Complex Frequency Response Functions Research Articles

Stable parameters identification for rational approximation of Single degree of freedom frequency response function of semi‐infinite medium

AbstractThe rational approximation is an important tool for establishing a dynamic analysis model in time domain for semi‐infinite water and soil. It accurately represents the interaction between the semi‐infinite medium and an engineering structure. Current research on rational approximation method focuses mainly on the establishment of time‐domain analysis models. However, the stability, accuracy, and calculation efficiency of the rational approximation model cannot be guaranteed simultaneously. Based on linear system control theory, this work decomposes the continuous‐time rational approximation (CRA) and the discrete‐time rational approximation (DRA) into a combination of first‐ and second‐order subsystems, and the stability boundaries for the identified parameters are established according to the stability conditions of each subsystem. A genetic algorithm and a sequential quadratic programming algorithm are used to construct a parameter identification method that can ensure stability in the time domain. Through parameter identification of complex frequency response functions, it is demonstrated that the method proposed in this work can guarantee the stability and accuracy simultaneously, and the calculation efficiency is also substantially improved over that of the existing methods.

FRF-based finite element model updating for non-viscous and non-proportionally damped systems

All structures exhibit some form of damping, but the characterization of damping is not well-understood, and there is no universal damping model for the dynamic systems. Recently, model updating methods have been used to update or identify the damping matrix in dynamic systems. Most of the finite element updating methods assume viscous and proportional damping models for updating or identifying of damping matrix. In this paper, a new finite element model updating method is proposed in which the damping model is assumed as non-viscous and non-proportional. A parametric exponential non-viscous damping model has been used to model the damping in the dynamic system. The proposed method is the frequency response function (FRF)-based updating model, which updates the non-viscous and non-proportional damping matrix in the dynamic system. The effectiveness of the proposed damped finite element updating method is demonstrated by a numerical example and actual laboratory experiments. First, a numerical study is performed on a cantilever beam structure with non-viscous and non-proportional damping. The numerical study is followed by cases involving actual measured data. Joints and boundary conditions are assumed as a major source of damping, therefore joints and boundary conditions are modelled using relaxation functions and damping coefficients. The updated results have shown that the proposed damped element model updating method can be used to derive accurate models for the non-viscous and non-proportional damped systems. This is illustrated by matching complex FRFs obtained from the updated model with from the experimental data.

A novel amplitude-FRF based SDOF resonator parameter extraction method

Single degree-of-freedom (SDOF) resonators are commonly used in modeling of dynamical systems. The SDOF resonator parameter extraction methods utilize the complex form of the frequency response function (FRF), which may not be available. The half-power bandwidth does not require the complex FRF, but uses only three points of the amplitude response, making it ineffective when the FRF data is noisy or incomplete. In this study, a linear regression based SDOF resonator parameter extraction method is presented. The method applies weighted least squares to the manipulated resonator FRF. The performance of the proposed method is compared with least-squares regression based on inverse-squared FRF and the half-power bandwidth method, using noise and nonlinearity added FRF data and experimental data. It is concluded that the proposed method outweighs other methods using amplitude FRF.

Vibration fatigue dynamic stress simulation under non-stationary state

The methods of vibration fatigue stress simulation and strength evaluation under stationary state are quite complete, but the research on vibration fatigue under non-stationary state is insufficient. Based on the theory of structural dynamics and the principle of modal superposition method, the formula for calculating the dynamic stress of vibration fatigue under non-stationary state is deduced in this paper. The complex frequency response function (FRF) of each mode is calculated by finite element program, and the impulse response function (IRF) of each mode is obtained by inverse fast Fourier transform (IFFT) method, so as to realize the decoupling of each mode. Then the principle of modal superposition method is applied to time domain, considering the external multi-input loads of the structure, the dynamic stress caused by each external load can be obtained by convolution calculation based on the load and the corresponding IRF. The final dynamic stress can be obtained by superposition of the dynamic stress caused by each load. So, this method can easily calculate the effect of each mode or each external load on structural damage, which is helpful for the study of structural vibration fatigue failure reasons. Finally, the simulated dynamic stress under non-stationary state is compared with the experimental dynamic stress, and three indicators, which are average maximum stress, structural damage and frequency evolution, are taken into account in the comparison. Compared with the experimental data, the damage error calculated by the stress time history between simulation and experiment is 2.84%, the trend of stress time history obtained by simulation is almost the same as the experimental result, and the frequency evolution process under non-stationary state can also be reproduced well. The comparison results show that the proposed method in this paper can be used as an effective reference for structural vibration fatigue dynamic stress simulation under non-stationary state.

Identification of non-proportional structural damping using experimental modal analysis data

A verified computational model of a complex structure is crucial for reliable vibro-acoustic simulations. Mass and stiffness matrices of such a computational model may be constructed correctly, provided all the design information is available. Since it is an unknown, the damping matrix is usually populated through mathematical models based on some assumptions. In the current study, it is proposed to use the identified non-proportional structural damping matrix in the computational model. Structural damping matrix can be identified using the complex frequency response functions obtained from experimental modal analysis data. No matter what type of a damping mechanism a structure has; proportional or non-proportional, the frequency response functions of the system can be measured. First, the calculation procedure for the non-proportional structural damping matrix is explained. The damping matrix of an analytical model is identified successfully using the proposed procedure. The same procedure is then applied through a case study. Computational model of a test vehicle is constructed. Next, the test vehicle is subjected to a modal test to measure the frequency response functions of the structure. Incompleteness of the measured data and the requirements of the procedure are discussed, as well. The described procedure can be used in any model updating framework.

The frequency response function of the creep compliance

Motivated from the need to convert time-dependent rheometry data into complex frequency response functions, this paper studies the frequency response function of the creep compliance that is coined the complex creep function. While for any physically realizable viscoelastic model the Fourier transform of the creep compliance diverges in the classical sense, the paper shows that the complex creep function, in spite of exhibiting strong singularities, it can be constructed with the calculus of generalized functions. The mathematical expressions of the real and imaginary parts of the Fourier transform of the creep compliance of simple rheological networks derived in this paper are shown to be Hilbert pairs; therefore, returning back in the time domain a causal creep compliance. The paper proceeds by showing how a measured creep compliance of a solid-like or a fluid-like viscoelastic material can be decomposed into elementary functions with parameters that can be identified from best fit of experimental data. The proposed technique allows for a direct determination of the parameters of the corresponding viscoelastic models and leads to dependable expressions of their complex-frequency response functions.

Experimental Evaluation and Modeling of the Damping Properties of Multi-Layer Coated Composites

In this work, the dissipative properties of different coating solutions are compared and a beam mechanical model, taking into account of dissipative actions at the interface between different layers is proposed. The aim is to find optimal coatings to be employed in the production of composites with high damping properties. The investigated coating layers are obtained from different materials and production processes, and are applied on different metallic substrates. The composite specimens, in the form of slender beams, are tested by means of forced excitation dynamic measurements. Force and displacement experimental data, in a wide range of excitation frequencies, are used to estimate the system damping properties. Homogeneous, uncoated specimens are also tested for comparison. A specific identification procedure is used to identify the specimens stress-strain relationship in the frequency domain. The ratio of the imaginary part and the modulus of the specimen estimated complex frequency response function is considered as a measurement of the damping behaviour. A modified third order multi layered beam model, based on the zig-zag beam theory, is proposed. The model takes into account the contribution to the damping behaviour of the frictional actions and slipping at the interface between layers. Frictional actions are modelled by means of a complex, elasto-hysteretic contribution.

Estimating the Complex Impedance of an MUTC Photodetector

This paper derives, and then implements, a maximum likelihood estimation procedure for the equivalent circuit model parameters of a photodetector. Specifically, we consider a frequency domain identification procedure whereby the observed data take the form of a complex frequency response function, here the reflection coefficient, $S_{11}$ . Both linear least-squares and maximum likelihood methods are compared in their respective performance. Results are presented for various bias voltages and dc photocurrent.

Transfer path analysis: Current practice, trade-offs and consideration of damping

Current practice of experimental transfer path analysis is discussed in the context of trade-offs between accuracy and time cost. An overview of methods, which propose solutions for structure borne noise, is given, where assumptions, drawbacks and advantages of methods are stated theoretically. Applicability of methods is also investigated, where an engine induced structure borne noise of an automobile is taken as a reference problem. Depending on this particular problem, sources of measurement errors, processing operations that affect results and physical obstacles faced in the application are analysed. While an operational measurement is common in all stated methods, when it comes to removal of source, or the need for an external excitation, discrepancies are present. Depending on the chosen method, promised outcomes like independent characterisation of the source, or getting information about mounts also differ. Although many aspects of the problem are reported in the literature, damping and its effects are not considered. Damping effect is embedded in the measured complex frequency response functions, and it is needed to be analysed in the post processing step. Effects of damping, reasons and methods to analyse them are discussed in detail. In this regard, a new procedure, which increases the accuracy of results, is also proposed.

주파수 응답 데이터를 이용한 비비례 점성감쇠행렬 추정

Accurate identification of damping matrix in structures is very important for predicting vibration responses and estimating parameters or other characteristics affected by energy dissipation. In this paper, damping matrix identification method that use normal frequency response functions, which were estimated from complex frequency response functions, is proposed. The complex frequency response functions were obtained from the experimental data of the structure. The nonproportional damping matrix was identified through the proposed method. Two numerical examples (lumped-mass model and cantilever beam model) were considered to verify the performance of the proposed method. As a result, the damping matrix of the nonproportional system was accurately identified.

Analysis and Optimization of a Vibration Isolation Platform Based on 6-DOF Parallel Mechanism

In this paper a 6-DOF(Degree of Freedom) vibration isolation platform with structural configuration of 3CPS (3 Cylindrical-Prismatic-Spherical)-1UPS (1 Universal-Prismatic-Spherical) parallel mechanism is proposed. Firstly, the dynamic model is developed through virtual work principle. The transfer function and the complex frequency response function were obtained from the dynamical equation. Then, 36 amplitude frequency diagrams are obtained from the frequency response function. By analyzing the 36 amplitude frequency diagrams, 10 vibration coupling terms were chosen. The range of the spring stiffness coefficient and the range of the damping coefficient of the magneto-rheological damper are determined by analyzing the acceleration transmissibility of the 10 vibration coupling terms. At last, an integrated optimization combining structural and controller optimization is developed. The optimal parameters of the parallel mechanism are obtained through Genetic Algorithm (GA). This work is significant for the prototype design and verification of the vibration isolator.

Glenis Long's contribution to animal psychoacoustics

Glenis and I were psychology graduate students together in 1970-1971. She was interested in auditory psychophysics, but soon developed an interest in comparative psychophysics which she pursued in a post-doc and a faculty position in Germany, where she obtained a behavioral audiogram for the horseshoe bat and later studied masking in the same species. This audiogram was one of the very first psychophysical investigations of hearing in any bat and the very first for the horseshoe bat. It confirmed a rather complex frequency response function with sensitivity peaks at about 20 kHz and 60 kHz, and a very sharp peak at about 80 kHz. The masked thresholds indicated rather normal critical masking ratios (CR) except in a narrow region at about 80 kHz where the CRs are up to 15 dB lower than expected, suggesting a critical band of about 300 Hz at 80 kHz that is presumably used in Doppler-shift compensation. In 1981-1983, Glenis went on to investigate frequency and rate modulation discrimination in the chinchilla, for the first time, and to study tone-on-tone masking in the chinchilla. Since these early works, Glenis has maintained her comparative interests with studies on birds, kangaroo rats, frogs, and fleas on cats.

Parametric control of structural responses using an optimal passive tuned mass damper under stationary Gaussian white noise excitations

In this study, the structural control strategy utilizing a passive tuned mass damper (TMD) system as a seismic damping device is outlined, highlighting the parametric optimization approach for displacement and acceleration control. The theory of stationary random processes and complex frequency response functions are explained and adopted. For the vibration control of an undamped structure, the optimal parameters of a TMD, such as the optimal tuning frequency and optimal damping ratio, to stationary Gaussian white noise acceleration are investigated by using a parametric optimization procedure. For damped structures, a numerical searching technique is used to obtain the optimal parameters of the TMD, and then the explicit formulae for these optimal parameters are derived through a sequence of curve-fitting schemes. Using these specified optimal parameters, several different controlled responses are examined, and then the displacement and acceleration based control effectiveness indices of the TMD are examined from the view point of RMS values. From the viewpoint of the RMS values of displacement and acceleration, the optimal TMDs adopted in this study shows clear performance improvements for the simplified model examined, and this means that the effective optimization of the TMD has a good potential as a customized target response-based structural strategy.

A method for damping matrix identification using frequency response data

Accurate modeling of damping in structures is of great importance for vibration response analysis and control. This paper addresses the issue of identification of damping matrix of a structure by posing it as a finite element damping matrix updating problem. Many of the current updating approaches, dealing with updating of damping matrix, perform simultaneous updating of mass, stiffness and damping matrices. However, such a strategy is faced with numerical problems in practical implementation, since the magnitude of stiffness and mass matrix elements is generally much more than that of the damping matrix elements causing difficulties in accurate identification of the damping matrix. Some other approaches divide the process of updating of the mass and stiffness matrix and the damping matrix into two stages, but their application is restricted to structures with low levels of damping. This paper addresses these issues by developing an updating formulation that seeks to separate updating of the damping matrix from that of updating of the stiffness and the mass matrix. The proposed damping matrix updating method utilizes the concept of normal frequency response functions (FRFs) available in the literature. The method is formulated so as to reduce the difference between the complex FRFs, which can be measured in practice, and the normal FRFs, whose estimates can be obtained from the measured complex FRFs. The effectiveness of the proposed method is demonstrated through a numerical study on a simple but representative beam structure. The issue of coordinate incompleteness and robustness of the method under presence of noise is investigated. It is found that the proposed method is effective in the accurate identification of the damping matrix in cases of complete, incomplete and noisy data and is not limited by the level of damping in the structure.

Normal response function method for mass and stiffness matrix updating using complex FRFs

Quite often a structural dynamic finite element model is required to be updated so as to accurately predict the dynamic characteristics like natural frequencies and the mode shapes. Since in many situations undamped natural frequencies and mode shapes need to be predicted, it has generally been the practice in these situations to seek updating of only mass and stiffness matrix so as to obtain a reliable prediction model. Updating using frequency response functions (FRFs) has been one of the widely used approaches for updating, including updating of mass and stiffness matrices. However, the problem with FRF based methods, for updating mass and stiffness matrices, is that these methods are based on use of complex FRFs. Use of complex FRFs to update mass and stiffness matrices is not theoretically correct as complex FRFs are not only affected by these two matrices but also by the damping matrix. Therefore, in situations where updating of only mass and stiffness matrices using FRFs is required, the use of complex FRFs based updating formulation is not fully justified and would lead to inaccurate updated models. This paper addresses this difficulty and proposes an improved FRF based finite element model updating procedure using the concept of normal FRFs. The proposed method is a modified version of the existing response function method that is based on the complex FRFs.The effectiveness of the proposed method is validated through a numerical study of a simple but representative beam structure. The effect of coordinate incompleteness and robustness of method under presence of noise is investigated. The results of updating obtained by the improved method are compared with the existing response function method. The performance of the two approaches is compared for cases of light, medium and heavily damped structures. It is found that the proposed improved method is effective in updating of mass and stiffness matrices in all the cases of complete and incomplete data and with all levels and types of damping.

Propagation of flexural wave in periodic beam on elastic foundations

The propagation properties of flexural wave in the periodic beam on elastic foundations are studied theoretically. The wavenumbers and traveling wave characteristics in the beam on elastic foundations are analyzed. Basing on the equations of motion, the complex band structures and frequency response function are calculated by the transfer matrix method. And the Bragg and locally resonant gaps properties and the effects are researched. A gap with low frequency and wide range can exist in a beam on elastic foundations.

Distributed Absorber for Noise and Vibration Control

An approach to a wide-band frequency passive vibration attenuation is introduced in this paper. This aims to suppress noise and vibration of extended multimode objects like plates, panels and shells. The absorber is arranged in the form of a single-layer assembly of small inertial bodies (balls) being distributed and moulded within the light visco-elastic media (e.g. silicone resin). The absorber as a whole is embedded into object face covering the critical patches of the system surface. For the purpose of characterization, the authors introduced the complex frequency response function relating the volume velocity produced by the vibrating object surface (response) stimulated by a point-wise force (stimulus) applied to a particular point. The simulation and optimization of the main frequency characteristics has been performed using a full scale 3-dimensional Finite Element model. These revealed some new dynamic features of absorber's structures, which can contribute to vibration attenuation. A full-scale physical experimentation with synthesised absorber's structures confirmed the main results of simulation and has shown significant noise reduction over a staggering 0–20 kHz frequency band. This was achieved with a negligible weight and volume penalty due to the addition of the absorber. The results can find multiple applications in noise and vibration control of different structures. Some examples of such applications are presented.

Distributed Absorber for Noise and Vibration Control

An approach to a wide-band frequency passive vibration attenuation is introduced in this paper. This aims to suppress noise and vibration of extended multimode objects like plates, panels and shells. The absorber is arranged in the form of a single-layer assembly of small inertial bodies (balls) being distributed and moulded within the light visco-elastic media (e.g. silicone resin). The absorber as a whole is embedded into object face covering the critical patches of the system surface. For the purpose of characterization, the authors introduced the complex frequency response function relating the volume velocity produced by the vibrating object surface (response) stimulated by a point-wise force (stimulus) applied to a particular point. The simulation and optimization of the main frequency characteristics has been performed using a full scale 3-dimensional Finite Element model. These revealed some new dynamic features of absorber's structures, which can contribute to vibration attenuation. A full-scale physical experimentation with synthesised absorber's structures confirmed the main results of simulation and has shown significant noise reduction over a staggering 0–20 kHz frequency band. This was achieved with a negligible weight and volume penalty due to the addition of the absorber. The results can find multiple applications in noise and vibration control of different structures. Some examples of such applications are presented.

Further experience with model updating incorporating damping matrices

Most of the model updating techniques do not employ damping matrices and hence cannot be used for accurate prediction of complex frequency response functions (FRFs) and complex mode shapes. In this paper, response function method (RFM) is extended to deal with the complexity of FRF by updating damping matrices along with mass and stiffness matrices. The effectiveness of the damped FE model updating procedure is demonstrated by actual laboratory experiments of an F-shaped test structure. The updated results have shown that the damped RFM model updating procedure can be used to derive accurate model of the system. This is illustrated by matching of the complex FRFs obtained from the updated model with the experimental data.

Sensitivity analysis of frequency response functions of composite sandwich plates containing viscoelastic layers

In the scope of structural dynamics, sensitivity analysis is a very useful tool in a number of numerical procedures such as parameter identification, model updating, optimal design and uncertainty propagation. In this paper the formulation of first-order sensitivity analysis of complex frequency response functions (FRFs) is developed for composite sandwich plates composed by a combination of fiber-reinforced and elastomeric viscoelastic layers, in arrangements that are frequently used for the purpose of noise and vibration attenuation. Although sensitivity analysis is a well known numerical technique, the main contribution intended for this study is its extension to viscoelastic structures, which are characterized by frequency- and temperature-dependent material properties and, thus, require particularly adapted analytical and numerical procedures. Due to the fact that finite element discretization has become the most used method for dynamic analysis of complex structures, the sensitivity analysis addressed herein is based on such models, being computed from the analytical derivatives of the FRFs with respect to a set of design parameters, such as fiber orientations and layer thicknesses. Also, a procedure for evaluating the sensitivity of the FRFs with respect to temperature of the viscoelastic material is suggested. After discussion of various theoretical aspects, including a parameterization scheme of the structural matrices with respect to the design variables, first-order response derivatives are calculated for a composite plate with inherent structural damping, and for a composite sandwich plate with a viscoelastic core. The results are compared to those obtained from first-order finite-difference approximations.