Discrete Supports Research Articles

FORCED OSCILLATIONS OF INFINITE PERIODIC STRUCTURES. APPLICATIONS TO RAILWAY TRACK DYNAMICS

Forced steady-state oscillations of infinite periodic structures, caused by a moving harmonic force, are considered. One can see two periodicities in railway tracks whose periods are equal to the rail length and sleeper spacing, respectively. The first one causes variations in track pliability near rail joints as well as low frequency oscillations of moving vehicles. This allows one to consider the track as a Euler-Bernoulli beam, resting on uniform visco-elastic foundation. Resilient hinges present rail joints in the track model. The second track periodicity is usually considered along with high frequency track excitation. Taking this into account, the more precise model, that presents the rail as a discretely supported Timoshenko beam, is considered. Three discrete support models of different complexity are studied. The first one presents the rail support as a concentrated mass on a spring and a dashpot in parallel. In the second model, the support is a uniform unbending beam on uniform visco-elastic foundation. The sleeper bend is considered in the third support model. The power series method is used to account its bend. An influence of the rail shear deformations and the sleeper bend on the track frequency response is estimated.

Fast-Wave Analysis of an Inhomogeneously-Loaded Helix Enclosed in a Cylindrical Waveguide

The inhomogeneously-loaded helix enclosed in a metal cylindrical waveguide operated in the fast-wave regime is field-analyzed using the tape-helix model. The inhomogeneous discrete dielectric supports have been modeled by azimuthally smoothing them out into a number of continuous, homogeneous dielectric tube regions of appropriate effective relative permittivity values determined by the relative volumes occupied by the supports in these regions. While accounting for the radial inhomogeneity, the nonuniformity of the radial propagation constant over the structure cross section is also taken into consideration. Further, structured expressions amenable to easy computation are presented with the help of the relevant recurrence relations, and a method is suggested to use them in an iterative cycle for the desired convergence accuracy. The dispersion relation developed, as a special case, passes on to that obtained elsewhere for a simpler structure in which the dielectric supports for the helix was not considered. It is observed that the dielectric-support parameters become more effective in controlling the dispersion curve than the bare helix parameters. It is also found that the effect of the controlling parameters such as the helix pitch angle on the phase velocity and interaction impedance of the structure becomes less pronounced as the helix is brought closer to the waveguide wall. The presence of the dielectric in the structure increases the value of the interaction impedance to a reasonable value over the entire operating frequency band. The results in terms of the structure parameters, namely, helix pitch angle, helix-support permittivity, helix-to-waveguide wall separation and structure inhomogeneity presented here are rather general, and can be used for the structure design and performance evaluation as well as for the control of any space harmonic of interest in a fast-wave device such as the gyro-traveling-wave amplifier.

Fast-Wave Analysis of an Inhomogeneously- Loaded Helix Enclosed in a Cylindrical Waveguide

The inhomogeneously-loaded helix enclosed in a metal cylindrical waveguide operated in the fast-wave regime is field-analyzed using the tape-helix model. The inhomogeneous discrete dielectric supports have been modeled by azimuthally smoothing them out into a number of continuous, homogeneous dielectric tube regions of appropriate effective relative permittivity values determined by the relative volumes occupied by the supports in these regions. While accounting for the radial inhomogeneity, the nonuniformity of the radial propagation constant over the structure cross section is also taken into consideration. Further, structured expressions amenable to easy computation are presented with the help of the relevant recurrence relations, and a method is suggested to use them in an iterative cycle for the desired convergence accuracy. The dispersion relation developed, as a special case, passes on to that obtained elsewhere for a simpler structure in which the dielectric supports for the helix was not considered. It is observed that the dielectric-support parameters become more effective in controlling the dispersion curve than the bare helix parameters. It is also found that the effect of the controlling parameters such as the helix pitch angle on the phase velocity and interaction impedance of the structure becomes less pronounced as the helix is brought closer to the waveguide wall. The presence of the dielectric in the structure increases the value of the interaction impedance to a reasonable value over the entire operating frequency band. The results in terms of the structure parameters, namely, helix pitch angle, helix-support permittivity, helix-to-waveguide wall separation and structure inhomogeneity presented here are rather general, and can be used for the structure design and performance evaluation as well as for the control of anv space harmonic of interest in a fast-wave device such as the gyro-traveling-wave amplifier.

Rigorous tape analysis of inhomogeneously-loaded helical slow-wave structures

The inhomogeneously-loaded helical slow-wave structure is field analyzed by azimuthally smoothing out the discrete supports into a number of continuous dielectric tube regions of appropriate permittivity values. The analysis is made rigorous by removing the usual approximation of considering the radial propagation constant to be the same in the different structure regions; moreover, a tape-helix instead of a simpler sheath-helix model has been used to consider the space-harmonic effects. This renders the present analysis more general, accurate and capable of dealing with a wide range of structure parameters, and particularly makes it valid for large values of helix pitch angle, inhomogeneity and separation between the helix and the metal envelope of the structure. Further, structured expressions amenable to easy computation are presented with the help of the recurrence relations for various structure regions, and methods to use them effectively in an iterative cycle for the desired accuracy are suggested. The developed analytical expressions have also been checked by seeing that they pass on, as special cases, to those obtained earlier by making simplifying assumptions. In this comparison, care has been taken to account for the helix tape thickness by considering a free-space gap equal to the difference between the mean helix radius and the inner radius of the dielectric helix support. The results for the propagation characteristics of an inhomogeneously-loaded helical structure presented here are rather general and can readily be used for the structure design, performance evaluation, and control of any space harmonic of interest in a practical device.

New method for deriving eigenvalue rate with respect to support location

The authors present a new method to derive the formulas of eigenvalue rate with respect to in-span support location using the generalized variational principle of the Rayleigh quotient. Following the ideas of this note, the eigenvalue rate, with respect to such variables as location of discrete or continuous in-span supports, the location of concentrated mass, location of elastic support, and location of a substructure, can be derived without any conceptual difficulties

Discretely Supported Rails Subjected to Transient Loads

SUMMARY In this paper, the vertical displacement of a railway rail due to a travelling vertical point load of variable intensity is calculated. The model of the track is largely due to Grassie, albeit that the sleepers are modelled more realistically. The rail is modelled as an Euler beam. The most salient features of the paper are that the travelling of the variable load and the irregular discrete support of the sleepers are taken fully into account. The complex Fourier transform is used throughout this paper.

Response and stability of a stochastic beam-column using stochastic FEM

A beam-column resting on continuous Winkler foundation and discrete elastic supports is considered. The beam-column is of variable cross-section and the variation of sectional properties along the axis of the beam-column is deterministic. Young's modulus, mass per unit length and distributed axial loadings of the beam-column have a stochastic distribution. The foundation stiffness coefficient of the Winkler model, the stiffnesses of discrete elastic supports, stiffnesses of end springs and the end thrust, are all considered as random parameters. The material property fluctuations and distributed axial loadings are considered to constitute independent, one-dimensional uni-variate homogeneous real stochastic fields in space. The foundation stiffness coefficient, stiffnesses of the discrete elastic supports, stiffnesses of end springs and the end thrust are considered to constitute independent random variables. Static response, free vibration and stability behaviour of the beam-column are studied. Hamilton's principle is used to formulate the problem using stochastic FEM. Sensitivity vectors of the response and stability parameters are evaluated. Using these statistics of free vibration frequencies, mode shapes, buckling parameters, etc., are evaluated. A numerical example is given.

Response of Railway Track on Nonlinear Discrete Supports

SUMMARY The field tests revealed that the relation between the vertical rail deflections and the wheel load under various environments may be linear or nonlinear. The quasistatic nonlinear rail response under vertical wheel loads - the rail deflections, the reaction of supports and the bending moments of the rail with uniformly spaced discrete supports is presented. The stiffness of the fasteners, sleepers and roadbed is included by a set of springs. The nonlinear finite element procedure is applied, using the Lusas computer program. A parametric study is carried-out to examine the effect of the geometric support nonlinearities on deformation and stresses of the track. The nonlinearity is attributed mainly to the play between the rails, the sleepers and the ballast and to the bending of sleepers while they take full bearing. The linear and nonlinear solutions are compared.

Vibration of composite plates with internal supports

The free vibration of rectangular laminated composite plates with arbitrary support conditions along the edges, internal line supports and discrete point supports are studied using the Rayleigh-Ritz method. Polynomial approximation functions are selected to satisfy all essential boundary conditions along the edges of the plate and to vanish along line supports parallel to the co-ordinate axes. Straight line supports at an angle from the co-ordinate axes and curved line supports are modeled by introducing several point supports along the line. Zero displacement constraints at the point support locations are enforced using the Lagrange multiplier technique. The plate constitutive equations are expressed in terms of stiffness invariants and the fundamental natural frequency is maximized by selecting the appropriate lay-up. Several examples are presented to illustrate the versatility of the approach and provide results not previously available. The influence of the number of plies in the laminate, lay-up, material properties and plate aspect ratios are investigated.

The Stability of Flow Over Periodically Supported Plates-Potential Flow

The linear stability of uniform potential flow over compliant plates resting on a periodic foundation is studied. The foundation is made up of a one-dimensional periodic array of discrete spring supports. A Fourier Transform-based approach is used to treat the multi-harmonic dynamic response of the compliant plate. The effects of support interval, support properties and plate properties on the stability of the flow are examined in relation to established results for plates on a continuous non-periodic spring foundation. An increase in the support interval leads to a complex proliferation of unstable modes and is generally destabilizing.

Dynamic Analysis of Active‐Control, Cable‐Stayed Guideway

The effects of cable‐tension control on guideway deflection of a planar, cable‐stayed support structure for moving vehicles is examined. The structure studied is referred to as double stayed, as it incorporates stay cables both above and below the guideway in a symmetric arrangement. Use of double stay cables allows static guideway alignment to be determined and maintained by cable pretension. A typical guideway section is modeled as a simply supported beam with discrete elastic supports at the cable connections. The supporting pylon is treated as a cantilever beam with concentrated end loads. Fourier analysis and the method of assumed modes are used to solve the coupled equations of motion when the guideway is subjected to a moving constant load. Numerical results show that guideway deflection can be effectively reduced through tension control in the lower stay cables. Modification of lower‐cable tensions in response to external loading has the effect of increasing the load‐carrying ability of the pretension in upper cables.

Parametric instability of stochastic columns

Columns which have stochastically distributed Young's modulus and mass density and are subjected to deterministic periodic axial loadings are considered. The general case of a column supported on a Winkler elastic foundation of random stiffness and also on discrete elastic supports which are also random is considered. Material property fluctuations are modeled as independent one-dimensional univariate homogeneous real random fields in space. In addition to autocorrelation functions or their equivalent power spectral density functions, the input random fields are characterized by scale of fluctuations or variance functions for their second order properties. The foundation stiffness coefficient and the stiffnesses of discrete elastic supports are treated to constitute independent random variables. The system equations of boundary frequencies are obtained using Bolotin's method for deterministic systems. Stochastic FEM is used to obtain the discrete system with random as well as periodic coefficients. Statistical properties of boundary frequencies are derived in terms of input parameter statistics. A complete covariance structure is obtained. The equations developed are illustrated using a numerical example employing a practical correlation structure.

Linear dynamics and active control of an elastically supported travelling string

In this paper, the linear dynamics and active control of a string travelling with uniform velocity is presented. Discrete elastic supports are introduced along the length of the string. Finite element formulation is adopted to obtain the governing equations of motion. The velocity of translation introduces gyroscopic terms in the system equations. The effect of translation and the discrete elastic supports on the free vibration solution is studied. The solution is utilized in actively controlling the string vibrations due to an initial disturbance. The control, affected in modal space, is optimal with respect to a quadratic performance index. Numerical results are presented to demonstrate the effectiveness of the control strategy in regulating the travelling string vibrations.

Modeling of practical multi-octave-band helical slow-wave structures of a traveling-wave tube for interaction impedance

A theoretical model is developed for evaluating the interaction impedance of practical helical slow-wave structures which are anisotropically and/or inhomogeneously loaded specially for multioctave bandwidths. The discrete supports have been azimuthally smoothed out into a number of dielectric tubes of different permittivity values, while the metal vanes have been modeled by an axially conducting cylinder suitably located in the structure. >

The inhomogeneous dielectric loading effects of practical helix supports on the interaction impedance of the slow-wave structure of a TWT

The helical slow-wave structure of a traveling-wave tube is usually supported by wedge-shaped bars or rods or circular or rectangular cross section. Such supports present an inhomogeneous loading of the helix caused by the variation of the effective permittivity of the helix surrounding with the radial coordinate. In one of the approaches to the analysis of such structures for dispersion characteristics, the discrete supports are azimuthally smoothed out into continuous dielectric-tube regions, the number of such tubes being increased for converging results. This method is extended for the interaction impedance of the helix supported by a number of dielectric bars/rods of rectangular or circular cross section, the whole enclosed in a metal envelope. The method gives more accurate results than that given by a single tube, particularly for situations in which the relative permittivity and radial thickness of the dielectric supports and the frequency of operation are high. The analysis developed is rather general.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Linear bifurcation of perfect column-supported cylinders: Support modelling and boundary conditions

Large elevated silos and tanks are generally supported on a number of columns. The discrete supports of the columns give rise to high stresses adjacent to the column terminations. In particular, very high meridional compressive stresses arise above each column termination and these can lead to buckling of the shell at a load much lower than that for a uniformly supported shell. This paper presents a study of the linear bifurcation buckling behaviour of column-supported thin elastic perfect cylinders using the finite element method. Cylinders on both rigid and flexible column supports are studied, and the effects of varying several shell parameters are examined. The effects of the top and bottom edge boundary conditions are also investigated. It is well recognised that the linear bifurcation load is only an upper bound on the actual elastic buckling load, because the effects of large deflections and initial imperfections are ignored. Nevertheless, our understanding of the buckling behaviour of these structures is so limited that the results obtained here provide a significant starting point from which more elaborate studies may follow.

Minimum stiffness of optimally located supports for maximum value of column buckling loads

The problem of maximizing the buckling load of Bernoulli-Euler columns, with their continuously distributed selfweight taken into account, is considered in cases where the stiffnesses and positions of discrete lateral spring supports are used as design variables. Necessary conditions for optimum support stiffness and position are derived, and they are discussed in the light of some classical results. Several numerical examples are presented.

Theoretical and Experimental Solutions of Cooling‐Tower‐Soil system

In recent years, there have been many papers on dynamic properties of cooling towers. However, the bottoms of the tower shells were assumed as a fixed boundary, or discrete column systems of towers were considered as elastic boundary conditions. In the present paper, tower shells and discrete supports are treated as before. Furthermore, a dynamic model of an infinite soil is developed, and dynamic properties of the whole system of the cooling tower have been calculated. Model tests of free vibrations of cooling towers on different simulated soils are investigated. The numerical and experimental results are in agreement. The results show that the elasticity of the soil reduces the natural frequencies of the cooling tower and the reduced value depends on the stiffness of the soil. Some of the mode shapes of the cooling tower are changed much as compared to a built-in tower.

Free vibration of an orthogonally stiffened cylindrical shell, part I: Discrete line simple supports

The hierarchical finite element method is used to establish the stiffness and mass matrices of a cylindrically curved rectangular panel. Some natural frequencies and modes of two such panels, each with different boundary conditions, are then determined. Excellent agreement if found between this work and that of other investigators. These stiffness and mass matrices are then combined with periodic structure theory to analyze an orthogonally stiffened cylindrical shell. This analysis is formulated for a “plane wave” type of motion which is characterized by different propagation constants in the axial and circumferential directions. The governing equations of free vibration are then solved as a matrix eigenvalue problem for the frequencies at which particular waves will propagate. Results are presented in the form of phase-constant surface stacks, and clearly show the qualitative effects of varying the major shell parameters b a , R a and R h .

Free vibrations of periodically point-supported rectangular plates

The basis and formulation of a new approach, namely, the flexibility function approach based on simulating point supports on free edges as zeros of a flexibility function representing a fictitious elastic restraint distribution over the edges, was recently presented [1,2]. Its application to a variety of problems of free vibrations of point-supported rectangular plates was presented subsequently [3,4]. The present study provides an extension of its application to the problem of periodically point-supported plates. It is shown that by the use of a simple flexibility function the problem of periodically point-supported plates can be easily analyzed. Numerical results for a range of problem parameters have been obtained which bring out the effect of the number of point supports on the natural frequencies, edge displacements and edge reactions. They also provide a quantitative estimate of the errors associated with the approximation of a large number of discrete supports by a continuous support condition or vice versa. Results have also been obtained for some typical problem parameters by using the impulse function approach based on representing the reactions at the point supports as Fourier series expansion of the impulse function. The comparison of the results clearly illustrates the fact that the flexibility function approach gives results which converge faster, requires significantly less computation time and provides accurate results.