Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation

The free in-plane vibrations of circular rings with wheel-plates as generalised elastic foundations are studied using analytical methods and numerical simulations. The three-parameter Winkler elastic layer is proposed as a mathematical model of the foundation. The effects of rotary inertia and shear deformation are included in the analytical model of the system. The motion equations of systems are derived on the basis of the thin ring theory and Timoshenko?s theory. The separation of variables method is used to find general solutions to the free vibrations. Elaborated analytical models are used to determine the natural frequencies and the natural mode shapes of vibrations of an arbitrarily chosen set of simplified models of aviation gears and railway wheels. The eigenvalue problem is formulated and solved by using a finite element representation for each simplified model. The results for these models are discussed and compared. The proposed solutions are verified by experimental investigation. It is important to note that the solutions proposed here could be useful to engineers dealing with the dynamics of aviation gears, railway wheels and other circular ring systems.     

Application of AFF and HPM to the systems of strongly nonlinear oscillation

This study is concerned with two well-known systems of nonlinear oscillators. One of them is a system consists of a mass grounded two springs which one of those springs is linear and the other is nonlinear, and the other one is dealt with nonlinear large amplitude free vibrations of a slender cantilever beam with a rotationally flexible root and carrying a lumped mass at an intermediate position along its span. The main objective was to obtain highly accurate analytical solutions for free vibration of conservative oscillators with inertia and static type cubic nonlinearities. Two methods are studied to analyze the dynamic system behavior. One method makes use of a set of Amplitude–Frequency Formulation (AFF) and the other method applies Homotopy Perturbation Method (HPM). Taking advantage of some simple mathematical operations on these methods, we can obtain their natural frequencies. The computed results agree with those analytical and numerical results given in the literatures. The results indicate that the present analysis is accurate, and provide us a unified and systematic procedure which is simple and more straightforward than the other similar methods.     

Dominant frequencies of train-induced vibrations

This paper investigates the dynamic characteristics of ground vibrations induced by moving vehicles including the mass rapid transit system, high-speed train railway, and general railway on bridges, embankments, and in tunnels using field experiments and theoretical solutions. The results indicate that train-induced ground vibrations at the trainload dominant frequencies are significantly large for both subsonic and supersonic train speeds, and the vibrations from carriage natural frequencies and engine frequencies are minor. For trains moving on bridges, the resonant vibrations are serious when the natural frequencies of the bridge are close to the trainload dominant frequencies, but resonance does not occur when the carriage natural frequencies and trainload dominant frequencies match. The trainload dominant frequency and its influence factor can be computed using the two simple equations deducted in this paper.     

On computation of random response of a clamped plate

The free vibrations of elastically connected circular plate systems with elastically restrained edges and initial radial tensions are investigated analytically. By using the equations developed for the general n-plate system, the plate systems consisting of three and two identical plates with identical boundary conditions and a uniform radial tension are treated in detail. Both axisymmetric and non-axisymmetric vibrations are considered. Attention is directed to the influence of the radial tension and the elastic edge constraints on the first nine eigenvalues and the corresponding natural frequencies of the systems.     

Investigating the effect of interphase and surrounding resin on carbon nanotube free vibration behavior

The free vibration analysis of a carbon nanotube (CNT) embedded in a volume element is performed using 3D finite element (FE) and analytical models. Three approaches consist of molecular and continuum mechanics FE methods and continuum analytical method are employed to simulate the CNT, interphase region and surrounding matrix. The bonding between CNT and polymer is treated as non-perfect bonding using van der Waals and triple phase material interaction in first and second approaches. In analytical approach a perfect bonding is assumed between nanotube and matrix. First, natural frequencies of CNT under different boundary conditions and aspect ratios are obtained by three approaches and the results are compared with published data. The results show the frequency response variations of CNT in GHz to THz range. Subsequently, vibration behaviors of CNT/polymer are evaluated and the results revealed the importance of interphase region role in the performance of nanocomposites. The results also showed the convergence of the natural frequencies for 1–2.5% of CNT volume in high aspect ratios using three methods, so that the interphase effects is negligible. In addition, it is observed that the molecular method due to interphase role has proper performance in vibration behavior investigation of volume elements.     

A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions

The subject of this paper is the development of a general solution procedure for the vibrations (primary resonance and nonlinear natural frequency) of systems with cubic nonlinearities, subjected to nonlinear and time-dependent internal boundary conditions—this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements. The equations of motion form a set of nonlinear partial differential equations with nonlinear, time-dependent, and coupled internal boundary conditions. The method of multiple timescales, an approximate analytical method, is applied directly to each partial differential equation of motion as well as coupled boundary conditions (i.e. on each sub-domain and the corresponding internal boundary conditions for a continuous system with intermediate elements) which ultimately leads to approximate analytical expressions for the frequency-response relation and nonlinear natural frequencies of the system. These closed-form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system. Moreover, the suggested solution procedure is applied to a sample problem which is discussed in detail.     

FREE TRANSVERSE VIBRATIONS OF AN ELASTICALLY CONNECTED COMPLEX BEAM-STRING SYSTEM

The present work is devoted to theoretical vibration analysis of a complex mixed continuous system. Undamped free transverse vibrations of an elastically connected beam-string system are considered. Solutions of the problem are formulated by using the modal expansion method. Two infinite sequences of the natural frequencies corresponding to two possible kinds of vibration motions: synchronous and asynchronous are determined. In a numerical example, illustrating the theory presented, the effect of string tension force on the natural frequencies of the system is investigated in detail.     

Free transverse vibrations of an elastically connected rectangular plate-membrane complex system

This paper theoretically analyzes undamped free transverse vibrations of an elastically connected rectangular plate-membrane system. Solutions of the problem are formulated by using the Navier method. Natural frequencies of the system in the form of two infinite sequences are determined. Normal mode shapes of vibration expressing two kinds of vibration, synchronous and asynchronous, are presented. The initial-value problem is also solved. In a numerical example, the effect of membrane tension on the natural frequencies of this mixed system is discussed.     

FREE VIBRATIONS OF FUNCTIONALLY GRADED PIEZOCERAMIC HOLLOW SPHERES WITH RADIAL POLARIZATION

By introducing displacement functions as well as stress functions, two independent state equations with variable coefficients are established from the three-dimensional equations of a radially inhomogeneous spherically isotropic piezoelastic medium. By virtue of the laminated approximation method, the state equations are then transformed into the ones with constant variables in each layer, and the state variable solutions are presented. Based on the solutions, linear algebraic equations about the state variables only at the inner and outer spherical surfaces are derived by utilizing the continuity conditions at each interface. Frequency equations corresponding to two independent classes of vibrations are finally obtained from the free surface conditions. Numerical calculations are presented and the effect of the material gradient index on natural frequencies is discussed.     

On the limiting of vibration amplitudes by a sequential friction-spring element

The possibility of using an additional sequentially connected friction spring element in order to reduce vibration amplitudes both for the self-excited oscillations and for the forced vibrations is discussed in the paper. The analysis is based on the averaging technique for systems with “slave variables” and demonstrates two main effects: damping during slipping in the additional element and fast switching between different natural frequencies due to alternating sticking/slipping phases. Analytic predictions for the oscillations’ amplitudes are obtained as steady state solutions of the equations governing slow motions of the system. The obtained analytic results enable optimal choice of friction in order to achieve maximal damping effect in case of the forced vibrations. The reasonable choice of the friction by the self-excited vibrations is a compromise between the acceptable amplitude and the robustness of the corresponding limit cycle. The asymptotic results are confirmed by numeric simulations.     

AN ANALYTICAL APPROACH TO DETERMINING THE DYNAMIC CHARACTERISTICS OF A CYLINDRICAL SHELL WITH CLOSING CRACKS

The paper is devoted to developing mathematical models of the elastic oscillations of a cylindrical shell with surface closing cracks. The respective forms of shell vibrations have been chosen to represent various types of damage of the shell. In the case of dispersed and single-surface damage, the transverse shell vibrations are simulated. The cycle of vibrations is assumed to be subdivided into two parts, in one of them the damaged surface fibers are compressed so closing the cracks and negating their influence. For the second part, the cracks are open, so their influence is taken into account. The problem is solved in a piecewise linear with different frequencies and amplitudes at each vibrations cycle interval. The vibration parameters are calculated by means of Relay's energy conservation method and are represented by analytical expressions, the system being assumed to be conservative. The functions determining the vibration process are decomposed by a Fourier analysis using the averaged frequency, the coefficients of the resulting series being obtained as analytical expressions. Vibrodiagnostic functions, which enable the geometrical parameters of the cracks to be determined depending on the geometry of the shell and type of damage, have been plotted.     

Mathematical Models for the Propagation of Stress Waves in Elastic Rods: Exact Solutions and Numerical Simulation

In this work, the Bishop and Love models for longitudinal vibrations are adopted to study the dynamics of isotropic rods with conical and exponential cross-sections. Exact solutions of both models are derived, using appropriate transformations. The analytical solutions of these two models are obtained in terms of generalised hypergeometric functions and Legendre spherical functions respectively. The exact solution of Love model for a rod with exponential cross-section is expressed as a sum of Gauss hypergeometric functions. The models are solved numerically by using the method of lines to reduce the original PDE to a system of ODEs. The accuracy of the numerical approximations is studied in the case of special solutions.     

Vibration modelling of helical springs with non-uniform ends

Helical springs constitute an integral part of many mechanical systems. Usually, a helical spring is modelled as a massless, frequency independent stiffness element. For a typical suspension spring, these assumptions are only valid in the quasi-static case or at low frequencies. At higher frequencies, the influence of the internal resonances of the spring grows and thus a detailed model is required. In some cases, such as when the spring is uniform, analytical models can be developed. However, in typical springs, only the central turns are uniform; the ends are often not (for example, having a varying helix angle or cross-section). Thus, obtaining analytical models in this case can be very difficult if at all possible. In this paper, the modelling of such non-uniform springs are considered. The uniform (central) part of helical springs is modelled using the wave and finite element (WFE) method since a helical spring can be regarded as a curved waveguide. The WFE model is obtained by post-processing the finite element (FE) model of a single straight or curved beam element using periodic structure theory. This yields the wave characteristics which can be used to find the dynamic stiffness matrix of the central turns of the spring. As for the non-uniform ends, they are modelled using the standard finite element (FE) method. The dynamic stiffness matrices of the ends and the central turns can be assembled as in standard FE yielding a FE/WFE model whose size is much smaller than a full FE model of the spring. This can be used to predict the stiffness of the spring and the force transmissibility. Numerical examples are presented.     

A sensitivity analysis of the effects of interconnection joint size,flexibility, and inertia on the natural frequencies of Timoshenko frames

The free vibrations of frame structures are influenced by the geometry, stiffness, and inertia of interconnection joints. The effects of generalized joint properties on the natural frequencies and mode shapes are studied for a wide range of natural frequencies by modeling the structure as a Timoshenko continuous system with discretized joints. Dynamic slope-deflection equations are used in the analysis, adapted to the boundary conditions imposed by joints with axial length, axial and rotary stiffness, and inertia. Beam/column axial deformation is also included. Frequency curves are presented for a wide range of beam/column and joint properties to establish the relative importance of model parameters on system free vibrations.     

Response characteristics of local vibrations in stay cables on an existing cable-stayed bridge

This paper examines local parametric vibrations in the stay cables of a cable-stayed bridge. The natural frequencies of the global modes are obtained by using a three-dimensional FE model. The global motions generated by (1) sinusoidal excitations using exciter, (2) a traffic loading, and (3) an earthquake are analyzed by using the modal analysis method or the direct integration method. The local vibration of stay cable is calculated by using a model in which inclined cable is subjected to time-varying displacement at one support during global motions. This paper describes the properties of the local vibrations in stay cables under these dynamic loadings by using an existing cable-stayed bridge.     

Exact solutions for free vibration of shear-type structures with arbitrary distribution of mass or stiffness.

In this paper, shear-type structures such as frame buildings, etc., are treated as nonuniform shear beams (one-dimensional systems) in free-vibration analysis. The expression for describing the distribution of shear stiffness of a shear beam is arbitrary, and the distribution of mass is expressed as a functional relation with the distribution of shear stiffness, and vice versa. Using appropriate functional transformation, the governing differential equations for free vibration of nonuniform shear beams are reduced to Bessel's equations or ordinary differential equations with constant coefficients for several functional relations. Thus, classes of exact solutions for free vibrations of the shear beam with arbitrary distribution of stiffness or mass are obtained. The effect of taper on natural frequencies of nonuniform beams is investigated. Numerical examples show that the calculated natural frequencies and mode shapes of shear-type structures are in good agreement with the field measured data and those determined by the finite-element method and Ritz method.     

Free vibrations of two rods connected by multi-spring-mass systems

This study deals with the longitudinal free vibrations of a system in which two rods are coupled by multi-spring-mass devices. The dynamics of this system are coupled through the motion of the masses. By using the transfer matrix method and considering the compatibility requirements across each spring connection position, the eigensolutions (natural frequencies and mode shapes) of this system can be obtained easily for different boundary conditions. The characteristic equation encompasses a function of the eigenvalues, the location of the spring connection positions, the ratio of the stiffness of the springs, the ratio of the lengths of the rods, the ratio of the sectional properties of the rods and the suspended masses. Some numerical results are presented to demonstrate the method proposed in this article.     

A computationally efficient finite element model with perfectly matched layers applied to scattering from axially symmetric objects

A frequency-domain finite-element (FE) technique for computing the radiation and scattering from axially symmetric fluid-loaded structures subject to a nonsymmetric forcing field is presented. The Berenger perfectly matched layer (PML), applied directly at the fluid-structure interface, makes it possible to emulate the Sommerfeld radiation condition using FE meshes of minimal size. For those cases where the acoustic field is computed over a band of frequencies, the meshing process is simplified by the use of a wavelength-dependent rescaling of the PML coordinates. Quantitative geometry discretization guidelines are obtained from a priori estimates of small-scale structural wavelengths, which dominate the acoustic field at low to mid frequencies. One particularly useful feature of the PML is that it can be applied across the interface between different fluids. This makes it possible to use the present tool to solve problems where the radiating or scattering objects are located inside a layered fluid medium. The proposed technique is verified by comparison with analytical solutions and with validated numerical models. The solutions presented show close agreement for a set of test problems ranging from scattering to underwater propagation.     

Frequency coalescence and mode localization phenomena: A geometric theory

Physical systems, the natural frequencies of which depend parametrically on certain quantities, may exhibit the phenomenon of frequency coalescence, when two or more natural frequencies become equal; or the phenomenon of avoided crossings, when two or more frequencies approach rapidly and then avoid crossing each other. In the process, the natural modes undergo significant variations. It is shown that both phenomena can be described in a unified manner from the singularities of the mapping from the complex parameter plane into the complex frequency plane. In particular, the formation of a saddle point in the frequency plane and a branch point in the parameter space is the basis for explaining the properties associated with these phenomena. The question of sensitivity of the system response near such singularities is subsequently examined. It is shown that the development of a branch point singularity in the complex parameter plane can cause large or even infinite sensitivity of the system about this point, depending on the distance of the branch point from the real axis in the parameter space. It is further shown that mode localization is also associated with the appearance of a branch point near the real axis of the complex parameter plane, which causes the large sensitivity that characterizes the phenomenon. The theory is applied to examples of conservative and dissipative systems undergoing free vibrations.     

THE EFFECT OF ROTATION UPON THE NATURAL FREQUENCIES OF A MASS-SPRING SYSTEM

When a mass-spring system vibrates it does so with frequencies characteristic of the system. If the system as a whole now undergoes a rotational motion then these characteristic frequencies will change from their non-rotational values. It is the purpose of this paper to show how these changes may be calculated for a specified system and, in particular, to investigate the role in these changes of both the system and the rotational parameters. A system of N masses linked sequentially by springs in tension is allowed to vibrate about an equilibrium configuration both radially and transversely upon a smooth turntable. If the turntable is stationary then the radial and transverse vibrations are independent of each other, provided the amplitudes of vibration are sufficiently small. There are then N natural frequencies of vibration for each mode. However, when the turntable rotates then the Coriolis effects give rise to an interaction between the two modes of vibration, and there are now 2 N natural frequencies for the combined vibrations. If the rate of rotation is “small” then the two modes are almost separated and it is possible to discuss the “essentially radial” or “essentially transverse” mode of vibration each of which has N natural frequencies. It is these natural frequencies which are considered in this work, in particular their dependence upon the rotation rate and upon the tension in the springs (when in the static configuration). In a previous paper, it was shown that if only radial vibrations are allowed (by admitting say a guide rail) then all the natural frequencies decrease, with increasing rotation rate, from their static values. It is shown that the opposite is the case here in that the “essentially radial” natural frequencies increase with increasing rotation rate. This is due to the Coriolis interaction with the transverse vibrations. The “essentially transverse” frequencies are also found and the nature of their dependence discussed. Also included in the analysis is the effect on the frequencies of the (weak) coupling between the motion of the masses and the rotation of the turntable as a consequence of the conservation of angular momentum. In addition to treating N being finite the limiting case of an infinite number of masses is considered to determine the natural frequencies of vibration of a continuous stretched string undergoing rotation.