexible multistage rotor systems have a variety of engineering applications. Vibration optimization is important to the improvement of performance and reliability for this type of rotor systems. Filling a technical gap in the literature, this paper presents a virtual bearing method for optimal bearing placement that minimizes the vibration amplitude of a flexible rotor system with a minimum number of bearings. In the development, a distributed transfer function formulation is used to define the optimization problem. Solution of the optimization problem by a real-coded genetic algorithm yields the locations and dynamic coefficients of bearings, by which the prescribed operational requirements for the rotor system are satisfied. A numerical example shows that the proposed optimization method is efficient and accurate, and is useful in preliminary design of a new rotor system with the number of bearings unforeknown.

Frame structures are widely used in engineering applications, especially in space structures. For special use such as shape and vibration control of such structures, piezoelectric patches are usually placed on the beam surfaces to form active frame structures. To perform shape control or vibration control tasks, modeling methods for the formed active frame structures need to be studied. This paper develops a new distributed model of an active frame structure composed of multilayer piezoelectric beam components. First, the governing equations of a beam, bonded with piezoelectric patches, are developed via the generalized Hamilton principle, by considering the transverse shear strain. Then, the analytical solutions of the governing equations and the generalized element stiffness matrix are derived through the distributed transfer function formulation. Finally, the analytical solution of the entire system is obtained by the technique for assembling element stiffness matrix. In numerical simulations, buckling and vibration of an active frame structure are both studied. In addition, a novel Improved Ant Lion algorithm is proposed for optimal design of the frame structures. The optimization examples confirm that the proposed algorithm is more efficient than other existing popular algorithms such as Genetic Algorithm (GA) and Ant Lion Optimization (ALO) algorithm.

A magnet can affect the vibration of the metal strings in a musical instrument. We show that the magnetic field from a magnetic pickup can cause a frequency splitting of a metal guitar string’s normally degenerate transverse vibration modes, leading to a beat note in the resulting sound. The amount of frequency splitting induced by the magnet depends on the product of the induced magnetization in the ferromagnetic string, and the magnet’s spatial gradient at the position of the string, and is on the order of a few Hz. We apply free vibration theory to the string to obtain an eigenvalue problem, which we solve using a distributed-transfer function method. This method accurately predicts the natural frequencies of the vibrating guitar string in a non-uniform magnetic field. Videos of a high-E string vibrating with and without a nearby magnet will be shown.

Flexible multistage rotor systems that are supported by water-lubricated rubber bearings (WLRBs) are seen in various engineering applications. Vibration analysis is important to design and performance of such dynamic systems. In the past, due to the lack of reliable models of WLRBs, vibration analysis of this type of rotor systems has not been well addressed. In this paper, a method for modeling and analysis of WLRB-supported multistage rotor systems is proposed. In the development, a new model of WLRBs is integrated with a distributed transfer function formulation, which eventually yields accurate results on the eigensolutions, critical speeds, and steady-state responses of WLRB-supported rotor systems. The proposed method is illustrated in a numerical study on a three-stage rotor system. It is shown that the proposed method provides a useful tool for optimal design of flexible multistage rotor systems with WLRBs.

A piezoelectric forceps actuator (PFA) is developed for potential use in minimally invasive surgery, biological cell diagnosis and semiconductor industrial applications. This article is concerned with modelling, vibration analysis and boundary feedback control of the actuator. In the development, the PFA is viewed as a slightly-curved beam laminated with piezoelectric layers. By generalised Hamilton's principle, the dynamic governing equations of the actuator are derived. The PFA control system that consists of the piezoelectric composite curved beam, a sensor and control logic is formulated by using distributed transfer functions. With the transfer function formulation, the natural frequencies, mode shapes and frequency response of the actuator are then predicted and a stabilising boundary feedback control law is developed. In addition, a characteristic performance analysis of the PFA is conducted, to relate the system parameters and multimorph configuration to the natural frequencies, damping and piezoe...

A beam structure carrying multiple moving oscillators is a mathematical model for various engineering applications, including rapid transit systems. With many moving oscillators having different speeds and varying inter-distances, the number of oscillators on the structure is time-varying, which inevitably complicates the beam–oscillator interactions. Consequently, the order of a mathematical model for the coupled beam–oscillator system changes with time, with many possibilities. Because of this, it is extremely difficult, if not impossible, for a conventional method to determine the dynamic response of a beam structure carrying many moving oscillators. In the literature, previous investigations have been limited to a beam structure with only one moving oscillator, which may not totally capture the physical behaviors of a structure with many moving oscillators, as seen in certain engineering applications. Developed in this work is a new semi-analytical method that can systematically handle arbitrarily many moving oscillators in both modeling and solution. In the development, an extended solution domain (ESD) is defined and based on the ESD a generalized assumed-mode method is devised. This modeling method completely resolves the issue of changing order in mathematical modeling. Because the proposed method makes use of the exact eigenfunctions of the beam structure (instead of traditional admissible functions), it renders determination of the dynamic response of a coupled beam–oscillator system highly accurate and efficient. The proposed method is demonstrated in several numerical examples. Furthermore, in a benchmark problem, it is shown that for the same accuracy in computation, the elapsed computation time used by the proposed method is just 3.3% of the time required by the finite element method.

A semi-analytical solution method, called the Finite Difference–Distributed Transfer Function Method, is developed for static and dynamic problems of two-dimensional elastic bodies composed of multiple rectangular subregions. In the development, the original two-dimensional elasticity problem is first reduced into a one-dimensional boundary-value problem by finite difference; the exact solution of the reduced problem is then obtained by using the distributed transfer functions of the elastic continuum. The proposed technique, which combines the simplicity of finite difference and the closed form of analytical solutions, is capable of handling arbitrary boundary conditions, delivers highly accurate solutions for static and dynamic problems, and is computationally efficient. The proposed method is illustrated on a square region and an L-shaped region.

A new analytical method is developed for transient vibration analysis of stepped systems composed of distributed components like elastic bars, flexible shafts and taut strings, and lumped masses. The method, with a distributed transfer function formulation and a residue formula for inverse Laplace transform, gives the exact transient response of a stepped system with any number of components and subject to arbitrary external, boundary and initial excitations. The proposed method does not depend on system eigenfunctions, is able to accurately predict jumps in stress and strain distributions, and is numerically efficient as its utility only requires simple operations of two-by-two matrices.

A new analytical solution method is developed for problems of conduction heat transfer in one-dimensional multilayer composite bodies. In this method, the original governing equations are first cast in a spatial state form in the Laplace transform domain; the solution of the state equation is then obtained in terms of distributed transfer functions. With a new residue formula, inverse Laplace transform of the s-domain solution yields exact transient temperature and heat flux. Exact steady-state solutions are also obtained with the transfer function formulation. The proposed method is numerically efficient as it only requires simple operations of two-by-two matrices.

Meso∕micro grasping of tiny soft objects such as biological tissues, which ranges from hundreds to thousands of micro-millimeters in dimension, plays a significant role in the fields of tele-surgery, minimally invasive surgery (MIS), and biomedical instrumentation. Recently, the authors proposed a novel piezoelectric forceps actuator (PFA), which is capable of grasping delicate soft objects. One of the advantages of the PFA over conventional MIS forceps lies in that it can be remotely controlled to achieve precision deflection and grasping force. Furthermore, it does not have any moving parts such as gears and hinges, and hence avoids problems in operation like friction, backlash, lubrication, leakage, and sterilization. In this paper, a mathematical model of the PFA is derived, based on which genetic algorithm (GA) is applied to optimize the grasping force-deflection relationship of the actuator. The model developed is experimentally verified on a prototype of the PFA.

This paper is a sequel to the work discussed in Pesterev et al. (Journal of Sound and Vibration, in press). In that paper, it was suggested that the technique to determine the effect of a local road surface irregularity on the dynamics of a vehicle modelled as a linear multi-degree-of-freedom system relies on the so-called pothole dynamic amplification factor (DAF), which is a complex-valued function specific to the irregularity shape. This paper discusses the companion problem of how to determine the DAF function for an irregularity represented as a superposition of simpler ones. Another purpose of this paper is to demonstrate the application of the pothole DAF functions technique to finding a priori estimates of the effect of irregularities with a repeated structure. Specifically, we solve the problem of finding the conditions under which the dynamic effect of two identical potholes located one after another is greater than that due to the single pothole. We also find the estimate for the number of periods of a periodic irregularity that are sufficient in order to consider the oscillator response as steady state. The discussions are illustrated by numerical examples.

A technique is developed to assess the dynamic contact forces arising after passing road surface irregularities by a vehicle modelled as a general linear MDOF system. The equations governing vibration of a vehicle moving along an uneven profile are, first, transformed to the state-space form and, then, to a system of uncoupled first order complex differential equations. For a local roadway irregularity described functionally, solutions of all equations are found analytically and expressed in terms of a unique function of one complex variable, the so-called pothole dynamic amplification factor, which is specific to the irregularity shape. The solutions obtained are combined to give dependencies of the harmonic components of the contact forces arising after the passage of the irregularity on the vehicle speed and irregularity dimensions. The problem is shown to be decomposed into separate calculation of vehicle and pothole-specific data. The technique developed is not specific to a particular vehicle model or an irregularity shape: the vehicle model is represented by its mass, stiffness, and damping matrices, and the replacement of one irregularity by another simply requires replacement of one dynamic amplification factor function by another. The latter are derived in Appendix A for several pothole configurations. The discussion is amply illustrated by examples of the application of the technique to the calculation of the tire forces for two simple vehicle models and several potholes of different shape.

A new structural modeling and analysis method, the distributed transfer function method, is presented for application to gossamer space structures. The distributed transfer function method uses distributed transfer functions, instead of shape function used by traditional finite element solvers, to represent the displacement field. The distributed transfer function method maintains the modeling flexibility of the finite element method, so that it is capable of modeling multibody complex structures, but it requires much fewer nodes and results in a significant reduction of computational time. The distributed transfer functions give rise to closed-form analytical solutions of both displacement and strain fields. As a result, the distributed transfer function method only decomposes a structure at those points where multiple components are connected, to keep each component as large as possible. Gossamer space structures are generally composed of several long booms and large membranes. Therefore, the distributed transfer function method can be used to model a gossamer structure with a small number of unknowns and matrices of low order. It offers very accurate results with high computational efficiency. The distributed transfer function method is applied to investigate the sensitivity of buckling strength of an inflatable/rigidizable boom to the variations in bending stiffness.

An analytical technique, called the Effective Stiffness Method (ESM), is developed for static buckling analysis of one-dimensional continua. Based on a distributed transfer function formulation, the ESM delivers exact buckling solutions for constrained stepped columns, eccentrically loaded columns, and columns with initial deflection. The method also obtains highly accurate buckling solutions for columns with arbitrary non-uniform cross-section. The ESM treats different column configurations in a systematic manner, and is numerically efficient. Unlike conventional analytical methods, the proposed method does not use series of continuous functions. The capabilities and advantages of the ESM are demonstrated in numerical examples.

The problem of the vibration of a beam subject to a travelling force is considered. The purpose of the study is to develop simple tools for finding the maximum deflection of a beam for any given velocity of the travelling force. It is shown that, for given boundary conditions, there exists a unique response–velocity dependence function. A technique to determine this function is suggested, which is based on the assumption that the maximum beam response can be adequately approximated by means of the first beam mode. To illustrate this, the maximum response function is calculated analytically for a simply supported (SS) beam and constructed numerically for a clamped–clamped beam. The effect of the higher modes on the maximum response is investigated, and the relative error of the one-mode approximation for a SS beam is constructed. The estimates obtained substantiate the assumption about adequacy of the one-mode approximation in a wide range of velocities; in particular, the relative error in the neighborhood of the velocity that results in the largest response is less than one percent.

Asymptotic behavior of the solution of the moving oscillator problem is examined for large and small values of the spring stiffness for the general case of non-zero beam initial conditions. In the limiting case of infinite spring stiffness, it is shown that the moving oscillator problem for a simply supported beam is not equivalent, in a strict sense, to the moving mass problem. The two problems are shown to be equivalent in terms of the beam displacements but are not equivalent in terms of stresses (the higher order derivatives of the two solutions differ). In the general case, the force acting on the beam from the oscillator is shown to contain a high-frequency component , which does not vanish and can even grow when the spring coefficient tends to infinity. The magnitude of this force and its dependence on the oscillator parameters can be estimated by considering the asymptotics of the solution for the initial stage of the oscillator motion. It is shown that, for the case of a simply supported beam, the magnitude of the high-frequency force depends linearly on the oscillator eigenfrequency and velocity. The deficiency of the moving mass model is principally that it fails to predict stresses in the supporting structure. For small values of the spring stiffness, the moving oscillator problem is shown to be equivalent to the moving force problem. The discussion is amply illustrated by results of numerical experiments.

In this paper three fundamental issues regarding modeling and analysis of wrinkled membranes are addressed. First, a new membrane model with viable Young's modulus and Poisson's ratio is proposed, which physically characterizes stress relaxation phenomena in membrane wrinkling, and expresses taut, wrinkled and slack states of a membrane in a systematic manner. Second, a parametric variational principle is developed for the new membrane model. Third, by the variational principle, the original membrane problem is converted to a non-linear complementarity problem in mathematical programming. A parametric finite element discretization and a smoothing Newton method are then used for numerical solution. The proposed membrane model and numerical method are capable of delivering convergent results for membranes with a mixture of wrinkled and slack regions, without iteration of membrane stresses. Three numerical examples are provided.

Based on the introduction of a nonnegative adjustable parameter into Poisson's ratio, a new numerical method is proposed to solve numerically the partially wrinkled membrane problems with arbitrary complex shapes. Three fundamental issues in this new method are addressed. First, a nonnegative adjustable parameter is introduced into the constitutive law, automatically representing unwrinkled (taut) and wrinkled states of a membrane. Second, a parametric variational principle is developed for the new membrane model. Third, by the variational principle, the original membrane problem is converted to a nonlinear mathematical programming problem. The variational principle and the mathematical programming formulation lay a foundation for efficient numerical analysis of wrinkled membrane structures using finite element discretization. Three typical examples are given to show the effectiveness of this method.

Flexible multistage rotating systems that are supported by water-lubricated rubber bearings (WLRB) have many engineering applications, including power generation, mining, water services, waste treatment, oil and gas industries, and industrial processes. Due to the flexibility of rotating shafts in these applications, dynamic modeling and vibration analysis is essentially important to optimal design and reliable operation of this kind of rotor systems. This paper presents a new model of WLRBs, with the focus on determination of bearing dynamic coefficients. Conventional bearing models are normally of pointwise type, and are invalid for WLRBs that have a large length-to-diameter ratio. The bearing model proposed in this work considers spatially distributed bearing forces, and for the first time, addresses the issue of mixed lubrication, which involves interaction effects of shaft vibration, elastic deformation of rubber material and fluid film pressure. With the bearing model, the dynamic response of a flexible multistage rotating system with WLRBs is described by a Distributed Transfer Function Method (DTFM). The steady-state response of the rotor system due to unbalanced mass is then computed by the DTFM, and compared with experimental data, yielding the dynamic stiffness coefficients of WLRBs. It is shown that the proposed bearing model and DTFM formulation is useful in vibration analysis and optimal design of WLRB-supported flexible multistage rotor systems.

A semi-analytical method is developed for static and dynamic analyses of three-dimensional elastic solids composed of prismatic subregions. Based on a prism distributed transfer function formulation, the method is capable of synthesizing multi-body continua, and at the same time maintains the high accuracy and efficiency of analytical solutions. The proposed method is unique in that it systematically generates closed-form, semi-analytical regional solutions that can be used for multi-body assembly. Unlike many analytical approaches, the method does not depend on any series expansion, and is valid for arbitrary external loads and boundary conditions. Numerical examples are provided to justify the advantages of the proposed method.

The problem of calculating the dynamic response of a one-dimensional distributed parameter system carrying multiple moving oscillators is examined. A solution procedure is suggested that reduces the problem to the integration of a system of linear ordinary differential equations governing the time-dependent coefficients of the series expansion of the response in terms of the eigenfunctions of the continuous structure. The program implementation of the solution procedure is discussed and numerical results are presented. Numerical illustrations clearly demonstrate that the incorporation of even one additional oscillator into the model makes the dynamics of the system vibration considerably more complicated.

This article, examines the vibrational characteristics of annular plates by using the three-dimensional elasticity theory. It aims to raise the quality of the investigation beyond that provided by the two-dimensional plate theories by resorting to a full three-dimensional analysis. A polynomials–Ritz model based on sets of orthogonally generated polynomial functions to approximate the spatial displacements of the plates in cylindrical polar coordinates is presented. The model is then used to extract the full vibration spectrum of natural frequencies and mode shapes. The vibration responses due to the variations of boundary conditions and thickness are investigated. Frequency parameters and three-dimensional deformed mode shapes are presented in vivid graphical forms. The accuracy of the method is validated through appropriate convergence and comparison studies.

This paper presents accurate three-dimensional elasticity solutions for free vibration of circular plates. The derivation of a linear frequency equation based on an exact three-dimensional, small-strain, linearly elastic principle is detailed. The solution to this problem is made possible by using the Ritz method with a set of orthogonal polynomial series to approximate the spatial displacements of the circular plate in cylindrical polar coordinates. The perturbation of frequency responses due to the variations of boundary conditions and thickness is investigated. First known frequency parameters and three-dimensional deformed mode shapes are presented in vivid graphical forms. The accuracy of these results are verified by appropriate convergence studies and, when possible, are checked with existing solutions.

A semi-analytical analysis of flexible plates with curved boundaries is presented. Through a co-ordinate transformation, the original spatial domain of the plate is mapped onto a rectangular region, where the isoparametric strip distributed transfer functions of the plate are introduced. The transfer function formulation yields closed-form predictions of the displacement, stresses and eigensolutions of the plate. The accuracy and efficiency of the proposed approach is demonstrated in numerical examples. Copyright © 1999 John Wiley & Sons, Ltd.

Conventional modal analyses are not applicable to structural systems with time delays because time delays destroy any possible orthogonality among system eigenvectors. A new series solution method is proposed for transient response analysis of a class of delayed linear structural systems. In this method, inverse Laplace transform is combined with a root locus sensitivity analysis, leading to a relation between the transfer function residues and eigensolutions of the delayed system. With this relation, closed-form time-domain response can be accurately and efficiently estimated in series of eigenvectors, without involving transfer function singularities and s-domain integration, which are usually encountered in inverse Laplace transform. The proposed method is illustrated on a gyroscopic dynamic system under delayed feedback control.

A distributed transfer function synthesis is developed for modelling and analysis of rotor systems assembled from multiple flexible and rigid components. The method delivers highly accurate and closed-form analytical solutions, and is capable of treating non-self-adjoint effects, general boundary conditions and multi-body coupling. It is shown that the proposed method provides a useful analysis tool for many problems in rotor dynamics.

Shape Memory Polymer Composites (SMPC) structures, due to their ability to be packed into compact volumes and then later on transformed back to their original shapes, are an ideal solution for deployment problems of large light-weight space structures. Although much research on constitutive laws and qualitative analysis of mechanics of SMPC structures has been carried out, little has been reported in the literate about the development of dynamic equations for SMPC structures. This paper documents a macroscopic model for shape fixation and shape recovery processes of an SMPC cantilever beam, with special attention given to quasi-static states describing the shape fixation process. In the development, a nonlinear geometric model is combined with a temperature-dependent constitutive law. Based on this beam model, numerical results are obtained through use of finite difference approximation and Newton’s method in iterative solution. Additionally, the dynamic equations of the SMPC cantilever beam are devised.

Multibody distributed dynamic systems are seen in many engineering applications. Developed in this investigation is a new analytical method for a class of branched multibody distributed systems, which is called the augmented distributed transfer function (DTFM). This method adopts an augmented state formulation to describe the interactions among multiple distributed and lumped bodies, which resolves the problems with conventional transfer function methods in modeling and analysis of multibody distributed systems. As can be seen, the augmented DTFM, without the need for orthogonal system eigenfunctions, produces exact and closed-form solutions of various dynamic problems, in both frequency and time domains.

Although quaternions are singularity-free in modeling and analysis of rigid bodies in three-dimensional motion, description of torques may lead to unbounded response of a quaternion-based model. This paper gives theorems on the conditions of torque-induced singularity in four coordinate systems: inertial frame, body frame, Euler basis, and dual Euler basis. According to the theorems, torques applied in an inertial frame or a body frame or a Euler basis will never cause unbounded motion; torques applied in a dual Euler basis, however, may lead to unbounded motion.