Solution For Frequency Response Research Articles

Frequency Response for Sweep-Up and Sweep-Down of Piezoelectric Energy Harvester

PurposeThe aim of this work is to study the frequency response of linear and nonlinear piezoelectric vibration energy harvester (pVEH) based on Duffing-type nonlinearity and Kirchhoff’s law, and the effect of the effective electromechanical coupling on the frequency response of nonlinear pVEH.MethodsThe analytical technique based on the harmonic balance method (HBM) is used to obtain the mechanical and electrical frequency responses of linear and nonlinear pVEH, and the Jacobian matrix is used to analyze the solutions’ stability. The Runge–Kutta fourth-order method as a numerical technique is used to solve the ordinary differential equations of pVEH.ResultsThe analytical calculations are completed, and the frequency response solution of displacement of pVEH is contained in real and complex values and those of electrical responses are presented. The harmonic balance results are compared with numerical simulations for various hardening nonlinear springs. Simple analytical expressions for the jump points are suggested for the sweep-up and sweep-down frequencies of pVEH.ConclusionsThe analytical analysis shows that the jump-down frequency ratios have definite behaviors for each low and high effective electromechanical coupling of hardening-nonlinear pVEHs. Moreover, the jump-up frequency ratio is directly proportional to the effective electromechanical coupling of hardening nonlinear pVEHs.

A three-machine equivalent system frequency response model and its closed-form solution

The frequency response (FR) of the large-scale power system shows significant frequency spatial distribution characteristics (FSDC), still using uniform frequency models may cause large errors in the FR solution, and the FSDC should be preserved in the modeling. In this paper, a three-machine equivalent system frequency response (TM-SFR) model is proposed to preserve the FSDC by the parameter equivalent aggregation method, including load, network, and generator. The TM-SFR model is further simplified into a solvable forced vibration equation by loop-opening and approximation, and its closed-form solution (CFS) is obtained by the modal superposition method, to analytically solve the FR following a major disturbance. Case studies are introduced to verify the performance of the TM-SFR model and its CFS over the WSCC 9-bus system, the New England 39-bus system, and the IEEE 118-bus system. The results prove that the proposed model can preserve the intermachine frequency oscillation caused by the FSDC in various scenarios, and improve the FR solution accuracy at high solution speed, which can provide a more reliable basis for frequency stability emergency control.

Nonlinear vibration analysis of cracked pipe conveying fluid under primary and superharmonic resonances

The nonlinear vibration response of the pipe conveying fluid with a breathing transverse crack, under primary and secondary excitations is investigated. Attention is concentrated on the superharmonic resonance which is the most sensitive phenomena for small crack detection. The crack is modeled with a transverse partial cut of the pipe wall thickness. The equivalent bending stiffness of the cracked pipe is derived based on an energy method and the breathing crack model is used to simulate the transition state between the fully open and the fully closed states. Based on Von Karman’s nonlinear geometric assumption and Euler–Bernoulli beam theory, the nonlinear geometric partial differential equations due to stretching effect, have been derived. The fractional viscoelastic model and the plug flow assumptions are used to model the damping and fluid flow. The coupled nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by the Galerkin method. The method of multiple scales is adopted to analyze steady-state solutions for the primary and superharmonic resonances. A parametric sensitivity analysis is carried out and the effects of different parameters, namely the geometry and location of the crack, nonlinear geometric parameter, fluid flow velocity, mass ratio, fractional derivative order and retardation time on the frequency-response solution are examined. In general, this paper shows that the superharmonic response amplitude measurement is useful tool for damage detection of pipes conveying fluid in the early stages.

Applications of an isogeometric indirect boundary element method and the importance of accurate geometrical representation in acoustic problems

This study presents the performance and the potential use of an isogeometric indirect variational boundary element method (IGiBEM) for industrial acoustics applications. The method is validated against semi-analytical frequency response solutions for a basic problem. After being validated, the performance of the IGiBEM is tested and benchmarked against the conventional indirect boundary element method (iBEM) for geometrically complex industrial applications. A vibrating car hood and a loudspeaker model with a vibrating woofer are used as industrial numerical tests. A non-isoparametric iBEM is included in the benchmarks to distinguish the contribution of the discretization versus the geometrical errors to overall error. The tests indicate that the geometrical errors play a more important role in problems involving complex geometries and high-frequency excitations. The study also demonstrates the advantages of IGiBEM over the conventional iBEM regarding the accuracy and computational efficiency for geometrically complex problems.

Exact solution for frequency response of sandwich microbeams with functionally graded cores

Based on the Euler–Bernoulli beam model and the modified strain gradient theory, the size-dependent forced vibration of sandwich microbeams with a functionally graded (FG) core is presented. The equation of motion and the corresponding classical and nonclassical boundary conditions are derived using the Hamilton’s principle. An exact solution of the governing equation is developed for sandwich beams with various boundary conditions and subjected to an arbitrarily distributed harmonic transverse load. Finally, parametric studies are presented to investigate the effects of geometric ratios, length scale parameters, power index, boundary conditions, layup, and thickness of the FG layer on the frequency response of clamped and simply supported microbeams. Numerical results show that in the case of clamped microbeams, the essential and natural size-dependent boundary conditions have a significant effect on the resonance frequency and transverse deflection of microbeams. Also, it is seen that an optimal layup (without change in total volume of each material) can significantly improve the frequency characteristics of sandwich microbeams.

Analytical frequency response solution for composite plates embedding viscoelastic layers

In this work analytical damped free-vibration and frequency response solutions are obtained for the analysis of composite plates structures embedding viscoelastic layers. On the basis of the Principle of Virtual Displacements, Layer-Wise models related to linear up to fourth order variations of the unknown variables in the thickness direction are treated. Analytical solutions using the Navier procedure are presented for the analysis of isotropic, cross-ply composite and simply-supported plate structures. The modelization of multilayered structure materials takes into account the composite material properties and the frequency dependence of the viscoelastic material. Various external loads are considered: closed form solution for bi-sinusoidal pressure, constant distributed pressure and concentrated loads. Several analyses are carried out to validate and demonstrate the accuracy and efficiency of the present formulation for the study of viscoelastic plates, taking into account different lamination sequences and different plate aspect ratios.

Geometrically Nonlinear Random Vibration Responses of Laminated Plates Subjected to Acoustic Excitation

An equivalent linearization technique has been incorporated into NASTRAN to predict the geometrically nonlinear random vibration responses of laminated plates subjected to acoustic excitation. The nonlinear equations of motion of laminated plates are obtained based on the classical laminated plate theory and large deflection finite element formulation. The existing internal variables in the modal frequency response solution sequence of NASTRAN are called directly by the direct matrix abstraction program, and the direct matrix abstraction program implements the iterative procedure for determining the modal equivalent linear stiffness matrix. In this way, the usual manner of computing variables is retained; that is, these variables are determined internally within NASTRAN using information provided in the bulk data file written by PATRAN. The numerical results for a thin, simply supported, aluminum plate are in good agreement with previously reported data, which confirms the proposed method has reasonable precision and high efficiency. Then, the proposed method is applied for analyzing the geometrically nonlinear random responses of thin, simply supported, laminated plates. It is shown that the geometric nonlinearity plays an important role in the vibration response of laminated plates, particularly at high acoustic pressure loading.

Coupled bending–torsional frequency response of beams with attachments: exact solutions including warping effects

This paper deals with the coupled bending–torsional vibrations of beams carrying an arbitrary number of viscoelastic dampers and attached masses. Exact closed analytical expressions are derived for the frequency response under harmonically varying, arbitrarily placed polynomial loads, making use of coupled bending–torsion theory including warping effects and taking advantage of generalized functions to model response discontinuities at the application points of dampers/masses. In this context, the exact dynamic Green’s functions of the beam are also obtained. The frequency response solutions are the basis to derive the exact dynamic stiffness matrix and load vector of a two-node coupled bending–torsional beam finite element with warping effects, which may include any number of dampers/masses. Remarkably, the size of the dynamic stiffness matrix and load vector is $$8\times 8$$ and $$8\times 1$$ , respectively, regardless of the number of dampers/masses and loads along the beam finite element.

A Review of Frequency Response Solution for Type - 3 Wind Turbines Using Energy Storage Device

This paper identifies the current approaches to provide the frequency response from type 3 (Doubly Fed Induction Generator based) wind turbines to support the grid during power imbalance. It discusses the use of energy storage devices for providing support during frequency deviation. The paper mainly focuses on the approach to connect the energy storage with the wind turbine through the DC coupling capacitor and use the already available wind turbine AC/DC/AC converter for power conversion. The process to choose an energy storage device and appropriately sizing it for desired support is stated. It also discusses the wind turbine controller strategy modification to properly operate the energy storage device and provide required support. Finally, the study shows simulation results indicating the validation of connecting energy storage through DC coupling capacitor and the updated control algorithm method in different operating condition.

A harmonic pulse testing method for leakage detection in deep subsurface storage formations

AbstractDetection of leakage in deep geologic storage formations (e.g., carbon sequestration sites) is a challenging problem. This study investigates an easy‐to‐implement frequency domain leakage detection technology based on harmonic pulse testing (HPT). Unlike conventional constant‐rate pressure interference tests, HPT stimulates a reservoir using periodic injection rates. The fundamental principle underlying HPT‐based leakage detection is that leakage modifies a storage system's frequency response function, thus providing clues of system malfunction. During operations, routine HPTs can be conducted at multiple pulsing frequencies to obtain experimental frequency response functions, using which the possible time‐lapse changes are examined. In this work, a set of analytical frequency response solutions is derived for predicting system responses with and without leaks for single‐phase flow systems. Sensitivity studies show that HPT can effectively reveal the presence of leaks. A search procedure is then prescribed for locating the actual leaks using amplitude and phase information obtained from HPT, and the resulting optimization problem is solved using the genetic algorithm. For multiphase flows, the applicability of HPT‐based leakage detection procedure is exemplified numerically using a carbon sequestration problem. Results show that the detection procedure is applicable if the average reservoir conditions in the testing zone stay relatively constant during the tests, which is a working assumption under many other interpretation methods for pressure interference tests. HPT is a cost‐effective tool that only requires periodic modification of the nominal injection rate. Thus it can be incorporated into existing monitoring plans with little additional investment.

Nonlinear vibration analysis of harmonically excited cracked beams on viscoelastic foundations

The frequency response of a cracked beam supported by a nonlinear viscoelastic foundation has been investigated in this study. The Galerkin method in conjunction with the multiple scales method (MSM) is employed to solve the nonlinear governing equations of motion. The steady-state solutions are derived for the two different resonant conditions. A parametric sensitivity analysis is carried out and the effects of different parameters, namely the geometry and location of crack, loading position and the linear and nonlinear foundation parameters, on the frequency-response solution are examined.

A one-dimensional model for dynamic analysis of generally layered magneto-electro-elastic beams

A new one-dimensional model for the dynamic problem of magneto-electro-elastic generally laminated beams is presented. The electric and magnetic fields are assumed to be quasi-static and a first-order shear beam theory is used. The electro-magnetic problem is first solved in terms of the mechanical variables, then the equations of motion are written leading to the problem governing equations. They involve the same terms of the elastic dynamic problem weighted by effective stiffness coefficients, which take the magneto-electro-mechanical couplings into account. Additional terms, which involve the third spatial derivative of the transverse displacement, also occur as a result of the piezoelectric and/or piezomagnetic behavior. It is also shown that the magneto-electric inputs can be treated as equivalent external axial forces and bending moments per unit length. Free vibrations and frequency response solutions are presented to validate the model by comparing the present results with those found in the literature or obtained by finite element analysis. Based on the successful validation of the model, new results for free vibrations of functionally graded magneto-electro-elastic beams are presented.

Dynamic Modeling and Analysis of Tooth Profile Modification for Multimesh Gear Vibration

AbstractThis work studies the effects of tooth profile modification on multimesh gearset vibration. The nonlinear analytical model considers the dynamic load distribution between the individual gear teeth and the influence of variable mesh stiffnesses, profile modifications, and contact loss. The proposed model yields better agreement than two existing models when compared against nonlinear gear dynamics from a finite element/contact mechanics benchmark. These comparisons are made for different loads, profile modifications, and bearing stiffness conditions. This model captures the total and partial contact losses demonstrated by finite element. Perturbation analysis based on the proposed model finds approximate frequency response solutions for the case of no total contact loss due to the optimized system parameters. The closed-form solution is compared with numerical integration and provides guidance for optimizing mesh phasing, contact ratios, and profile modification magnitude and length.

Convected field analysis of flat panels response to turbulent boundary layer induced excitation

Abstract The main objective of the work was the development of a procedure allowing the study of the effect of fluid loading over the vibration response of a plate driven by a TBL excitation field. The reasons for such analysis are concerned with the limitations of the main numerical codes available on the market when dealing with the subject, and the need for the extension of the existing predictive capabilities of deterministic methodologies, when facing with the stochastic pressure distribution over a flat plate. The availability of such specific tools is judged as mandatory for the improvement and/or better classification of the design criteria for the enhancement of modern aircraft acoustic comfort. After recalling and rearranging the theoretical concepts applicable to the aeroelastic behaviour of flat plates, to be used as a reference for the validation process, two different numerical approaches are herein presented. A first procedure was elaborated by using a vibro-acoustic coupling formulation, in which an appropriate convected field Green function was employed, and by relying on the modal information provided by FEM based numerical tools. An alternative numerical procedure was developed by employing the aeroelastic module of NASTRAN commercial code, in which the baffle condition was sought and obtained throughout proper adaptation and tuning of the input parameters. Both procedures were built up to be applicable to any baffled flat plate, including sandwich and composite panels. A two-ways validation process, based on the assumption of point forces loading conditions, was carried out by introducing the fluid back reaction contribution associated to the convected field in the frequency response solution of the first procedure. The evidence of the convected field effect on the panel response was highlighted, mainly associated to a reduction of the investigated panel stiffness properties due to aeroelastic interactions. A preliminary study on stochastic loading conditions response was performed by considering a “rain on the roof” excitation, while the final investigation, focused on the TBL induced excitation field, confirmed the potentiality of the proposed procedures in dealing with the subject. The relevance of a correct estimation of the convected field contribution on the plate response in terms of both qualitative and quantitative aspects is pointed out, specially for what concerns the attenuation of a plate vibration energy.

Mixture Diffusion in Nanoporous Adsorbents: Development of Fickian Flux Relationship and Concentration-Swing Frequency Response Method

A frequency response method using concentration variation is developed theoretically and experimentally and applied to investigate mixture diffusion in nanoporous adsorbents. The method is based on periodically time-varying species feed concentrations with a constant total molar inlet flow rate. It can be used without the need of a carrier gas. A mathematical model is formulated considering nanopore diffusion, surface barrier resistance, external film resistance, and axial dispersion. The related analytical solutions for frequency response are derived. For nanopore diffusion, a theory for nonconstant mixture Fickian diffusivity with cross-terms is developed from irreversible thermodynamics and shows that mixture Fickian diffusivities can be expressed as the product of corrected diffusivities and a thermodynamic factor that accounts for concentration dependence. The number of unknown variables for Fickian diffusivities is the same as the number for Onsager coefficients or Maxwell−Stefan diffusivities. Adso...

Experimental and Analytical Estimation of Loss Factors by the Power Input Method

The Power Input Method (PIM) is used both experimentally and a nalytically to estimate the system loss factor for sandwich panels with various configurations of Constrained Layer Damping (CLD) treatments over a broad frequency range. The experimental power input method is applied to both uniformly and non -uniformly da mped structures. Results are compared with results from other experimental methods. A new analytical power input method is proposed for evaluating the loss factor of built -up structures, based on the finite element model with assigned properties of the con stituents. The new analytical power input method is evaluated by comparison with the commonly used Modal Strain Energy (MSE) method. Instead of making an approximate correction of the constant material properties, this analytical power input method directl y takes into account the frequency -dependent material properties of the viscoelastic material using the MSC/NASTRAN direct frequency response solution. Results of experimental and analytical methods are presented, compared and discussed. It is shown that: 1) all three currently available experimental methods yield consistent results, while the power input method gives damping estimation other than just at several discrete frequencies basically in the low frequency range; 2) both the analytical power input m ethod and the modal strain energy method yield consistent results with the experimental power input method. Furthermore, both experimental and analytical power input methods are used to investigate how loss factors change as the excitation position change. This shows another merit of the analytical power input method, because analytical modal methods cannot take into account the change of excitation position.

Transfer Functions of One-Dimensional Distributed Parameter Systems by Wave Approach

An alternative analysis technique, which does not require eigensolutions as a priori, for the dynamic response solutions, in terms of the transfer function, of one-dimensional distributed parameter systems with arbitrary supporting conditions, is presented. The technique is based on the fact that the dynamic displacement of any point in a waveguide can be determined by superimposing the amplitudes of the wave components traveling along the waveguide, where the wave numbers of the constituent waves are defined in the Laplace domain instead of the frequency domain. The spatial amplitude variations of individual waves are represented by the field transfer matrix and the distortions of the wave amplitudes at point discontinuities due to constraints or boundaries are described by the wave reflection and transmission matrices. Combining these matrices in a progressive manner along the waveguide using the concepts of generalized wave reflection and transmission matrices leads to the exact transfer function of a complex distributed parameter system subjected to an externally applied force. The transient response solution can be obtained through the Laplace inversion using the fixed Talbot method. The exact frequency response solution, which includes infinite normal modes of the system, can be obtained in terms of the complex frequency response function from the system’s transfer function. This wave-based analysis technique is applicable to any one-dimensional viscoelastic structure (strings, axial rods, torsional bar, and beams), in particular systems with multiple point discontinuities such as viscoelastic supports, attached mass, and geometric/material property changes. In this paper, the proposed approach is applied to the flexural vibration analysis of a classical Euler–Bernoulli beam with multiple spans to demonstrate its systematic and recursive formulation technique.

Analytical solution for frequency response of equipment in laterally loaded multistorey buildings

An analytical and closed-form frequency response of equipment mounted on multistorey buildings subjected to horizontal ground motion is proposed. In this study, the dynamics of the equipment and the building is expressed as a state-flow graph model, in which the interaction effect between the equipment and the building is considered. Based on the graph model, the analytical results for the frequency response of the acceleration of the equipment and the internal force in the support are derived. One of the advantages of this method is that the closed-form solutions of the frequency response expressed by polynomial form will be easily examined by analytical and numerical computations without complex operation. Moreover, the dynamic of the primary and secondary systems and their dynamic interaction are expressed separately in the derived formula. Thus most of the items in the formula need not be computed repeatedly for different supports of the equipment in design.

Evaluation system for residual vibration from HDD mounting mechanism

An evaluation system for calculating the residual vibration caused by a HDD (Hard Disk Drive) mounting system is presented. It contains both the elements for measurement and analysis. The PES (Position Error Signal) of an ideally balanced rigid rotary actuator depends on the rotational vibration of the HDD, specifically at lower frequencies. First, this evaluation technique obtains a measurement of the transfer function of the HDD rotational acceleration over actuator current. Next, using the measured power spectrum of the actuator in combination with the HDD's tracking error transfer function, the response of the entire HDD and mount system is calculated in the frequency domain. Finally the time domain system response is calculated by computation of the Inverse Fast Fourier Transform (IFFT) of the frequency response solution, assuming a random phase of the input force. Good correlation is obtained between measured and calculated system responses. Experimental result show that the dominant residual vibration is caused by the mounting mechanism response and that both the HDD and HDD mount system need to be evaluated and understood as a system, in order to achieve higher TPI (Track per inch) in the future.

Frequency response analysis for multicomponent diffusion in adsorbents

Frequency response behaviour of diffusion of multiple sorbates in adsorbents is theoretically investigated. A mathematical model describing diffusion of n components with both equilibrium and diffusional interference is developed and analytical solutions for frequency response and moments are derived. It is shown that, for the model system CO2 and C2H6 in 4A zeolite, the fastest diffusing component can be strongly affected by other slower diffusing components and its partial-pressure frequency response curves exhibit a roll-up phenomenon induced exclusively by the diffusional interference. The frequency response curves of the total pressure generally have multi-modal forms as a result of multicomponent diffusion; the peak positions and heights within these responses, which are closely related to the diffusivities, are found to be sensitive to the extent of equilibrium and diffusional interference. Comparison of these resonance frequencies obtained from pure-component and multicomponent experiments can therefore give a good indication of the interference between the components. The effects of film resistance and heat dissipation, which may be important for fast diffusing systems, are also discussed.