Frequency Response Function Of System Research Articles

Enhancing accuracy and robustness in FBS decoupling through Tikhonov regularization and unit balancing

Frequency-based Substructuring (FBS) enables the prediction of transfer functions for an assembled system based on those of a single-unit system. Conversely, FBS decoupling derives the frequency response function (FRF) of a single-unit system from the assembled system. This approach is valuable when obtaining the transfer functions of a system is challenging due to boundary conditions and deformation conditions. FBS decoupling can enhance accuracy by using additional indicator sensors. However, this often results in significant noise due to the ill-posed FRF matrix inversion. In this paper, Tikhonov regularization is employed to reduce noise-induced errors. Furthermore, this paper addresses the issue of inadequate Tikhonov regularization caused by the inherent scaling differences between the rotational and translational components within the FRF matrix. To rectify this challenge, a unit balancing method is proposed to mitigate the influence of scaling differences. This method enhances the accuracy and reliability of FRF matrix inversion processes, contributing to enhanced noise reduction and robustness in practical applications.

Research on The Influence of Continous Damping Control Shock Absorber on Frequency Response Function of Spindle Coupled Road Simulation Test Bench

When the semi-active suspension is tested based on the 6-DOF spindle coupled road simulation test bench, the traditional test method is no longer applicable, because the continous damping control shock absorber of the original vehicle controller cannot be used and the overall nonlinear of the system is stronger. At present, there are few related engineering applications in the industry. This paper introduces the current reproduction technology of semi-active suspension on the 6-DOF spindle coupled road simulation bench and researches the difference of the system frequency response function by excitation of continous damping control shock absorber with different control current.

Virtual vibration test rig for fatigue analysis of dozer push arms

AbstractTo obtain accurate fatigue life results for construction machinery components, acquiring load spectra is crucial, as their authenticity and validity directly determine the precision of the analysis. In working conditions, component attitudes change continuously, but they remain static on the vibration test rig (VTR), so the acquired target signals should match with the actual component attitudes in the driving signal generation. This paper proposes an efficient and economical simulation‐based virtual VTR for fatigue analysis of dozers. First, the relationship between the push arm rotation angle and the cylinder stroke is established, since the cylinder strokes can be measured easily in data acquisition experiments. Second, load decomposition is used to determine the attitude relationship between virtual VTR conditions and actual conditions, and target signals are calculated based on this attitude relationship and measured data. According to the system's frequency response function, the driving signals are iterated until the system's response signals converge with the target signals. Finally, the iteratively obtained load spectra are utilized for fatigue life analysis. The results show that the virtual VTR can effectively and accurately obtain the results of fatigue analysis and has engineering application significance.

Identification of modal parameters of soil specimen based on impact force

This study used vibration testing signals of soil samples under external loading to identify modal parameters (including natural frequencies and damping ratios) with different compaction degrees. Based on these parameters, a novel approach was proposed for reliable roadbed vibration compaction control and compaction process optimization. The experimental section utilized six soil samples with varying compaction degrees as experimental subjects, using the hammering method as the excitation mode. Subsequently, the frequency response function and modal parameters of the sample system were obtained through the acquisition, analysis, and parameter identification of samples’ acceleration signals. Firstly, samples with compaction degrees ranging from 88 % to 97 % primarily exhibited three modes, with the second modal frequency response displaying the weakest amplitude, and the fundamental mode being the dominant one. Additionally, parameter identification results revealed that the fundamental modal frequency exhibited a significant negative exponential growth with increasing compaction degree, while the second and third modal frequencies showed significant linear growth. Furthermore, the average damping ratio also demonstrated a tendency toward linear change with increasing compaction degree. Finally, the feasibility of modal parameters being actively used in practical engineering is discussed. Consequently, this study aimed to propose an indicator system for accurately assessing the bearing level of compacted soils from a modal dynamics perspective and to integrate modal dynamic indicators with density-class indicators into further optimization design work on road compaction processes.

Modal identification from turbulence response based on improved frequency domain decomposition

Turbulence excitation is an unavoidable form of excitation in flutter flight tests, and it is also a necessary and effective excitation method in high-speed flights, dives, other high-risk test, and high-frequency modal information mining. However, because the turbulence excitation signals are unmeasurable in the time domain, although the turbulence response contains rich and valuable flutter test information, the randomness and low quality of the data often cause difficulties in modal analysis. Therefore, an improved frequency domain decomposition algorithm for turbulence response processing is proposed in this paper. First, due to the irrelevancy of the atmospheric excitation, the power spectral density function matrix of the multi-channel turbulence response is subjected to singular value decomposition. There is a mathematical relationship between the maximum singular value curve and the system frequency response function. Second, the modal assurance criteria are used to calculate the maximum singular value of a single-degree-of-freedom system. Finally, an orthogonal polynomial method is applied to fit the maximum singular value curve, and the system identification is performed directly in the frequency domain. The simulated data and a certain type of aircraft flutter flight test data are used to verify the proposed method, and the results confirm the effectiveness and engineering applicability of the method developed in this work.

Vibration and force properties of an actuator formed from a piezoelectric stack within a frame structure

This paper examines the vibration and force properties of an actuator formed from a piezoelectric stack within a frame structure. Such configuration avoids the use of the traditional “earth-connected” support and allows the convenient attachment of the actuator to structures, such as power transformer tanks. The steady-state sensitivity of the actuator is analyzed, which comprised a linear frequency response function of the actuator system and a steady-state line frequency spectra of the transducer force at the frequency of the input voltage and its odd harmonics. The sensitivity is first analyzed by examining the forces transmitted to a rigid base structure through the stack and two legs of the frame when the actuator is excited by a single-frequency voltage input. The magnitude of the steady-state sensitivity at the frequency of the input voltage is approximately 15 dB higher than that at the other frequencies. The force transmitted through the piezoelectric stack is approximately twice as large as that through the two frame legs and with opposite phase. Next, the properties of transmitted forces when the rigid base structure is replaced by a beam and a plate of infinite size are also investigated. The interaction between the actuator and the elastic base structure has little effect on the magnitudes of the transmitted forces except at the resonace frequency of the actuator structure. However, the induced structural response is greatly affected and should be considered when describing the excitation capacity of the actuator. This capacity could be illustrated by the power flow between the actuator and the base structure and by the spatial distribution of the structural vibration induced by the actuator.

Filter Approximations for Random Vibroacoustics of Rigid Porous Media

Abstract An approximate efficient stochastic dynamics technique is developed for determining response statistics of linear systems with frequency-dependent parameters, which are used for modeling wave propagation through rigid porous media subject to stochastic excitation. This is done in conjunction with a filter approximation of the system frequency response function. The technique exhibits the following advantages compared to alternative solution treatments in the literature. First, relying on an input–output relationship in the frequency domain, the response power spectrum matrix is integrated analytically for determining the stationary response covariance matrix, at zero computational cost. Second, the proposed filter approximation facilitates a state-variable formulation of the governing stochastic differential equations in the time domain. This yields a coupled system of deterministic differential equations to be solved numerically for the response covariance matrix. Thus, the nonstationary (transient) response covariance can be computed in the time domain at a relatively low computational cost. Various numerical examples are considered for demonstrating the accuracy and computational efficiency of the herein developed technique. Comparisons with pertinent Monte Carlo simulation (MCS) data are included as well.

Tool Condition Monitoring Based on Nonlinear Output Frequency Response Functions and Multivariate Control Chart

Tool condition monitoring (TCM) is a key technology for intelligent manufacturing. The objective is to monitor the tool operation status and detect tool breakage so that the tool can be changed in time to avoid significant damage to workpieces and reduce manufacturing costs. Recently, an innovative TCM approach based on sensor data modelling and model frequency analysis has been proposed. Different from traditional signal feature-based monitoring, the data from sensors are utilized to build a dynamic process model. Then the nonlinear output frequency response functions, a concept which extends the linear system frequency response function to the nonlinear case, over the frequency range of the tooth passing frequency of the machining process are extracted to reveal tool health conditions. In order to extend the novel sensor data modelling and model frequency analysis to unsupervised condition monitoring of cutting tools, in the present study, a multivariate control chart is proposed for TCM based on the frequency-domain properties of machining processes derived from the innovative sensor data modelling and model frequency analysis. The feature dimension is reduced by principal component analysis first. Then the moving average strategy is exploited to generate monitoring variables and overcome the effects of noises. The milling experiments of titanium alloys are conducted to verify the effectiveness of the proposed approach in detecting excessive flank wears of solid carbide end mills. The results demonstrate the advantages of the new approach over conventional TCM techniques and its potential in industrial applications.

An investigation of active enclosed-space noise control with a nonlinear loudspeaker

Active control algorithms are widely used in enclosed-space noise control with the nonlinearities of secondary-source loudspeakers ignored. Inspired by nonlinear energy transfer resulting from nonlinear stiffness, this work concerns using a nonlinear loudspeaker to improve enclosed-space noise control. An enclosed-space active noise control system with a nonlinear loudspeaker is modeled. Based on the model, an approximate frequency response function of the noise control system is derived analytically via the harmonic balance method and verified using numerical methods. A globally consistent and bounded Lyapunov function is derived. The effects of the mass, nonlinear stiffness, volume, and area ratio on the noise reduction are also explored. Both analytical and numerical results demonstrate that the improved nonlinear loudspeaker is effective in improving the low-power characteristics of the loudspeaker, increasing noise reduction and improving fault-tolerant and robustness of the active noise control system.

Data-driven fractional order feedback and model-less feedforward control of a XY reluctance-actuated micropositioning stage.

This paper proposes a compound data-driven control method to solve the problems of low damping resonance, different dynamic properties, and hysteresis in the large-range compliant micropositioning stage driven by a Maxwell reluctance actuator. First, in order to verify the proposed control algorithm, a reluctance-actuated, XY compliant micropositioning stage is constructed according to the principle of operation of a reluctance actuator. Second, in order to eliminate the influence of complex dynamics on the controller design, a fractional order proportional-integral feedback controller is designed using a data iterative feedback turning algorithm. Third, the finite impulse response feedforward filter is optimized using experimental data, and the on-line inverse estimation of the system frequency response function and its iterative feedforward compensation are carried out to further eliminate the influence of light damping resonance. Finally, the proposed control method is used for tracking the experiment and compared with other methods. The experimental results show that the proposed control method can better meet the requirements of high precision, fast speed, and strong anti-interference ability for large stroke micro/nanopositioning and tracking.

Quantifying frequency containment reserve using cross-entropy frequency-constrained contingency-state-analysis model

With the increasing penetration of converter-interfaced generators, the frequency containment reserve (FCR) from conventional generators keeps going down, leading to a potential risk of frequency instability under contingencies. Consequently, Converter-interfaced generators are required to provide FCR and participate in the corrective rescheduling. Nevertheless, how to assess the FCR and quantify the adequacy of FCR under contingencies is a big challenge in modern new power system. To address this challenge, a cross-entropy-based frequency-constrained contingency-state-analysis (FC-CSA) model is proposed in this paper. Notably, both frequency control (FC) of units (i.e., conventional synchronous generators and converter-interfaced generators), and under frequency load shedding (UFLS) are incorporated in the primary frequency response. Then a unified system frequency response (SFR) function representing frequency dynamic is derived. This SFR function is extracted and reformulated as a group of mixed-integer linear constraints and participates in the traditional CSA model. Moreover, a set of frequency dynamic indexes, i.e., Expectation of UFLS risk, Expectation of FCR from conventional and converter-interfaced generators, is extended to depict the FCR that the power system requires. These indexes are calculated by the FC-CSA in a cross-entropy-based monte carlo simulation (CE-MCs). Case studies on a modified IEEE 6-bus test system and IEEE 118-bus test system are carried out to demonstrate the effectiveness of the proposed FC-CSA model.

On the Forced Vibration of Bending-Torsional-Warping Coupled Thin-Walled Beams Carrying Arbitrary Number of 3-DoF Spring-Damper-Mass Subsystems

This paper presents an analytical transfer matrix modeling framework for the forced vibration of a bending-torsional-warping coupling Euler-Bernoulli thin-walled beam carrying an arbitrary number of three degree-of-freedom (DOF) spring-damper-mass (SDM) subsystems. The thin-walled beam is divided into a series of distinct sub-beams whose ends are connected to the SDM subsystems. The transfer matrix for each sub-beam is developed based on the exact shape functions of the bending-torsional-warping coupling Euler-Bernoulli theory. Each SDM system is modelled by a set of effective springs based on the dynamic condensation method. The governing matrix equation is formulated based on the compatibility conditions of the placement and the force at the common interfaces of two adjacent sub-beams. Then, a closed-form expression for the frequency response function of the thin-walled beam system is proposed. The results computed by the proposed method achieve good agreement with those obtained by the conventional finite-element method, which shows the accuracy and reliability of the proposed method. The effects of the system parameters on the vibration transmission and vibration isolation properties of the thin-walled beam system are studied. The presented method can simultaneously consider arbitrary number of SDM subsystems and boundary conditions. Furthermore, none of the associated matrices are larger than 12 × 12, which provides a significant computational advantage.

Frequency-domain method for non-stationary stochastic vibrations of train-bridge coupled system with time-varying characteristics

When a train passes over a bridge, the vibrations of the vehicles and the bridge are the result of non-stationary stochastic processes due to the time-dependent characteristics of the coupled vehicle-bridge system. The aim of this study is to generalize the frequency domain method to investigate the non-stationary random vibration of the coupled system subjected to the excitation of track irregularities with consideration of time-dependent characteristics. To illustrate the method, a three-span simply supported bridge traversed by a single railway vehicle is adopted as an example. The time-dependent frequency response function (FRF) of the coupled system is theoretically derived through solving ordinary differential equations with variable complex coefficients, and the perturbation method is adopted to improve the calculation efficiency. By combining this with Priestley’s Evolutionary Spectra theory, the evolutionary power spectral density (PSD) of the non-stationary random response of the system is then derived. The transitions will occur when the wheels cross the joints between each bridge span and between the bridge and the adjacent roadway. By adopting mode shapes of the full structure, the change of states of the vehicle crossing multiple bridge spans and moving onto the roadway can be solved as a continuous process without separation. The proposed method is validated by comparisons with the Monte Carlo method, showing higher accuracy and efficiency when calculating the time-varying standard deviation of the response. It is found that the vibration of the vehicle is approximately stationary but with large variance due to the random track irregularities, while the bridge vibration follows a strongly non-stationary process with small randomness and is more related to the moving mass effect.

ИДЕНТИФИКАЦИЯ ДИНАМИЧЕСКОЙ СИСТЕМЫ С ЗОНОЙ НЕЧУВСТВИТЕЛЬНОСТИ УПРУГОГО ЭЛЕМЕНТА И СУХИМ ТРЕНИЕМ ПО ГОДОГРАФАМ

The study objective is to increase the efficiency of identifying parameters of dynamic systems that have an elastic element with a dead zone and dry friction in their composition according to experimental data. The problem to which the paper is devoted is to develop an identification algorithm that allows defining the parameters of the frequency response function of a nonlinear dynamic system for all experimentally obtained points of the system frequency curve. The research methods are based on the theory of dynamical systems, in particular on harmonic linearization. The novelty of the work is in applying modules of modifying deviations of the curve points of the system model from the corresponding experimentally obtained system curve points as a measure of proximity for the system curves and the model of the sum of squares. The study result is an algorithm for identifying a nonlinear system, which is reduced to solving a system of equations linear to unknown parameters of the transfer function of the system model. Computing experiment showed that the errors in defining the values of the system parameters with a dead zone of an elastic element and a non-linearity of dry friction type are approximately equal to the measurement errors of the experimental readings of this system curve. The main conclusion of the study is that the developed algorithm should be used to solve practical problems of identifying nonlinear systems, which include an elastic element with a dead zone and dry friction.

Response adjustable performance of a visco-elastomer sandwich plate with harmonic parameters and distributed supported masses under random loading

Vibration control of composite structures with distributed masses under random loadings is a significant issue. Adjustability of dynamic characteristics including response spectrum peaks and valleys is important for structural vibration control. The vibration control design in space contains structure and conformation designs which combination results in periodic composite structures. In the present paper, spatial periodicity control design is proposed. Stochastic response adjustable performance of a visco-elastomer sandwich plate with harmonic distribution of geometrical and physical parameters and distributed supported masses under random base motion loading is studied. Both facial layer thickness and core layer modulus of the sandwich plate are considered as harmonic distribution in length and width directions as well as periodically distributed masses. Partial differential equations of coupling motions of the sandwich plate system are derived and converted into ordinary differential equations for multi-mode coupling vibration. Generalized stiffness, damping, and mass coefficients are functions of the harmonic distribution parameters. An analysis solution with frequency response function and response spectral density expressions of the sandwich plate system is obtained. Numerical results are given to show the response adjustable performance through the harmonic geometrical and physical parameters and distributed masses. The results have a potential application to stochastic vibration control or dynamic optimization design of smart composite structure systems.

Asymptotic analysis of multiple mode structures equipped with multiple tuned mass dampers

The design of a mitigation dissipative system composed of several tuned mass dampers (TMDs) is not a simple task. As soon as several structural modes or several additional degrees-of-freedom are considered, the analytical response of the coupled system is not tractable anymore. The generic problem involving any number of structural modes and any number of attached tuned mass dampers is considered in this paper. The appropriate re-scaling of the problem is formulated in order to highlight the governing scales in the problem. An asymptotic analysis in the frequency domain, similar to the multiple timescale spectral analysis, is developed. The resulting expression for the frequency response function of the coupled system is expressed in a lower dimensional space, which offers the possibility to derive simple and accurate analytical solutions in other cases than the well-known single-mode single-TMD case. Although the major contribution of this paper consists in the derivation of this analytical formulation, two illustrations are also provided. The first one concerns the well-known single-mode single-TMD case, which illustrates the benefits of the proposed formulation and validates the resulting approximations against established literature. The second example concerns a single-mode double-TMD, where the proposed approximation is used to discuss the accuracy with which the two damping units need to be tuned in order to provide reasonable optimality with respect to mitigation performances.

Flexible interface models for force/displacement field reconstruction applications

Frequency response functions (FRF) are valuable structure characterizations that are used for the study of noise and vibration, load identification, modal analysis or model updating. They are usually estimated as mobilities by exciting the structure and measuring the normalized response for each frequency. However, rotational degrees of freedom (DOF) are typically difficult or impossible to measure directly. Previous studies have shown that rotational DOFs can be critical for the successful application of different techniques, such as dynamic substructuring. Consequently, different approaches have been proposed to take them into account, such as the finite differences method. Frequency Based Substructuring (FBS) is a dynamic substructuring technique that allows for the coupling of the frequency response functions of separate subsystems in order to synthesize the coupled system's FRFs without the physical assembly. This technique can be sensitive to inconsistencies between the substructures or incomplete information in the interface characterization. The inclusion of rotational degrees of freedom in the FRFs of the interfaces for the application of substructuring has been extensively discussed in literature, but there are still some limitations in frequency due to the assumptions that are required, especially the rigidity assumption of the interfaces. The goal of this paper is to present an experimental approach to relax the rigidity assumption and increase the complexity of the force and displacement field characterization of interfaces. This would extend the value of the estimated FRFs to higher frequency ranges, in which the local deformations start to have an important effect. To particularize the general approach, different possible models are proposed and discussed, highlighting their benefits and disadvantages. Some of these models are validated numerically and experimentally. Finally, the performance of the approach for substructuring applications is also analyzed on a numerical model.

Active Vibration Control of Helicopter Fuselage with Large Dynamic Modeling Errors

Active vibration control is an effective method for vibration control of a helicopter fuselage. However, when there are large dynamic modeling errors of a control system, the effectiveness of active vibration control may decrease, or even the control may diverge by using the existing harmonic control algorithms. Large modeling errors will result in inaccurate frequency response functions in the control system. In this paper, a new frequency response function updating-harmonic identification algorithm and its convergence condition have been proposed for effective active vibration control of a helicopter fuselage with large modeling errors through identifying the frequency response function of the control system in real time. In addition, to improve the accuracy of identifying frequency response function and further improve the effect of active vibration control, a dynamic harmonic control weight matrix algorithm is also proposed in this paper. The active vibration control simulations and experimental validation on a dynamically similar and geometrically scaled structure of a helicopter fuselage verify that the proposed algorithm in this paper can achieve faster speed of convergence and better effect of vibration control when there are large dynamic modeling errors over the existing harmonic control algorithms.

Influence of frequency resolution in case of frequency response function measurement in structural dynamics

Frequency resolution is an essential parameter in acoustical testing, even if we are using numerical or experimental method, for example when determining frequency response function (FRF) of a dynamic mechanical system, or executing modal analysis based on the FRFs. Finer resolution leads to more accurate results, at the expense of longer calculation/measurement process and larger data size. This parameter is generally set based on rules of thumb, prior practice or with big margin for safety. This results in waste time and data storage if the required frequency resolution is overestimated, or even significant errors in the results, if it is underestimated. Present paper offers a direct, method for the conscious determination of optimal frequency resolution. It is based fully on theoretical considerations, and investigates amplitude and phase distortion at resonances as target parameters. Beside defining the steps of the process, it is tested on a real structure, and the results are presented as well, proving the applicability and the appropriateness of the method. With this method, development engineers get a practical tool for adjusting the parameters of dynamic measurements and simulations.

A Novel Method for Fault Diagnosis of the Two-Input Two-Output Nonlinear Mass-Spring-Damper System Based on NOFRF and MBPCA

For fault diagnosis of the two-input two-output mass-spring-damper system, a novel method based on the nonlinear output frequency response function (NOFRF) and multiblock principal component analysis (MBPCA) is proposed. The NOFRF is the extension of the frequency response function of the linear system to the nonlinear system, which can reflect the inherent characteristics of the nonlinear system. Therefore, the NOFRF is used to obtain the original fault feature data. In order to reduce the amount of feature data, a multiblock principal component analysis method is used for fault feature extraction. The least squares support vector machine (LSSVM) is used to construct a multifault classifier. A simplified LSSVM model is adopted to improve the training speed, and the conjugate gradient algorithm is used to reduce the required storage of LSSVM training. A fault diagnosis simulation experiment of a two-input two-output mass-spring-damper system is carried out. The results show that the proposed method has good diagnosis performance, and the training speed of the simplified LSSVM model is significantly higher than the traditional LSSVM.