Formulation of Multiple Belt System and its Dynamic Analysis

This paper presents a method for analyzing the dynamic characteristics of driving systems consisting of multiple belts and pulleys. First, the algorithm which derives the linear equations of motion of arbitrary multi-coupled belt systems is shown. Secondly, by using the algorithm, the computer program which formulates the equations of motion and calculates the transient responses of the belt system is presented. The fundamental idea of the algorithm is as follows: Complicated belt systems consisting of multiple belts and pulleys are regarded as combinations of simple belt systems consisting of a single belt and some pulleys. Therefore, the equations of motion of the belt systems can be derived by the superposition of the equations of motion of the simple belt systems. By means of this method, the responses of arbitrary multi-coupled belt systems can be calculated. Finally, to verify the usefulness of this method, the simulation results are compared with the experimental results.

Vibration Localization in Band/Wheel Systems: Theory and Experiment

An investigation of the localization phenomenon in band/wheel systems is presented. The effects of tension disorder, interspan coupling, and translation speed on the confinement of the natural modes of free vibration are investigated both theoretically and experimentally. Two models of the band/wheel system dynamics are discussed; a simple model proposed by the authors [1] and a more complete model originally proposed by Wang and Mote [9]. The results obtained using the simple interspan coupling model reveal phenomena (i.e., eigenvalue crossings and veerings and associated mode localization) that are qualitatively similar to those featured by the more complex model of interspan coupling, thereby confirming the usefulness of the simple coupling model. The analytical predictions of the two models are validated by an experiment. A very good agreement between the experimental results and the theoretical ones for the simple model is observed. While both the experimental observations and the theoretical predictions show that a beating phenomenon takes place for ordered stationary and axially moving beams, beating is destroyed (indicating the occurrence of localization) when any small tension disorder is introduced especially for small interspan coupling (i.e., when localization is strongest).

On the Free Vibration of Mono-Coupled Periodic Distributed Parameter Systems

A new approach to the exact solution is given for the free vibration of a periodic structure comprised of a multiplicity of identical linear distributed parameter substructures, closely coupled through identical linear springs. The method used is an extension of a classical result for periodic discrete systems.

Localization of Vibration in Disordered Multi-Span Beams With Damping

The combined effects of disorder and structural damping on the dynamics of a multi-span beam with slight randomness in the spacing between supports are investigated. A wave transfer matrix approach is chosen to calculate the free and forced harmonic responses of this nearly periodic structure. It is shown that both harmonic waves and normal modes of vibration that extend throughout the ordered, undamped beam become spatially attenuated if either small damping or small disorder is present in the system. The physical mechanism which causes this attenuation, however, is one of energy dissipation in the case of damping but one of energy confinement in the case of disorder. The corresponding rates of spatial exponential decay are estimated by applying statistical perturbation methods. It is found that the effects of damping and disorder simply superpose for a multi-span beam with strong interspan coupling, but interact less trivially in the weak coupling case. Furthermore, the effect of disorder is found to be small relative to that of damping in the case of strong interspan coupling, but of comparable magnitude for weak coupling between spans. The adequacy of the statistical analysis to predict accurately localization in finite disordered beams with boundary conditions is also examined.

Theoretical and Experimental Analysis of the Forced Response of Sagged Cable/Mass Suspensions

This study examines the forced response of a sagged elastic cable supporting an array of discrete masses. Such systems arise, for instance, in ocean engineering applications employing cable hydrophone arrays. The excitation considered is harmonic and normal to the cable and may, for instance, approximate prescribed environmental loading. An asymptotic model is presented that describes the linear forced response of a cable/mass suspension having small equilibrium curvature. Closed-form expressions for the Green’s function to an associated boundary-value problem are obtained using a transfer matrix formulation. The derived Green’s function is utilized to construct integral representations for steady-state response under boundary and/or domain excitation. Solutions obtained for a variety of domain loading distributions demonstrate the utility and efficiency of this solution strategy. The theoretical response predictions are verified through experimental measurements of the natural frequency spectrum and frequency response of laboratory cable/mass suspensions.

Free and Forced Localization in a Nonlinear Periodic Lattice

Free and forced localized periodic motions in an infinite nonlinear periodic lattice are analytically investigated. The lattice consists of weakly coupled identical masses, each connected to the ground by a nonlinear stiffness. In order to study the localized motions of the discrete system a continuoum approximation is assumed, and the ordinary differential equations of motion are replaced by a single nonlinear partial differential equation. The time-periodic solutions of this equation are then obtained by an averaging method, and their stability is examined using an analytic linearized method. It is shown that localized periodic motions of the lattice correspond to standing solitary solutions of the partial differential equation of the continuous approximation. For the free lattice, localized free motions occur when the coupling stiffnesses forces are much smaller than the nonlinear effects of the grounding stiffnesses. Moreover, these free localized motions are detected in the perfectly periodic nonlinear lattice, i.e., even in the absence of structural disorder (a feature which is an essential prerequisite for linear mode localization). When harmonic forcing is applied to the chain, localized, non-localized, and chaotic motions occur, depending on the spatial distribution and the magnitude of the applied loads. A variety of spatially distributed harmonic loads and analytic expressions for the resulting localized motions of the chain are derived.

Distributed Transfer Function Synthesis of Complex Flexible Systems

This paper presents a new analytical and numerical method for modeling and synthesis of complex flexible systems (CFS) that are multiple continua combined with lumped parameter systems. In the analysis, the CFS is first divided into a number of subsystems; the distributed transfer functions of each subsystem are determined in exact and closed form by a state space technique. The CFS is then assembled by imposing displacement compatibility and force balance at the nodes where the subsystems are interconnected. With the distributed transfer functions and the transfer functions of the constraints and lumped parameter systems, exact, closed-form formulation is obtained for various dynamics and vibration problems. The method does not require a knowledge of system eigensolutions, and is valid for non-self-adjoint systems with inhomogeneous boundary conditions. In addition, the proposed method is convenient in computer coding, and suitable for computerized symbolic manipulation.

Localization of Helical Flexural Waves on an Irregular Cylindrical Shell

Small amounts of irregularity are known to produce vibration localization in one-dimensional chains, but the effects on two and three dimensional systems are typically much weaker. We have considered a system with both one and two dimensional aspects, an axisymmetric, irregularly ribbed fluid-loaded cylindrical shell. A new formulation of this problem is given and each mode of the system is shown to be approximately equivilent to a nearest neighbor coupled chain with six degrees of freedom — corresponding to three left going and three right going traveling waves. Using these results all the azimuthal modes have been shown to be localized simultaneously and thus vibrational energy on an irregular cylindrical shell is Anderson localized. The slow flexural waves have been shown to localize separately from the fast waves provided the helical angle of the flexural wave is greater than about Ω1/25°, and for smaller helical angles, exponential localization is expected for distances less than a critical wave mixing length. Our results indicate that strong localization effects occur for the helical flexural waves yielding very short localization lengths of order a single rib spacing.

On the Identification of Localized Losses of Stiffness in Structures

In damage detection it is common to use measured modal data and a mathematical model in connection with system identification. The part of the system undergoing the largest stiffness decrease is defined to contain damage. This approach is very sensitive to measurement errors. The measurement errors are much larger for mode shape functions than for the eigen-frequencies. The errors in the mode shapes are often of the same order of magnitude as the variations due to damage leading to poor results in damage detection. Thus, the use of the mode shape functions themselves instead of their small damage induced variations would dearly be preferable. In this paper we examine the relation between the changes in the eigenfrequencies, the local stiffness losses and the mode shape functions of the undamaged system. This relation is then utilized in a damage detection procedure.

Forced Harmonic Response of a Continuous System Displaying Eigenvalue Veering Phenomena

Prior studies of self-adjoint linear vibratory systems have extensively explored the phenomenon of veering of eigenvalue loci that depict the dependence of natural frequencies on a system parameter. The present work is an exploration of the effect of such phenomena on the response of a continuum to harmonic excitation. The focus of the analysis is the prototypical system of a two-span beam with a strong torsional spring at the intermediate pin support. The results of an exact eigenvalue analysis, not previously disclosed, are used to perform a modal analysis of the steady-state response of the beam to a harmonic concentrated force applied to the middle of one span. The analysis is used to identify situations in which the forced response is localized to one span, as well as the degree to which the location and magnitude of the peak displacement displays parameter sensitivity.