Harmonic Point Force Research Articles

Fundamental study on the difference between materials and vibration conditions in piezoelectric vibration power generation using electromechanical coupling phenomena

Conventionally, vibration energy and sound energy are emitted as noise and vibration. We have proposed a system that converts the vibration of the plate into energy. In this study, a circular plate of constant thickness, on which a piezoelectric element was installed, was adopted as the host structure of an energy-harvesting system. The plate was supported at its elastic end, which was intermediate between simple and clamped supports, and was subjected to a harmonic point force. The piezoelectric performance of the system was highly dependent on the vibration characteristics of the plate. Regarded as one of the representative vibrational characteristics, the natural frequency of the plate was affected by the materials, excitation positions, and vibration modes of the plate. Therefore, the plate materials varied the natural frequency, thus affecting the conversion of vibration energy into electrical energy through the piezoelectric effect. In this study, we investigated plates made of selected materials commonly used in industry and tested different excitation conditions. As the result, we clarified that the proposed system is almost unaffected by the material of the plates, and it was shown to be useful.

Vibro-acoustic response and sound transmission loss of sandwich plates with tunable Poisson's ratio core in a hygrothermal environment

In this study, vibro-acoustic response and sound transmission loss of sandwich plates with tunable Poisson's ratio (TPR) core in a hygrothermal environment are investigated. The sandwich plate is taken as a moderately thick plate and various plate theories are used to examine the vibroacoustic behavior. First, the equations of motion of a sandwich plate in the hygrothermal environment are derived based on Hamilton's principle. Then, the solution to free vibration for a simply supported sandwich plate is presented. Next, we obtain the sound power level (SPL) for the point harmonic force exerted on the surface of the sandwich plate using the Rayleigh integral and the sound transmission loss (STL) as the harmonic sound wave is incident on the plate surface. Finally, numerical examples are conducted and discussed. The effects of the core-to-thickness ratio, plate theories, parameters of the TPR core, hygrothermal environment, and incident elevation angle on the SPL and STL of the sandwich plate are analyzed. This study will contribute to the design and manufacturing of lightweight structures in the aviation industry.

Steady state responses of an infinite beam resting on a tensionless visco-elastic foundation under a harmonic moving load

In this paper we study the steady state responses of an infinite beam resting on a tensionless visco-elastic foundation under a harmonic point force moving with a constant speed. The equation of motion is formulated with respect to a coordinate system moving with the point load. The beam and the foundation are defined as in contact when both the deflection and the “contact force” are negative. We replace the infinite beam by a finite beam with sufficient length and expand the beam deflection into a harmonic series. Newmark method is used to integrate the discretized equation. For a tensionless foundation, the contact condition between the beam and the foundation changes with time. Therefore, the matrices involved in the discretized equation of motion must be updated at every time step. In the bifurcation diagram we change the excitation frequency and record the beam deflection at the loading point via Poincaré sampling. When the dimensionless amplitude of the harmonic force is high, say 15, one can observe periodic, quasi-periodic, and chaotic solutions. We have also found that reducing the amplitude of the harmonic component or adding a time-invariant component can simplify the bifurcation diagram to include only periodic solutions.

Vibration Control of Functionally Graded Panels using Parallel Resonators

Functionally graded materials (FGM) are often an integral part these days in many engineering applications, such as, nuclear structural components, spacecraft and marine structures, thermal barrier coatings used for military applications, etc. These structures are also susceptible to dynamic loads varying from harmonic to impulse type of loadings which are in the form of rotating engines, sudden blasts and others. These loadings often pose serious threats to the structural systems by inflicting fatigue damages or by driving the system in tune with its resonating frequency that eventually lead to the complete collapse of the structure. Therefore, a vibration control strategy needs to be devised to protect these structures from unwanted vibrations due to the external loading. A passive vibration control strategy is proposed in the present research work to control the vibration response of a flat panel made of functionally graded material. At first, the FG plate is numerically modelled using the finite element (FE) method to calculate its response due to a point harmonic force. Ceramic (Alumina) is used for the top part of the FG plate while the bottom is made of metal (Aluminium) and the material property is smoothly varied from ceramic to metal using the power law distribution. Then, several resonators consisting of spring-mass system and parallel to each other are attached to both sides of the panel to isolate the response in the resonating frequency ranges. The FE model for the FG plate with resonator is developed and the controlled vibration response is obtained. The controlled response indicates that the resonators are efficient to produce band-gaps in the resonating frequency regime compared to the bare FG plate.

Robust optimised design of 3D printed elastic metastructures: A trade-off between complexity and vibration attenuation

In this work, a strategy for optimal design of mechanical metastructure is proposed taken into account uncertainties arising from additive manufacturing. A locally resonant Π-shaped beam with parallel plate-like insertions and two cantilever mass resonators at each unit cell is manufactured through a selective laser sintering process. The variability of the material properties introduced by the additive manufacturing procedure is experimentally obtained. Given that such manufacturing approaches are predominantly employed for producing complex metastructure architectures, it can significantly compromise the optimality of the design. A transfer matrix approach is employed to propagate variability at a structural level and predict the structural receptance due to a point harmonic force in the finite length metastructure. Then, the mass ratio of the metastructure is optimised for maximising vibration attenuation considering different numbers of added resonators and relative masses. A cost function is introduced in the classical robust design approach in order to favour designs with least complexity, represented by the number of added resonators. It is exhibited in several cases that the robustly optimal design is away from the deterministic optimal one, emphasising the relevance of the proposed approach in the optimisation of complex and locally resonant structures. Moreover, it is shown that the frequency range of interest plays a major role on the derived optimal design for each number of implemented resonators. The presented results show that even small variability in the Young’s modulus of up to 3% and in the mass density of up to 1% can still affect the robustness of the optimal design for locally resonant metastructure as due to the consequent mistuning of the added resonators.

Reconstruction of vibratory field and structural intensity of vibrating plates using moving sensors

This paper investigates the reconstruction of the vibratory field and structural intensity of transversely vibrating plates using a single moving sensor. Two kinds of sensors are considered; a moving microphone traversing an arbitrary path and a laser vibrometer scanning an arbitrary measurement line. The corresponding measurement time histories are used to reconstruct the dynamic displacement field. The plate displacement is approximated by a series of appropriate functions with unknown coefficients. The field is reconstructed using pseudoinverse procedure and structural intensity diagrams are extracted to identify energy sources, sinks, and streamlines. To investigate the validity of the proposed method, numerically simulated time histories along with experimental data are employed. In the simulation part, as a case study, a rectangular plate excited by a harmonic point force and attached to a dashpot is considered. Then, the simulated measurement data are extracted using the structural model. Next, adding random noise to the simulated measurement data, different numerical simulations are conducted and the effects of sensor traverse path, approximation series length, excitation frequency, sampling rate, noise level, and sensor speed are studied. In the experimental part, sound pressure radiated by a vibrating rectangular plate is measured on two arbitrary traverse paths using a microphone. The measured data are employed to reconstruct the vibratory field. The presented results confirm the potentials of the proposed method to reconstruct single-harmony vibratory fields. Since the proposed method works with different arbitrary paths, it allows the application of manually moving devices to perform vibrational measurements which can provide great practical benefits.

Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity

In many practical engineering situations, a source of vibrations may excite a large and flexible structure such as a ship’s deck, an aeroplane fuselage, a satellite antenna, a wall panel. To avoid transmission of the vibration and structure-borne sound, radial or polar periodicity may be used. In these cases, numerical approaches to study free and forced wave propagation close to the excitation source in polar coordinates are desirable. This is the paper’s aim, where a numerical method based on Floquet-theory and the FE discretision of a finite slice of the radial periodic structure is presented and verified. Only a small slice of the structure is analysed, which is approximated using piecewise Cartesian segments. Wave characteristics in each segment are obtained by the theory of wave propagation in periodic Cartesian structures and Finite Element analysis, while wave amplitude change due to the changes in the geometry of the slice is accommodated in the model assuming that the energy flow through the segments is the same. Forced response of the structure is then evaluated in the wave domain. Results are verified for an infinite isotropic thin plate excited by a point harmonic force. A plate with a periodic radial change of thickness is then studied. Free waves propagation are shown, and the forced response in the nearfield is evaluated, showing the validity of the method and the computational advantage compared to FE harmonic analysis for infinite structures.

Effect of acoustic radiation from circular plate installed with piezoelectric element on electricity generation performance

This study examined the electricity generation performance of an energy-harvesting system, in which a circular plate with a piezoelectric element was excited by a harmonic point force. The theoretical electricity generation efficiency, which does not involve acoustic radiation from the plate, is greater than the experimental efficiency in such a power generation system, and the discrepancy between the theoretical and experimental efficiencies increases with the plate thickness. A sound field is formed in the atmosphere surrounding the plate, whose vibration compresses the medium on the plate surface and causes a reaction force to the plate. In such a system, it is difficult to completely avoid the effect of acoustic radiation; therefore, the abovementioned effect must be examined to estimate the electricity generation characteristics of the system. The generation characteristics are estimated not only by the radiation resistance that is involved in the acoustic radiation, but also by the reaction force caused by the sound pressure on the plate. Consequently, the power reduced by the reflection of the reaction force and sound pressure decreased because of the increasing flexural rigidity of the plate when the plate thickness increased while the radius remained constant. However, because the radiation resistance increases with the plate thickness, the acoustic radiation energy, except for the work exerted by the sound pressure, is emitted to the atmosphere. To utilise such acoustic radiation energy, a new electricity generation system comprising a steel cylinder with two plates at both ends was used, where a sound field was formed within the cylinder. The system performance was estimated in terms of the electricity generation efficiency by comparing it with that of the abovementioned plate system. As a result, although the thin plate was preferable to increase the efficiency because of the weakened acoustic radiation, the thick plate was useful to harvest vibration energy, which was unused normally, because of enabling to apply mechanical–acoustic coupling between the plate vibrations and the internal sound field.

Plane wave finite element model for the 2-D phononic crystal under force loadings

Direct numerical simulations of the dynamic response of the phononic crystal to point or distributed forces via conventional numerical methods are still formidable currently due to large number of DOFs involved in the simulations. In this study, based on the plane wave expansion (PWE) method, a plane wave finite element (PWFE) model for the 2-D phononic crystal which is infinite in the horizontal direction and finite in the vertical direction and subjected to a point harmonic force is proposed. To this end, the point harmonic force is decomposed into the wavenumber domain (WND) components by the Fourier transform first. The variables of the 2-D phononic crystal under the WND force component are expanded into a series of plane waves along the horizontal direction, and the variables associated with each plane wave are discretized by the FE method in the vertical direction. By using the virtual work principle, FE equations for the plane waves of the phononic crystal are established. With the FE equations for the plane waves, the eigenvalue equation for the displacement of the plane waves and wavenumber is formulated. Superposing the eigenvectors of the above eigenvalue equation yields the displacement for the plane waves due to the WND force component. Synthetization of the contributions from all the plane waves and inversion of the Fourier transform with respect to the wavenumber via the contour integration method generate the response of the phononic crystal to the point force. Comparison of present results with those due to the transfer matrix method validates the proposed model. Also, presented numerical results suggest that the dynamic response of the phononic crystal to a point harmonic force exhibits remarkable resonance phenomenon.

Wave transmission and vibration response in periodically stiffened plates using a free wave approach.

There are various structures constructed with periodically stiffened thin plates. Vibration prediction of such structures is not easy compared to the structures comprised of uniform plates only due to the mathematical complexity stemmed from the periodic nature. This study provides the analytic method to predict the wave transmission at junctions connecting two semi-infinite periodic structures and the response of a finite periodic structure to an external harmonic point force. The same theoretical framework is employed for dealing with both phenomena. First, free wave solutions are obtained by solving the governing equation for the bending motion of a periodically stiffened, infinite plate using the spatial Fourier Transform and the Floquet's theorem. Then, the free wave solutions are linearly superposed, and the linear coefficients are calculated by applying the appropriate boundary conditions. Numerical simulation is conducted. In dealing with the periodic finite structure, the result is compared with that by the finite element analysis. It is revealed that the periodic nature of the structures affects both the energy transmission and the vibration response of the periodically stiffened plates.

Forced vibration of a bi-axially pre-stressed plate subjected to a harmonic point force and resting on a rigid foundation

This paper models the harmonically forced vibration of a bi-axially pre-stressed plate resting on a rigid foundation using the three-dimensional linearized theory of elastic waves in initially stressed bodies. This model assumes that the material is linearly elastic, homogenous, and isotropic. The model of this system is numerically evaluated using the finite element method simulations. The numerical results illustrate the influence of several system parameters on the dynamic responses of the stresses acting at the interface between the plate and its rigid foundation. The simulations show that the aspect ratio of the plate determines how the initial stress influences the behavior of the system.

Sound radiation from a finite perforated panel set in a rigid baffle: A fully coupled analysis

A fully coupled formulation is presented for sound radiation from a finite flexible perforated panel set in an infinite rigid unperforated baffle. The panel is excited by a point harmonic force and radiates sound into the acoustic medium on both sides of the baffle. Prior to coupling with the fluid, the change in the inertial and stiffness properties of the panel due to perforations and their effect on the in vacuo natural frequencies and mode shapes are accounted for using the receptance method. The coupled acoustic and structural equations are formulated using the 2-D wavenumber transforms. Since the formulation is in terms of the panel velocity, a certain square root singularity that typically appears in the denominator is avoided. The self- and inter-modal coupling coefficients are evaluated either analytically or numerically. The radiated power, panel mean quadratic velocity and radiation efficiency plots are obtained for different perforation ratios for both center and off-center excitations. At first, our formulation is validated against an unperforated case using a study available in the literature. Next, for our perforated panel case, we assume that both the half-spaces are filled with the acoustic fluid. It is observed that the radiation efficiency decreases with the increase in the perforation ratio, irrespective of the surrounding acoustic medium. However, for the water-loaded panel, the radiation efficiency is found to be lower than when the panel is immersed in air. It is found that for a light fluid like air, a one-way coupled formulation is adequate. The natural frequencies of a water-loaded perforated panel are also presented. It is found that for a given mode, the natural frequency increases with increase in the perforation ratio.

Mindlin 판 이론을 적용한 단순지지 단면 접수평판의 음향방사효율 수치해석

Acoustic radiation efficiency is a crucial factor to estimate Underwater Radiated Noise (URN) of ships accurately. This paper describes a numerical method to analyse acoustic radiation efficiency of one side-wetted rectangular Mindlin plate with simply supported boundaries excited by a harmonic point force. Transverse displacements of plate and acoustic radiation pressures are evaluated by the mode superposition method. The acoustic radiation efficiencies analyzed by both Mindlin and thin plate theories show little differences at monopole and corner modes of low frequency regions but relatively large differences at edge and critical modes of high frequency regions. Especially, the critical frequency with the highest acoustic radiation efficiency evaluated by the Mindlin plate theory is higher than that of thin plate theory. In addition, the acoustic loading effect of fluid also increases bending wave-number of plate and its critical frequency. Finally, the acoustic radiation characteristics of plates with different aspect ratios and thicknesses through numerical analyses are investigated and discussed.

Energy flow finite element analysis of general Mindlin plate structures coupled at arbitrary angles

Energy Flow Finite Element Analysis (EFFEA) is a promising tool for predicting dynamic energetics of complicated structures at high frequencies. In this paper, the Energy Flow Finite Element (EFFE) formulation of complicated Mindlin plates was newly developed to improve the accuracy of prediction of the dynamic characteristics in the high frequency. Wave transmission analysis was performed for all waves in complicated Mindlin plates. Advanced Energy Flow Analysis System (AEFAS), an exclusive EFFEA software, was implemented using MATLAB®. To verify the general power transfer relationship derived, wave transmission analysis of coupled semi-infinite Mindlin plates was performed. For numerical verification of EFFE formulation derived and EFFEA software developed, numerical analyses were performed for various cases where coupled Mindlin plates were excited by a harmonic point force. Energy flow finite element solutions for coupled Mindlin plates were compared with the energy flow solutions in the various conditions.

An examination of the concept of driving point receptance

In the field of vibration, driving point receptance is a well-established and widely applied concept. However, as demonstrated in this paper, when a driving point receptance is calculated using the finite element (FE) method with solid elements, it does not converge as the FE mesh becomes finer, suggesting that there is a singularity. Hence, the concept of driving point receptance deserves a rigorous examination. In this paper, it is firstly shown that, for a point harmonic force applied on the surface of an elastic half-space, the Boussinesq formula can be applied to calculate the displacement amplitude of the surface if the response point is sufficiently close to the load. Secondly, by applying the Betti reciprocal theorem, it is shown that the displacement of an elastic body near a point harmonic force can be decomposed into two parts, with the first one being the displacement of an elastic half-space. This decomposition is useful, since it provides a solid basis for the introduction of a contact spring between a wheel and a rail in interaction. However, according to the Boussinesq formula, this decomposition also leads to the conclusion that a driving point receptance is infinite (singular), and would be undefinable. Nevertheless, driving point receptances have been calculated using different methods. Since the singularity identified in this paper was not appreciated, no account was given to the singularity in these calculations. Thus, the validity of these calculation methods must be examined. This constructs the third part of the paper. As the final development of the paper, the above decomposition is utilised to define and determine driving point receptances required for dealing with wheel/rail interactions.

Electricity Generation System with Piezoelectric Element Using Acoustic Radiation Energy

In this study, a circular plate that is installing a piezoelectric element at its center is adopted as energy-harvesting system and is subjected to a harmonic point force. Because this system cannot avoid the influence of its acoustic radiation, the influence is considered theoretically using the equation of plate motion taking into account its radiation impedance and is estimated by the electricity generation efficiency, which is derived from the ratio of the electric power in the electricity generation and the mechanical power supplied to the plate. As a result, the efficiency is suppressed by the acoustic radiation from the plate, so that the efficiencies are so different in whether to take into consideration the radiation impedance or not. Because those results are verified by the electricity generation experiment and radiation acoustic energy has a hopeful prospect for improving the performance of this system, mechanical-acoustic coupling is used to make the most of the acoustic energy. Therefore, a cylinder that has the above plates at both ends is also adopted as the electricity generation system and mechanical-acoustic coupling is caused between the plate vibrations and an internal sound field into the cylindrical enclosure by subjecting one side of each plate to a harmonic point force. Then, the effect of coupling is evaluated by comparing with the efficiencies in the electricity generation system of only plate. Specifically, because the radiation impedance increases with the plate thickness, i.e., with the natural frequency of the plate, it is demonstrated that the effect of coupling becomes remarkable with increasing the thickness on the electricity generation efficiency.

Low-frequency sound radiation of infinite orthogonally rib-stiffened sandwich structure with periodic subwavelength arrays of shunted piezoelectric patches

In order to minimize the low-frequency sound radiation of an infinite orthogonally rib-stiffened sandwich structure, the periodic subwavelength arrays of shunted piezoelectric patches are introduced. Based on the piezoelectric shunt technique and effective medium method, the panels and piezoelectric patches are equivalent to two homogeneous facesheets. Then, considering all the inertia terms of the rib-stiffeners, a complete theoretical model is built for harmonic point force excitation by using Kirchhoff’s thin plate theory. The vibration displacements of the facesheets in wavenumber space are solved by Fourier transform technique to complete the prediction of vibroacoustic responses. Furthermore, the correctness and effectiveness of the present model are verified by sequentially using published analytical models, simulation results and theoretical predictions in strict accordance with two prerequisites. At last, the influence laws of key influencing parameters on the research structure are investigated. All the results demonstrate that the proposed structure can effectively attenuate the structural radiation.

Solutions of vibration problems for thin infinite plates subjected to harmonic loads

New closed form solutions for harmonic vibrations of infinite Kirchhoff plates subjected to a constant harmonic ring load, a constant harmonic circular load and an alternating harmonic circular load are derived. Two different approaches are used to define the closed form solutions. The first approach uses the integration of the harmonic point force and the addition theorem for Bessel functions, while the second approach applies the Hankel transform to solve the inhomogeneous partial differential equation of the Kirchhoff plate theory. The new closed form particular solutions can especially be used in Trefftz like methods and extend their field of application.

The Relationship between Structural Intensity and Sound Field Characteristics of Cylindrical Shells

It is an important subject in many engineering areas to recognize the vibro-acoustic behaviors of vibrational structures. Many researches have been done to study this characteristics of plane structure with structural intensity (SI) analysis. This paper extends the subject to study this subject by structural intensity approach in cylindrical shell structure and present a method of controlling the sound energy transmission. Structural intensity, a powerful method, can not only quantitatively reflect power flow strength, but also can provide the energy flow directions and distributions. The structural intensity patterns vary under different load and damping conditions, and thus affects the energy level of radiation sound field in cylindrical shell. In this paper, Numerical simulation system was built to represent the simply supported thin cylindrical shell coupled vibration and acoustical radiation structure which excited by harmonic normal point force. We get the corresponding results of structural intensity distribution, sound pressure and acoustic intensity of internal radiation sound field by adjusting damper positions, and calculating the sound field energy. The characteristics of SI of thin cylindrical shell are investigated, and the relationships between SI and sound field are established. We hope that our works could provide researchers new ideas for noise reduction of similar shell structures in potential applications.

Electricity Generation Characteristics of Energy-Harvesting System with Piezoelectric Element Using Mechanical-Acoustic Coupling

This paper describes the electricity generation characteristics of a new energy-harvesting system with piezoelectric elements. The proposed system is composed of a rigid cylinder and thin plates at both ends. The piezoelectric elements are installed at the centers of both plates, and one side of each plate is subjected to a harmonic point force. In this system, vibration energy is converted into electrical energy via electromechanical coupling between the plate vibration and piezoelectric effect. In addition, the plate vibration excited by the point force induces a self-sustained vibration at the other plate via mechanical-acoustic coupling between the plate vibrations and an internal sound field into the cylindrical enclosure. Therefore, the electricity generation characteristics should be considered as an electromechanical-acoustic coupling problem. The characteristics are estimated theoretically and experimentally from the electric power in the electricity generation, the mechanical power supplied to the plate, and the electricity generation efficiency that is derived from the ratio of both power. In particular, the electricity generation efficiency is one of the most appropriate factors to evaluate a performance of electricity generation systems. Thus, the effect of mechanical-acoustic coupling is principally evaluated by examining the electricity generation efficiency.