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A new form-finding method, namely the Fixed Nodal Position Method (FNPM), is developed for design of geometric configuration and internal force distribution for a class of truss structures with a requirement of high shape or surface accuracy. Different from existing form-finding methods, which usually follow a stress-first-and-displacement-later procedure, the FNPM first assigns nodal coordinates for a truss structure, and then determines the internal force distribution of the structure by an optimization process. The highlight of the FNPM is that the prescribed nodal coordinates are unchanged during the form finding process. This special feature of fixed nodal positions allows the proposed method to place the nodes of a truss structure at desired locations, satisfying complicated constrains and yielding high shape/surface accuracy as required. Moreover, because the assignment of geometric configuration (nodal coordinates) and the determination of internal force distribution are undertaken separately, the computational effort in the FNPM-based form finding is much less than that by a conventional method. The proposed method is applied to two large deployable mesh reflectors of 217 nodes and 865 nodes, respectively, and show high efficiency and accuracy in form finding.

A double-beam system, which is a structure composed of two parallel beams that are interconnected by a viscoelastic layer, is seen in many engineering applications. Vibration analysis is essentially important for the safe and reliable operation, and optimal design of such dynamic systems. This paper presents an analytical method, the distributed transfer function method (DTFM), for modeling and vibration analysis of double-beam systems with arbitrary beam linear densities and flexural rigidities , and general boundary conditions. Exact closed-form analytical solutions for natural frequencies, mode shapes, and steady-state responses to periodic excitations are determined. The proposed method is applicable to a double-beam system with lower beam being fully, partially, or not supported by an elastic foundation. Through numerical study, the accuracy and efficiency of the proposed method are validated, and the effects of the stiffness, length, and location of an elastic foundation are investigated. It is shown that the DTFM is a useful tool for optimal design of elastically connected double-beam systems.

In traditional form-finding of a deployable mesh reflector (DMR), the nodes of the DMR mesh are placed on the desired working surface and the surface accuracy of the DMR is measured either by the deviation of the nodes from the desired working surface or by the deviation of the mesh from its best-fit surface. Placement of nodes on working surface and inaccurate measures of surface accuracy cause non-negligible surface errors that cannot be further reduced. To deal with these issues and to further improve surface accuracy of DMRs, a new mesh geometry design method, called the Projecting Surface Method (PSM), is presented in this paper. The highlight of the PSM is that it purposely places the nodes of a DMR off its working surface, to achieve higher surface accuracy. To this end, a direct RMS error measuring the deviation of a DMR mesh from its desired working surface is introduced and a projecting surface for hosting the nodes of the DMR mesh is defined. By the direct RMS error and projecting surface, an optimization process produces a mesh geometry with its best-fit surface closest to the desired working surface, leading to significant surface error reduction. As shown in numerical examples of DMRs with 37, 271 and 817 nodes, the PSM can reduce surface errors by 50% or more. The proposed method is usable with existing form-finding methods for further improvement of surface accuracy of DMRs.

Flexible multistage rotor systems with water-lubricated rubber bearings (WLRBs) have a variety of engineering applications. Filling a technical gap in the literature, this effort proposes a method of optimal bearing placement that minimizes the vibration amplitude of a WLRB-supported flexible rotor system with a minimum number of bearings. In the development, a new model of WLRBs and a distributed transfer function formulation are used to define a mixed continuous-and-discrete optimization problem. To deal with the case of uncertain number of WLRBs in rotor design, a virtual bearing method is devised. Solution of the optimization problem by a real-coded genetic algorithm yields the locations and lengths of water-lubricated rubber bearings, by which the prescribed operational requirements for the rotor system are satisfied. The proposed method is applicable either to preliminary design of a new rotor system with the number of bearings unforeknown or to redesign of an existing rotor system with a given number of bearings. Numerical examples show that the proposed optimal bearing placement is efficient, accurate and versatile in different design cases.

Conventional analytical methods for transient heat conduction solutions, due to complicated derivations and formidable calculations, have been limited to composite bodies with two or three layers. Developed in this work is a new analytical solution method for transient heat conduction in hollow composite cylinders with an arbitrary number of layers and subject to general boundary conditions. In this method, a distributed transfer function formulation gives an s-domain solution of a composite cylinder and inverse Laplace transform of the s-domain solution via a newly derived residue formula yields the transient solution of the cylinder in an explicit and closed form. Unlike conventional analytical methods, the proposed method does not require different derivations for different cylinder configurations (such as number of layers, boundary conditions and thermal resistance at layer interfaces). Because it only involves two-by-two matrices in the solution process, the proposed method is highly efficient in computation, as demonstrated in numerical examples.

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.

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.

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.

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.

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.

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 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.

A new technique, called the strip distributed transfer function method, is developed for static and dynamic analysis of two-dimensional elastic bodies that are composed of multiple rectangular subregions. The method is capable of modeling elastic regions of complex geometry and arbitrary boundary conditions, delivers highly accurate semi-analytical solutions and saves tremendous computer storage. In the analysis, a complex elastic region is first divided into a number of subregions; each subregion is then divided into finite strips. Through introduction of strip distributed transfer functions, the response of every subregion is presented in a semi-exact and closed form; the whole region is systematically assembled from the subregions, leading to a dynamic equilibrium equation. Solution of the equilibrium equation yields the semi-exact displacements and stresses of the elastic body. The proposed method is illustrated on a square region and an L-shaped region, and compared with the finite element method.

Linear vibration of a coupled string–rigid bar system is investigated. Owing to dynamic interaction between the strings and the rigid bar, the coupled system is not self-adjoint in its original equations of motion. Consequently, the classical modal analysis cannot be applied. This study develops an augmented formulation for the eigenvalue and boundary value problems of the coupled system. In this formulation, orthogonal eigenfunctions are obtained, and the general response of the coupled string–bar system is represented by an eigenfunction series. Green’s function formula for the coupled system is derived by two methods: modal analysis and Laplace transform. The analytical predictions about the system eigensolutions and response are illustrated in numerical simulation.

This paper studies vibration control of a class of gyroscopic systems by direct velocity feedback. Of special interest is the stabilization of a gyroscopic system of many modes via a smaller number of sensors and actuators. A matrix decomposition and a co-ordinate transformation are proposed for stability analysis and controller design. Based on the information about the eigensolutions of the uncontrolled system, the asymptotic stability of the control system is determined. Also, a root locus sensitivity coefficient is defined as an index of damping introduced to each mode by the velocity feedback.

The natural frequencies of combined gyroscopic systems are studied through use of a transfer function method. The characteristic equation of the combined system is determined in terms of the transfer functions of the component systems and the transfer functions describing the combination. With the characteristic equation, the natural frequencies of three specific combined systems are analyzed. It is shown that the natural frequencies of the combined systems alternate with those of the uncombined component systems.

Transfer functions of constrained/combined dynamic systems contain all of the information required to determine the system response and stability. This paper presents an exact closed form method for evaluating transfer functions of a class of constrained/combined vibrating continua. In the analysis, the type of the continuous system, boundary conditions, constraints and combined subsystems are treated in a unified manner. The method is valid for both self-adjoint and non-self-adjoint systems, and is convenient for computer coding.