Time-harmonic Force Research Articles

Dynamic response of vertically loaded rectangular barrettes in multilayered viscoelastic soil

Rectangular barrettes have been increasingly used as foundations for many infrastructure projects, but the vertical vibration of a barrette has been rarely addressed theoretically. This paper presents an analysis method of dynamic response for a rectangular barrette subjected to a time-harmonic vertical force with the aid of a modified Vlasov foundation model in multilayered viscoelastic soil. The barrette-soil system is modeled as a continuum, the vertical continuous displacement model for the barrette and soil is proposed. The governing equations of the barrette-soil system and the boundary conditions are obtained and the vertical shaft resistance of barrette is established by employing Hamilton's principle for the system and thin layer element, respectively. The physical meaning of the governing equations and shaft resistance is interpreted. The iterative solution algorithm flow is proposed to obtain the dynamic response of barrette. Good agreement of the analysis and comparison confirms the correctness of the present solution. A parametric study is further used to demonstrate the effects of cross section aspect ratio of barrettes, depth of soil column, and module ratio of substratum to the upper soil layers on the complex barrette-head stiffness and the resistance stiffness.

Uncertainties in the attenuation performance of a multi-frequency metastructure from additive manufacturing

Additive manufacturing has been used to propose several designs of phononic crystals and metamaterials due to the low cost to produce complex geometrical features. However, like any other manufacturing process, it can introduce material and geometrical variability in the nominal design and therefore affect the structural dynamic performance. Locally resonant metamaterials are typically designed such that the distributed resonators have the same natural frequency or, in the case of rainbow metastructures, a well-defined spatial profile. In this work, the effects of the break of periodicity caused by additive manufacturing variability on the attenuation performance of a multi-frequency metastructure is investigated. First, an experimental investigation on the manufacturing tolerances of test samples produced from a Selective Laser Sintering process are assessed and variability levels are used to propose a random field model for the metastructure. Subsequently, the stochastic model is used to investigate the vibration suppression performance of broadband multi-frequency metastructures. An analytical model based on a transfer matrix approach is used to calculate transfer receptance due to a point time harmonic force in a finite length metastructure, which is composed of evenly spaced non-symmetric resonators attached to a beam with Π-shaped cross-section. This design creates a multi-frequency metastructure, i.e. band gaps in more than one frequency band. Individual samples of the random fields are used to show that the mistuned resonators can change the vibration attenuation performance of the metastructure and that even small levels of variability, given by less than 1% for the mass and less than 3% for the Young’s modulus can have a significant effect on the overall vibration attenuation performance of the metastructure when considered together. It is also shown that different spatial profiles can have a significant effect on the vibration attenuation performance in both band gaps. Therefore, the modelling of the uncertainty metastructures has to take into account the spatial correlation of the properties of the metastructure resonators. The obtained results are expected to be useful for further robust design in mass produced industrial applications.

Brushless wound rotor synchronous machine with third-harmonic field excitation

This manuscript discusses a novel harmonic current excitation technique for a wound rotor synchronous machine based on control of the time harmonic magneto-motive force (MMF) for brushless operation. Two inverters are used to feed the armature winding in such a way that at a time, one inverter supplies the current to armature winding. The two inverters are 180° phase-shifted. The inverter assignment is based on time-division multiplexing technique to generate continuous armature currents containing reasonable third-harmonic component beside fundamental component. This third-harmonic current component generates pulsating MMF and is induced in the rotor harmonic winding; the induced current is rectified and applied to rotor field winding to generate rotor field. The rotor field interacts with armature fundamental current component for torque generation. Finite element analysis is performed to verify the idea.

Dynamic response of a bi-axially pre-stressed bi-layered plate resting on a rigid foundation under a harmonic force

This study aims to investigate the forced vibrations caused by a time-harmonic force from a pre-stressed bi-layered plate resting on a rigid foundation under the action of a time-harmonic pointwise loading. Our investigation was conducted according to a piecewise homogeneous body model utilizing the three-dimensional linearized theory of elastic waves in initially stressed bodies. Throughout this study, we assumed that there is complete contact between the plate and the rigid foundation. The purpose of this study is threefold: the development of a mathematical model to investigate the dynamic response of the pre-stressed bi-layered plate, the analysis of the frequency response of the plate under consideration, and finally, demonstrating the relationship between the initial stress and the dimensionless frequency of the plate. We solved the mathematical model by employing the finite element method. We present our numerical results on the dynamic behavior of the plate. In particular, we have shown that an increase in the values of the aspect ratio of a plate under fixed thickness leads to a decrease in the normal stress resonance values.

Micro-beam resonator parametrically excited by electro-thermal Joule’s heating and its use as a flow sensor

We study parametric resonance (PR) of double-clamped micro-beams that are electro-thermally actuated by a time-dependent Joule’s heating and cooled by a steady air flow. The developed model demonstrates applicability of such device as a bifurcation-based flow velocity sensor. An AC electric current through the beam induces a time-harmonic compressive force that leads to parametric excitation of the structure. Convective cooling due to the air flow affects the location of the parametric transition curves on the driving voltage–frequency plane. The flow velocity can be obtained by measuring the frequency corresponding to the steep amplitude transition of the response. The device is modeled as an Euler–Bernoulli beam with an axial force parameterized by the electric current. The heat transfer problem is solved analytically; the heat flux due to the air flow is calculated using empirical correlations. The behavior of the beam is studied numerically, by means of finite differences, and analytically, using an approximate single degree of freedom Galerkin model, reduced to the Mathieu–Duffing equation. We show that while the PR always emerges at the driving voltage/current below the critical static buckling value, practical realization of the purely electro-thermal parametric excitation is challenging and is highly influenced by the device dimensions and quality factors. We evaluate the parameters required to assure the PR and demonstrate, using the model, feasibility of the suggested flow-sensing approach in the devices of realistic dimensions.

A Mathematical Model for Forced Vibration of Pre-Stressed Piezoelectric Plate-Strip Resting On Rigid Foundation

A mathematical model to investigate the dynamic response of a piezoelectric plate-strip with initial stress under the action of a time-harmonic force resting on a rigid foundation is presented within the scope of the three-dimensional linearized theory of electro-elasticity waves in initially stressed bodies (TLTEEWISB). The governing system of equations of motion is solved by employing the Finite Element Method (FEM). The numerical results illustrating the dependencies of different problem parameters are investigated. In particular, the influence of a change in the value of the initial stress parameter on the dynamic response of the plate-strip is discussed.

Extensional vibration characteristics and screening of polarization charges in a ZnO piezoelectric semiconductor nanofiber

We study extensional vibrations of a ZnO piezoelectric semiconductor fiber driven by a time-harmonic end force. A theoretical analysis is performed using a one-dimensional model based on the phenomenological theory of piezoelectric semiconductors consisting of the equations of piezoelectricity coupled to the continuity equation of electrons. A series of resonance frequencies are identified. It is found that these frequencies are lower than those of the corresponding piezoelectric dielectric fiber because the mobile charges in the semiconductor screen the effective polarization charges and thus weaken the piezoelectric stiffening effect. Various electromechanical fields at resonances are calculated and examined. Numerical results show that the effective surface polarization charge plays a more dominant role than the effective body polarization charge. The mobile charges can screen the body polarization charges greatly, but not the surface polarization charges.

Dynamic Green's functions for multiple circular inclusions with imperfect interfaces using the collocation multipole method

This paper presents a semi-analytical approach to solve anti-plane dynamic Green's functions for an elastic infinitely extended isotropic solid (matrix) containing multiple circular inclusions with imperfect interfaces. A linear spring model with vanishing thickness is employed to character the imperfect interface. The multipole expansions of anti-plane displacement of the matrix and inclusion, induced by a time-harmonic anti-plane line force located in the matrix or in the inclusion, are expanded by using Hankel and Bessel functions, respectively. The imperfect interface condition is satisfied by uniformly collocating points along the interface of each inclusion. For the imperfect interface condition, the normal derivative of the anti-plane displacement with respect to a non-local polar coordinate system is developed without any truncation error for multiply-connected domain problems. For the case of one circular inclusion, the proposed quasi-static stress field matches well with the analytical static solution. The proposed quasi-static stress fields containing two and three circular inclusions are critically compared with those calculated by static analysis using the finite element method. Finally, extensive studies are presented to investigate the effects of the frequency of excitation, imperfect interface and separation between inclusions on the dynamic Green's functions.

The influence of the initial strains of the highly elastic plate on the forced vibration of the hydro-elastic system consisting of this plate, compressible viscous fluid, and rigid wall

The hydro-elastic system consisting of a pre-stretched highly elastic plate, compressible Newtonian viscous fluid, and the rigid wall is considered and it is assumed that on the plate a lineal-located time-harmonic force acts. It is required to investigate the dynamic behavior of this system and determine how the problem parameters and especially the pre-straining of the plate acts on this behavior. The elasticity relations of the plate are described through the harmonic potential and linearized (with respect to perturbations caused by external time-harmonic force) form of these relations is used in the present investigation. The plane-strain state in the plate is considered and the motion of that is described within the scope of the three-dimensional linearized equations of elastic waves in elastic bodies with initial stresses. The motion of the fluid is described by the linearized Navier-Stokes equations and it is considered the plane-parallel flow of this fluid. The Fourier transform with respect to the space coordinate is applied for a solution to the corresponding boundary-value problem. Numerical results on the frequency response of the interface normal stress and normal velocity and the influence of the initial stretching of the plate on this response are presented and discussed. In particular, it is established that the initial stretching of the plate can decrease significantly the absolute values of the aforementioned quantities.

Effect of imperfect bonding on the dynamic response of a pre-stressed sandwich plate-strip with elastic layers and a piezoelectric core

In this paper, we consider the dynamic response of a pre-stressed sandwich plate-strip with a piezoelectric core and elastic layers under the action of a time-harmonic force resting on a rigid foundation. The investigation is carried out within the framework of the piecewise homogeneous body model by utilizing the exact equations of motion and relations of the linear theory of electro-elasticity. It is assumed that there is a shear-spring-type imperfect contact between the layers, but a complete contact between the plate-strip and the rigid foundation. A mathematical model of the problem is constructed, and the governing equations of motion are solved by employing the finite element method (FEM). Numerical results illustrating the influence of a change in the value of the shear-spring parameter on the dynamic response of the plate-strip are then presented.

Electric potential and carrier distribution in a piezoelectric semiconductor nanowire in time-harmonic bending vibration

We study a ZnO piezoelectric semiconductor nanowire in bending vibration with shear deformation for energy harvesting application. The wire is a cantilever fixed at one end and is driven by a time-harmonic transverse shear force at the other end. A theoretical analysis is performed using one-dimensional equations based on the phenomenological theory of piezoelectric semiconductors consisting of the momentum equation, the charge equation of electrostatics, and the conservation of charge for holes and electrons. An analytical solution is obtained. The distributions of the electric potential and carrier concentration near resonances are calculated. The fields at the first resonance are qualitatively similar to the static fields. At the second and higher resonances the fields reverse their directions when crossing the nodal points of the vibration modes. The results obtained are fundamental to the development and optimization of energy harvesters and other devices based on the bending vibration of ZnO nanowires.

Thermoelastic damping in bilayered microbar resonators with circular cross-section

It is always a challenge to determine the Thermoelastic damping (TED) in bilayered microbars precisely. In this paper, a model for TED in the bilayered and cantilevered microbar was proposed, in which the total damping was derived by calculating the energy evanished in each layer. The distribution of temperature in the bilayered microbar with a thermodynamically ideal boundary receiving a time-harmonic force is obtained. An infinite summation for the computing of TED in the bilayered slender microbars under axial loading is presented, and the convergence rate of it is discussed. There are little differences between the results computed by our model and that by finite element method (FEM).

Effect of Initial Stress on the Dynamic Response of a Multi-Layered Plate-Strip Subjected to an Arbitrary Inclined Time-Harmonic Force

The forced vibration of a multi-layered plate-strip with initial stress under the action of an arbitrary inclined time-harmonic force resting on a rigid foundation is considered. Within the framework of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB), a mathematical modelling is presented in plane strain state. It is assumed that there exists the complete contact interaction at the interface between the layers and the materials of the layer are linearly elastic, homogeneous and isotropic. The governing system of the partial differential equations of motion for the considered problem is solved approximately by employing the Finite Element Method (FEM). Further, the influence of the initial stress parameter on the dynamic response of the plate-strip is presented.

Coupled BE-FE Scheme for Three-Dimensional Dynamic Interaction of a Transversely Isotropic Half-Space with a Flexible Structure

The response of structures bonded to the surface of a transversely isotropic half-space (TIHS) under the effect of time-harmonic forces is investigated using a coupled FE-BE scheme. To achieve this end, a Finite Element program has been developed for frequency domain analysis of 3D structures, as the first step. The half-space underlying the structure is taken into consideration using a Boundary Element technique that incorporates half-space surface load Green’s functions for a transversely isotropic medium. Next, the two programs are combined using a direct coupling algorithm and the final program is obtained. To validate the results, some benchmark problems are solved with the FE and the BE programs, separately and then the coupled technique is checked with the results of some special cases for which the solutions are available in the literature. At the end, a parametric study is carried out on several common types of structures to study the effects of the degree of anisotropy of transversely isotropic soil medium on the dynamic behavior of the structure. Moreover, the effect of soil-structure interaction (SSI) on the natural vibration frequency of the structures is also studied.

Forced vibration of the elastic system consisting of the hollow cylinder and surrounding elastic medium under perfect and imperfect contact

The bi-material elastic system consisting of the circular hollow cylinder and the infinite elastic medium surrounding this cylinder is considered and it is assumed that on the inner free face of the cylinder a point-located axisymmetric time harmonic force, with respect to the cylinder\'s axis and which is uniformly distributed in the circumferential direction, acts. The shear-spring type imperfect contact conditions on the interface between the constituents are satisfied. The mathematical formulation of the problem is made within the scope of the exact equations of linear elastodynamics. The focus is on the frequency-response of the interface normal and shear stresses and the influence of the problem parameters, such as the ratio of modulus of elasticity, the ratio of the cylinder thickness to the cylinder radius, and the shear-spring type parameter which characterizes the degree of the contact imperfectness, on these responses. Corresponding numerical results are presented and discussed. In particular, it is established that the character of the influence of the contact imperfection on the frequency response of the interface stresses depends on the values of the vibration frequency of the external forces.

Dynamic response of a pre-stressed bi-layered plate-strip subjected to an arbitrary inclined time-harmonic force

Within the scope of the piecewise homogeneous body model with utilizing of the three dimensional linearized theory of elastic waves in initially stressed bodies the dynamical stress field problem in a bi-layered plate-strip with initial stress under the action of an arbitrary inclined timeharmonic force resting on a rigid foundation is investigated. The concrete materials such as a pair of Aluminum and Steel are selected. It is assumed that there exists a complete contact interaction between the layers. The mathematical modeling of the problem under consideration is carved out, and the governing system of the partial differential equations of motion is approximately solved by employing Finite Element Method. The numerical results related to the influence of certain parameters on the dynamic response of the plate-strip are presented.

Anti-plane time-harmonic Green’s functions for a coated circular inhomogeneity in a piezoelectric medium with spring- or membrane-type imperfect interfaces

Two-dimensional anti-plane time-harmonic dynamic Green’s functions for a coated circular inhomogeneity in an infinitely extended matrix with spring- or membrane-type imperfect interfaces are presented. The inhomogeneity, coating and matrix are all assumed to be piezoelectric and transversely isotropic. By using the Bessel function expansions, explicit solutions for the electromechanical fields induced by a time-harmonic anti-plane line force and line charge located in the unbounded matrix, the annular coating and the circular inhomogeneity are derived. The present solutions can recover the anti-plane Green’s functions for some special cases, such as the dynamic or quasi-static Green’s functions of piezoelectricity with perfect interfaces, as well as the dynamic or quasi-static Green’s functions for a two-phase composite with perfect or imperfect interfaces. By means of detailed discussions, selected calculated results are graphically shown to demonstrate the dependence of the electromechanical fields on the circular frequency and the interface properties as well as the coating and size of the inclusion.

A self-consistent formulation for the sensitivity analysis of finite-amplitude vortex shedding in the cylinder wake

We use the adjoint method to compute sensitivity maps for the limit-cycle frequency and amplitude of the Bénard–von Kármán vortex street in the wake of a circular cylinder. The sensitivity analysis is performed in the frame of the semi-linear self-consistent model recently introduced by Mantič et al. (Phys. Rev. Lett., vol. 113, 2014, 084501), which allows us to describe accurately the effect of the control on the mean flow, but also on the finite-amplitude fluctuation that couples back nonlinearly onto the mean flow via the formation of Reynolds stress. The sensitivity is computed with respect to arbitrary steady and synchronous time-harmonic body forces. For a small amplitude of the control, the theoretical variations of the limit-cycle frequency predict well those of the controlled flow, as obtained from either self-consistent modelling or direct numerical simulation of the Navier–Stokes equations. This is not the case if the variations are computed in the simpler mean flow approach overlooking the coupling between the mean and fluctuating components of the flow perturbation induced by the control. The variations of the limit-cycle amplitude (that falls out the scope of the mean flow approach) are also correctly predicted, meaning that the approach can serve as a relevant and systematic guideline to control strongly unstable flows exhibiting non-small, finite amplitudes of oscillation. As an illustration, we apply the method to control by means of a small secondary control cylinder and discuss the obtained results in the light of the seminal experiments of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, pp. 71–107).

Analyzing excitation forces acting on a plate based on measured acoustic pressure.

This paper presents a theoretical study on "seeing" through an elastic structure to uncover the root cause of sound and vibration by using nearfield acoustical holography (NAH) and normal modes expansion. This approach is of generality because vibro-acoustic responses on the surface of a vibrating structure can always be reconstructed, exactly or approximately. With these vibro-acoustic responses, excitation forces acting on the structure can always be determined, analytically or numerically, given any set of boundary conditions. As an example, the explicit formulations for reconstructing time-harmonic excitation forces, including point, line and surface forces, and their arbitrary combinations acting on a rectangular thin plate in vacuum mounted on an infinite baffle are presented. The reason for choosing this example is that the analytic solutions to vibro-acoustic responses are available, and in-depth analyses of results are possible. Results demonstrate that this approach allows one to identify excitation forces based on measured acoustic pressures and reveal their characteristics such as locations, types and amplitudes, as if one could "see" excitation forces acting behind the plate based on acoustic pressure measured on the opposite side. This approach is extendable to general elastic structures, except that in such circumstance numerical results must be sought.

Inverse identification of time-harmonic loads acting on thin plates using approximated Green’s functions

In this paper, a non-iterative technique is proposed for the transverse load identification on Kirchhoff plates using approximate Green’s functions (AGFs). In this way, we firstly employ the recently introduced meshless method to construct the AGFs, as the combination of a series of Trefftz basis, i.e. Exponential basis functions (EBFs), and the fundamental solutions of the governing equation. As will be explained, using a proper set of EBFs, as well as a collocation technique, enables us to construct the AGFs for different types of domain shape and boundary conditions. In the second step, a set of artificially generated results, in the absence of realistic experimental results, are used to express the plate’s response field, i.e. deflection or velocity fields, as a series of AGFs through a collocation technique. It will be shown how the constant coefficients of the response series are related to the intensity of the reconstructed force at a set of selected points. The proposed method is capable of constructing both distributed and concentrated loads with desirable accuracy. This ability is shown in the solution of three sample problems of the static and time-harmonic force recovery.