The innovation and advancement of pipeline leakage detection technology are crucial for ensuring water supply safety and the effective utilization of resources. Currently, transient wave-based leakage detection methods face challenges such as signal attenuation in complex pipeline systems, difficulties in identifying multiple leakage points, and insufficient real-time performance, which limit their localization accuracy and reliability. Therefore, this paper revisits the fundamental principles which form the basis for transient-based leak detection methods, systematically exploring the influence of leakage on the characteristics of the frequency response function (FRF) and its intrinsic relationship with the standing wave-leak interactions through experimental verification and theoretical analysis. The experiments reveal that leakage significantly alters the peaks and amplitudes of the FRF, with factors such as leakage volume, background pressure, transient pressure, and leakage location affecting its characteristics. Changes in leakage location can cause shifts in the FRF peak patterns through the standing wave-leak interactions, this phenomenon is similar to the Bragg effect, but the mechanism is not entirely the same. This study successfully constructs a frequency-domain mathematical model of transient wave-leakage coupling, innovatively introducing the precise and efficient Trikha-Vardy-Brown (TVB) unsteady friction model combined with viscoelastic theory to accurately simulate FRF characteristics under different pipe materials and operating conditions. Additionally, the study employs the matched-field processing (MFP) method to achieve accurate estimation of the location and size of single and multiple leaks, the single leakage error is within 0.6 m for steel pipes, within 4 m for acrylic pipes, and the multiple leakage errors are all within 4 m. In conclusion, this study confirms the effectiveness and accuracy of transient wave propagation technology in pipeline leakage detection, providing a new technical approach for the rapid diagnosis and localization of water pipeline leaks.

Milling is a well-established manufacturing process thanks to its accuracy and surface finishing, yet tool vibrations limit these qualities. At the core of many approaches dealing with vibrations the tool-tip frequency response function (FRF) is found allowing the computation of the appropriate cutting conditions for accuracy and stability. The evaluation of the tooltip FRF is often based on experimental modal tests which require additional sensors to be mounted on the machine tool as well as devices to excite the tool. However, the reliability of these methods is strictly related to the test setup and the expertise of the operator performing the measurement. This paper presents a methodology to evaluate the tooltip FRF starting from the surface location error (SLE) without requiring additional sensors or experienced operators. The method is based on a frequency domain cutting force model coupled with the SLE measured at different spindle speeds using on-machine measuring probe. The methodology was experimentally tested to check its effectiveness and limitations.

Abstract Accurate experimental and analytical identification of Frequency Response Functions (FRFs) of bio-based composites such as a tobacco composite panel (TCP) is challenging due to their anisotropy and heterogeneity, and variability in damping. Conventional experimental modal analysis (EMA) procedures, specially designed for isotropic materials, cannot be directly adopted to TCP. Similarly, the finite element (FE) modelling approach is limited by difficulties in representing damping, which often result in large discrepancies in both resonance and off-resonance FRF amplitudes. This study proposes a framework combining optimised EMA configurations with calibrated FE modelling to achieve accurate FRF prediction of TCP. Comparative EMA setups using different orientation suspensions are investigated and the outcomes evaluated. Vertical suspension at two corners is identified as the most suitable configuration due to its ability to reduce constraint interference. FE FRFs are compared with the EMA counterparts, and predictive accuracy is evaluated using the Modal Assurance Criterion (MAC) and Mean Squared Error (MSE). Damping calibration, through which modal damping values are tuned in the FE model to correlate EMA off-resonance amplitude trends, yields good agreement between the FE and EMA FRFs. The findings show that damping significantly affects the characteristics of off-resonance and resonance FRF peaks. The proposed framework establishes a validated and practical approach for accurate FRF prediction of TCP, enabling its broader use in engineering applications.

A method for identifying inertia properties of vibration structure from experimental frequency response functions

Acoustic modeling of bulky transformers for far-field noise prediction and treatment faces the challenges of computational inefficiency and inaccuracy due to the large model size and complex environment. In this paper, a transfer path analysis (TPA) model is built for two substation transformers by constructing virtual sound sources (VSSs) at a distance in front of the transformer surfaces. A real beamforming regularization (RBR) algorithm is proposed to solve the VSS signals by introducing the beamforming response to regularize the minimization task for the inverse problem. Unlike the original beamforming regularization that uses Green’s functions, the RBR employs measured frequency response functions between the VSSs and near-field receivers to calculate the beamforming response. The noise at far-field receivers 30 m away from the transformers is then predicted by the TPA model using the solved VSSs and validated against the measurement. The proposed TPA-RBR method is computationally efficient and accurate, considering that the maximum prediction error of the prominent noise components is within 3 dB. The noise contributions from different transmission paths are evaluated to provide guidance for further noise treatment. Our study also has potential applications in the localization of sound sources shielded by bulky obstacles.

Additively manufactured thermoplastics are increasingly employed in vibration-sensitive applications across the automotive, biomedical, and aerospace industries, where mechanical performance under dynamic loading is critical. This study provides a novel integration of Dynamic Mechanical Thermal Analysis (DMTA) with frequency response function (FRF) evaluation to capture the sub-glass-transition viscoelastic behaviour of 3D-printed ABS and PLA within 5–100 Hz range, a comparative approach not previously reported in the literature. The narrow 35–60 °C range was selected to focus on sub-Tg behaviour relevant to service conditions at which applications include vibration control in automotive panels, biomedical housings, and electronic enclosures. Unlike prior DMTA-only studies, this work combines FRF and DMTA to provide novel frequency-domain insights. The results reveal distinct thermomechanical responses: ABS demonstrates superior damping near its glass transition temperature due to its amorphous structure, whereas PLA exhibits stable stiffness and frequency-independent damping at lower temperatures, attributed to its semi-crystalline morphology. Key findings reveal that ABS exhibits a peak damping factor (tanδ ≈ 0.25) around 55 °C, while PLA maintains a higher storage modulus at lower temperatures (while PLA retains a storage modulus of ~ 1.2E + 08 Pa at 35 °C) but shows a sharp decline beyond 50 °C, limiting its use in thermally elevated conditions. These insights elucidate structure–property relationships in 3D-printed polymers and inform material selection for vibration-damping applications requiring thermal and frequency stability within sub-Tg operating regimes.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-26846-9.

Torsional vibrations in rotating machinery cause mechanical wear, electronic malfunctions, and a reduction in service life, particularly in high-speed industrial systems such as rotors. This study presents the development and integration of a Tuned Mass Damper (TMD) designed to mitigate damage to a die-cutting system. A theoretical model is formulated, demonstrating how an auxiliary mass coupled to a rotor absorbs energy at a designated frequency. Frequency response function analysis identifies torsional resonances, which are validated through a multibody model providing modal shapes and overall dynamic behavior. The design is carried out in strict compliance with the constraints and limitations of a real packaging machine. The TMD employs anti-vibration mounts, selected and tuned to deliver a required torsional stiffness based on finite element analysis used to determine their optimal radial placement. Experimental testing confirms theoretical predictions: the added inertia significantly reduced the first resonance peak and attenuated rotary torque oscillations, thereby improving the system’s dynamic response. These findings highlight passive torsional damping as a robust and effective approach to improving the rotor’s dynamic response and reducing alternating stresses, which predictively contributes to enhanced operational reliability and reduced machine downtime.

Industrial robots provide a flexible and cost-effective solution for machining large workpieces, but their application is often limited by inherent structural properties. Serial articulated robots, in particular, exhibit relatively low, configuration-dependent stiffness, making them susceptible to vibrations during milling that impair the final surface topography. To address this, this paper presents a predictive model for surface topography that integrates the robot’s posture-dependent dynamics with tool vibration. The proposed method begins by establishing an ideal kinematic model of the cutting edge trajectory. Next, the posture-dependent frequency response function is predicted using an inverse distance-weighted algorithm, and the dynamic parameters of the dominant vibration mode are identified. The resulting tool vibration displacements are then calculated by solving the system’s dynamic model and are integrated into the cutting edge’s swept surface. Milling experiments were conducted to validate the model, demonstrating strong agreement between the predicted and measured surface topography.

A vibration-based protocol is presented for identifying barely visible impact damage (BVID) in type-IV composite-overwrapped pressure vessels (COPVs). A 1 kJ hemispherical-tip strike was applied to a fully pressurized vessel, which was subsequently depressurized and characterized by free–free experimental modal analysis over a 168-point grid. The frequency response functions (FRFs) at the impact meridian exhibited distinct peaks near 3.70, 4.34, and 4.90 kHz with larger amplitudes and lower coherence than at the diametrically opposite meridian, indicating local circumferential stiffness loss. A detailed finite element model of the liner, bosses, and carbon/epoxy overwrap was updated by idealizing a cylindrical sub-volume with a 90% reduction in orthotropic stiffness. The pristine and “damaged” numerical modal sets agreed closely (mean frequency error < 2%), and for most of the first 60 modes, the diagonal Modal Assurance Criterion (MAC) remained ≥ 0.90. However, in several nearly degenerate circumferential mode pairs, the diagonal MAC dropped to 0.49–0.88 because the local asymmetry rotated the eigenvectors within a common subspace, showing that classical MAC alone cannot expose such early-stage defects. Radial displacement scan-lines provided the missing spatial resolution. Modes whose antinodal regions intersect the dent showed pronounced local amplitude bulges and slight angular shifts in the peak toward the impact site, whereas modes with a nodal line across the damage were virtually unchanged. The combined use of FRF asymmetry, MAC screening, and scan-line deformation profiling localized the impact to the correct circumferential sector with centimeter-scale resolution along the scan ring, yielding predictive signatures for rapid, non-pressurized in situ assessment of impacted COPVs after depressurization.

ABSTRACT Non-destructive vibration evaluations for detecting defects in additively manufactured parts have become a key research focus because they can identify flaws across an entire structure without targeting specific areas. Among different vibration-based techniques, one effective approach involves comparing the frequency response functions (FRFs) of healthy and defective parts to determine defect locations. Although vibration mode shapes are often used for this purpose, their application in additively manufactured components is complex, as small defects require high-frequency measurements and fine modal analysis. This study demonstrates that, by using the frequency response function curvature (FRFC) method, it is possible to detect even small defects at low frequencies, before the first resonance occurs. This characteristic provides a major practical advantage. To verify this, the researchers conducted modeling and numerical simulations on a thin plate with three cracks of varying depths and then performed experimental modal testing on a steel plate fabricated by selective laser melting, containing the same cracks. The results showed that, at low frequencies, the FRFC method accurately located cracks with 50%, 30%, and 10% damage intensities in simulations and achieved less than one percent error in experiments, confirming its precision and reliability for non-destructive evaluation.

Computational models serve as useful complements to physical experiments, but they require validation to build confidence in their applicability. This study outlines the validation of biomechanical models for mysticete sound reception, specifically using experiments involving an instrumented gray whale skull exposed to underwater sound. Detailed descriptions of the models are provided. The models were evaluated using a set of similarity metrics applied to both measured and computed frequency response functions. While high-quality agreement was not achieved, the models corresponded reasonably well with observed experimental data. A sensitivity analysis examined the models’ responses to variations in input material properties. Although these changes in material properties influenced model response, they accounted for only modest changes in similarity. A more significant challenge to achieving higher accuracy was the mismatch between the acoustic waves generated in experiments and the models’ assumption of plane wave loading. Despite this, the models successfully captured important biomechanical behavior, such as the enhancement of motion of the tympanic bullae relative to the basicranium. Model validation remains an ongoing endeavor, and this study represents an initial step.

In the present study, the theoretical framework underlying interferometric measurement using multimode lasers—forming the basis of laser vibrometry—is comprehensively described. A numerical simulation of the Doppler signal for the Laser Doppler Vibrometer (LDV) is conducted, and the technique for extracting vibration velocity via zero-crossing counting is detailed and subsequently verified through experimental assessment. In addition, two critical optical challenges in this field, namely the determination of the optimal object distances and laser focusing, are addressed. To overcome these issues, a novel set-up is proposed that incorporates systematic adjustments of the reference arm length along with active auto-focusing, facilitated by a laser rangefinder. Moreover, to validate the performance of the proposed device and facilitate comparison with conventional contact accelerometers, several frequency response function (FRF) tests were conducted under different conditions. The test cases included laser measurements when the accelerometer was not mounted on the plate, measurements obtained directly from the accelerometer, laser measurements when the beam was directed onto the accelerometer, and laser measurements when the beam was directed onto the plate. Comparative analysis of the data acquired via the accelerometer and the LDV confirmed the accuracy and reliability of the LDV set-up. Moreover, the measurements revealed that mass loading from the accelerometer caused detectable shifts in the natural frequencies of the 1-mm-thick galvanized plate—shifts that typically require a correction step often treated as modal analysis noise in the literature. This underscores the advantage of LDV’s non-contact nature in preserving the structure’s true vibrational characteristics of the sample and eliminates the need for such corrective procedures.

The development of NVH (Noise, Vibration, and Harshness) characteristics in vehicles is facing new challenges with the widespread utilization of electric drivetrains. This shift introduces new requirements in several areas, such as reduced noise and vibration levels, the need for advanced nonlinear characterization methods, and tuning/masking the typically more prominent tonal noise components. More accurate simulation and measurement techniques are essential to meet these demands. This study focuses on the experimental frequency response function (FRF) testing of electric drivetrain components, specifically on potential phase errors caused by inappropriate measurement settings. The influencing parameters and their quantitative effects are analyzed theoretically and demonstrated using real measurement data. A novel numerical approach, termed Maximum Phase Error Analysis (MPEA), is introduced to systematically quantify the largest potential phase errors due to arbitrary alignment between resonance frequencies and discrete spectral lines. MPEA enhances the robustness of phase accuracy assessment, especially critical for lightly damped systems and closely spaced resonance peaks. Based on the findings, optimal testing parameters are proposed to ensure phase errors remain within a predefined limit. The results can be applied in various dynamic testing scenarios, including durability testing and rattling analysis.

Purpose To guarantee smooth transmission and develop digital twins for status monitoring of belt drive systems (BDSs), this research aims to develop a physical model and perform dynamic analysis for BDSs. Design/methodology/approach A mathematical model of a three-pulley BDS is first developed in both the time and frequency domains, considering the hysteresis characteristics of the automatic tensioner. Then, a harmonic balance and alternating frequency/time domain (HB-AFT) method is proposed to calculate the dynamic characteristics of BDSs. For dynamic analysis, the frequency response function (FRF) and energy dissipation characteristics of automatic tensioners, as well as their influencing factors, are calculated and analyzed. Finally, the digital twins development and fault diagnosis methods of BDSs are proposed using the dynamic analysis results. Findings The dynamic characteristics of the BDS calculated by the HB-AFT method and traditional numerical methods are compared, and the comparison results show that the HB-AFT method has a calculation speed more than 40 times faster than the numerical iteration method. The FRF and energy dissipation characteristics of tensioners are related to the torsional vibration amplitude of the driving pulley and the sliding frictional torque of the tensioner, and the fault in tensioners can be detected by their oscillation amplitude calculated by digital twins. Originality/value The proposed HB-AFT method can be used to calculate the dynamic characteristics of BDSs with high accuracy and efficiency, while considering the hysteresis characteristics of the tensioner. It can also be used for status monitoring and fault diagnosis in digital twins of BDSs.

Since the support struts of the helicopter main gearbox are the main transmission path for the excitation force from the main rotor to the fuselage, vibration transmission can be suppressed effectively by embedding actuators in the support struts. A new helicopter active vibration isolation system with hydraulic actuators embedded into the helicopter main gearbox support struts is proposed. A dynamic model of the helicopter active vibration isolation system with an eight-strut centralized configuration is established using the frequency response function synthesis method. Based on this model, the characteristics of vibration transmission in the system are analyzed. Considering the influence of coupling crosstalk of control channels, a feedback global multichannel (FGMC) adaptive controller based on the filtered-x least mean square (FxLMS) algorithm is proposed. The simulation and experimental results indicate that the convergence rate and stability of an FGMC controller are better than those of conventional decentralized multichannel (DMC) FxLMS controllers. In the experiment, the root-mean-square value attenuations are over 63% and 40% for the FGMC and DMC controllers, respectively. The experimental results of variable amplitude disturbing force further verify that the FGMC controller has strong convergence and robustness.

In the development of domestic refrigeration compressors, optimizing the housing for acoustic performance is essential. Numerical simulations are commonly used to accelerate the design process, however, experimental validation is necessary to ensure the reliability of the results. Although modal analysis is often employed for validation, it does not fully capture the structure's dynamic response under excitation. Therefore, this research proposes a validation approach based on frequency response functions (FRFs), using an impact hammer to excite the structure and an accelerometer and microphone to measure the response. This method allows for the determination and validation of the dynamic response of the structure in terms of both vibration and acoustics. Different parts of the housing were excited, and the accelerometer and microphone were positioned in specific positions of the structure to capture dynamic response of the top, bottom and sides of the housing. Finally, this approach could, successfully, identify the structural modes and the harmonic response, validating behavior observed in the numerical simulation. The validation ensures that numerical simulations accurately represent the real system's dynamic behavior. This reliability enables future studies to optimize the housing geometry, aiming to minimize the noise generated by the compressor and improve its acoustic performance.

Due to structural characteristics and connection dimensions, the dynamic characteristics of dual metal rubber clamps (DMRCs) show significant differences in bolt connection direction and opening direction. Accurately identifying the dynamic parameters of DMRC in different directions is of great significance for analyzing the dynamic characteristics and vibration control of aero-engine piping systems. This paper takes a DMRC-double straight pipe structure as the research object and establishes a dynamic model of this structure based on the finite element method as the mechanical parameter identification model of DMRCs. A refined simulation mechanism is adopted in the model to reflect the dynamic characteristics of the DMRC. The DMRC is simplified into four concentrated mass blocks and four spring-damping groups to simulate its mass, stiffness, and damping effects. Each spring-damping group consists of a linear spring, a rotational spring, and a damper. The four groups of springs are further divided into two directional groups to simulate the stiffness and damping effects in the opening direction and bolt connection direction, respectively. Four concentrated mass blocks are applied to the four nodes of the pipe to simulate the mass effect of DMRCs. Based on the dynamic model of the pipeline structure mentioned above, the synchronous identification algorithms and procedures for multiple mechanical parameters of DMRCs are proposed, aiming to minimize the deviation of natural characteristic indicators (natural frequency and peak of frequency response function) obtained through testing and model simulation. This method can synchronously identify linear stiffness, rotational stiffness, and damping in different directions. Finally, the effectiveness of the identification method is verified through experiments.

Viscoelastic materials like natural rubber (NR) and poly naphthalene sulphonate (PNS) are commonly used in damping and sealing applications. This study aimed to estimate their damping parameters using aluminum as the base material. The materials were prepared into an unconstrained sandwich beam, known as an Oberst beam, to assess their damping behavior under temperature variations. The beams were subjected to sinusoidal base excitation (1 g) over a frequency range of 20 Hz to 1000 Hz at temperatures of 40°C, 60°C, and 80°C. Notable behavior was observed at 40°C and 80°C, with both materials exhibiting good damping below 40°C and softening above 80°C. The 3 dB method was applied to estimate damping characteristics at the first three fundamental natural frequencies, with frequency response functions (FRFs) captured at the beam tip. While the first natural frequency showed minimal frequency shifts, the second and third natural frequencies revealed more pronounced changes in damping parameters, particularly for NR. Temperature increased the damping ratio and loss factor for both materials, indicating higher energy dissipation, while the quality factor decreased. NR exhibited significant reductions in damping efficiency at higher temperatures, especially at the second and third natural frequencies, whereas PNS showed more stable behavior. These findings suggest NR is ideal for high-damping applications at elevated temperatures, while PNS is better suited for stable damping across a wider temperature range.

Abstract An analytical method is presented to find the stability domain of PID controller parameters for nonlinear time-delay systems by utilizing the Volterra series and its generalized frequency response function (GFRF), and all the parameters in the derived region can stabilize the system. According to the conditions for closed-loop L 2 stabilization, the procedures are derived to determine the PID controller stabilization region. The simulation results show the accuracy of the stability domain. Meanwhile, this proposed method preserves the nonlinear characteristics and is suitable for nonlinear time-delay systems with arbitrary-order nonlinearity. It also brings convenience to the tuning of PID parameters, providing a new idea for the practical design of nonlinear time-delay systems.

The paper proposes a metamaterial foundation to reduce the damage caused by the seismic waves on the buildings. Metamaterials are periodic structures composed of repeated unit cells of hard core and flexible cladding. Their ability to create low-frequency band gaps due to Bragg reflections and local resonance makes them ideal for seismic isolation. In the present work, a scaled model of the metamaterial foundation (1:80 of the original structure) has been investigated experimentally and computationally. A harmonic excitation is applied to the base of metamaterial foundation during the experiments on a vibration shaker. Frequency response functions (FRF) are measured for the reduction of vibration amplitude from the base to the top layer of metamaterial foundation. Experiments and simulations show that increasing the number of layers provides higher wave attenuation in the band gap region. Further, study of building on such a novel foundation is analysed. A frame is placed on the metamaterial foundation and the response has been studied. The metamaterial foundation is found to be very effective in reducing the vibration on frame/building. Based on the results and effectiveness of the scaled model, it is proposed that the larger metamaterial foundation can attenuate the seismic waves range from 6.6 Hz to 10.7 Hz.