Free vibrations of a tire as a toroidal membrane

A tire is modeled as a toroidal membrane under internal pressure and mounted on a rim, to investigate its free vibration characteristics using a 12 d.o.f. rectangular membrane finite element. Such a modeling is valid if the tire is assumed to be incapable of supporting any weight in the absence of internal pressure. To verify the formulations of the membrane finite element, a flat rectangular membrane subject to in-plane loads and a circular cylindrical membrane under internal pressure are first analyzed. Analytical solutions for these cases are also derived. The analytical and numerical results are in good agreement. A toroidal membrane under internal pressure, assumed to model a low pressure tire, is studied next. Both the analytical derivation and the finite element solutions are presented. For the analytical solution the equations of motion yield a complicated differential equation for which an approximate solution is obtained by assuming that the parallel circle radius is constant as in the case of a bycycle wheel. The finite element solution successfully predicts the symmetrical and the twisting modes of vibration documented by other researchers, and is also in good agreement with the analytical results. The present formulations are useful to obtain a good first approximation of the free vibration response of a tire.     

Optical analysis of vibrations in continuous media

Optical methods of vibration analysis are described which are independent of assumptions associated with theories of wave propagation. The principles involved are reviewed and several variations of the methods used are demonstrated with examples of application to the solution of three problems not solved before. These problems are of special interest in the field of underwater technology. Procedures for scaling data obtained with photoelastic models to predict the behavior of prototypes are illustrated. Difficulties involved in extending optical methods of complete analysis to three-dimensional problems which have no planes of symmetry and to studies of compressional wave propagation in fluids are discussed.     

Linear and nonlinear vibrations analysis of viscoelastic sandwich beams

In this paper, numerical models are proposed for linear and nonlinear vibrations analyses of viscoelastic sandwich beams with various viscoelastic frequency dependent laws using the finite element based solution. Real and various complex eigenmodes approaches are investigated as Galerkin bases. Based on harmonic balance method, simplified and general approaches are developed for nonlinear vibration analysis. Analytical frequency-amplitude and phase-amplitude relationships are elaborated based on the numerically computed complex eigenmodes. The equivalent loss factors and frequencies as well as the forced harmonic response and phase curves are performed for sandwich beams with various boundary conditions and frequency dependent viscoelastic laws.     

On the vibration transmission of a rolling tire on a suspension system due to periodic tread excitation

Several methods of modeling the vibration excitation of an automotive tire by its tread pattern were developed, ranging from a very simplified approach with an equivalent single point, single line scanning procedure to multi-point, multi-line scanning. This results in a periodic model, since even for a so called random tread pattern there is a periodicity corresponding to one tire revolution. Various tread profiles are explored and response spectra for forces and displacement at the suspension point are obtained and discussed. Receptances formulated in terms of the free tire shell modes are utilized. To contrast the force input type models of the tread excitation with the displacement type input of long road waves, the latter is also discussed briefly.     

Theoretical modeling and characteristic analysis of moving-train induced ground vibrations

An integrated train-track-subsoil dynamic interaction model of moving-train induced ground vibration is developed on the basis of vehicle dynamics, track dynamics and the Green's functions of subsoil. The model takes account of the vibrations of vehicle components, the quasi-static axle loads and the dynamic excitations between the wheels and track. The analyzed results from an example show that the ground vibration characteristics have a close relationship with train speed and soil properties; the dynamic responses excited by wheel-track irregularity have big influence on the high frequency components of ground vibration; with the increase of distance to the track, the ground acceleration has the tendency of decrease, and the relevance of acceleration curves and train excitation becomes less obvious; the intersections of moving load speed-lines and subsoil dispersion curves are some resonance frequencies that cause the amplification of ground vibrations; there exists a critical speed for moving train that is close to the minimum velocity of the Rayleigh's wave in the subsoil.     

Vibration analysis of automobile tire due to bump impact

The improvement of car driving comfort is requested according to development of automobile manufacturing technology. Therefore, many experts have been studying automotive vibration of normally driving cars to improve automotive driving comfort. In this paper, vibration characteristics of an automobile tire that passes a cleat are analyzed. The model for automobile tire is verified through numerical analysis and experimental method. The automobile tire is assumed to 7-degree of freedom system, and the effect on the proposed tire design factor is considered. This analysis result proved that the proposed tire design factor reduced tire and wheel vibration energy.     

Finite element analysis of the axisymmetric vibrations of cylinders

A finite element analysis is developed for finite and infinite solid or hollow cylinders in axisymmetric vibration. The elements themselves are solid or annular cylinders, and have 16 degrees of freedom. Results are given for the propagation constants of solid and hollow infinite cylinders, and excellent agreement is found with those from the exact Pochhammer theory and Mindlin and McNiven's three-mode theory. Frequency spectra are presented for the symmetric and antisymmetric modes of solid and hollow finite rods. Excellent agreement is found with experimental results, and this suggests that some of the results obtained from the three-mode theory by McNiven et al., and in particular the frequency of the end mode, are in error by more than 10%. Details of the finite element inertia and stiffness matrices appear in an appendix.     

Experiments and analysis on chaotic vibrations of a shallow cylindrical shell-panel

Detailed experimental results and analytical results are presented on chaotic vibrations of a shallow cylindrical shell-panel subjected to gravity and periodic excitation. The shallow shell-panel with square boundary is simply supported for deflection. In-plane displacement at the boundary is elastically constrained by in-plain springs. In the experiment, the cylindrical shallow shell-panel with thickness 0.24 mm, square form of length 140 mm and mean radius 5150 mm is used for the test specimen. All edges around the shell boundary are simply supported by adhesive flexible films. First, to find fundamental properties of the shell-panel, linear natural frequencies and characteristics of restoring force of the shell-panel are measured. These results are compared with the relevant analytical results. Then, geometrical parameters of the shell-panel are identified. Exciting the shell-panel with lateral periodic acceleration, nonlinear frequency responses of the shell-panel are obtained by sweeping the frequency of periodic acceleration. In typical ranges of the exciting frequency, predominant chaotic responses are generated. Time histories of the responses are recorded for inspection of the chaos. In the analysis, the Donnell equation with lateral inertia force is introduced. Assuming mode functions, the governing equation is reduced to a set of nonlinear ordinary differential equations by the Galerkin procedure. Periodic responses are calculated by the harmonic balance method. Chaotic responses are integrated numerically by the Runge-Kutta-Gill method. The chaotic responses, which are obtained by the experiment and the analysis, are inspected with the Fourier spectra, the Poincaré projections, the maximum Lyapunov exponents and the Lyapunov dimension. It is found that the dominant chaotic responses of the shell-panel are generated from the responses of the sub-harmonic resonance of order and of the ultra-sub-harmonic resonance of order. By the convergence of the maximum Lyapunov exponent to the embedding dimension, the number of predominant vibration modes which contribute to the chaos is found to be three or four. Fairly good agreements are obtained between the experimental results and the analytical results.     

Group theoretical analysis of lattice vibrations in molecular crystals

Computer programs have been developed by Warren and Worlton, for the external modes in ionic crystals where the principal axes of the radicals of groups coincide with the crystallographic axes. In this paper, we discuss the generalisation of these computer programs for molecular crystals where the principal axes do not coincide with the crystallographic axes and for molecular crystals with linear molecules having a redundant rotational degree of freedom. Our results are discussed for the q→O modes of solid NH₃, N₂ and CS₂.     

Computer analysis of coherent electric vibrations in biological systems

The Fröhlich condensed state may be created in the system of vibration modes if the energy supply is above the critical one. A model of a vibration system of a limited number of modes in the frequency range 0·1–4 THz with uniform density of state function and limited energy in the condensed state (saturation) is used for computer simulation. Energy stored in particular modes is calculated as a function of energy supply and energy leakage to the heat bath. The energy stored in the condensed state may be greater than the thermal energy of the system. In the saturated condensed state the stored energy is independent of energy supply. In the vicinity of the critical energy supply the change of the supply of the order of 10% switches on or off the condensed state. The interaction between systems with condensed states leads to interaction forces. Dependent on various conditions these forces may be attractive or repulsive.     

Quenching of self-excited vibrations: one- and two-frequency vibrations

An analysis is presented of the possibility of initiation of two-frequency self-excited vibrations, and it is shown on this basis, and amply documented by analogue solutions, that the occurrence of such a phenomenon is rather exceptional in the systems being examined. It also is shown that unless the value of the coefficient of additional damping is very close to that lying on the boundary of the stability of the equilibrium position, it will have little effect on reducing the amplitude of self-excited vibrations even in the case of optimum tuning.     

An analysis of lattice vibrations of ordered Fe3Al

The frequencies of the normal modes of vibration of ordered Fe₃Al are classified according to the irreducible representations of the groups of the wave vector for points and lines of special symmetry.A simple model, with interactions extending up to the second neighbours is used to evaluate the coefficients of the dynamical matrix. The coefficients are given in terms of nine force constants.     

Maxi-element analysis for free vibrations of orthotropic shells of revolution

The theory of free vibration for orthotropic shells of revolution with arbitrary homogeneous boundary conditions is developed. The essence of the method is to decompose the overall shell into a number of so-called cylindrical, conical, and plate “maxi-elements”. Since the eigenfunctions of each of the individual maxi-elements are analytically determined directly from the solution of the governing differential equations, the procedure has the advantage of requiring significantly fewer elements compared with the usual finite element or finite difference procedures. For the conical shell and the plate, the method of solution is novel, while the solution for the cylindrical shell has been published by the first author. The versatility and accuracy of the method is shown through the inclusion of a number of examples which present the excellent correlation with test results and other numerical schemes.     

Chemical surface analysis by Raman spectroscopy utilizing dimer vibrations

We report on the investigation of reconstructed semiconductor surfaces by Raman spectroscopy from dimer vibrations. Localized modes of Te dimers on the (100) surface of the zinc blende II-VI semiconductor BeTe allow the analysis of Te- and Be-rich surface structures, as well as oxidation effects. The Te-rich surface exhibits one dimer-vibration mode at 165 cm(-1), while two modes appear at the Be-rich surface (157 and 188 cm(-1)). The mode assignment as dimer vibrations is underscored by their symmetry properties and by frozen phonon calculations, yielding mode frequencies and eigenvectors. This approach opens up a new field of surface chemistry analysis by dimer-vibration spectroscopy.     

Stability analysis of a nonlinear rotating blade with torsional vibrations

This paper discusses the stability of a spinning blade having periodically time varying coefficients for both linear model and geometric nonlinear model. To obtain a reduced nonlinear model from nodal space, a standard modal reduction procedure based on matrix operation is developed with essential geometric stiffening nonlinearities retained in the equation of motion. For the linear model, the stability chart with various spinning parameters of the blade is studied via the Bolotin method, and an efficient boundary tracing algorithm is developed to trace the stability boundary of the linear model. For the geometric nonlinear model, the method of multiple time scale is employed to study the steady state solutions, and their stability and bifurcations for the periodically time-varying rotating blade. The backbone curves of steady-state motions are achieved, and the parameter map for stability and bifurcation is developed.