Steady-state Frequency Response Research Articles

Exact solution for the electrical response of a constant phase element with a series resistance to linear voltage sweep

A resistance Rs in series with a constant phase element (CPE) of frequency-dependent impedance given by the power law function Zc(s)=1/(Cαsα) is commonly used for the analysis of steady-state frequency response data exhibiting non-purely capacitive behavior. This is the case in most (bio)(electro)chemical systems including dielectrics, batteries, supercapacitors, capacitive deionization units, biological tissues and bioelectrodes. Passing to the time domain, the current, voltage and charge of the system are governed by differential equations with non-integer, fractional-order operators. The purpose of this study is to provide the exact analytical expressions for the electrical response of an Rs-CPE model under linear sweep voltammetry with the use of the Laplace transform method. The electrical variables are expressed in terms of special functions regularly encountered with fractional calculus such as the Fox’s H-function and the Mittag-Leffler function, and can be used for modeling non-ideal devices as well as extracting their characteristic parameters.

A Ball-Contacting Dynamic Vibration Absorber with Adjustable Stiffness and Nonlinear Characteristics

Structural vibration has always been a major concern in the engineering field. A dynamic vibration absorber in the form of contacts with adjustable stiffness (CDVA) offers effective vibration suppression and can improve conventional dynamic vibration absorbers with high sensitivity to frequency deviation and difficulty in adjusting the frequency. In this research, first, based on the theoretical model of the contact between a rubber ball and an inner cone, the feasibility of changing the axial contact state to change the structure’s natural frequency was verified using an ANSYS simulation. A theoretical model of the static contact stiffness between the ball and the inner cone was constructed using Hertzian contact theory and Hooke’s law, and a theoretical model of the cubic nonlinear elastic restoring force was used to characterize the stiffness properties of the rubber ball during compressive rebound. The steady-state frequency response equations of the main vibration structure were derived using the averaging method in conjunction with the two-degree-of-freedom dynamics model, and the stability of the solutions to the frequency response equations was obtained in conjunction with the stability determination criterion. Then, the impact of the CDVA’s design parameters on the nonlinear dynamic response of the primary vibration structure was simulated and analyzed. The resulting findings can serve as guidance for designing dynamic vibration absorber parameters. Based on the principles of ball-inner cone contact, a dynamic vibration absorber structure was proposed. A design test was conducted to verify the correctness of the contact stiffness model, and an experimental study was carried out to investigate the law of change in the dynamic stiffness and damping of the principle structure of CDVA under dynamic excitation conditions. Finally, the vibration test platform of the solidly supported beam structure was constructed, and vibration suppression tests of the CDVA in different compression states were conducted to investigate the tunability and feasibility of CDVA vibration suppression. The results showed that the dynamic vibration absorber had good vibration absorption characteristics and could be used for single-mode vibration suppression of multimodal main structures.

Nonlinear forced vibration analysis of PFG-GPLRC conical shells under parametric excitation considering internal and external resonances

Comprehending the vibration dynamics of porous functionally graded-graphene platelet reinforced composite (PFG-GPLRC) structures is vital for accurate predictions and reliability in practical applications. This study addresses gaps in nonlinear dynamics, instability, and frequency response research within truncated PFG-GPLRC conical shells under parametric loading and ½ subharmonic and 1:1 internal resonance. To achieve this, three porosity distributions in metal foam (uniform, non-uniform symmetric, and non-uniform asymmetric) are considered, along with various graphene platelet dispersion patterns (GPL-O, GPL-V, GPL-U, GPL-A, and GPL-X) within the matrix. These considerations lead to a comprehensive conical shell model. Utilizing the first-order shear deformation theory and von-Karman's assumptions, stress-strain relations are extracted, yielding nonlinear motion equations for the truncated conical shell. Employing Galerkin's method and considering simply supported boundaries, two-degree-of-freedom equations of motion are derived. The research culminates in steady state frequency responses obtained through perturbation theory and the multiple scales method, encompassing ½-subharmonic excitation resonance and 1:1 internal resonance. Bifurcation points are analysed to highlight the influence of parametric excitation and system instabilities. A parametric study underscores the significance of porosity and graphene platelets within the metal foam in relation to system instability, revealing their intricate impact on PFG-GPLRC structure behavior.

Coupled Thermomechanical Dynamics of Phase Transitions in Shape Memory Alloys and Related Nonlinear Phenomena

The aim of this paper is to study the coupled non-linear dynamics of the behaviour of Shape Memory Alloys (SMA) under harmonic excitation. The thermomechanical model developed by Falk and based on Landau theory is used to describe the coupled thermomechanical behaviour of the SMA mass-damper-bar system. Fully coupled mechanical and thermal field equations are established to reveal the strong coupling phenomena arising from the phase transition. Then, the harmonic balance method is used to obtain the steady-state frequency responses. The effect of thermomechanical coupling on the frequency and time response curves is taken into account by modifying the heat transfer coefficient. The results show that deformation and temperature provide a stable sinusoidal response. The analysis of the results leads to different ways of controlling the nature and extent of the coupled and non-linear response of SMA-based oscillators. Comparison of the analytical results with experimental data from the literature validates the coupled thermomechanical nonlinear model. These results can be used effectively to control the external vibrations of various systems.

Hybrid Model-Based BESS Sizing and Control for Wind Energy Ramp Rate Control

This paper presents a hybrid model constituting dynamic smoothing technique and particle swarm optimization techniques to optimally size and control battery energy storage systems for wind energy ramp rate control and power system frequency performance enhancement. In today’s modern power system, a high-proportion renewable energy grid is inevitable. This high-proportion renewable energy grid is a power system with abundant integration of renewable energy resources under the presence of energy storage tools. Energy storage tools are integrated into such power systems to balance the fluctuation and intermittence of renewable energy sources. One of the requirements in a high-proportion renewable energy grid is the fractional power balance between generation and load. One of the requirements set by power system regulators is the generation variation between two time points. A power producer is mandated to satisfy the ramp rate requirement set by the grid owner. This paper proposes dynamic smoothing techniques for initial size determination and particle swarm optimization based on optimal sizing and control of battery energy storage systems for ramp rate control and frequency regulation performance of a power system integrated with a large percentage of wind energy systems. Wind energy data taken from Zhangjiakou wind farm in China are used. The results indicate that the battery energy storage system improves the ramp rate characteristics of the wind farm. In addition, the virtual inertia capability of the battery energy storage system enabled the transient and steady-state frequency response of the test power system to improve significantly.

Retooling results of a massive database of home theater measurements

In 2010 the first results of a large-scale measurement program on home theaters was reported in an AES presentation. More than 1,000 rooms were measured for impulse response from each of a minimum of six loudspeakers to at least four (usually more) listening locations. However, computer processing capacity limited the initial report to 275 rooms. Room acoustics and sound system information was reported: RT and its std. dev. with frequency, room volume statistics for those rooms, and transient and steady-state frequency response for some example rooms were shown. From these few rooms an important item of interest emerged: the Schroeder frequency did not show as a boundary of response deviations between lower and higher frequencies—the average deviation from the average response was more constant with frequency than expected. This work will be briefly reviewd. Now in 2022 a new look at this data plus that which was gathered in the intervening years may be studied with much more advanced math tools such as machine learning so much more can be understood. This presentation is interim describing work to be done and soliciting comments from practitioners.

Steady-state and transient vibration analysis of a thermally loaded power law-based functionally graded rotor system with induced porosities

The dynamic analysis of a power law-based functionally graded (FG) rotor-bearing system with induced porosities for non-uniform porosity distributions has been performed using the finite element method. A novel porous rotor element considering the effects of transverse shear deformation, gyroscopic moments, translational and rotational inertial has been developed for non-uniform porosity distributions based on the Timoshenko beam theory. Material properties are graded based on the power law along the radial direction of the FG shaft. The nonlinear temperature distribution law has been considered in conjunction with the power law to vary the temperature across the cross-section of the FG shaft. Material is graded in such a way that the stainless steel is comprised at the core of the FG shaft and the outer periphery of the FG shaft is made of Zirconia. The steady-state and transient time and frequency responses of a thermally loaded porous FG rotor-bearing system have been computed for both types of uneven porosity distribution using the Houbolt time marching technique and Fast Fourier Transform, respectively. The free vibration analysis has also been performed on the porous FG rotor-bearing system for various parameters such as power law index, the volume fraction of porosity and porosity distribution under different thermal gradients. The reduction of critical speed with increased amplitude is observed from the steady-state and transient frequency responses with an increase in volume fraction of porosity and temperature gradients

Modal analysis of viscoelastic three-dimensional rotating beam with generic tip mass

In this paper, the nonlinear dynamic modelling, free vibration and nonlinear analysis of three-dimensional viscoelastic Euler-Bernoulli beam undergoing hub motion incorporating substantial tip mass is accomplished. The kinetic, and potential energy of the system are derived in terms of velocities of general point on the link and center of gravity of the tip mass expressed in multi-floating co-ordinate systems. Hamilton's principle is used to obtain the governing equations of motion and linearly coupled boundary conditions. The material of the beam is considered as viscoelastic constituted of Kelvin-Voigt rheological model. The mass attached at the terminal end of the beam is assumed to have eccentricity in axial, lateral, as well as transverse directions. Free vibration analysis is performed on the linearized system model to obtain the transcendental eigenfrequency equation. Further, the dynamic equations of motion of beam are discretized using the obtained mode shapes and the response of system under rotary motion of hub is investigated. The steady state solutions and frequency response curves exhibiting bi-stable and tri-stable regions are obtained using method of multiple scales. The bifurcation diagrams of the system are studied for resonance conditions when the frequency of the rotary motion becomes equal or nearly equal to the normalized beam frequencies. The saddle node and pitchfork bifurcations exhibiting multiple solutions and jump phenomena are observed and investigated to avoid catastrophic failure of the system. The numerical simulations of modal parameters of the system, nonlinear characteristics, and their parametric dependency is discussed thoroughly.

Steady State and Transient Vibration Analysis of an Exponentially Graded Rotor Bearing System Having a Slant Crack

The dynamic behaviour of a slant-cracked exponentially graded (EG) rotor-bearing system has been investigated using the finite element method for flexural vibrations. A two nodded EG rotor element has been developed based on the Timoshenko beam theory. Local flexibility coefficients (LFCs) of a slant-cracked EG shaft element have been derived using fracture mechanics concepts to develop the stiffness matrix of a cracked EG element. The steady-state and transient vibration responses of cracked and uncracked rotor systems have been simulated using the Houbolt time marching method. When a crack is present in the shaft, the subharmonic frequency peaks are centred on operating speed in the steady-state frequency responses, whereas on critical speed in the transient frequency responses at an interval frequency corresponding to the torsional frequency. It has been found that the crack parameters such as crack depth and location, temperature gradients and torsional frequencies have a significant influence on natural frequencies and dynamic responses, which could be implemented for efficient rotor crack detection methodologies.

Assortment of Dispatch Strategies with the Optimization of an Islanded Hybrid Microgrid

In this work, the optimization of an off-grid micro-hybrid system is evaluated. This is conducted with the estimation of the proper sizing of each element and the steady-state voltage, frequency, and power responses of the microgrid. Kangaroo Island in South Australia is considered to be the test case location and the grid incorporates solar PV (photo-voltaic), diesel generator, battery storage, and wind turbine. Optimal sizing of the studied microgrid is carried out for four various power dispatch techniques: (i) cycle charging (CC), (ii) generator order (GO), (iii) load following (LF), and (iv) combined dispatch (CD). The proposed off-grid micro-hybrid is optimized for three performance indices; minimal Levelized Cost of Energy (LCOE), CO2 emission, and Net Present Cost (NPC). Using iHOGA (improved hybrid optimization by genetic algorithm), microgrid optimization software, all the above-mentioned dispatch strategies have been implemented and following this, MATLAB/Simulink platform has been used for the steady-state studies. The results show that the LF strategy is the utmost optimum dispatch technique in terms of the studied performance indices i.e. considering the optimal size and voltage and frequency responses. The results obtained from these studies provide a pathway for the estimation of the resource-generation-load combination for the islanded off-grid microgrid for its optimal operation with the various dispatch strategies.

Computational homogenization of locally resonant acoustic metamaterial panels towards enriched continuum beam/shell structures

Locally resonant acoustic metamaterial (LRAM) panels are excellent candidates for the low-frequency flexural wave attenuation in thin structures. To enable the efficient analysis and design of LRAM beam/shell structural elements for practical applications, a computational homogenization method for modelling wave propagation phenomena in LRAM panels is presented in this work. The approach is based on the notion of a relaxed separation of scales, which tailors the methodology to the phenomena governed by local resonators embedded in a host medium. The macroscopic LRAM panel is modelled as a thin continuum beam/shell described by proper structural kinematics and momentum balance relations. At the microscale, a LRAM unit cell is considered with lamina-wise in-plane boundary conditions and zero out-of-plane tractions, representing the free top and bottom surfaces of the macroscopic LRAM panel. Under the assumption of linear elasticity, in the relaxed separation of scales regime, the microscale unit cell problem can be represented by the superposition of a static and a dynamic problem, hereby enabling a significant model reduction. As a result, the macroscale effective material properties can be computed once and for all off-line for a given unit cell. In addition, a new macroscale evolution equation emerges describing the effect of the microscale internal dynamics on the macroscopic fields through the introduction of new macroscale enrichment variables, which reflect the modal amplitudes of localized microscale modes. The proposed homogenization methodology reveals a high level of accuracy and numerical efficiency compared to the reference direct numerical simulation (DNS) for all relevant analyses: computation of real and complex dispersion spectra for an infinite LRAM panel, as well as steady-state frequency response and transient behaviour for a finite LRAM panel.

Estimating experimental dispersion curves from steady-state frequency response measurements

Dispersion curves characterize the frequency dependence of the phase and the group velocities of propagating elastic waves. Many analytical and numerical techniques produce dispersion curves from physics-based models. However, it is often challenging to accurately model engineering structures with intricate geometric features and inhomogeneous material properties. For such cases, this paper proposes a novel method to estimate group velocities from experimental data-driven models. Experimental frequency response functions (FRFs) are used to develop data-driven models, which are then used to estimate dispersion curves. The advantages of this approach over other traditionally used transient techniques stem from the need to conduct only steady-state experiments. In comparison, transient experiments often need a higher-sampling rate for wave-propagation applications and are more susceptible to noise.The vector-fitting (VF) algorithm is adopted to develop data-driven models from experimental in-plane and out-of-plane FRFs of a one-dimensional structure. The quality of the corresponding data-driven estimates is evaluated using an analytical Timoshenko beam as a baseline. The data-driven model (using the out-of-plane FRFs) estimates the anti-symmetric (A0) group velocity with a maximum error of 4% over a 40 kHz frequency band. In contrast, group velocities estimated from transient experiments resulted in a maximum error of 6% over the same frequency band.

Numerical investigation into optical and electronic performance of crystal orientation-dependent InGaAs/InP near-infrared laser

The energy band dispersion profile and their dependence on crystal orientation for III-V zinc-blende compound quantum wells (QW) have earned significant attention in recent years due to reduced band mixing effects emerging from increased energy separation between valence subbands. So QW like InGaAs grown in non-conventional orientation seems to be quite optimistic to serve as active region in near-infrared optoelectronic applications due to small piezoelectric polarization, improved confinement of high energy states and increased favorable spectral range than traditional (100)-oriented growth. Here, crystal orientation-dependent electronic and optical performance of InGaAs-InP laser subjected to 1.60% compressive strain and emitting around 2 µm is studied numerically after working out an eight-band k.p Hamiltonian with the help of finite difference method and Tensor rotation technique. An equivalent circuit model using three-level laser rate equations is employed here to reveal optical output power and steady state frequency response. It is noticed that the wave functions of hole are more confined in (111) orientation than (100) orientation. Also, there is a noteworthy reliance of the energy band gap, optical gain spectra and output lasing power profile on change in crystal orientation. The estimated gains are found to be 4250, 3900, 3555, 3210 and 2950 cm−1 in (111), (100), (131), (110), and (113) orientations. The topmost lasing power and lowest threshold current are noted to be 56.4mW and 4.5 mA, respectively, in the (111) crystal orientation. Further, the highest optical emission point is observed to be moved towards longer wavelength for the alteration in crystal orientation from (131) to (111).

Harmonic Standing-Wave Excitations of Simply-Supported Thick-Walled Hollow Elastic Circular Cylinders: Exact 3D Linear Elastodynamic Response

The forced-vibration response of a simply-supported isotropic thick-walled hollow elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surfaces is studied within the framework of linear elastodynamics. Exact semi-analytical solutions for the steady-state displacement field of the cylinder are constructed using recently-published parametric solutions to the Navier-Lame equation. Formal application of the standing-wave boundary conditions generates three parameter-dependent $6\times6$ linear systems, each of which can be numerically solved in order to determine the parametric response of the cylinder's displacement field under various conditions. The method of solution is direct and demonstrates a general approach that can be applied to solve many other elastodynamic forced-response problems involving isotropic elastic cylinders. As an application, and considering several examples, the obtained solution is used to compute the steady-state frequency response in a few specific low-order excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a unique subset of the natural frequencies of the simply-supported cylinder. The considered problem is of general theoretical interest in structural mechanics and acoustics and more practically serves as a benchmark forced-vibration problem involving a thick-walled hollow elastic cylinder.

Transient and steady-state viscoelastic contact responses of layer-substrate systems with interfacial imperfections

This paper reports the development of a novel semi-analytical model for solving the transient and steady-state contact responses of a rigid sphere sliding/rolling on a viscoelastic layer-elastic substrate system. The displacement transmissions at the layer-substrate interface are affected by spring-like or dislocation-like defects. The analytical transient and steady-state viscoelastic frequency response functions (FRFs) are derived from the elastic solutions with imperfect interfaces. Instead of using the integration form of the creep function, viscoelastic modulus Ε(ω) is directly incorporated into the viscoelastic FRFs by a frequency-velocity transform that links the time-related frequency, ω, and sliding velocity, V, with the space-related frequency number, m, i.e. ω=−mV. The solutions are so formulated that fast numerical techniques, such as the conjugate gradient method (CGM) and the discrete convolution-fast Fourier transform (DC-FFT) algorithm, can be incorporated for computation efficiency. The developed model is employed to investigate the effects of layer thickness, modulus, sliding velocity, and the degree of interface imperfection on the viscoelastic contact response of the material system, including pressure distributions, displacements, viscoelastic dissipation, and subsurface stresses.

Frequency‐constrained unit‐commitment using analytical solutions for system frequency responses considering generator contingencies

The frequency responses of a power system following a generator outage are closely related to the commitment states of generators, which describe the droop and inertial characteristics and system pre-contingency power dispatch. This study presents a new model to estimate the transient and steady state frequency responses of two-area power systems at any moment following the loss of an online generator. In the proposed model, the deviations of tie-flow, area frequencies and rate of change of area frequencies as well as the extreme values of these functions and their associated time of occurrence following the outage of a generation unit are derived as modified analytical functions. A new approach is also proposed to linearise the modified extreme deviations based on pseudo-Boolean function theorem. In addition, a frequency-constrained unit commitment model with modified frequency-dependent constraints incorporating primary frequency regulation reserve is presented as a mixed-integer linear programming problem. The results indicate that the proposed method gives acceptable approximations for post-contingency responses and can ensure power system security from frequency viewpoint.

Vibration analysis of sandwich beams with viscoelastic coating described by fractional constitutive equation

In this study, a dynamic model of a sandwich beam with viscoelastic coating based on a fractional constitutive equation was developed. The fractional model is a generalized model of sandwich beams and can be modified into integer models. The wave propagation method was extended for vibration analysis of fractional sandwich beams. Further, the influence of fractional order and size parameters on the dynamic characteristics of the sandwich beam was investigated, and the steady-state frequency response of the laminated beam was calculated. The results can provide a theoretical basis for the size design of laminated beams with damping requirements.

Correction to: Steady State Frequency Response Design of Finite Time Iterative Learning Control

Correction to: Steady State Frequency Response Design of Finite Time Iterative Learning Control

A Study of Nonlinear Behavior of Flexible Tool and Workpiece in Turning Operation With Regenerative Effect Under Internal and Primary Resonance Conditions

Abstract In this work, the nonlinear dynamic behavior of turning operation has been studied considering flexible tool and thin cylindrical workpiece. The system is taken as a two-degree-of-freedom system with nonlinear stiffness and is subjected to self-excited vibration because of the regenerative effect. The cutting force is considered to be a combination of a constant and periodic force in addition to the force due to the regenerative effect. The regenerative effect during turning operation is included in the mathematical model, resulting in a nonlinear delay differential equation. Here, the natural frequency of the workpiece is assumed to be closed to that of the tool, leading to 1:1 internal resonance. Further, the frequency of the time-varying cutting force is assumed to be closed to the natural frequency of the workpiece giving rise to primary resonance condition. The nonlinear responses and the stability of the tool and the workpiece have been determined using a higher-order method of multiple scales (MMS) under internal and primary resonance conditions. The solution of the equation of motion using the MMS is validated by comparing the solution obtained using the numerical method. The effect of the tool and workpiece stiffness nonlinearities on steady-state frequency responses and stability is investigated, and system parameters are also identified to have stable turning operation. This work will find applications in estimating the system parameters for chatter-free turning operation with a flexible tool and workpiece when their dynamic compliances are comparable.

Estimating dispersion curves from Frequency Response Functions via Vector-Fitting

Driven by the need for describing and understanding wave propagation in structural materials and components, several analytical, numerical, and experimental techniques have been developed to obtain dispersion curves. Accurate characterization of the structure (waveguide) under test is needed for analytical and numerical approaches. Experimental approaches, on the other hand, rely on analyzing waveforms as they propagate along the structure. Material inhomogeneity, reflections from boundaries, and the physical dimensions of the structure under test limit the frequency range over which dispersion curves can be measured. In this work, a new data-driven modeling approach for estimating dispersion curves is developed. This approach utilizes the relatively easy-to-measure, steady-state Frequency Response Functions (FRFs) to develop a state-space dynamical model of the structure under test. The developed model is then used to study the transient response of the structure and estimate its dispersion curves. This paper lays down the foundation of this approach and demonstrates its capabilities on a one-dimensional homogeneous beam using numerically calculated FRFs. Both in-plane and out-of-plane FRFs corresponding, respectively, to longitudinal (the first symmetric) and flexural (the first anti-symmetric) wave modes are analyzed. The effects of boundary conditions on the performance of this approach are also addressed.