Harmonic Line Load Research Articles

DYNAMIC RESPONSE OF A THREE-DIMENSIONAL FLUID CHANNEL BOUNDED BY AN ELASTIC FLOOR IN THE PRESENCE OF A SUBMERGED INCLUSION VIA BEM

Numerical methods for computing the three-dimensional pressure field in a flat fluid channel bounded either by a rigid boundary, an elastic semi-infinite medium or by a layer of sediment, subjected to incoherent line sources are presented. After verification Greens functions are incorporated in a Boundary Element Method (BEM) code that simulates the pressure variation inside the fluid channel in the vicinity of a rigid or elastic inclusion, avoiding the discretization of the fluid and solid channel interfaces. After the verification of the solution, the models developed are then used to simulate the pressure variation within the fluid channel in the presence of infinitely long rigid and elastic inclusions of differing sizes, when the channel is struck by a spatially-sinusoidal harmonic pressure line load. The results are then compared with those obtained when the channel floor is assumed to be rigid. Time domain results are given by means of inverse Fourier transforms, to help understand how the mechanical properties of the channel floor may affect the variation of the pressure field within the channel.

An explicit representation of steady state response of a beam on an elastic foundation to moving harmonic line loads

AbstractA closed‐form deflection response of a beam rest is presented in this paper using the integral transform method. The theory of linear partial differential equations is used to represent the deflection of beam subjected to a moving harmonic line load in integration form. The solution is finally carried out using the inverse Fourier transform. To evaluate the integration analytically, poles of the integrand are identified with the help of algebraic equation theory. Residue theorem is then utilized to represent the integration as a contour integral in the complex plane. Closed‐form deflections and numerical results are provided for different combinations of load frequency and velocity. Copyright © 2002 John Wiley & Sons, Ltd.

Response of clamped structural slabs subjected to a dynamic point load via BEM

This work computes the response of clamped slabs when subjected to spatially sinusoidal harmonic line loads via the Boundary Element Method (BEM). The formulation uses 2.5D Green’s functions for the steady state response of a homogeneous three-dimensional free solid layer formation of infinite extent, proposed earlier by the authors. The inclusion of these Green’s functions in the BEM formulation avoids the discretization of free horizontal surfaces, which contributes to the efficiency of the BEM model. Frequency and time responses have been computed for slabs with and without lateral confinements, for different thickness and varying spatially sinusoidal harmonic line loads.

Assessing the effect of a barrier between two rooms subjected to low frequency sound using the boundary element method

The boundary element method (BEM) has been used to compute the acoustic wave propagation through a single vertical panel, which separates two rooms, made of concrete, when one of the rooms is excited by a steady-state, spatially sinusoidal, harmonic line load pressure at low frequencies. This work focuses on how the connection of the panel to the ceiling affects the acoustic insulation provided by the wall. Perfect double-fixed partitions and acoustic barrier-type structures with differently-sized gaps between the ceiling and the barrier are studied. The BEM model is formulated in the frequency domain and takes the air-solid interaction fully into account. Insulation dips are localised in the frequency domain and identified with dips associated with both the wall's natural dynamic vibration modes and with those associated with the air in the rooms. The influence of the wall's thickness on acoustic insulation is also analysed. The computed results obtained with the acoustic barrier type structure are compared with those obtained by a rigid model. The importance of the rooms' surface conditions is assessed, modelling the rooms with cork.

ACOUSTIC INSULATION PROVIDED BY A SINGLE WALL SEPARATING TWO CONTIGUOUS TUNNELS VIA BEM

This paper describes the computation of the acoustic insulation provided by a single simple wall separating two contiguous tunnels when a steady state, spatially sinusoidal, harmonic line load pressure excites one of the tunnels. The Boundary Element Method (BEM), formulated in the frequency domain, is used to compute the acoustic pressure in the two tunnels. The insulation conferred by the wall is characterized, identifying the location of insulation dips in the frequency domain with those dips related to its own natural dynamic vibration modes and those related to the natural vibration modes of the tunnels. This model is also used to assess how the dimension of the rooms, and the rigidity and thickness of the wall affect acoustic insulation. The transmission loss results obtained via the BEM are then compared with those provided by simplified analytical models such as the mass law.

ACOUSTIC INSULATION OF SINGLE PANEL WALLS PROVIDED BY ANALYTICAL EXPRESSIONS VERSUS THE MASS LAW

Analytical solutions are presented for the calculation of the acoustic insulation provided by an infinite single panel wall when subjected to a spatially sinusoidal harmonic line load or a point pressure load (modelled as a superposition of spatially sinusoidal harmonic line loads). The method used does not entail limiting the thickness of the layer, as the Kirchhoff or Mindlin theory requires, and fully takes into account the coupling between the fluid (air) and the solid panel. All calculations are performed in the frequency domain. Time signatures are obtained by means of inverse Fourier transforms. Special attention is given to the limitations of the simplified models, which are not able to predict dips of insulation such as that due to the coincidence effect. It has been shown that, although time results may appear complicated, the arrival of various pulses at the receivers can be understood in terms of the travelling body pulses and guided waves. Simulated results have been computed for ceramic, concrete and glass walls of different thickness, when subjected to plane, linear and spherical waves. The insulation computed was found to be highly dependent on receiver position, given the interaction between the incident wave field and the directed reflected field on the wall, when the wall is struck by a cylindrical or a spherical pulse wave.

3D seismic response of a limited valley via BEM using 2.5D analytical Green's functions for an infinite free-rigid layer

This paper presents analytical solutions for computing the 3D displacements in a flat solid elastic stratum bounded by a rigid base, when it is subjected to spatially sinusoidal harmonic line loads. These functions are also used as Greens functions in a boundary element method code that simulates the seismic wave propagation in a confined or semi-confined 2D valley, avoiding the discretization of the free and rigid horizontal boundaries. The models developed are then used to simulate wave propagation within a rigid stratum and valleys with different dimensions and geometries, when struck by a spatially sinusoidal harmonic vertical line load. Simulations are performed in the frequency domain, for varying spatial wave numbers in the axial direction of the valley. Time results are obtained by means of inverse Fourier transforms, to help understand how the geometry of the valley may affect the variation of the displacement field.

3D scattering of waves by a cylindrical irregular cavity of infinite length in a homogeneous elastic medium

The three-dimensional (3D) wave field scattered by an irregular, cylindrical cavity of infinite length contained in a homogeneous elastic medium illuminated by a dilatational point load is obtained. This model is used to evaluate the effect of the cross-sectional geometry of the cavity on the waves propagating in its vicinity. It particularly highlights the identification of the normal modes excited both in the frequency and time domain. The solution is formulated using the boundary element method for a wide range of frequencies and spatially harmonic line loads, which are then synthesized to obtain the time responses. The 3D solution is obtained as a summation of two-dimensional responses for different axial wavenumbers. The responses in the frequency vs. axial-wavenumber domains are presented, allowing the recognition, identification, and physical interpretation of the variation of the wave field when five irregular cross-sections are used, namely a circle, an oval, a thin oval, a kidney and a boomerang.

Green's functions for 2.5D elastodynamic problems in a free solid layer formation

This work presents analytical Green's functions for the steady state response of a homogeneous three-dimensional free solid layer formation (slab) subjected to a spatially sinusoidal harmonic line load, polarized along the horizontal, vertical and z directions. The equations presented here are not only themselves very interesting but are also useful for formulating three-dimensional elastodynamic problems in a slab-type formation, using integral transform methods and/or boundary elements. The final expressions are validated by comparing them with the results obtained by using the Boundary Element Method solution, for which both free surfaces of the slab are discretized with boundary elements.

APPLICATIONS OF THE GREEN FUNCTIONS IN THE STUDY OF ACOUSTIC PROBLEMS IN OPEN AND CLOSED SPACES

This paper presents a set of analytical solutions (Green functions) for the steady state response of a homogenous acoustic three-dimensional space subjected to a point harmonic load or a spatially sinusoidal harmonic line load. The propagation medium is modelled with plane surfaces placed so as to reproduce spaces that vary from a simple half-space to a rectangular parallelepiped closed space. The final expressions are implemented to evaluate first the pressure field inside a rectangular parallelepiped room, whose walls allow different absorption coefficients. Then, the acoustic scattering of a three-dimensional sound source by an infinitely long rigid barrier in the vicinity of a tall building is evaluated using the boundary-element method (BEM), making use of the analytical solution for a spatially sinusoidal line source. The use of these Green functions allows the required BEM discretization to be limited to the surface of the barrier, avoiding the discretization of the tall building and ground. The calculations are performed in the frequency domain and time signatures are obtained by means of inverse Fourier transforms. Complex frequencies are used to attenuate the response at the end of the time frame, which is taken into account by rescaling the time response.

TIME–HARMONIC ELASTODYNAMIC GREEN FUNCTIONS OF PLATES FOR LINE LOADS

In this paper the Fourier transform is used to derive the elastodynamic Green function of a plate on a viscoelastic foundation subjected to impulse and harmonic line loads. The solution is first given as a convolution of the Green function of the plate. Poles of the integrand in the integral representation of the solution are identified for different cases of damping and load frequency. The Green function corresponding to an impulse line load is obtained and can be numerically computed. The theorem of residue is then utilized to evaluate the generalized integral of the Green function corresponding to a harmonic line load. This representation permits one to construct algorithms for the parameter identification of the inverse problem involved in a pavement non-destructive test. Validation of the result is partly verified by comparing the static solution of a point source obtained from this paper to a well-known result.

Green's function for two-and-a-half dimensional elastodynamic problems in a half-space

This paper presents an analytical solution, together with explicit expressions, for the steady state response of a homogeneous three-dimensional half-space subjected to a spatially sinusoidal, harmonic line load. These equations are of great importance in the formulation of three-dimensional elastodynamic problems in a half-space by means of integral transform methods and/or boundary elements. The final expressions are validated here by comparing the results with those obtained with the boundary element method (BEM) solution, for which the free surface of the ground is discretized with boundary elements.

Amplification of elastic waves due to a point source in the presence of complex surface topography

This paper first presents the amplification or de-amplification in the three-dimensional wave scattering generated by the half-space when subjected to a dilatational point load placed at some point in the medium, when the free surface is changed to simulate a two-dimensional ridge or a valley deformation. This model is then extended to include the presence of a long, circular, buried cavity. The solution is formulated using the boundary elements method for a wide range of frequencies and spatially harmonic line loads, which are subsequently used to obtain time series by means of (fast) inverse Fourier transforms into space–time. Our simulation results indicate larger wave amplification and de-amplification in the time and frequency domains when the topographic surface includes a smooth ridge and canyon deformations. The presence of a buried inclusion further complicates the wave field displacement pattern, including additional reflections.

Reduction of train-induced vibrations on adjacent buildings

In this paper, the procedure for deriving an infinite element that is compatible with the quadrilateral Q8 element is first summarized. Enhanced by a self mesh-expansion procedure for generating the impedance matrices of different frequencies for the region extending to infinity, the infinite element is used to simulate the far field of the soil-structure system. The structure considered here is of the box type and the soils are either homogeneous or resting on a bedrock. Using the finite/infinite element approach, a parametric study is conducted to investigate the effect of open and in-filled trenches in reducing the structural vibration caused by a train passing nearby, which is simulated as a harmonic line load. The key parameters that dominate the performance of wave barriers in reducing the structural vibrations are identified. The results presented herein serve as a useful guideline for the design of open and in-filled trenches concerning wave reduction.

3D scattering by multiple cylindrical cavities buried in an elastic formation

This paper presents the three-dimensional scattering field obtained when multiple cylindrical circular cavities of infinite length buried in a homogeneous elastic medium, are subjected to dilatational point loads placed at some point in the medium. The solution is formulated using boundary elements for a wide range of frequencies and spatially harmonic line loads, which are then used to obtain time series by means of (fast) inverse Fourier transforms into space-time. The method and the expressions presented are implemented and validated by applying them to a cylindrical circular cavity buried in an infinite homogeneous elastic medium subjected to a dilatational point load, for which the solution is calculated in closed form. The boundary elements method is then used to evaluate the wave-field elicited by a dilatational point load source in the presence of a different number of cylindrical oval cavities. Simulation analyses using this idealized model are then applied to the study of wave propagation patterns in the vicinity of these inclusions. The amplitude of the wavefield in the frequency vs axial-wavenumber domain is presented, allowing the recognition, identification, and physical interpretation of the variation of the wavefield.

A CLOSED-FORM SOLUTION OF A BERNOULLI-EULER BEAM ON A VISCOELASTIC FOUNDATION UNDER HARMONIC LINE LOADS

Fourier transform is used in this paper to solve the problem of steady state response of a beam on a viscoelastic foundation subjected to a harmonic line load. The solution is constructed in the form of the convolution of the Green function of the beam. The theorem of residue is employed to evaluate the generalized integral such that a closed-form solution can be achieved. All the different combinations of damping and vibration frequency are discussed and analytical solutions are presented. As a special case, the solution of the beam on a Winkler foundation is also discussed. The validation of the solution is verified by considering the static solution of the beam and comparing the degraded solution to a well-known result. The closed-form expression of the result can be used to construct algorithms for the inverse problems of non-destructive testing of pavement structures using vibration devices.

Frequency and Time Numerical Solutions of 3D Sound Propagation in Open and Closed Spaces

Problems in acoustics can often be solved with the aid of formulas or expressions known as Green's functions. These functions, or fundamental solutions, relate the field variations (such as pressures) in an acoustic medium caused by sound sources placed elsewhere in the medium. The two fundamental solutions most often used are for harmonic point loads in a three-dimensional, infinite, homogeneous space, and for a harmonic line load acting within two-dimensional spaces. These are chosen because their solutions are known in closed-form and they are relatively simple in structure. This paper presents a set of three-dimensional solutions applied when the space domain is modeled with plane barriers placed together to reproduce spaces that vary from a simple half-space to a parallelepiped closed space. It is assumed that the homogeneous three-dimensional space is subjected to 3D point sources and spatially sinusoidal, harmonic line sound loads. The resulting expressions are implemented to evaluate the field inside a rectangular space, whose walls allow different absorption coefficients. The time responses are obtained by means of Inverse Fourier Transforms. Complex frequencies are used to attenuate the response at the end of the time window. The effect of this attenuation is taken into account by re-scaling the time response.

Green's Functions for Two-and-a-Half-Dimensional Elastodynamic Problems

This paper presents a fully analytical solution, together with explicit expressions, for the steady-state response of a homogeneous 3D space subjected to a spatially sinusoidal, harmonic line load. In the literature, this problem is often referred to as the two-and-a-half-dimensional \Ifundamental solution\N or two-and-a-half-dimensional Green’s functions. These equations are of great usefulness in the formulation of 3D elastodynamic problems by means of integral transforms methods and/or boundary elements. The final expressions are validated here by applying the equations to the problem of a 3D point load, for which the solution is known in analytical form.

Dynamic Green's Function for SH-Wave in Multi-Dipping Layers Embedded in a Half-Space

ABSTRACTIn this paper, the recursive T-matrix formalism is developed to determine the dynamic Green's function for SH-wave in the multi-dipping layers with arbitrary shape embedded in a half-space. In each layer, the wave field generated by a harmonic line load can be separated into two parts: the source term and the complementary part. The source term is exactly the Green's function for SH-wave in two-dimensional half-space, and the complementary part which causes the SH-wave to satisfy the boundary conditions at the interface is determined by the wave function expansion method. Using the polar coordinates, the basis functions are constructed by Bessel typed cylindrical functions, and their orthogonality conditions at the corresponding interfaces can be derived by means of Betti's third identity. Applying the conditions of continuity at the interface, the recursive T-matrix formalism is developed for determining the expansion coefficients of the complementary part from the associated source dependent constants.

A parametric study of wave barriers for reduction of train-induced vibrations

In this paper, a finite/infinite element scheme developed previously by the authors is employed to investigate the effectiveness of three different wave barriers, i.e., the open trench, in-filled trench, and elastic foundation, in reducing the ground vibrations caused by the passage of trains. The mathematical model adopted herein is the two-dimensional profile that contains the cross-section of the railway, barrier and underlying soils, with the moving train loads simulated as a vertical harmonic line load. Concerning the effect of isolation, the geometric and material parameters of the three barriers investigated include the distance to rail, depth, width and thickness, damping ratio, shear modulus, mass density, Poisson's ratio, elastic modulus, etc. Also examined is the effectiveness of the three barriers with respect to different exciting frequencies. Conclusions are made regarding the selection of optimal parameter values for the three barriers in isolating the train-induced ground vibrations. © 1997 John Wiley & Sons, Ltd.