Frequency Response Function Research Articles

Fast calculation of vehicle-road coupled response based on moving frequency response function

Vehicle–road coupled system is inherently time–varying, and its responses are traditionally calculated using time–domain methods which involves significant computational effort. Aiming to improve the efficiency of response calculation for the coupled system, this paper proposes a fast calculation method in frequency domain, based on the newly developed moving frequency response function (FRF). Firstly, considering the vibration characteristics of an infinitely long road, the road response is straightforwardly expressed using the road impulse response function (IRF). Subsequently, the concept of the road moving IRF is proposed and derived with respect to the moving observation points. The moving FRF is then obtained by applying Fourier transform, which allows the responses of the road moving observation points to be established in frequency domain for fast calculation under moving loads. Furthermore, by analyzing the vehicle–road coupled vibrations, based on the vehicle FRF and road moving FRF, a formula for the vehicle–road coupling force is derived in frequency domain, along with an expression for the responses at the vehicle–road contact points. Finally, the approach is illustrated in numerical simulations of vehicle–road coupled systems, and its computational efficiency and accuracy are verified through comparison with currently popular methods.

Modeling the Dynamic Properties of Multi-Layer Glass Fabric Sandwich Panels.

Sandwich panels are key components of many lightweight structures. They are often subjected to time-varying loads, which can cause various types of vibrations that adversely affect the functionality of the structure. That is why it is of such importance to predict the dynamic properties of both the panels and the structures made of them at the design stage. This paper presents finite element modeling of the dynamic properties (i.e., natural frequencies, mode shapes, and frequency response functions) of sandwich panels made of glass fabric impregnated with phenolic resin. The model reproducing the details of the panel structure was built using two-dimensional, quadrilateral, isoparametric plane elements. Afterwards, the model was subjected to an updating procedure based on experimentally determined frequency response functions. As a result, the average relative error for natural frequencies achieved numerically was 5.0%. Finally, a cabinet model consisting of the analyzed panels was built and experimentally verified. The relative error between the numerically and experimentally obtained natural frequencies was on average 5.9%.

A Review of Chatter Stability in Thin-Walled Mirror Milling

This paper addresses the chatter problem in thin-walled part milling, reviewing the advancements in mirror milling technology for thin-walled parts. It provides an in-depth discussion of the fundamental causes of chatter during thin-walled part milling. A comparative analysis of the advantages and disadvantages of empirical, finite element, and analytical models for milling force in thin-walled parts is presented. The paper also discusses the experimental methods, finite element modal analysis, and structural coupling methods used to obtain the frequency response functions of workpieces and tools, comparing their respective strengths and weaknesses. Furthermore, time-domain and frequency-domain methods for stability prediction are analyzed and compared, with a discussion on the future prospects of chatter stability prediction in mirror milling.

Hysteretic Dynamic Modeling and Vibration Characteristics Analysis of Metal-Rubber Isolators

A novel hysteretic constitutive model (HCM) with four parameters was proposed to reproduce the nonlinear elasticity and dry friction characteristics of metal–rubber isolators (MRIs). The HCM accurately characterized the nonlinear hysteretic behavior due to its intrinsic relationship with the geometric properties of the hysteresis loop. A physics-based parameter estimation strategy was developed to provide reasonable initial iteration values. Three MRIs with different design stiffnesses were tested to investigate their hysteretic behavior under various excitation levels. The HCM accurately reproduced the entire experimental hysteresis loops using only a subset of the experimental data. This exhibited higher precision and scalability compared to the classical dynamic model. Friction damping forces were employed to distinguish the stick-slip states of the three MRIs. Additionally, the identified parameters were used to simulate the steady-state vibration response of a complex MRI system. The resulting frequency response functions (FRFs) agreed well with the experimental data. The proposed model can effectively capture the amplitude-dependent effects in MRI systems with various design stiffnesses, enabling more accurate predictions of vibration responses.

Influence of bolt preload degradation on nonlinear vibration responses of jointed structures

The clamping performance of bolted joints was gradually degrading under long-term vibration loads, which induced a great impact on the stick–slip friction contact behavior of bolted joint interfaces. In this paper, a time-varying Iwan model was developed to reproduce the tangential hysteresis behavior of bolted joint interfaces by considering preload degradation. A three-degree-of-freedom mass–spring-damper oscillator was numerically simulated to investigate the influence of preload degradation on nonlinear vibration responses, as well as the experimental investigations of a bolted joint structure. The comparison results showed that the proposed model could well describe the effects of preload degradation on the stick–slip friction contact behavior of bolted joint interfaces. The numerical results revealed the nonlinear effects of preload degradation on the frequency response functions (FRFs), which demonstrated good agreement with experimental results.

Active Vibration Control Performance Comparison Based on Middle Pedestal Stiffness Using a Mobility Model and the Narrowband Fx-LMS Technique

Vibrations generated from equipment mounted on ships radiate into the water and affect covert operation capabilities. Accordingly, various studies are being conducted to reduce vibration transmitted from mounted equipment. In this study, a system consisting of mounting equipment, a 3-axis active mount, a middle pedestal, and a lower mount of the middle pedestal was modeled using a finite element analysis program, and a mobility model was constructed by calculating the frequency response function between the positions required for analysis. The error signal (primary path) obtained using the mobility model and the response at the operating point by the control force of the actuator (secondary path) are applied to the narrowband Fx-LMS algorithm for vibration control, and the control performance was compared. Through coupling analysis of the middle pedestal, the control influence according to the rigidity of the middle pedestal was analyzed. As a result of the control simulation, the time required for vibration control was controlled approximately 6 times faster in the model, with increased stiffness of the middle pedestal, and the vibration reduction performance was predicted to improve by a minimum of 0.9 dB and a maximum of 13.3 dB. Through this study, a simulation model that can provide a guide for the design of the middle pedestal of a ship was obtained, and it is expected that it can be utilized for a preliminary design review before manufacturing the middle pedestal of a ship.

Enhancing reliability in estimating modal parameters for automotive disk Brakes: A comprehensive approach using the Eigensystem Realization algorithm and additional criteria

This paper investigates the challenges in accurately identifying and assessing vibration characteristics in automotive disk brakes, which, due to their geometric symmetry, exhibit closely spaced vibration modes. Understanding the damping characteristics is crucial due to their significant impact on dynamic stability and the potential onset of brake squeal, as damping directly influences the stability threshold and its uneven distribution can destabilize the system. Therefore, this study primarily focuses on the identification of modal parameters with an emphasis on the damping of disk brakes. Traditional Experimental Modal Analysis (EMA) techniques, which rely heavily on Frequency Response Function (FRF) measurements, encounter challenges in distinguishing closely spaced modes, thereby affecting the reliability of parameter estimation, particularly damping. These algorithms are primarily based on FRF measurements and are of multi-input – single-output type (MISO). In comparison to Operational Modal Analyses (OMA) algorithms, which are widely used in civil engineering and are of the multi-input − multi-output type and contain statistical tools for assessing the uncertainty of the estimate, EMA provides limited possibilities to assess the quality of estimating parameters. The primary contribution of this work is demonstrating that the Eigensystem Realization Algorithm (ERA), supplemented with indicators such as Extended Modal Amplitude Coherence (EMAC), Modal Phase Collinearity (MPC), and the Consistent Mode Indicator (CMI), offers a robust tool for accurate damping estimation. Additionally, the application of ERA to data originally acquired for frequency-based methods, specifically the Rational Fraction Polynomial (RFP) method, allows for a direct comparison of these two EMA algorithms using the same dataset. The results demonstrate that the proposed identification process effectively distinguishes system modes from those influenced by noise, providing a pathway to optimize test data acquisition and improve the accuracy of modal parameter estimation, especially in systems with closely spaced modes. The method can also serve as a tool for optimizing the acquisition of test data to improve the estimation of modal parameters, by varying the positions of input (SIMO) or by transitioning to MIMO identification to enhance the observability and controllability of closely spaced modes of disk brake.

Characteristic of temporal light modulation reference meter and standard light source for calibration and verification

Abstract The measurement of the temporal light modulation (TLM) of a light source is gaining increasing importance owing to its detrimental effects on human vision and health. To establish traceability in TLM measurements, the National Institute of Metrology, China (NIM), established a TLM reference meter (TLMM) and standard TLM light source (STLS) for calibration and verification services. The TLMM and STLS employ detector- and source-based approaches, respectively, to define the standards for TLM quantities. The TLMM rapidly acquires light waveform data from the TLM light sources and calculates corresponding TLM quantities. Frequency calibration result is with an expanded uncertainty of 1 × 10−5, frequency response function is almost flat in range of 1 kHz–25 kHz, and response time is less than 8 × 10−6 s. The electrical noise was evaluated at approximately 1000:1–5000:1. The STLS generates an arbitrary TLM waveform by converting the virtual waveform into an actual LED output light waveform. For the frequency response function, the relative amplitude decreased by 1% with a frequency of 15 kHz, and response time was less than 5 × 10−6 s. The linearity of the frequency dependency was studied with ratio varied by approximately 0.19% when the periodic triangle excitation in frequency band from 1 Hz to 1 kHz. Experimental comparisons demonstrated that TLM quantities (frequency, percentage flicker, flicker index, short-term flicker severity P st LM , and stroboscopic effect visibility measure ( SVM )) are consistent between the TLMM and STLS devices on 14 typical TLM waveforms. For P st LM , most of the deviations were in the range of −0.019–0.079 (except for one with 0.182). For SVM , most of the deviations were in the range of −0.001–0.003 (except for one with 0.014).

A method for dynamic parameter identification of an industrial robot based on frequency response function

AbstractHaving accurate values of the dynamic parameters is necessary to characterize the dynamic behaviors of mechanical systems and for the prediction of their responses. To accurately describe the dynamic characteristics of industrial robots (IRs), a new method for dynamic parameter identification is proposed in this study with the goal of developing a real IR dynamics model that combines the multibody system transfer matrix method (MSTMM) and the nondominated sorting genetic algorithm‐II (NSGA‐II). First, the multibody dynamics model of an IR is developed using the MSTMM, by which its frequency response function (FRF) is calculated numerically. Then, the experimental modal analysis is conducted to measure the IR's actual FRF. Finally, the objective function of the minimum errors between the calculated and measured eigenfrequencies and FRFs are constructed to identify the dynamic parameters of the IR by the NSGA‐II algorithm. The simulated and experimental results illustrate the effectiveness of the methodology presented in this paper, which provides an alternative to the identification of IR dynamic parameters.

Fault tracing in multistage gearbox systems based on an improved transfer path analysis method

This study, has developed a novel structural dynamics decoupling method based on TPA. This approach effectively facilitates fault diagnosis and tracing in multistage gearbox systems. Initially, the research explores the relationship between the transfer function of systems with rigid link decoupling and the coupled system’s frequency response function. Compared with Huangfu et al. (2023) , the signal measurement points are reduced. The MWI (Yu et al., 2023) is employed to solve the inverse problem of bearing force identification. Subsequently, the decoupled frequency response and bearing forces are multiplied to calculate the path contribution, determining the dominant fault transfer path and thereby facilitating fault tracing in gearboxes. The effectiveness of this method is validated through numerical simulations and experimental studies, with fault characteristic enhancement exceeding 200%. To evaluate the method’s universality and robustness, experiments on diverse gearbox systems under varying conditions show enhanced fault characteristics across three fault types, confirming its generalizability.

Attenuation of Lamb waves in coupled-resonator viscoelastic waveguide

Guidance of elastic waves is one of the main applications of artificial crystal structures. The attenuation of the guided waves is, however, often overlooked, as most of the proposed waveguides only comprise ideal elastic materials. In this work, we study the propagation of evanescent Lamb waves guided in coupled-resonator viscoelastic waveguide (CRVW), with special attention to attenuation. CRVW is defined by considering a linear chain of coupled defect cavities in a phononic plate made of epoxy. The viscoelastic behavior of epoxy is characterized numerically by the Kelvin–Voigt (K–V) model. Based on finite element analysis, the complex band structure and the spectrum of frequency response function (FRF) are obtained. Due to viscosity, guided Lamb waves are spatially damped. Two theoretical models are devised to predict the displacement distributions inside and outside a bandgap for guided waves, respectively, considering either the first or the first two least evanescent Bloch waves identified in the complex band structure. A CRVW sample is fabricated and characterized experimentally by laser vibrometry. Evanescent Lamb waves are observed to be strongly confined along the waveguide and at the same time to decay rapidly along the waveguide axis. Experiments and numerical simulations are found to be in fair agreement. The present work is expected to inspire practical applications of highly confined viscoelastic phononic waveguides.

Parametric Study on the Effect of Rail Dampers on Track Decay Rate

Track decay rate (TDR), meaning the rate of attenuation of bending waves through the rail, is the most important indicator of a track’s dynamic characteristic impacting the rail noise emission. TDR depends on various parameters related to the construction of the track, and it can be increased using rail dampers. These are mechanical devices working on the principle of dynamic absorbers and are attached to the rail. This paper addresses the track with light rails needing improvements to reduce the rail noise emission using a particular rail damper with a mixed damping system (rubber–oil). The bending waves that propagate through the rail, the frequency response function of the rail, and TDR are investigated considering different scenarios regarding the parameters of the track: soft/stiff rail pad, tampered/settled ballast, and sleeper bay. To this end, an analytic model of the track featuring rail dampers consisting of an infinite Timoshenko beam with discrete attached oscillators is used. Numerical results show the possibility to increase TDR of railway track with light rails for both soft/stiff rail pads from 4 to 500 Hz up to 1250–1600 Hz using rail dampers with a mixed damping system.

Uncertainty sensitivity analysis for vibration properties of composite doubly-curved shallow shells using Kriging method

This paper presents a Kriging based global sensitivity analysis (GSA) method for the frequency response of displacements of composite doubly-curved shallow shells. A unified solution is utilized to develop the dynamic vibration formulation using the First-order Shear Deformation Theory (FSDT) and the Rayleigh–Ritz method. Kriging surrogate model is employed to substitute the frequency response function (FRF) of displacements. Ten parameters including materials and geometrical dimension are considered as input uncertain variables. A variance-based GSA method for dynamic model is employed to quantify the influence of each uncertain parameter. In addition, to avoid the computational burden of Monte Carlo simulation method (MCS), the presented sensitivity indices are computed analytically based on the Kriging mode, which further improves computational efficiency. Based on the convergence studies and comparison with traditional methods, the accuracy and efficiency of the present method are validated. The results shows that the frequency response of displacements exhibits greater sensitivity to changes in width, and thickness is more influential than others in the example from this article. Finally, the presented numerical results demonstrate vibration characteristics of different types of shells and observation points, which can also serve as a reference for further study on uncertainty-propagation analysis.

Modal analysis of key components of crusher based on digital simulation technology

The modal characteristics of the crusher rotor and shell constitute the crucial factors influencing vibration and noise. Based on the principle of simplification, the rotor component model was established. Through mesh optimization, the model accuracy and calculation efficiency can be ensured, and the calculation of natural frequency and modal shapes was completed based on ANSYS. To verify the accuracy of the finite element model, the modal test was carried out by the hammering method. Sensors were set in three different directions to obtain the frequency response function and the modal assurance criterion matrix mode confidence criterion. Using the same research method, the modal characteristics of the shell model were simulated and analyzed. The research results show that the modal parameters identified by the modal test are basically consistent with the simulation model. The natural frequencies of the rotor and the shell are quite different from the excitation frequency of the motor, and resonance problems will not occur when the crusher is proper functioning.

Modeling of Ball Screw–Nut Interface Stiffness With Wear (Ball Screw Wear Dynamics)

Abstract The effects of wear, preload loss, and missing balls on the dynamics of ball screw drives in machine tools are modeled and incorporated into the finite element model of the drive assembly for condition monitoring. The contacts between the ball–nut and ball–screw are modeled using Hertzian springs, whose stiffnesses vary as a function of the worn contact area. These contact stiffnesses are then transformed to the finite element nodes on the nut and screw. The frequency response functions at the motor shaft and table, which can be measured by commercial computer numerical control (CNC), are predicted for various faults at different positions of the table. The experimentally validated model demonstrates that the faults primarily affect the first coupled torsional-axial mode of the ball screw drive and can be utilized for automated condition monitoring of ball screw drives.

A Spatial-Frequency Approach to Point-Wise Frequency Response Function Estimation with Digital Image Correlation

The use of digital image correlation for modal analysis is becoming an appealing option thanks to its non-contact and full-field measurement process. However, frequency response function (FRF) estimation can be challenging due to the limited number of time domain data and heavy measurement noise. Thereby, the present work aims to propose a method which improves the estimation accuracy of point-wise FRFs. Firstly, a Gaussian-process-based spatial-frequency model is proposed, which makes use of the intrinsic properties of the FRF and the local spatial information of field measurements. Then, a Bayesian solution is developed, which is enforced by a stable and efficient numerical procedure. Finally, the effectiveness of the proposed method is verified by making a comparison with the spectral estimator through the use of simulated data, and it is further validated based on an experimental application.

Determination of the dynamic stiffness matrix of a rail damper and application to vibration decay rate

In this paper, the 6×6 dynamic stiffness matrix of a rail damper seen by a short section of rail is defined based on the force simplification theory, and a method to determine the matrix is presented. According to the method, frequency response functions (FRFs) between specific points on the rail as a rigid body are measured with hammer testing. These FRFs are then used to express the acceleration of the mass center of the rail as well as the rail's angular acceleration. As the third step, the equations of motion of the rail are established based on the laws of momentum and moment of momentum, from which the dynamic stiffness matrix can be worked out. The method is then applied to determine the dynamic stiffnesses of a typical damper, and the determined dynamic striffnesses are combined with an infinite periodic track model to stduy the effect of the damper on the verical and lateral vibration decay rates of the rail.

Automatic shock excitation system for dynamic calibration of six-axis force/torque sensor

The coupling behavior between the six-axis force/torque (F/T) sensor and the mechanical environment introduces variations in the dynamic parameters obtained through different calibration methods and devices, which poses challenges in proposing dynamic performance criteria. This paper presents a comprehensive automatic calibration system for determining the dynamic characteristics of the six-axis F/T sensor. Impulse force generators (IFGs), positioned at various orientations, apply three mutually orthogonal directional force and torque components to the sensor using a cruciform artifact. The system ensures repeatable impulse excitation by maintaining consistent spring compression in the IFG, measured by the displacement control system's grating ruler. The pulse width and frequency response function (FRF) are determined based on the response signal without measuring the input signal. The calibration frequency and the impact force magnitude can be controlled by selecting appropriate system parameters. Compared to manual impulse hammers, IFGs exhibit superior features such as high bandwidth, repeatability, and controllable force amplitude. They provide high-quality impulse excitation with calibration frequencies and forces up to 30 kHz and 2500 N or even higher, respectively. Furthermore, a linear relationship between peak value of impulse excitation and spring compression is studied to ensure effective excitation application. The consistency of dynamic calibration results of the six-axis F/T sensor verifies the effectiveness of this automatic calibration system.

Experimental and numerical analysis of the vibrational behaviour of metallic structures filled with damping materials

This work focuses on characterizing and modeling the vibrational behavior of hollow metallic structures filled with damping material. Vibration tests are first carried on a hollow structure, then two filling materials are considered: a cast viscoelastic material and a granular material composed of epoxy and polyurethane beads. Modal parameters (natural frequencies, modal damping and mode shapes) are extracted from the Frequency Response Functions with experimental modal analysis tools. Damping ratios greater than 5~\% are measured. Given the difference in the nature of the filling materials, dissipation mechanisms are different. The effectiveness of a damping material depends on the type of structural modes (bending, torsion or wall mode) to be damped, and dissipation in filling materials cannot be modelled by a constant loss factor or viscous damping. Consequently, a finite element modeling of the structure filled with viscoelastic material is proposed. Numerical modal parameters of the structure are correlated with experimental measurements. By extension, a bead mix material law, based on a fractional Zener model, is proposed, whose parameters are obtained by inverse identification.

Vibration analysis of 3D printed cymbals and Tzouras top plates

An integrated method of determining the vibration characteristics of musical instruments is here presented. The method includes modal measurements and holographic imaging, FEM modelling and simulations. Parts of musical instruments made of composite materials have been simulated, printed and measured. The vibrational response of 3D printed cymbals and Tzouras top plates, a Greek traditional stringed musical instrument, is experimentally measured and computationally simulated. The PLA printed parts vary in geometry and thickness. The frequency response functions, and the vibrational modes are presented and further studied in relation to the aforementioned geometry variations. The experimental findings provide inputs to FEM modeling and simulation parameters.