Time-harmonic Loading Research Articles

Fractional derivative modeling for dynamic response of layered viscoelastic media under moving time-harmonic loads

The dynamic response of layered viscoelastic media under moving time-harmonic loads is an important issue in transportation geotechnics. However, the existing research often has limitations in terms of the constant creep rate or time-independent relaxation modulus. To fill this gap, this study introduces a fractional Zener model to simulate the viscoelastic properties of media. First, the governing equations of the elastic medium are converted into ordinary differential equations using an integral transform. Subsequently, by introducing the fractional viscoelastic model and extended precise integration method, a solution to the dynamic response of layered viscoelastic media under moving time-harmonic loads is obtained. After comparing the results with those in existing literature, the reliability and accuracy of the proposed solution were verified. Finally, a parametric analysis was performed to further explore the influence of load velocity and fractional order on the dynamic response of the viscoelastic media.

Electromechanical Vibration Response of Pre-stressed Bi-layered Piezoelectric Plate Under a Harmonic Mechanical Force

Abstract Purpose This study presents the outcomes of a finite element analysis (FEA) of forced vibrations by a time-harmonic loading of a bi-layered piezoelectric plate with two-axially pre-stressed layers. Methods The investigation is conducted based on the following assumptions: (i) the resultant system is resting on a rigid foundation, (ii) each layer is poled along the direction perpendicular to the free surface, (iii) a complete contact state exists at the interface of the plane in the plate, and (iv) the initial stress state at the layers is modeled based on the three-dimensional linearized theory of elasticity for solids under initial stress (TLTESIS). First, we describe nonlinear governing equations of motion and boundary-contact conditions for the dynamical model of the current system and then apply a linearization and non-dimensionalization procedure to the problem under consideration. In terms of Hamilton principle, a finite element model (FEM) is developed based on the weak form. Results and Conclusions The proposed and validated FEM approach can help to address several issues in the piezoelectric structure of finite lengths, either pre-stressed or not. In particular, we present an investigation of the effects of changing problem factors on the dynamic behavior as well as the frequency response of the composite plate. The numerical results demonstrate that the stress transition across the interface of the layers plays a key role in the resonance mode of the system, in both a quantitative sense and a qualitative sense.

Coupling dynamic response of saturated soil with anisotropic thermal conductivity under fractional order thermoelastic theory.

In this paper, a two-dimensional (2D) thermo-hydro-mechanical dynamic (THMD) coupling analysis in the presence of a half-space medium is studied using Ezzat's fractional order generalized theory of thermoelasticity. Using normal mode analysis (NMA), the influence of the anisotropy of the thermal conduction coefficient, fractional derivatives, and frequency on the THMD response of anisotropy, fully saturated, and poroelastic subgrade is then analyzed with a time-harmonic load including mechanical load and thermal source subjected to the surface. The general relationships among the dimensionless physical variables such as the vertical displacement, excess pore water pressure, vertical stress, and temperature distribution are graphically illustrated. The NMA method does not require the integration and inverse transformation, increases the decoupling speed, and eliminates the limitation of numerical inverse transformation. The obtained results can guide the geotechnical engineering construction according to different values of load frequency, fractional order coefficient, and anisotropy of thermal conduction coefficient. This improves the subgrade stability and enriches the theoretical studies on thermo-hydro-mechanical coupling.

Static bending and forced vibration analyses of a piezoelectric semiconductor cylindrical shell within first-order shear deformation theory

This paper examines the mechanically induced electric potential and charge redistribution in a piezoelectric semiconductor cylindrical shell employing first-order shear deformation theory. The two-dimensional governing equations and corresponding boundary conditions are simultaneously derived through a principle of virtual work method and the fundamental lemma of the calculus of variation. For the application of electronic devices, static bending and forced vibration analyses of an open cylindrical shell under locally distributed normal forces are analytically solved with the derived theoretical model. The effects of doping level on electric potentials and mechanical displacements are presented. An interesting result shows that doping levels can alter the peak position of the zeroth-order electric potential. In addition, the first three order natural frequencies of the shell under time-harmonic load are identified, and the doping level is found to have an inhibiting effect on the first natural frequency. All the numerical results are beneficial for optimizing the design and performance of cylindrical shell-shaped sensors and energy harvesters.

Study of a time-harmonic load on a Kirchhoff–Love plate with modified thermoelasticity theory using higher-order memory-dependent derivatives

Study of a time-harmonic load on a Kirchhoff–Love plate with modified thermoelasticity theory using higher-order memory-dependent derivatives

Elastodynamic response of an orthotropic layered elastic half-space to time-harmonic loading.

The elastodynamics of an orthotropic half-space coated by a thin orthotropic layer is theoretically investigated in this article. We newly propose explicit expressions of free Rayleigh waves in a layered half-space that are dependent on only one unknown constant representing amplitude. The main contribution is on deriving, in a simple manner, the theoretical predictions of far-field Rayleigh wave motion arising from time-harmonic loads using elastodynamic reciprocity theorems. These are the very first closed-form exact solutions found for the forced motion of Rayleigh waves in a layered half-space of orthotropic materials. To demonstrate the theoretical results, computation of Rayleigh wave motion in a jointed rock, including a layer of quartz-schist and a half-space of soil, is considered. We present the phase and group dispersion curves superimposed with the amplitude spectra that provide useful information on wave modes, frequencies, and displacement amplitudes. The inclusion of the amplitude spectra in the dispersion curves is a significant improvement over other dispersion curves currently available in the literature. The analytical predictions are compared with numerical results found by finite element analysis, and they show excellent agreement for the cases of a uniform distributed load and a varying distributed load both applied over a strip on the layer surface. The calculations obtained in the current study could generally be very useful for applications in seismology and materials characterization of coated structures.

Three-dimensional dynamic Green’s functions in exponentially graded transversely isotropic tri-material composites

This paper presents an analytical formulation for deriving the three-dimensional (3D) elastodynamic Green’s functions of functionally graded transversely isotropic tri-material composite under time-harmonic loading. With the aid of a complete set of displacement potentials, Fourier expansions, and Hankel integral transforms, displacement and stress components of 3D point-load, patch-load, and ring-load are obtained in the form of complex-plane infinite line-integrals. By virtue of a reliable and fast numerical scheme, i.e., the contour integration method, they are numerically treated, and its accuracy is achieved by comparison with some special cases. Finally, some numerical results are selected to demonstrate the influences of the material inhomogeneity and vibration frequency on the displacement and stress components. Particularly, the dependency of the stress transfer process at the interface of the mediums on the degree of inhomogeneity is presented, which is of high importance in evaluating the performance of composite materials.

Thermal analysis of thermal barrier coating systems under transient and time harmonic thermal loads

In this study, the temperature fields in a thermal barrier coating system (TBC) under various types of thermal loadings, including steady and transient thermal loads, is investigated. TBCs are exposed to a variety of harsh thermal loadings, including thermal shock and periodic thermal loads. Therefore, the need to achieve the temperature distribution in different positions of the TBCs, especially in the protective coating is very crucial. Non-uniform Thermal boundary conditions are considered for both external temperature and heat flux distributions. The coating and substrate are not stationary at the onset of transient thermal load. Using the solution of steady-state problem as initial condition, the transient temperature fields within the coating-substrate system is obtained. The dual-phase lag (DPL) heat conduction is utilized to scrutinize the transient response of the TBC system. Fourier and Laplace integral transforms have been used to solve the temperature field equations. The effect of concentrated thermal shock and steady harmonic thermal load on the temperature fields of coating and substrate is investigated. The results demonstrate that the steady state and transient temperature fields strongly depend on the type of thermal load, the thickness and thermal properties of the coating, and the loading parameters. The results indicate that the effect of concentrated thermal shock is felt after at least 8.33 s at distance 1.2 cm far from the top surface of ceramic coating. The effect of concentrated thermal shock is eminent adjacent to point (x=0,y=0) of the free boundary of coating, but after a longitudinal distance of 2 cm drastically reduced. The ceramic coating is able to relieve the amplitude of temperature oscillations up to 1280C at a vertical distance of 3mm from the free boundary of coating.

Mass minimization with conflicting dynamic constraints by topology optimization using sequential integer programming

In this paper mass minimization of hysteretically damped structures subjected to static and time-harmonic loading is studied via the Topology Optimization of Binary Structures (TOBS) method. Elements are removed or added to the finite element model of a structure in every iteration based on the solution to an integer linear program (ILP). The ILP is constructed from the sensitivity information of the objective function and the constraints which are in the form of the static and dynamic compliance. The proposed methodology is demonstrated on a 2D clamped–clamped beam and compared with published results for a 2D cantilever beam. The optimization starts from the full design domain and solutions with low mass that fulfill the constraints for a range of different bounds are found. The results also indicate that the mass is much more sensitive to changes in the static compliance constraint than in the dynamic compliance constraint. The effect of mass and upper bound of the constraints on the dynamic compliance at the fundamental resonance frequency is also studied, though no clear conclusions can be drawn. Finally the sensitivity information at the converged topology is studied and it is shown that the algorithm converges because the structural regions that are non-critical for the different constraints do not overlap. • The TOBS method is used to solve forced vibration problems. • Tightening the dynamic compliance constraint forces a change in topology. • At optimality, non-critical areas for different constraints lack overlap. • Minimum mass is more sensitive to static compliance constraints than dynamic.

Modeling the effect of the inclination angle on the dynamic response of a biaxially pre-stressed plate

In this study, I report on an investigation of the forced vibrations procured by an arbitrary angled time-harmonic loading from a plate based on a rigid foundation. The study was formulated according to the three-dimensional linearized theory of elasticity for solids under initial stress (TLTESIS). It was assumed throughout the investigation that there is a rigid clamped state between the system and the rigid ground; further, it was assumed that the plate was exposed to biaxially static initial stresses. Given this, a mathematical model was developed, and then solved using a three-dimensional finite element method (3D-FEM). Presented are numerical investigations that illustrate the influence of changes in the inclination of the force, as well as other important factors such as dimensionless frequency parameters, on the dynamic behavior of the system. In particular, the results indicate that the effect the initial stresses have on the dynamic stress distribution character increases with the aspect ratio but decreases with the thickness ratio.

Effect of memory dependent derivative and variable thermal conductivity in cantilever nano-Beam with forced transverse vibrations

This research work studies the effect of variable thermal conductivity on transversely isotropic thermoelastic (TIT) homogeneous nonlocal clamped-free beam. The beam undergoes forced vibrations due to two types of heat sources i.e. exponential time-varying and time-harmonic load. The mathematical model has been formed for a 2-D problem using a heat equation with two temperature (2T) involving memory-dependent derivatives (MDD) and Euler Bernoulli beam theory (EBBT). Laplace transforms are used to solve the problem. The dimensionless expressions for a thermal moment, axial stress, lateral deflection, and temperature change are calculated for these two types of forced vibrations. MATLAB software is used for programming and numerical data for cobalt nanobeam has been considered. The influence of variable thermal conductivity and kernel function of MDD has been depicted graphically on the physical quantities such as temperature change, lateral deflection, axial stress thermal moment, for the forced vibrations. Specific cases have also been mentioned.

Time-harmonic analysis of multiple interface cracks in two dissimilar FGM half-planes: In-plane problem

In this study, the dynamic stress intensity factors (DSIFs) at the tip of several cracks located at the interface between two half-planes made of functionally graded materials (FGMs) are calculated using the Distributed Dislocation technique (DDT). In this paper, the in-plane time-harmonic loading is considered and it is assumed that the properties of the FGMs follow the law of exponential function. First, the problem of two dissimilar half-planes that are weakened by an interface Volterra type climb and glide edge dislocation is solved using the Fourier transform, and then, using the DDT, singular integral equations with a singularity of the Cauchy type are extracted. To calculate the DSIFs at the cracks tips, the resulting singular integral equations are numerically discretized and the dislocation densities on the crack surfaces are obtained. Second, the graphs are presented in the numerical results section to show the effects of change in the parameters such as non-homogeneous constant, Poisson ratio, crack length, frequency and interaction between the cracks on DSIFs.

Analysis of dynamic response of two-dimensional orthotropic layered media with imperfect interfaces between layers

Although solving the dynamic response of layered media has always been a major concern in the engineering field, the problem of interlayer contact of layered media has seldom been addressed in the related studies. Thus, this paper proposes an algorithm to analyze the dynamic displacement and stress responses of two-dimensional layered media with imperfect interfaces between layers at an arbitrary point under time-harmonic loads. The Fourier transform was used to convert the dynamic equation of the generalized plane strain problem from the frequency-spatial domain to the frequency-wavenumber domain. In addition, combined with the introduction of dual variables, an integration algorithm with high precision was employed to solve the state equation. Based on the displacement response in the frequency-wavenumber domain, the dynamic displacement and stress responses at an arbitrary point were obtained using the inverse Fourier transform. This algorithm not only considers the transverse isotropic properties of layered media but also considers the contact problem at the interface between the adjacent layers of the layered media with different properties. In addition, the harmonic load can be applied to both the surface and inside of the medium. The accuracy of the proposed algorithm was verified by comparing the calculated data with the numerical results, and the influence of the imperfect interface parameters on the dynamic response of layered media was analyzed in detail. Finally, the mechanism of the impact of the imperfect interface between the layers on the dynamic displacement and stress responses of layered media was examined.

Fundamental solution of general time-harmonic loading over a transversely isotropic, elastic and layered half-space: An efficient and accurate approach

Time-harmonic loading over layered elastic half-spaces has applications in various science and engineering fields. While various approaches have been proposed in solving the related boundary-value problems, in this paper, we propose a new approach, which is based on the novel Fourier-Bessel series system of vector functions and the dual variable and position method (DVP). While the DVP method was proposed recently and verified to be computationally stable and efficient, the Fourier-Bessel series system of vector functions is newly introduced. Similar to the cylindrical system of vector functions, the normal (dilatational) and shear (torsional) deformations (waves) can be separated and solved in terms of the LM- and N-types of the new vector function system. The new formulation is coded, and the corresponding algorithm/program is applied to a couple of cases. It is shown that, by comparing previous approaches, this new series system of vector functions is equally accurate, but much more computationally powerful. Since it is substantially time saving in calculation, it is hopeful that this new approach would have broad applications related to transient response and inverse problems in elastodynamics of layered systems.

A 3D BEM-FEM approach using layered transversely isotropic half-space Green's functions in the frequency domain for SSI analyses

In this paper, a numerical approach based on a three-dimensional (3D) standard coupled boundary element method-finite element method (BEM-FEM) formulation in the frequency domain is presented. The approach allows studying the dynamics response of a structure bonded to the surface of a layered transversely isotropic half-space subjected to time-harmonic loading. The FEM is used to model the structure, and the BEM is employed for the soil, using layered half-space Green's functions, by which the radiation condition is implicitly satisfied. The BEM procedure is improved by implementing exponentially gradient (EG) elements to capture the singular behavior of tractions at the edges and corners of the soil-structure interface. The proposed approach is verified by comparison with existing semi-analytical and numerical results. Afterward, the dynamic response of a building on a layered transversely isotropic half-space subjected to lateral loading is studied, and the effect of soil layering on the response of the structure and interface tractions is demonstrated.

Elastodynamic Solutions of a Finite Soil Layer under Interior Distributed Actions

Through a method of displacement potentials, Fourier series, and Hankel integral transformation, the generalized solutions of an elastic layer resting on a rigid base under arbitrary, distributed, buried, and time-harmonic loads are developed in this study. With the proposed solution, the specific results for two kinds of uniform distributions as a kind of fundamental solutions in the interaction analysis of media and inclusions by the method of boundary integral equations are included as illustrations. Finally, numerical examples involving surface and buried patch loads are presented to validate the solutions and examine the effects of the thickness of the elastic layer. The results show that the proposed solution can cover the classical half-space solution by taking enough large thickness of the elastic layer (e.g., the ratio of the layer thickness beneath the load to the load radius ≥ 50 ) and the surface load solution by setting the load depth to zero; the underlying rigid base has significant and complex influence on the dynamic response of the thin layer due to wave reflections, which needs to be considered in the design and practice of related engineering.

Real Time Harmonic Load Identification Based on Fast Fourier Transform and Artificial Neural Network

Harmonics at nonlinear loads have different values for each load. These characteristics can be used as identification of the type of load from the harmonic value produced from nonlinear load. Therefore, this paper proposes harmonic load identification using Fast Fourier Transform (FFT) and Artificial Neural Network (ANN). In order to obtain harmonic values, a prototype of measuring instruments is used, while the method for obtaining harmonic values is FFT. The harmonic value obtained will be used as training data and testing data in ANN. Then, the type of training used as the classification of load type is the Levenberq Marquardt. The input of this method is a harmonic value of three types of nonlinear loads with seven combinations. The training process is carried out in MATLAB. Then, the weight and the bias values of each neuron are obtained and programmed in microcontroller. In order to validate the proposed algorithm, testing is conducted by three nonlinear load combinations. The results show that the proposed algorithm has good results in load identification with the highest accuracy of 99.94%.

Influence functions for a 3D full-space under bilinear stationary loads

This manuscript brings the derivation of influence functions for a three-dimensional full-space under bilinearly-distributed time-harmonic loads. The differential equations describing the medium are decomposed in terms of uncoupled vector fields. A double Fourier transform allows the system of equations to be solved algebraically in the transformed space, where the bilinear-loading boundary conditions are imposed. The resulting displacement and stress solutions are presented in terms of double improper integrals to be evaluated numerically. The manuscript brings selected results from the evaluation of these solutions. These influence functions can be used in boundary element models of elastodynamic problems to yield computationally-efficient solutions and improved representation of sharply-varying contact traction fields.

Dynamic Response of an Inhomogeneous Elastic Pile in a Multilayered Saturated Soil to Transient Torsional Load

In this paper, we solve the dynamic response of an inhomogeneous elastic pile embedded in a multilayered saturated soil and subjected to a transient torsional load via a semianalytical method. To portray the inhomogeneity of the pile and the stratification of surrounding soil, the pile-soil system is subdivided into Nth layers along the depth direction in view of the variation of shear modulus or cross-sectional dimension of the pile or differences in soil properties. Then, the vibration displacement solution with undermined constants for any saturated soil layer subjected to the time-harmonic torsional load is obtained by virtue of the separation of variables scheme. To establish the connection of adjacent longitudinal soil layers, the circumferential contact traction at the interface of the adjacent layers is treated as the distributed Winkler subgrade model independent of the radial distance. Then, by utilizing the continuity conditions of the pile-soil system and the method of recursion typically used in the transfer function technique, the torsional impedance of the pile top can be derived in the frequency domain. By virtue of inverse Fourier transform and convolution theorem, the velocity response of an inhomogeneous pile subjected to a transient half-sine exciting torque and embedded in a layered saturated soil is gained in the time domain. Finally, selected numerical results are gained to investigate the influence of typical defects in pile and soil layering on the velocity response of the pile top in the time domain.

In-plane stress analysis of multiple parallel cracks in an orthotropic FGM medium under time-harmonic loading

This study aimed to investigate the fracture behavior of a cracked orthotropic FGM medium when exposed to an in-plane harmonic load. The mechanical feature of the medium exponentially varied along the y-axis. Using the Fourier transform method, stress was analyzed by assessing Volterra edge dislocation in orthotropic FGM media. The obtained solution was then applied to derive Cauchy singular integral equations to analyze multiple cracks. The mentioned equations were numerically solved to determine dislocation density functions on the crack face which were then used to evaluate dynamic mixed stress intensity factors. The influence of crack interaction, non-homogeneity, and load frequency on the dynamic stress intensity was numerically illustrated through examples.