Characterisation of rotating machinery dynamics within limited frequency intervals using modal analysis

Modal parameters, i.e., modal frequencies, damping ratios, and mode shapes of a two disk-shaft system are estimated in Multiple Input/Multiple Output (MIMO) scheme. The response at the output degrees of freedom (dof) of the considered structure due to the excitation at the input dofs is estimated from theoretical analysis. The generated theoretical data is used to estimate the Frequency Response Function (FRF) matrix, which relates the output (vibration) at response points with the input at the excitation points in the frequency domain. The corresponding Impulse Response Function (IRF) matrix, which relates response and excitation in the time domain, is obtained by Inverse Fast Fourier Transform (IFFT) of the FRF matrix. The unified Matrix Polynomial Approach (UMPA) is employed in the frequency and time domains with the estimated FRF and IRF matrices, respectively, to estimate the desired modal parameters. The obtained results are compared with results from Theoretical Modal Analysis (TMA).

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.

Inverse combustion force estimation based on response measurements outside the combustion chamber and signal processing

To meet the demand of providing basis for improving triaxial vibration system, it is necessary to extract the overall Frequency Response Function (FRF) matrix of triaxial vibration system from its digital model. The mechanical parts of triaxial vibration system is connected tightly which satisfy rigid connection assumption. The Jetmudsen FRF method is used to construct the overall frequency response function matrix of triaxial vibration system. The electrodynamic shaker (referred to as the shaker) is an important component of the three-axis vibration system, and its function is to provide the test bench with an excitation force in a certain direction. Therefore, it is necessary to define a large number of internal nodes on the shaker. The FRF matrix of shaker (including origin FRF matrix of internal nodes and joints, cross-point FRF matrix of internal nodes and joints) should be obtain to carry out the construction of overall FRF matrix of triaxial vibration system. It is difficult because the size of FRF matrix of shaker is too large. So a way to simplify the distributed loads exerting on the structure into multi-point concentrated loads is proposed, it can be used to obtain the simplified digital model of the electrodynamic shaker. Based study above, the overall frequency response function matrix of the triaxial vibration system is obtained, and the digital modeling of the triaxial vibration system is completed.

Most of the mechanical systems are composed of different subsystems coupled by several links. Any excitation acting on the system is divided into several internal forces which propagate through these links or so called transfer paths. This paper presents the use of experimental transfer path analysis in identifying the transmission paths for a heavy duty truck in order to estimate the vibration and noise transmitted from the cabin and engine mounts. The most challenging part of the TPA analysis is estimation of the internal operational forces so that the total response can be predicted accurately. At the circumstances where direct measurement of the operational forces is impossible, especially for complex structures, a common approach to address the problem is based on a measured frequency response function (FRF) matrix and a set of operational responses. The main problem of this approach is the inversion of the FRF matrices which can be ill-conditioned. Once the internal operational forces are estimated, the vibration or noise response for the selected location in the truck can be calculated. To validate the predicted results, coherence of the collected data and the condition numbers of FRF matrices are investigated so that the accuracy of the predicted results can be quantified for the frequency band of interest. The predictions of the total response are compared with the experimentally measured data such that the coherence and condition number related observations are validated.

An important procedure in transfer path analysis (TPA) is to measure the frequency response functions (FRFs) of the decoupled passive subsystem. The classical TPA method obtains the passive subsystem’s FRFs by direct measuring when the system is disassembled. The main shortcoming of the classical method to measure the FRFs is that it is time-consuming due to the necessity to dismount the active part. In this paper, a novel method is proposed to estimate the passive subsystem’ FRF matrix without disassembling the coupled mechanical structure. The key idea of this method is that the effect of a coupled subsystem will be canceled out if the links which connect this subsystem with the other one have no deformation, since the coupled systems influence each other only through the links which can be regarded as combinations of connecting springs and dampers. Following this idea, the expression of the passive subsystem’s FRF matrix can be deduced from the entire system’s FRF matrix directly. The proposed method in this paper is called the virtual decoupling method, since the decoupling is not ‘real’ but ‘virtual’. Obviously, the actual decoupling procedure is avoided so that the shortcoming mentioned above is overcome. The method is validated by a numerical model and a finite element model.

For a hydraulic or electro-dynamic shaker test rig, the goal of the iterative control is to reproduce the structural deformations of vehicles, measured under normal operating conditions for a vehicle driven on public roads or test tracks. The approach is based on the correct calculation of the excitation signals applied to the actuators of the multi-axis test rig. The conventional iterative control approach adjusts drive waveforms according to the particular Frequency Response Function (FRF) matrix, measured prior to the test and fixed. To ensure the system stability, the feedback gain should be small, which will lead to a very slow convergence rate. To overcome these defects, an adaptive inverse model based on the filtered-X least mean-square (LMS) algorithm is proposed to continuously improve the identification precision of the inverse characteristic of the FRF, which is called the impedance function of the system under test. And the frequency domain LMS algorithm is introduced to make sure the inverse characteristic is not critical to the iterative gain and thus allow for a bigger feedback gain. The experimental results indicate that the adaptive control strategy can not only solve the system instability problem arising from the improper feedback gain selection in the conventional iteration process, but also has the higher convergence rate.

Multiple Input Multiple Output (MIMO) vibration testing provides the capability to expose a system to a field environment in a laboratory setting, saving both time and money by mitigating the need to perform multiple and costly large-scale field tests. However, MIMO vibration test design is not straightforward oftentimes relying on engineering judgment and multiple test iterations to determine the proper selection of response Degree of Freedom (DOF) and input locations that yield a successful test. This work investigates two DOF selection techniques for MIMO vibration testing to assist with test design, an iterative algorithm introduced in previous work and an Optimal Experiment Design (OED) approach. The iterative-based approach downselects the control set by removing DOF that have the smallest impact on overall error given a target Cross Power Spectral Density matrix and laboratory Frequency Response Function (FRF) matrix. The Optimal Experiment Design (OED) approach is formulated with the laboratory FRF matrix as a convex optimization problem and solved with a gradient-based optimization algorithm that seeks a set of weighted measurement DOF that minimize a measure of model prediction uncertainty. The DOF selection approaches are used to design MIMO vibration tests using candidate finite element models and simulated target environments. The results are generalized and compared to exemplify the quality of the MIMO test using the selected DOF.

Aircraft Flutter Modal Parameter Identification Using a Numerically Robust Least-squares Estimator in Frequency Domain

Computational modal analysis performed using finite element analysis is an established method for estimating the modal parameters of complex structures. This work aims to provide insight into the techniques necessary to accurately model structural components within an aircraft by performing computational modal analysis on a rear-mounted pump support structure. The results of this analysis will be used to further aid in the understanding of the vibratory transmission path from the support structure to the aircraft cabin. The frequency response functions were used to validate the finite element model via a visual and analytical comparison against experimental data. A modal frequency response analysis was used to estimate the dynamic response at a discrete set of points on the structure. A model validation study showed an excellent correlation between the experimental and computational results for frequencies between 100–2000 Hz with an average percent difference of the centre frequencies of 9.4%. The agreement between the two sets of results across the 20–4000 Hz bandwidth deteriorated slightly to 11.7 %. Modifications were made to the webbing of the pump support yoke as well as the in-board and out-board isolator plates. These tests confirmed that differences between the frequency responses of the original and modified finite element models can be justified using modal analysis theory. It was found that these particular structural design changes did not significantly influence the modal response of the structure. The methodology presented in this work provides an outline for future modelling and analysis of aircraft structural designs.

Traditional H∞ norm depicts peak system gain over infinite frequency range, but many applications like filter design, model order reduction and controller design etc. require computation of peak system gain over specific frequency interval rather than infinite range. In present work, new computationally efficient techniques for computation of H∞ norm over frequency limited interval are proposed. Proposed techniques link norm computation with maximum singular value of the system in limited frequency interval. Numerical examples are incorporated to validate the proposed concept.

Equivalent linear modal parameter estimation for a nonlinear aeroelastic system

Inverse substructuring method has been recently proposed and applied for inverse analysis of the dynamical response of product transport system. The component-level frequency response functions (FRFs) and the coupling dynamic stiffness for facilitating the cushioning packaging design are all predicted from only the system-level FRFs. However, the system-level FRFs from coupling degree of freedoms may not be measured accurately because of the difficulties of vibration excitation and response measurement for the coupled interface between packaged product and vehicle within the limited accessible space. The aim of this paper is to develop a new FRF-based indirect inverse substructuring method for the analysis of the dynamic characteristics of a three-substructure coupled product transport system without measuring system-level FRFs at the coupling degree of freedoms. By enforcing the dynamic equilibrium conditions at the coupling coordinates and the displacement compatibility conditions, a closed-form analytical solution to inverse sub-structuring analysis of multi-substructure coupled product transport system is derived based on the relationship of easy-to-monitor component-level FRFs and the system-level FRFs at the coupling coordinates.. The proposed method is validated by a lumped mass-spring-damper model, and the predicted coupling dynamic stiffness is compared with the direct computation, showing exact agreement. Then, the FRF tests of a physical prototype of multi-substructure coupled product transport system are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely. The proposed method may help to obtain the main controlling factors and contributions from the various structure-borne paths for product transport system. Copyright © 2014 John Wiley & Sons, Ltd.

When structures are excited by random force excitation the circle fits of the data around resonance are usually poor. The structural parameter estimates, which result from this fit, are usually erroneous. No matter how elegant the circle or the multimodal fit, the results will be poor if the frequency response function (FRF) is a poor representation of the actual structural response. In general for the random excitation case, this is the case. The conventional fast Fourier transform (FFT) method which is used to estimate the frequency response function, is given by H1 (f) = Gxy/Gxx. This produces poor results when the coherence of the data falls in the resonance region. A drop in the coherence usually indicates noise at the input of the structure for this case. H1 (f) is quite sensitive to such noise giving erroneous estimates. This paper investigates an alternative method for computing the frequency response function, H2 (f) = Gyy/Gyx, and its impact on the accuracy of the circle fit procedure used in modal analysis. This new estimator is not sensitive to input noise like the currently used H1 (f). H2(f) provides the best estimate at or around resonance even in the presence of noise on the input signal. If one defines the average percentage fit error in the circle fit operation as 100 times the average radial deviation of the data points from the radius of the statistically fit circle divided by the fit circle radius, one can compute the circle fit accuracy for each of the proposed methods of data treatment. Typically, the percentage fitting error for H1 (f) might be 10 percent while the fitting error for H2 (f) using exactly the same data will be 0.5 percent. Thus, the proposed method eliminates long-standing system analysis errors through the use of a simple revision of the way the data are treated in the FFT processor around the resonance regions.

Identification of modal parameters of unmeasured modes using multiple FRF modal analysis method