Experimental Frequency Response Research Articles

Improved Bayesian Model Updating Method for Frequency Response Function with Metrics Utilizing NHBFT-PCA

To establish a high-fidelity model of engineering structures, this paper introduces an improved Bayesian model updating method for stochastic dynamic models based on frequency response functions (FRFs). A novel validation metric is proposed first within the Bayesian theory by using the normalized half-power bandwidth frequency transformation (NHBFT) and the principal component analysis (PCA) method to process the analytical and experimental frequency response functions. Subsequently, traditional Bayesian and approximate Bayesian computation (ABC) are improved by integrating NHBFT-PCA metrics for different application scenarios. The efficacy of the improved Bayesian model updating method is demonstrated through a numerical case involving a three-degrees-of-freedom system and the experimental case of a bolted joint lap plate structure. Comparative analysis shows that the improved method outperforms conventional methods. The efforts of this study provide an effective and efficient updating method for dynamic model updating based on the FRFs, addressing some of the existing challenges associated with FRF-based model updating.

Non-linear frequency response analysis for assessment of the ageing history of lithium ion batteries: A combined simulation and experimental approach

In this work, we present a novel approach for identifying the ageing history of lithium-ion batteries based on experimental nonlinear frequency response analysis (NFRA) measurements. A regression model, trained on simulated NFRA data, is shown to be capable of quantifying degradation modes such as solid electrolyte interphase (SEI) growth, lithium plating, and loss of active material (LAM) with no a-priori knowledge of the cell’s historical duty. Our analysis, combining experimental and simulation approaches, demonstrates NFRA’s potential as a powerful tool for ageing diagnosis by capturing various degradation modes. Changes in NFRA response through life exhibit strong correlations with ageing paths, particularly in the frequency range of 0.2 to 10 Hz. Observations highlight a strong influence of the state of charge on the resultant NFRA response, emphasizing that measurements at a single open circuit voltage (OCV) and harmonics values from a single frequency are insufficient for comprehensive characterization. This analysis underscores the need for correlating NFRA at multiple OCVs and frequencies for detailed ageing assessment. Evaluation on commercially relevant cells enhanced the models’ reliability for industrial applications. This quantitative, data-driven approach using NFRA holds potential to enhance battery management strategies, extend lifespan and improve confidence in second-life applications of batteries. Future work should focus on improving regression analysis robustness, reducing dimensionality, and broadening testing conditions.

Active Vibration Damping Using Rotor Torque Control in Electric Aircraft

Vibration damping in aerospace structures can decrease the likelihood of failures and instabilities and improve passenger comfort. This paper introduces the novel idea of damping vibration using the electric proprotors on aircraft without compromising flight control. The equations of motion of a cantilevered beam with a propeller at the tip driven by an electric motor are obtained using Hamilton’s principle, solved analytically in the frequency domain, and approximately in the time domain. Feeding back the beam tip angular rate to the motor torque is shown to asymptotically stabilize all transverse beam vibration modes. The overall vibration control consists of the inner rate feedback damping loop with an outer rotor speed control loop. Experimental frequency response and step response validate the models and show that the closed-loop damping in the first mode is three times higher than open loop with less than 1% rotor speed change for a 3% initial tip displacement. Theoretical results give good agreement with experiments. Parametric studies based on a 12 kg quadcopter indicate that if the rotor inertia is sufficiently large the proposed control can provide 8% damping with minimal impact on rotor speed.

Identification of the tip mass parameters in a beam-tip mass system using response surface methodology

Abstract In this study, a response surface based approach is introduced to determine the physical parameters of the tip mass of a beam – tip mass system, such as mass, mass moment of inertia and coordinates of the centre of gravity with respect to the beam end point. To this end, first, a difference function was formulated based on the differences between the peak frequencies and peak amplitudes of the experimental and analytical frequency response functions. Later, observation points were established in the design space using orthogonal arrays, and a response surface was developed using the difference function values at these points. Next, the tip mass parameters were determined by minimizing the response surface with genetic algorithm and particle swarm optimization as well as fmincon, a gradient-based solver of the Matlab program. For comparison purposes, those parameters were obtained by also direct minimization of the difference function with the same algorithms. It was concluded that the tip mass parameters were successfully determined within reasonable error limits by the response surface method with less computational burden. Finally, the effect of design space width on the response surface quality is demonstrated numerically.

Dynamic soil–structure interaction of a three-span railway bridge subject to high-speed train passage

In this study, the influence of Soil-Structure Interaction (SSI) on the dynamic behavior of a three-span concrete slab railway bridge with integrated retaining walls is investigated. The bridge is subjected to controlled excitations using a hydraulic actuator with different frequencies and load amplitudes. A 3D model of the railway bridge-soil system is implemented and calibrated using the experimental frequency response functions at each sensor location. A soil-free model is also created to compare with the calibrated model. It is observed that the dynamic behavior of the railway bridge is substantially altered by the presence of the surrounding soils, and neglecting SSI can lead to underestimation and inaccurate results. Additionally, the calibrated model is used for further train-passage analyses. For the studied bridge, neglecting SSI increases the maximum acceleration response of the bridge during high-speed train passages from 5.5 m/s2 up to 14.5 m/s2. It is also shown that the response of the bridge during train passage is predominantly influenced by its first bending mode, with higher modes inducing no discernible effect. Finally, parametric studies are performed in order to study the uncertainties related to the soil properties.

Vibrational timber characterization through the use of model updating

The characterization of orthotropic materials has challenged the vibration and acoustics community for quite some time. Complex composite materials such as wooden structures require attention to factors including moisture, grain boundaries in addition to macroscopic features. Here we devise a basic model developed by measuring the vibrational response in two separate axes to determine the material characteristics of a timber dowel. A proposed benchtop procedure utilises vibrometers and accelerometers to gather data before the updating process, for which, FEMtools was used. Based on the input material parameters, uncovered by previous studies, provide a starting point for the model updating procedure where experimental mode shapes and frequency responses are correlated to the finite element model. With the focus on radiata pine, the results show radial and tangential values converge similar to previous literature but with variation in the longitudinal direction and shear planes. Overall, this study provides a solid foundation to the characterization process of orthotropic materials like timber which can be further expanded into fields of structural health monitoring, damage detection and potential use in digital twins. The authors acknowledge the support of the Australian Research Council Linkage Project LP200301196.

Fractional-Order Models of Damping Phenomena in a Flexible Sensing Antenna Used for Haptic Robot Navigation

In this paper, two types of fractional-order damping are proposed for a single flexible link: internal and external friction, related to the material of the link and the environment, respectively. Considering these dampings, the Laplace transform is used to obtain the exact model of a slewing flexible link by means of the Euler–Bernoulli beam theory. The model obtained is used in a sensing antenna with the aim of accurately describing its dynamic behavior, thanks to the incorporation of the mentioned damping models. Therefore, experimental data are used to identify the damping phenomena of this system in the frequency domain. Welch’s method is employed to estimate the experimental frequency responses. To determine the best damping model for the sensing antenna, a cost function with two weighting forms is minimized for different model structures (i.e., with internal and/or external dampings of integer- and/or fractional-order), and their robustness and fitting performance are analyzed.

Quantification of the human–structure interaction effect through full-scale dynamic testing: The Folke Bernadotte Bridge

An analytical expression for the frequency response function of a coupled pedestrian-bridge system is presented and evaluated using an experimental measurement campaign performed on the Folke Bernadotte Bridge in Stockholm, Sweden. A finite element model and the modal models that consider the human–structure interaction effect are calibrated with respect to the measurements. The properties of the spring–mass–damper model representing the pedestrians were identified, considering the different structural modes of the system. Good agreement was obtained between the experimental and theoretical frequency response functions. A sensitivity analysis of the obtained solution was performed, validating the determined analytical expression for the frequency response function of the coupled pedestrian-bridge system that takes into account the human–structure interaction effect.

Determination of the stiffness of ecological asphalt mixtures with non-destructive impact resonance tests

The objective of this study is to characterize the linear viscoelastic behavior of ecological asphalt mixtures with non-destructive impact resonance testing. Two specimens of each of the four investigated mixtures were produced and tested in laboratory: a reference mixture (REFMIX), a mixture containing a modified binder with 30% of Kraft lignin in mass of total binder (LIGNIN), a mixture with 20% of reclaimed asphalt pavement (RAP) in mass of total aggregates (RAP20), and a mixture with 50% of RAP (RAP50). The specimens were tested at five temperatures (-20 °C, 0 °C, 10 °C, 20 °C, and 30 °C) with an impact resonance equipment to obtain experimental Frequency Response Functions (FRF’s). Subsequently, a back-calculation procedure was used to determine two linear viscoelastic properties (absolute value and phase angle of the complex modulus) from FRF’s measurements using finite element method (FEM) simulations. The results indicated a good repeatability of the test, with results from all specimens presenting the same trend, a decrease of the modulus and an increase of the phase angle with increasing temperature. Results also showed that the modified mixtures are stiffer than the reference mixture at each temperature. Kraft lignin has the greatest influence in increasing the stiffness in comparison with the reference mixture, followed by the addition of 50% and 20% of RAP. It was concluded that the addition of ecological materials, such as lignin and RAP, directly affects the viscoelastic behavior of bituminous materials, which can be beneficial for many applications.

Multi-beam piezoelectric systems by means of dynamically equivalent stiffness concept

Energy harvesting devices allow to obtain forms of energy present in nature and to convert them into electrical energy. One way of generating energy from mechanical vibrations is by using beams of piezoelectric materials. This paper proposes an alternative methodology for characterizing the dynamic behavior of a vibrating composite system composed of a cantilever steel base beam (primary system) and a piezoelectric beam attached to it. The approach involves representing the piezoelectric beam using an equivalent dynamic stiffness at its base. This simplifies the mathematical representation of the compound system and enables the system dynamics to be described solely in terms of the generalized coordinates of the primary system, which is advantageous in optimization environments where a reduced number of equations can facilitate analysis. To determine the equivalent dynamic stiffness, different mathematical models of one and multiple degrees of freedom are presented, including the description of the polyamide base of the piezoelectric sheet. An inverse problem is used to identify system parameters, and the energy generation over a wide range of frequencies is analyzed. Experimental frequency response functions of the voltage–acceleration type are obtained to validate numerical findings, demonstrating that the proposed methodology is a cost-effective alternative for parameter identification or optimal design in energy generation.

The dynamic coupling between the pulse wake mixing strategy and floating wind turbines

Abstract. In recent years, control techniques such as dynamic induction control (often referred to as “the pulse”) have shown great potential in increasing wake mixing, with the goal of minimising turbine-to-turbine interaction within a wind farm. Dynamic induction control disturbs the wake by varying the thrust of the turbine over time, which results in a time-varying induction zone. If applied to a floating wind turbine, this time-varying thrust force will, besides changing the wake, change the motion of the platform. In light of the expected movement, this work investigates if applying the pulse to a floating wind turbine yields similar results to that of the pulse applied to bottom-fixed turbines. This is done by considering first the magnitude of motions of the floating wind turbine due to the application of a time-varying thrust force and secondly the effect of these motions on the wake mixing. A frequency response experiment shows that the movement of the floating turbine is heavily frequency dependent, as is the thrust force. Time domain simulations, using a free-wake vortex method with uniform inflow, show that the expected gain in average wind speed at a distance of 5 rotor diameters downstream is more sensitive to the excitation frequency compared to a bottom-fixed turbine with the same pulse applied. This is due to the fact that, at certain frequencies, platform motion decreases the thrust force variation and thus reduces the onset of wake mixing.

Linear Time-Invariant Model Identification Algorithm for Mechatronic Systems Based on MIMO Frequency Response Data

This article describes a new frequency- domain multi-input multioutput linear time-invariant (LTI) system identification algorithm for accurate model construction suitable for motion control systems. The proposed method can capture the effects of time-delay, lightly damped poles (structural resonances), as well as highly damped complex or real poles, and direct or derivative-like terms. The effectiveness of the algorithm has been validated using experimental frequency response measurements obtained from different types of motion control mechanisms. The proposed algorithm has also been compared with “ <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tfest</monospace> ” and “ <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">modalfit</monospace> ” functions in MATLAB. Over these methods, nearly two orders of magnitude improvement is observed in the closeness of the model prediction, in terms of root mean square of the frequencywise modeling error.

Nonlinear analysis of a typical nonlinear beam structure based on experimentally extracted dynamical characteristics

Measuring nonlinear characteristics of a flexible structure is difficult and complicated since the nonlinear phenomena often exhibits several special features such as unstable vibration and bifurcation. The purpose of this paper is to propose a novel computing method for extracting dynamical characteristics of a typical nonlinear beam structure. Through comparing with the experimental data derived from modal experiments, the accuracy and feasibility of nonlinear normal modes function can be validated. Afterwards, the dimensionless dynamical equation that can indeed analyze nonlinear characteristics of the system is deduced, and the associated nonlinear frequency response is calculated based on Harmonic Balance method. According to comparisons between numerical simulations and experimental harmonic frequency response of the structure from a swept-sine experiment, the accuracy of the deduced dimensionless dynamical equation can be certified and the nonlinear analysis of the system is believed to be realized. Finally, the nonlinear attractor and bifurcation phenomena are discussed, and the stable condition is validated based on the Lorenz-like system.

Small-Signal Modeling of Phase-Shifted Digital PWM in Interleaved and Multilevel Converters

In this article, small-signal modeling of digital pulsewidth modulators (DPWMs) used in multicell voltage source converters (VSCs) is addressed. In addition to sampling and computation, DPWM introduces delay, which impairs VSC's dynamic performance and robustness. In order to take into account the influence of modulation delay, an accurate small-signal representation of DPWM is necessary. Here, modeling of multisampled bipolar and unipolar phase-shifted DPWMs for single-, double-, and multi-update strategies is presented. The simplest multilevel modulation of single-cell full-bridge VSCs, unipolar DPWM, is also covered by the analysis. The derived operating-point-dependent small-signal DPWM models are verified using simulated and experimental frequency response measurements up to four times the Nyquist frequency. Comparisons are also made with the models conventionally considered in the literature. Additionally, an approximate method is presented to model the influence of dead time on DPWM's small-signal dynamics. For the purposes of showcasing the importance of the proposed DPWM models, high-frequency admittance of a VSC employing multisampled multiupdate unipolar DPWM is modeled and verified in simulations and experiments.

Closed-loop identification of multi-axis active magnetic bearing systems based on decentralized and decoupling control

In response to the open-loop instability of magnetic bearing systems, this paper develops a closed-loop identification method to obtain an accurate plant model for controller design. The identification method is based on a decentralized and decoupling control architecture, which allows the decoupling of the multi-input-multi-output (MIMO) system into two single-input-single-output (SISO) systems for simplifying the identification process. The identification uses a pseudo-random binary sequence as the excitation signal. A parameterized model is obtained through a parameter estimation algorithm based on the output error model. To ensure that the model prediction error can converge during the computation process, a filter is incorporated into the estimation algorithm. The filter can be synthesized systematically by solving a set of linear matrix inequalities. The performance of the proposed closed-loop identification method is firstly verified by simulations. Then it is implemented on a five-axis magnetic bearing platform and the parameterized model obtained from the identification is compared to the experimental frequency response.

Design and Implementation of Robust Resonant Controller for Piezoelectrically Driven Micro-Lens Actuator

MEMS based piezoelectrically driven micro-lens actuator experiences resonance frequency shift with changing actuation voltage and exhibits low level of damping. In this paper, we present the design and experimental implementation of a resonant controller, robust against the variation in dynamics, to enhance the damping at the first resonance frequency. The controller is designed using an identified model of the micro-lens actuator from experimental frequency response data at a fixed actuating voltage. The closed-loop stability is guaranteed using the mixed negative-imaginary and small-gain approach. The experimental results demonstrate a damping of 21dB of the first resonance mode. The designed controller achieved the similar level of damping for the shift in resonance frequency due to the change in the actuation voltage. The settling time, due to a step change, of the piezoelectric micro-lens actuator is reduced to 7ms from 120ms. Such quick settling time is necessary for the fast auto-focusing functionality of the piezoelectrically driven micro-lens actuator in a miniaturized camera module. [2022-0137]

Magnetic Force-Assisted Nonlinear Three-Dimensional Wideband Energy Harvester Using Magnetostrictive/Piezoelectric Composite Transducers.

This paper presents a nonlinear magnetoelectric energy harvester which has the potential to harvest vibrational energy over a wide bandwidth in arbitrary motion directions. Three springs with equal intersection angles are adopted to absorb the multi-directional vibration energy. Magnetic interaction between the magnets and ME transducers allows the nonlinear motion with enhanced harvesting frequency range. Very good agreement is observed between the numerical and experimental open-circuit voltage output frequency response curves. The experimental results show that the harvester can harvest vibrational energy in an arbitrary direction, exhibiting a further bandwidth of 5.2 Hz. This study provides a new solution to effectively use the magnetoelectric energy harvester for multi-directional and bandwidth vibrational energy scavenging in the surrounding environment.

Flutter Boundary Prediction Based on Structural Frequency Response Functions Acquired from Ground Test

Establishing an accurate, fast, and low-risk flutter boundary prediction method is of great significance for flight vehicle design. In this paper, a ground flutter boundary prediction method (GFBP) based on experimental structural frequency response functions (FRFs) is proposed. A low-order multi-input multi-output (MIMO) aeroelastic system is established by combining the structural FRFs acquired from a ground test and the calculated unsteady aerodynamic FRFs in physical coordinates. The multivariable Nyquist criterion is used to predict the flutter boundary. A fixed-root aluminum plate wing is selected as the research model. A GFBP experiment is carried out for the wing’s normal state, leading-edge clump weight state, and trailing-edge clump weight state. The feasibility and accuracy of the proposed method are verified by comparison with theoretical flutter results, in which the errors of flutter speed and frequency in the test statistics are no more than 1.7%. In a simulation model established by the proposed method, Monte Carlo simulation is used to study the influence of deviations in the mode frequency and damping of the structural FRFs and deviations in the positions of excitation and measurement points in the ground test. The experiment and simulation results show that the proposed method can predict the flutter boundary accurately with accurate positions of excitation and measurement points, and it has good robustness to deviations in the mode frequency and amplitude of the structural FRFs.

Analyze the Performance of Headphones at Different Prices Through Survey and Frequency Response Experiment

Headphones on the market have a large price range, but is there a linear relationship between their performance and price? The article introduces an experiment to analyse the performance of headphones at different prices. Both blind listening survey and frequency response experiment were conducted to discover the relationship between headphone performance and price.

A New Crack Detection Method Based on the Correlation Analysis of Frequency Response Function Curvature Differentiation (FRFCD)

Structural inherent dynamic characteristics would change when damage occurs, and this fact is the theoretical basis of damage detection method based on vibration features. As one of structural inherent properties, frequency response function (FRF) contains rich information and has more potential in damage detection. On the basis of crack detection researches, a new method based on the change of FRF is proposed in this paper. First, the single-degree-of-freedom (SDOF) system is used to illustrated the change of dynamic features caused by cracks represented by changing stiffness, and it is found that the change of stiffness leads to the obvious nonlinear variation of frequency response function curvature (FRFC) and frequency response function curvature differentiation (FRFCD) near the natural frequency. To describe this phenomenon, the Pearson correlation coefficient is employed to analyze correlation relationship of FRF, FRFC and FRFCD between the undamaged and the damaged system. The results demonstrate that absolute values of these coefficients approach from 1 to 0 with the increase of stiffness change. The rail with cracks is taken as the research object, and relationships between the crack size and the correlation coefficients of FRF, FRFC and FRFCD are investigated through the combination of finite element simulation and frequency response experiment. The correlation coefficients of FRF, FRFC and FRFCD are close to 1 when the structure is intact. When cracks occur, the absolute value of the correlation coefficient will gradually trend from 1 to 0, and the sensitivity of FRFCD correlation coefficient to crack is much higher than that of FRF and FRFC. The selection of calculated frequency bands containing different modes and the damping of the structure would affect the absolute value of FRFCD correlation coefficient, but will not change the characteristics of high sensitivity of FRFCD and the relationship between the coefficient and crack size. Besides, noise signals would enhance the nonlinear relationship between responses of intact structure and the structure with cracks, and the FRFCD correlation coefficient value is influenced accordingly.