Frequency Response Function Matrix Research Articles

Explicit frequency response functions of discretized structures with uncertain parameters

A novel procedure for deriving approximate explicit expressions of the frequency response function (FRF) matrix of linear discretized structures with uncertain parameters is presented. The following main steps are required: (i) to decompose the deviation of the structural matrices with respect to their nominal values as sum of rank-one matrices; (ii) to derive the so-called Rational Series Expansion (RSE) which provides an approximate explicit expression of the FRF holding for any uncertainty model. The potentials of the RSE are demonstrated within the interval framework by determining the region of the modulus of the FRF of structures with uncertain-but-bounded parameters.

A hybrid expansion method for frequency response functions of non-proportionally damped systems

This study is aimed at eliminating the influence of the higher-order modes on the frequency response functions (FRFs) of non-proportionally viscously damped systems. Based on the Neumann expansion theorem, two power-series expansions in terms of eigenpairs and system matrices are derived to obtain the FRF matrix. The relationships satisfied by eigensolutions and system matrices are established by combining the two power-series expansions. By using the relationships, an explicit expression on the contribution of the higher-order modes to FRF matrix can be obtained by expressing it as a sum of the lower-order modes and system matrices. A hybrid expansion method (HEM) is then presented by expressing FRFs as the explicit expression of the contribution of the higher-order modes and the modal superposition of the lower-order modes. The HEM maintains original-space without having to use the state-space equation of motion such that it is efficient in computational effort and storage capacity. Finally, a two-stage floating raft isolation system is used to illustrate the effectiveness of the derived results.

Impact force identification with pseudo-inverse method on a lightweight structure for under-determined, even-determined and over-determined cases

Force identification using inverse technique is important especially when direct measurement through force transducer is not possible. Considering the effects of impact excitation force on the integrity of a lightweight structure, impact force identification has become the subject of several studies. A methodology utilising Operating Deflection Shape (ODS) analysis, Frequency Response Function (FRF) measurement and pseudo-inverse method to evaluate the dynamic force is presented. A rectangular plate with four ground supports was used as a test rig to simulate the motions of a simple vehicle body. By using the measured responses at remote points that are away from impact locations and measured FRFs of the test rig, unknown force locations and their time histories can be recovered by the proposed method. The performance of this approach in various cases such as under-determined, even-determined and over-determined cases was experimentally demonstrated. Good and bad combinations of response locations were selected based on the condition number of FRF matrix. This force identification method was examined under different response combinations and various numbers of response locations. It shows that in the over-determined case, good combination of response locations (i.e. low average of condition number of FRF matrix) and high number of response locations give the best accuracy of force identification result compared to under-determined and even-determined cases.

Vibration modeling and modification of cutting platform in a harvest combine by means of operational modal analysis (OMA)

Vibration in combine cutting platform causes increase in grain loss, reduction of vehicle lifetime and driver’s comfort, and also affects the machine working precision. In order to determine vibration behavior of a mechanical structure, the model which relates acting forces on the structure with the resulting responses of it plays a significant role. For a great deal of mechanical structures, only response data are measurable while the actual loading conditions are unknown. Therefore, the system identification process will need to build on the output-only data. The use of such processes allows the identification of modal models of structures which are excited by unknown ambient vibration. In this contribution, controlled vibration experiments were conducted on cutting platform of a combine in order to recognize and reduce its vibration problem. A finite element model is constructed representing the vibration behavior of the cutting platform. Also, the frequency domain decomposition (FDD) technique is used to estimate the modal parameters of vibrating structures in operational conditions. Then, the finite element model is successfully updated by comparing the estimated results of the operational modal analysis. It is found that there is a resonance condition around 50Hz. Then, the mass change strategy is used to estimate the scaling-factor and frequency response function (FRF) matrix. Finally, the resonance condition and subsequently vibration of cutting platform is reduced by the structural modification technique. As a result of this modification, natural frequency of fifth mode shape (50Hz) of the combine cutting platform is shifted to 48Hz.

Theoretical Research on Two Improved Optimal Sensor Placement Methods

Optimal sensor placement technique plays a key role in structural health monitoring and structural vibration control. Based on the advantages and disadvantages of effective independence (EI) and modal kinetic energy (MKE) methods, two improved optimal sensor placement methods which are Effective Independence - Average Acceleration Amplitude (EI-AAA) method and Effective Independence - Modal Kinetic Energy (EI-MKE) method are proposed in this paper. Firstly the formulas are deduced from modal expansion of multiple-input-multiple-output (MIMO) displacement frequency response function matrix. Then a computational simulation of steel cross beam structure has been implemented to demonstrate the feasibility of the two improved methods above. The obtained optimal sensor locations using the two improved methods are compared with those gained by EI method and MKE method. Finally six classical comparison criteria are employed to demonstrate the advantage and disadvantage of these four methods. The results showed that some innovations proposed in this paper are effective and reliable. The two improved optimal sensor placement methods (EI-AAA method and EI-MKE method) can not only make the truncated mode shapes as linearly independent as possible but also enable the measured modal kinetic energy to maintain the maximum value.

Improved approximate methods for calculating frequency response function matrix and response of MDOF systems with viscoelastic hereditary terms

As we know, it is difficult and unnecessary to obtain all the eigenpairs of a large-scaled viscoelastic (nonviscous or hysteretic) damping systems, which means that the mode truncation scheme is generally used and the mode-truncated error is therefore introduced. This study is aimed at eliminating the influence of the unavailable modes on the dynamic response of MDOF systems with viscoelastic hereditary terms. The energy dissipation terms of the system depend on the past history of motion via convolution integrals over some kernel functions. Therefore, the system is a nonviscously damped system, which has been considered as the most generalized damping model within the scope of a linear mechanical analysis. To approximate frequency response function (FRF) matrix and response without using the unavailable modes, we suggest two methods, which attempt to approximate the influence of the unavailable modes in terms of the lower modes and system matrices by using the first one or two terms of Neumann expansion of the contribution of the unavailable modes. In contrast with the FRF matrix approximated in terms of the first two terms of Neumann expansion, these procedures cannot be extended to further high order terms since all of them will be affected by the frequency-dependent variation of damping matrix from previous terms. Finally, an example is shown that the two presented methods can make the mode-truncated error reduce and may be used to approximate the influence of nonviscous modes contributed to FRF matrix due to the fact that the nonviscous modes are difficult to be obtained accurately even if a small scaled model is used for some eigensolution methods.

Estimation of Cutting Force Model Coefficients with Regularized Inverse Problem

In this paper, a method for estimation of cutting force model coefficients is proposed. The method makes use of regularized total least squares to identify the cutting forces from the measured acceleration signals and the frequency response function (FRF) matrix. An original regularization method is proposed which is based on the relationship between the harmonic components of the cutting forces. Numerical tests are performed to evaluate the effectiveness of the method. The method is compared with unregularized methods and common Tikhonov regularization combined with GCV and L-curve methods. It was found that the proposed method provides more accurate estimates of the cutting force coefficcients than the unregularized method and common regularization techniques. Furthermore the influence of acceleration measurement errors, FRF matrix errors and FRF matrix conditioning on the accuracy of the estimated coefficients is investigated. It was concluded that FRF matrix errors influence the most the accuracy of the results.

Stability analysis of aeroelastic systems based on aeroelastic FRF and condition number

Purpose – The purpose of this paper is to analyze the stability of aeroelastic systems using aeroelastic frequency response function (FRF).Design/methodology/approach – The proposed technique determines the instability boundary of an aeroelastic system based on condition number (CN) of aeroelastic FRF matrix or directly from FRFs data.Findings – Stability margins of typical section and hingeless helicopter rotor blade in the subsonic flow regimes (quasi‐steady and unsteady models) are determined using proposed techniques as two case studies.Originality/value – The paper introduces a technique which is applicable not only when aerodynamic and structure analytical models are available but also when there are experimental models for structure and/or aerodynamics, such as impulse response functions data or FRFs data. In other words, the main advantage of the proposed method, besides its simplicity and low memory requirement, is its ability to utilize experimental data.

Bounds for the stationary stochastic response of truss structures with uncertain-but-bounded parameters

The aim of the present paper is to determine the region of the probabilistic characteristics of the stationary stochastic response (mean-value vector, power spectral density function and covariance matrix) of truss structures with uncertain-but-bounded parameters under stationary multi-correlated Gaussian random excitation via interval analysis. The main steps of the proposed procedure are: i) to express the stiffness, damping and mass matrices of the structural system as linear functions of the uncertain-but-bounded parameters; ii) to split the probabilistic characteristics of the nodal interval stationary stochastic response, evaluated in the frequency domain, as sum of the midpoint and deviation values; iii) to evaluate in explicit approximate form the parametric interval frequency response function matrix. The effectiveness of the presented procedure is demonstrated by analyzing a truss structure with uncertain-but-bounded axial stiffness and lumped masses subjected to stationary multi-correlated Gaussian wind excitations.

Stochastic optimal control analysis of a piezoelectric shell subjected to stochastic boundary perturbations

The stochastic optimal control for a piezoelectric spherically symmetric shell subjected to stochastic boundary perturbations is constructed, analyzed and evaluated. The stochastic optimal control problem on the boundary stress output reduction of the piezoelectric shell subjected to stochastic boundary displacement perturbations is presented. The electric potential integral as a function of displacement is obtained to convert the differential equations for the piezoelectric shell with electrical and mechanical coupling into the equation only for displacement. The displacement transformation is constructed to convert the stochastic boundary conditions into homogeneous ones, and the transformed displacement is expanded in space to convert further the partial differential equation for displacement into ordinary differential equations by using the Galerkin method. Then the stochastic optimal control problem of the piezoelectric shell in partial differential equations is transformed into that of the multi-degree-of-freedom system. The optimal control law for electric potential is determined according to the stochastic dynamical programming principle. The frequency-response function matrix, power spectral density matrix and correlation function matrix of the controlled system response are derived based on the theory of random vibration. The expressions of mean-square stress, displacement and electric potential of the controlled piezoelectric shell are finally obtained to evaluate the control effectiveness. Numerical results are given to illustrate the high relative reduction in the root-mean-square boundary stress of the piezoelectric shell subjected to stochastic boundary displacement perturbations by the optimal electric potential control.

Damage detection from the variation of parameter matrices estimated by incomplete FRF data

It is not easy to experimentally obtain the FRF (Frequency Response Function) matrix corresponding to a full set of DOFs (degrees of freedom) for a dynamic system. Utilizing FRF data measured at specific positions, with DOFs less than that of the system, as constraints to describe a damaged system, this study identifies parameter matrices such as mass, stiffness and damping matrices of the system, and provides a damage identification method from their variations. The proposed parameter identification method is compared to Lee and Kim's method and Fritzen's method. The validity of the proposed damage identification method is illustrated in a simple dynamic system. An accurate dynamic finite element model of a structure is very important for structural design and analysis. The modes of vibration and frequency response of the finite element model are compared to experimental measurements to test its accuracy. If a discrepancy between the two is found, the analytical model should be modified to satisfy the experimental measurements, and then the model should be updated for subsequent simulation and design studies. An FRF matrix of the full set of DOFs measured by experiments is used to predict parameter matrices of the dynamic system. The FRFs have been used directly to update condensed analytical models for obtaining the proper modal model. A single FRF measured at several frequencies, along with a correlated analytical model of the structure in its original state, is used for updating structural parameters. There have been many attempts to update the unknown physical parameters directly from the FRFs (Mottershead and Stanway 1986). It has been reported (Lee and Shin 2002) that FRF data provide more information than modal data, as the latter are extracted from a very limited frequency range related to resonance. Friswell and Penny (1990) proposed an approach to reduce the model order so that the stated estimation process reduces to a least squares problem based on the FRFs. Fanning and Carden (2004) presented a method for detecting added mass in structural systems from a single FRF measured at several frequencies in an identification algorithm. Cha and

Modal parameters of a flexible disk-flexible shaft system from simulated data

A global multiple-input multiple-output modal parameter scheme is employed with the frequency response function (FRF) matrix, to accurately estimate modal parameters of the closely coupled modes of a flexible disk-flexible shaft system from simulated data. The Lagrangian approach is combined with the assumed mode method to derive the governing equations of motion, where the uncoupled disk and shaft modes are employed as the assumed functions. The corresponding state space equations are obtained, which were integrated to provide response at points of interest (output, or response degrees of freedom) due to excitation at desired points (input, or excitation degrees of freedom). Random noise is added to the obtained time signal to simulate actual vibration measurements. The excitation and response time signals are used to generate the corresponding FRF matrix, used in the modal parameter estimation scheme. The extracted modal parameters are found to be in excellent agreement with theoretical results.

Bayesian force reconstruction with an uncertain model

This paper presents a Bayesian approach for force reconstruction which can deal with both measurement noise and model uncertainty. In particular, an uncertain model is considered for inversion in the form of a matrix of frequency response functions whose modal parameters originate from either measurements or a finite element model. The model uncertainty and the regularization parameter are jointly determined with the unknown force through Monte Carlo Markov chain methods. Bayesian credible intervals of the force are built from its posterior probability density function by taking into account the quantified model uncertainty and measurement noise. The proposed approach is illustrated and validated on numerical and experimental examples.

Electric potential response analysis of a piezoelectric shell under random micro-vibrationexcitations

The response characteristics of a spherically symmetric piezoelectric shell under randomboundary micro-vibration excitations are analyzed and calculated. The equation for electricpotential is integrated radially to obtain the electric potential as a function ofdisplacement, so that the differential equations for the piezoelectric shell with electrical andmechanical coupling are converted into an equation only for the displacement. Thedisplacement transformation is constructed to convert the random boundary conditionsinto homogeneous ones, and the transformed displacement is expanded in space to furtherconvert the partial differential equation for the displacement into ordinary differentialequations using the Galerkin method. The equations represent a multi-degree-of-freedomdynamic system with an asymmetric stiffness matrix under random micro-vibrationexcitations. The frequency-response function matrix, power spectral density matrix andcorrelation function matrix of the system response are derived from these equations basedon the theory of random vibration. The expressions of mean-square displacement,stress and electric potential of the piezoelectric shell are finally obtained andillustrated by numerical results for random micro-vibration excitations. The randomelectrical and mechanical coupling properties, in particular the relations betweenboundary electric potential responses and micro-displacement excitations, areexplored.

Extraction of artificial boundary frequencies for damage identification

This paper introduces some recent progress in a study which is aimed at incorporating the so-called artificial boundary condition (ABC) frequencies for damage identification. The ABC frequencies are those corresponding to the natural frequencies of the system with additional pin supports, but may be extracted from specially configured incomplete frequency response function matrix of the original structure without the need of physically imposing the additional supports. A particular focus of this paper is placed on the actual extraction of such frequencies from physical experiments and the associated data processing and analysis. Results will demonstrate that it is possible to extract the first few ABC frequencies for a variety of boundary conditions with 1-2 artificial pin supports in a beam or a slab structure.

An accurate substructural synthesis approach to random responses

An accurate substructural synthesis method including random responses synthesis, frequencyresponse functions synthesis and mid-order modes synthesis is developed based on rigorous substructure description, dynamic condensation and coupling. An entire structure can firstly be divided into several substructures according to different functions, geometric and dynamic characteristics. Substructural displacements are expressed exactly by retained mid-order fixed-interfacial normal modes and residual constraint modes. Substructural interfacial degree-of-freedoms are eliminated by interfacial displacements compatibility and forces equilibrium between adjacent substructures. Then substructural mode vibration equations are coupled to form an exact-condensed synthesized structure equation, from which structural mid-order modes are calculated accurately. Furthermore, substructural frequency-response function equations are coupled to yield an exact-condensed synthesized structure vibration equation in frequency domain, from which the generalized structural frequency-response functions are obtained. Substructural frequency-response functions are calculated separately by using the generalized frequency-response functions, which can be assembled into an entire-structural frequency-response function matrix. Substructural power spectral density functions are expressed by the exact-synthesized substructural frequency-response functions, and substructural random responses such as correlation functions and meansquare responses can be calculated separately. The accuracy and capacity of the proposed substructure synthesis method is verified by numerical examples.

Micro-vibration suppression of equipment supported on a floor incorporating magneto-rheological elastomer core

In this study, magneto-rheological elastomers (MREs) are adopted to construct a smart sandwich beam for micro-vibration control of equipment. The micro-vibration response of a smart sandwich beam with MRE core which supports mass-concentrated equipment under stochastic support-motion excitations is investigated to evaluate the vibration suppression capability. The dynamic behavior of MREs as a smart viscoelastic material is characterized by a complex modulus dependent on vibration frequency and controllable by external magnetic fields. A frequency-domain solution method for the stochastic micro-vibration response of the smart sandwich beam supporting mass-concentrated equipment is developed based on the Galerkin method and random vibration theory. First, the displacements of the beam are expanded as series of spatial harmonic functions and the Galerkin method is applied to convert the partial differential equations of motion into ordinary differential equations. With these equations, the frequency-response function matrix of the beam–mass system and the expression of the velocity response spectrum are then obtained, with which the root-mean-square (rms) velocity response in terms of the one-third octave frequency band can be calculated. Finally, the optimization problem of the complex modulus of the MRE core is defined by minimizing the velocity response spectrum and the rms velocity response of the sandwich beam, through altering the applied magnetic fields. Numerical results are given to illustrate the influence of MRE parameters on the rms velocity response and the response reduction capacity of the smart sandwich beam. The proposed method is also applicable to response analysis of a sandwich beam with arbitrary core characterized by a complex shear modulus and subject to arbitrary stochastic excitations described by a power spectral density function, and is valid for a wide frequency range.

Variance Estimation for the Identified Force Spectra Based on Matrix Inversion Method

To improve noise and vibration quality of mechanical structures, it is important to identify excitation force spectra and the positions at which the force acts. Matrix inversion method is the most frequently used technique for input identification. In this method, excitation force spectra are identified from the inverse problem based on the frequency response function (FRF) and acceleration, which is measured under operational conditions. Although the matrix inversion method is popular because of its simplicity, the identified force can be influenced by measurement error. Therefore, the identification accuracy should be statistically evaluated. To address this issue, in this paper, a variance estimation method of the force spectra identified by the matrix inversion method is proposed. The variance is estimated on the basis of the variance and covariance of FRF, and the proposed method is validated via a numerical model. In addition, the variance estimation method is applied to a vibration test to compare the reliability of the force spectra determined using the estimated variance with the condition number of the FRF matrix, which is also known as an error-propagation indicator of the input identification. Finally, it is proved that the proposed method is suitable for accurately evaluating the identified force.

Comparison of decentralized velocity feedback control for thin homogeneous and stiff sandwich panels using electrodynamic proof-mass actuators

Theoretical and experimental work is presented to compare the effect of decentralised velocity feedback control on thin homogeneous and sandwich panels. The decentralised control system consists of five control units, which are composed of a proof-mass electrodynamic actuator with an accelerometer underneath its footprint and an analogue controller. The stability of the feedback loops is analysed by considering the sensor-actuator open-loop frequency response function of each control unit and the eigenvalues of the fully populated matrix of open-loop frequency response functions between the five sensors and five actuators. The control performance is then analysed in terms of the time-averaged total kinetic energy and total sound power radiated by the two panels. The results show that for a stiff sandwich panel higher stable feedback gains can be implemented than on a thin homogeneous panel of comparable weight per unit area. Moreover the implementation of decentralised velocity feedback can offset some of the undesirable sound transmission properties of lightweight sandwich structures by efficiently reducing structural vibration and sound power radiation in the mid audio frequency range.

A Method for Forced Response Analysis of Mistuned Bladed Disks With Aerodynamic Effects Included

A method has been developed for high-accuracy analysis of forced response levels for mistuned bladed disks vibrating in gas flow. Aerodynamic damping, the interaction of vibrating blades through gas flow, and the effects of structural and aerodynamic mistuning are included in the bladed disk model. The method is applicable to cases of high mechanical coupling of blade vibration through a flexible disk and, possibly shrouds, to cases with stiff disks and low mechanical coupling. The interaction of different families of bladed disk modes is included in the analysis providing the capability of analyzing bladed disks with pronounced frequency veering effects. The method allows the use of industrial-size sector models of bladed disks for analysis of forced response of a mistuned structure. The frequency response function matrix of a structurally mistuned bladed disk is derived with aerodynamic forces included. A new phenomenon of reducing bladed disk forced response by mistuning to levels that are several times lower than those of their tuned counterparts is revealed and explained.