Frequency Response Function Analysis Research Articles

Sensitivity analysis of frequency response functions with imaginary parts decoupling based on multicomplex-step perturbation

Imaginary perturbation is used in the complex step differentiation method to compute first-order derivatives, widely known as an effective approach for sensitivity analysis in structural dynamics. However, coupling of imaginary parts occurs in the damped frequency response functions when employing this method. To mitigate this coupling, a novel approach for sensitivity analysis based on multicomplex-step perturbation is proposed in this paper, for sensitivity analysis of frequency response functions (FRFs) in structural dynamics. The structural parameters are perturbed in multicomplex domain, the dimensions of structural matrices are expanded using the Cauchy Riemann matrix representation, the equation of motion for sensitivity analysis in frequency domain is transformed to matrix operation in field of real numbers, imaginary term will not exist in the equation of motion for sensitivity analysis, the imaginary part of the frequency response function and the imaginary part of the perturbation are decoupled, the structural frequency response functions and the corresponding sensitivities are obtained from the dimension-expanded equation of motion. A truss structure and a solar wing are adopted to verify the accuracy of the proposed method. Results show that the sensitivity of FRFs can be effectively calculated using the proposed method. Compare to the finite difference method, the proposed method is not depended on the step-size selection procedure. The multi-order and mixed-order sensitivity matrices, especially Hessian matrix can also be obtained using the proposed method.

CT-FEM of the human thorax: Frequency response function and 3D harmonic analysis at resonance

Background and ObjectiveHigh-frequency chest wall compression (HFCC) therapy by airway clearance devices (ACDs) acts on the rheological properties of bronchial mucus to assist in clearing pulmonary secretions. Investigating low-frequency vibrations on the human thorax through numerical simulations is critical to ensure consistency and repeatability of studies by reducing extreme variability in body measurements across individuals. This study aims to present the numerical investigation of the harmonic acoustic excitation of ACDs on the human chest as a gentle and effective HFCC therapy. MethodsFour software programs were sequentially used to visualize medical images, decrease the number of surfaces, generate and repair meshes, and conduct numerical analysis, respectively. The developed methodology supplied the validation of the effect of HFCC through computed tomography-based finite element analysis (CT-FEM) of a human thorax. To illustrate the vibroacoustic characteristics of the HFCC therapy device, a 146-decibel sound pressure level (dBSPL) was applied on the back-chest surface of the model. Frequency response function (FRF) across 5–100 Hz was analyzed to characterize the behaviour of the human thorax with the state-space model. ResultsWe discovered that FRF pertaining to accelerance equals 0.138 m/s2N at the peak frequency of 28 Hz, which is consistent with two independent experimental airway clearance studies reported in the literature. The state-space model assessed two apparent resonance frequencies at 28 Hz and 41 Hz for the human thorax. The total displacement, kinetic energy density, and elastic strain energy density were furthermore quantified at 1 µm, 5.2 µJ/m3, and 140.7 µJ/m3, respectively, at the resonance frequency. In order to deepen our understanding of the impact on internal organs, the model underwent simulations in both the time domain and frequency domain for a comprehensive analysis. ConclusionOverall, the present study enabled determining and validating FRF of the human thorax to roll out the inconsistencies, contributing to the health of individuals with investigating gentle but effective HFCC therapy conditions with ACDs. This innovative finding furthermore provides greater clarity and a tangible understanding of the subject by simulating the responses of CT-FEM of the human thorax and internal organs at resonance.

A New Top-Mounted Shear-Hinge Structure Based on Modal Theory and Rubber-Pad Damping Theory

Steel-spring floating-slab tracks (SSFSTs) are widely used as efficient vibration-damping beds, and in China, they are mainly used in subways and municipal railroads. The shear hinge is an important component that improves the stability of the line, and field research has found that the top-mounted shear hinge (TMSH) undergoes varying degrees of damage, which indirectly affects the safety and stability of line operation. In this work, we studied the causes of damage to TMSHs, designed a new TMSH structure with a rubber-pad layer installed based on modal theory and rubber-pad vibration-damping theory, and proved that the new structure can reduce the occurrence of damage by comparing it with the original TMSH structure. The main aspects of this study are as follows: Firstly, the ultimate load capacity of the existing and new TMSH structures was checked by establishing a refined finite-element model. Then, modal analysis and frequency-response function analysis were carried out based on modal theory and frequency-response function theory to reveal the causes of TMSH damage and prove that the new structure can effectively delay damage. Finally, the modal and vibration patterns of the two structures were obtained via indoor hammering tests and compared with the simulation results. The results show that the two TMSH structures are in line with the strength requirements, and the existing TMSH damage mainly results from the resonance between its natural frequency and the high-excitation frequency of the floating slab under long-term cyclic train loading, causing high-frequency vibration fatigue damage. It is also demonstrated that the new structure can effectively reduce the natural frequency of the TMSH so that its value is located in the region of low vibration on the floating slab. The excitation vibration levels of the TMSH mounted on the curved section of the 4.8 m floating slab and the 3.6 m floating slab were reduced by 9 dB and at least 3 dB, respectively. After adding rubber pads located in the 400–3000 Hz floating-slab high-vibration-level region of the TMSH damage-prone parts, the amplitude reduction, including lateral excitation of damage-prone parts, resulted in a vibration amplitude reduction of more than 30 dB. However, the vertical excitation of the mid-end and rear-end bolts slightly increased their amplitudes, whereas the shear-rod amplitude was reduced by 48 dB, and the front-bolt amplitude was reduced by 5.28 dB. The natural frequency and vibration pattern obtained from the hammering test were consistent with the simulation results, and the reliability of our conclusions was verified from both experimental and simulation perspectives.

Fatigue analysis of coil springs in the primary suspension of a railway vehicle based on synthetic spectrum for time-varying vibration load

The fatigue life of coil springs is usually predicted with a stationary Gaussian vibration load and deterministic structural parameters. However, the obtained fatigue life is inconsistent with the observed fatigue life of fractured springs which varies within a wide range. The work aims to propose a method to predict the fatigue life of the coil spring by considering the time-varying vibration load, i.e., root mean square (rms) varies with time and the uncertainties of geometric parameters. First, a synthetic method for time-varying vibration loads is proposed. The time-varying load is decomposed into multiple stationary Gaussian short samples represented by their power spectral density (PSD). These PSDs are synthesized according to the distribution characteristics of spectral values, in which data that are non-Gaussian are processed with the Johnson system. Second, the influence of parameter uncertainties in the coil spring is studied by a Monte Carlo analysis of the stress frequency response function. Finally, the fatigue life is calculated and compared with the results predicted by using the measured stress. The results show that the synthetic spectrum has almost the same damage potential as the measured time-varying load. In comparison with results predicted from the measured stress, the synthetic spectrum gives much better estimates of the fatigue life of the coil spring than the average spectrum. Parameter uncertainties of coil springs significantly affect fatigue life and should be taken into account.

Uncertain frequency response analysis of clamp-pipe systems via the coordinate transformed polynomial chaos expansion

The frequency response function (FRF) that links structural responses and external excitations is required to realize the dynamic analysis of a clamp-pipe system. Due to random geometry and stiffness parameters, the FRF becomes a stochastic quantity that is represented by a surrogate model based on the polynomial chaos expansion (PCE) in this paper. Considering that an FRF trajectory is mainly controlled by resonance and anti-resonance frequencies, a reference frequency coordinate characterized by the controlling frequencies associated with a nominal value of input random variables is first determined. By reserving the original amplitude, an FRF sample is projected to the reference coordinate through a linear transform. Then, a PCE surrogate model with coefficients estimated by the Gauss-quadrature scheme is obtained, and the utility of an inverse transform allows one to obtain the predicted FRF result in the original domain. Together with estimations on the confidence intervals and extreme boundary results, engineering applications are demonstrated by the uncertain FRF analysis of straight and L-shape clamp-pipe systems.

Power Device Interface Characterization With Low-Cost Thermal System Identification

This article develops a technique, requiring no dedicated temperature sensing calibration step, to rapidly characterize transient heat transfer in packaged, power semiconductor components. It is presented as an alternative to traditional step response characterization methods by exploiting the phase delay metric native to frequency response function (FRF) analysis in the field of system identification. This article presents the principles of power device physics and transient waveform analysis to identify the design space in which FRF data extracted from experiments are robust. Electronic circuitry is introduced, providing the needed, periodic heat actuation, and a measurement strategy leveraging direct processing of a temperature-sensitive electrical parameter (TSEP). The developed method is applied to make a key measurement confirming the high-frequency-only thermal FRF sensitivity to component die-attach condition. The measurement is shown to align with an output from a partial differential thermal model embedding a single, corrective scaling factor. Overall, this article highlights the emergent opportunity to measure a packaged power device’s transient thermal impedance with standard lab equipment and the ongoing opportunity to realize converter degradation self-sensing.

Innovative optimization approach to improve noise vibration harshness and lightweight in lower control arm

This paper presents a multi-material design for a lightweight swing arm of a McPherson suspension. Starting from a stamped steel lower control arm (LCA) of a C-segment vehicle a hybrid arm made of a metal core and two Carbon Fiber Reinforced Plastics covers has been hand sketched, Computer Aided Design designed, and Finite Element Method tested. To fully exploit hybridization potential, it has developed a specific methodology which combines the topological optimization of the metal core, and the 3-phase optimization of the composite covers. Also, an innovative viscoelastic material has been included in the laminates to enhance the damping properties of the arm and a Frequency Response Function analysis has been performed. The results in terms of mass, stiffness and stress distribution have been compared with the original stamped steel LCA and with another hybrid LCA design. The final comparison has shown that with the redesign approach it has been possible to reduce the mass to a greater extent without affecting stiffness performance and stress distribution.

Prediction of Frequency Response Function for Cylindrical Thin-Walled Workpiece with Fixture Support Constraints

Auxiliary fixtures are widely used to enhance the rigidity of cylindrical thin-walled workpieces (CTWWs) in the machining process. Nevertheless, the accurate and efficient prediction of frequency response function (FRF) for the workpiece-fixture system remains challenging due to the complicated contact constraints between workpiece and fixture. This paper proposes an analytical solution for the comprehensive FRF analysis of the CTWW-fixture system. Firstly, based on the vector mechanics, the mode shape functions of the workpiece are presented using the classical theory of thin shell. The variable separation method is utilized to deal with the inter-mode coupling of the workpiece. Secondly, the motion equation of the CTWW with fixture constraints is established using analytical mechanics from the viewpoint of energy balance. Finally, the FRFs of the CTWW-fixture system are derived by means of modal superposition. Experimental modal tests verify that the predicted FRFs are in good agreement with the measured curves.

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.

Unsupervised Damage Identification Scheme Using PCA-Reduced Frequency Response Function and Waveform Chain Code Analysis

Mechanical machines face structural damage problems during their service life. Structural damage can severely affect safety and functionality of the structure and lead to economic loss. In this work, a damage identification scheme is developed by combining waveform chain code (WCC) analysis and hierarchical cluster analysis based on complex network theory. Waveform chain code analysis was carried out using the principal component analysis reduced frequency response function (PCA-reduced FRF), and the areas under the slope differential value curves were calculated as damage-sensitive WCC features. Unsupervised machine learning using hierarchical cluster analysis was then conducted on these damage-sensitive features. A rectangular Perspex plate was studied using the newly developed damage identification scheme as an example. Experimental results showed that the proposed scheme can successfully separate all the damage conditions from the undamaged state with 100% accuracy. In terms of damage severity and location identification, the proposed scheme is sensitive to detect damage severity with damage index as low as 0.17. In addition, combination of PCA-reduced FRF and mode shapes showed positive correlation between the magnitude of the resonant peak and the displacement of the impact point in identifying different damage locations of the plate.

The construction and comparison of damage detection index based on the nonlinear output frequency response function and experimental analysis

When the structures or parts are subjected to fatigue damage, its nonlinear characteristics will be characterized sensitively. The nonlinear characteristics of the system can be well described by the nonlinear output frequency response function deduced by the Volterra series. The nonlinear output frequency response function is the collection of frequency functions of various orders which contain different nonlinear information about the system. In other words, it is not intuitive when it is used directly for the detection of fatigue damage to structures or parts. Therefore, how to build an intuitive detection index will be the key to the application of the nonlinear output frequency response function in engineering practice. In this paper, based on the concept of information entropy, complexity and divergence, 4 detection indexes were proposed and the definition and estimation methods were given. Fatigue tests were carried out on the specimens by a fatigue test machine, and a series of specimens with different fatigue cycle times were obtained. The nonlinear output frequency response function was estimated by using the data of the hammer excitation measurement. The detection effect of 4 detection indexes and traditional detection index Fe was analyzed and compared. The results show that the proposed detection indexes are more sensitive and effective for detecting fatigue damage before microcracks formed.

Investigation of Concrete Floor Vibration Using Heel-Drop Test

In recent years, there is an increased in floor vibration problems of structures like residential and commercial building. Vibration is defined as a serviceability issue related to the comfort of the occupant or damage equipment. Human activities are the main source of vibration in the building and it could affect the human comfort and annoyance of residents in the building when the vibration exceed the recommend level. A new building, Madrasah Tahfiz located at Yong Peng have vibration problem when load subjected on the first floor of the building. However, the limitation of vibration occurs on building is unknown. Therefore, testing is needed to determine the vibration behaviour (frequency, damping ratio and mode shape) of the building. Heel-drop with pace 2Hz was used in field measurement to obtain the vibration response. Since, the heel-drop test results would vary in light of person performance, test are carried out three time to reduce uncertainty. Natural frequency from Frequency Response Function analysis (FRF) is 17.4Hz, 16.8, 17.4Hz respectively for each test.

Analytical Stability Criterion in Haptic Rendering: The Role of Damping

A new closed-form equation for stability analysis of a haptic device is presented using frequency response function analysis via continuous time model of the system. Desired impedance of the virtual environment (VE), time delay, and zero-order hold are considered, beside the continuous model of the haptic device. The developed equations provide critical virtual damping and stiffness of the VE versus the mass and viscous friction of the haptic device, sampling time, and time delay. Unlike prior work in this field, the developed equations are valid without any limitation on the values of the time delay and virtual damping. It is shown that they cover available well-known equations in literature for special cases of small values of virtual damping and time delay. The resulting equations are of practical usefulness in many fields, such as surgery simulation for avoiding instability during virtual tool interaction with high-stiffness VEs. The proposed analytical derivation can also be used for studying the effect of operator, sensors, actuator dynamics, and velocity filtering. Simulation and experimental results on the KUKA light weight robot show that the proposed criterion can accurately predict the stability boundaries.

High-frequency random vibration analysis of a high-speed vehicle–track system with the frequency-dependent dynamic properties of rail pads using a hybrid SEM–SM method

ABSTRACTA hybrid Spectral Element Method (SEM)–Symplectic Method(SM) method for high-efficiency computation of the high-frequency random vibrations of a high-speed vehicle–track system with the frequency-dependent dynamic properties of rail pads is presented. First, the Williams-Landel-Ferry (WLF) formula and Fractional Derivative Zener (FDZ) model were, respectively, applied for prediction and representation of the frequency-dependent dynamic properties of Vossloh 300 rail pads frequently used in China's high-speed railway. Then, the proposed hybrid SEM–SM method was used to investigate the influence of the frequency-dependent dynamic performance of Vossloh 300 rail pads on the high-frequency random vibrations of high-speed vehicle–track systems at various train speeds or different levels of rail surface roughness. The experimental results indicate that the storage stiffness and loss factors of Vossloh 300 rail pad increase with the decrease in dynamic loads or the increase in preloads within 0.1–10,000 Hz at 20°C, and basically linearly increase with frequency in a logarithmic coordinate system. The results computed by the hybrid SEM–SM method demonstrate that the frequency-dependent viscous damping of Vossloh 300 rail pads, compared with its constant viscous damping and frequency-dependent stiffness, has a much more conspicuous influence on the medium-frequency (i.e. 20–63 Hz) random vibrations of car bodies and rail fasteners, and on the mid- (i.e. 20–63 Hz) and high-frequency (i.e. 630–1250 Hz) random vibrations of bogies, wheels and rails, especially with the increase in train speeds or the deterioration of rail surface roughness. The two sensitive frequency bands can also be validated by frequency response function (FRF) analysis of the proposed infinite rail–fastener model. The mid and high frequencies influenced by the frequency-dependent viscous damping of rail pads are exactly the dominant frequencies of ground vibration acceleration and wheel rolling noise caused by high-speed railways, respectively. Even though the existing time-domain (or frequency-domain) finite track models associated with the time-domain (or frequency-domain) fractional derivative viscoelastic (FDV) models of rail pads can also be used to reach the same conclusions, the hybrid SEM–SM method in which only one element is required to compute the high-order vibration modes of infinite rail is more appropriate for high-efficiency analysis of the high-frequency random vibrations of high-speed vehicle–track systems.

The Researches on Damage Detection Method for Truss Structures

This paper presents an effective method to detect damage in truss structures. Numerical simulation and experimental analysis were carried out on a damaged truss structure under instantaneous excitation. The ideal excitation point and appropriate hammering method were determined to extract time domain signals under two working conditions. The frequency response function and principal component analysis were used for data processing, and the angle between the frequency response function vectors was selected as a damage index to ascertain the location of a damaged bar in the truss structure. In the numerical simulation, the time domain signal of all nodes was extracted to determine the location of the damaged bar. In the experimental analysis, the time domain signal of a portion of the nodes was extracted on the basis of an optimal sensor placement method based on the node strain energy coefficient. The results of the numerical simulation and experimental analysis showed that the damage detection method based on the frequency response function and principal component analysis could locate the damaged bar accurately.

Optimum design of vibratory electrode in micro-EDM process

Micro-electro discharge machining (EDM) has a crucial role in fabrication of parts such as molds. One of the most challenging aspects of EDM process relates to flushing of debris. Increasing the material removal rate causes significant enhancement in EDM process. In this study, new numerical approach for enhancing flushing is proposed. A model of vibratory electrode and dielectric flow pattern is validated with theoretical and experimental results. Next, the effect of vibratory electrode geometry and its material properties on flushing of debris is investigated. To this end, design of experiment method is applied to find optimum vibratory electrode shape and material. At first step, vibrational analysis is performed for electrodes to find mode shapes and their corresponding frequency utilizing the finite element method. Vibrational results have good agreement with theoretical ones that are presented in literatures. Next, frequency response function analysis is done for vertical excitation force to find out displacements of electrodes in longitudinal modes. The results of these simulations are utilized to construct 3D computational fluid dynamic (CFD) models to find velocity distribution on gap space between electrode and work piece. Full factorial design is used to evaluate electrode’s parameters. The results from these simulations are studied to achieve best electrode shape that is able to flush the gap flow significantly. Finally, dielectric flow regime for optimum electrode shape is surveyed.

An investigation of detection metrics for the remote acoustic sensing of mechanical changes in a vibrating plate

Acoustic radiation from a mechanical structure due to broadband forcing is inherently dependent on the structure’s material, geometry, and boundary conditions. Measurements of the radiated field can be used to detect mechanical changes (i.e., defects) when compared to known baseline measurements of the same structure. However, there are many available options for evaluating these changes, and each presents benefits and liabilities. In this presentation, several detection metrics are evaluated for the remote acoustic detection of mechanical changes in a 0.3-m-square by 3-mm-thick aluminum plate with 100 to 2000 Hz broadband forcing. The radiated acoustic field from the vibrating plate is recorded with a 15-element receiver array used to reconstruct an estimate of the plate’s acoustic response. This response is then quantitatively compared to previous baseline responses using cross-correlation, spectral analysis, wavelet analysis, frequency response function analysis, and other relevant detection techniques. The performance of each detection metric is evaluated and ranked by considering the minimum detectible size of a change, required a priori knowledge of the system, and ease of implementation. The feasibility of classification of the detected changes is also considered. [Sponsored by NAVSEA through the NEEC, and by the US DoD through an NDSEG Fellowship.]

스톡브리지 댐퍼 설계 파라미터 선정

The Stockbridge damper is used to control the aeolian vibration of a overhead transmission line due to the natural wind under a low velocity, between 1 m/s to 7 m/s. The damper model can be simply derived with several design parameters and the location of eigenvalues of design parameters are important to determine the efficiency of energy dissipation by excitation itself with two counterweights. First, the importance of resonance frequencies of Stockbridge damper was reviewed through the analysis of frequency response function of damper system. Then, the best selection of design parameters was investigated with the introduction of objected function that minimize the distance between the calculated eigenvalues and target frequency points. The best choice of design parameters was reviewed using the simulated results from the objective function and the effectiveness of selected design case was discussed at the point view of practical implementation.

Experiments on vibration control of a piezoelectric laminated paraboloidal shell

A paraboloidal shell plays a key role in aerospace and optical structural systems applied to large optical reflector, communications antenna, rocket fairing, missile radome, etc. Due to the complexity of analytical procedures, an experimental study of active vibration control of a piezoelectric laminated paraboloidal shell by positive position feedback is carried out. Sixteen PVDF patches are laminated inside and outside of the shell, in which eight of them are used as sensors and eight as actuators to control the vibration of the first two natural modes. Lower natural frequencies and vibration modes of the paraboloidal shell are obtained via the frequency response function analysis by Modal VIEW software. A mathematical model of the control system is formulated by means of parameter identification. The first shell mode is controlled as well as coupled the first and second modes based on the positive position feedback (PPF) algorithm. To minimize the control energy consumption in orbit, an adaptive modal control method is developed in this study by using the PPF in laboratory experiments. The control system collects vibration signals from the piezoelectric sensors to identify location(s) of the largest vibration amplitudes and then select the best two from eight PVDF actuators to apply control forces so that the modal vibration suppression could be accomplished adaptively and effectively.

Structural acoustics analysis and optimization of an enclosed box-damped structure based on response surface methodology

In this paper, the structural acoustics analysis and optimization of an enclosed box-damped structure are investigated by using the response surface methodology (RSM). Acoustic frequency response function analysis, i.e. a unit harmonic force imposing on structure and calculating the sound pressure in cavity, is applied to achieve the critical frequency. The acoustic sensitivity analysis of sound pressure level with respect to the thicknesses of damping layer panels are employed to identify the significant variables. With the help of faced central composite design, an efficient set of sample points are generated, and then the second order polynomial function of the sound pressure response at critical frequency is computed and verified by the adjusted coefficient of multiple determination. After the response surface function is verified, the effect of the thicknesses of damping layer panels on sound pressure is analyzed quantitatively, and the thicknesses of damping layer panels are further optimized to minimize the sound pressure response of the target node. The results indicate that, by using the RSM, the computational time for structural acoustic is saved and the optimization process is simple. The sound pressure of the target node is controlled effectively with less damping material used.