Mode shape expansion using perturbed force approach

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Abstract

A new approach for expanding incomplete experimental mode shapes is presented which considers the modelling errors in the analytical model and the uncertainties in the vibration modal data measurements. The proposed approach adopts the perturbed force vector that includes the effect of the discrepancy in mass and stiffness between the finite element model and the actual tested dynamic system. From the developed formulations, the perturbed force vector can be obtained from measured modal data and is then used for predicting the unmeasured components of the expanded experimental mode shapes. A special case that does not require the experimental natural frequency in the mode shape expansion process is also discussed. A regularization algorithm based on the Tikhonov solution incorporating the generalized cross-validation method is employed to filter out the influence of noise in measured modal data on the predictions of unmeasured mode components. The accuracy and robustness of the proposed approach is verified with respect to the size of measured data set, sensor location, model deficiency and measurement uncertainty. The results from two numerical examples, a plane frame structure and a thin plate structure, show that the proposed approach has the best performance compared with the commonly used existing expansion methods, and can reliably produce the predictions of mode shape expansion, even in the cases with limited modal data measurements, large modelling errors and severe measurement noise.

Introduction

Mode shape expansion has many applications, such as for test-analysis correlation study, for updating finite element models and for identifying damage in structures [1], [2], [3], [4]. In general, the analytical mode shapes obtained from finite element analysis contain a full set of degrees of freedom (DOF) from the analytical model. The measured data set of a dynamic test, however, is usually incomplete and only exists at the DOFs associated with the test points, because the measurements are often taken at a limited set of locations in selected coordinate directions [5], [6]. In addition, difficulties in measuring vibration modal data often arise in the cases associated with internal nodes and rotational DOFs. In many structural dynamics applications such as vibration-based model updating and damage identification [7], [8], [9], it is desirable to expand the reduced experimental data set onto the associated full finite element coordinate set. The alternative would be a model reduction process which destroys the original sparse pattern in mass and stiffness matrices and propagates modelling errors or structural damage all over the reduced mass and/or stiffness matrices.

Most mode shape expansion methods utilized today involve the use of the model reduction transformation matrix as an expansion mechanism to obtain the unmeasured mode components of the actual tested dynamic system. For example, the Guyan static expansion method [10] is based on the assumption that the inertial forces acting on the unmeasured DOFs can be ignored in the expansion process to provide a simple static transformation matrix for mode shape expansion. The Guyan static method may give accurate mode shape expansion estimates only when there are sufficient DOFs to represent the mass inertia of the actual tested dynamic system. The IRS expansion method, proposed by O’Callahan [11] and later extended by Friswell et al. [12], modifies the Guyan static condensation transformation matrix by adding a term to make some allowance for the ignored mass inertia associated with the unmeasured DOFs. The Kidder dynamic method [13] is similar to the Guyan static method except that the inertial forces at the unmeasured DOFs are considered at a particular frequency. The System Equivalent Reduction Expansion Process (SEREP) method [14] utilizes the analytical mode shapes to generate a transformation matrix between the measured DOFs and the unmeasured DOFs. The SEREP method could produce poor expansion estimates if the experimental mode shapes are not correlated well with the corresponding analytical mode shapes, which often happens in the cases with large modelling errors in the analytical model. In addition, the penalty method [15] uses a weighting variable as a measure of the relative confidence in the experimental mode shapes to produce mode expansion estimates by minimizing the modal strain energy. These existing expansion methods need information about the modal data (frequency and mode shape) or structural parameters (mass and stiffness) of the analytical model in mode shape expansion processes, but they do not consider the modelling errors due to the discrepancy in structural parameters between the analytical model and the actual tested dynamic system. Also, these existing methods do not include effective measures to filter out the influence of inevitable measurement uncertainties on the predictions of mode shape expansion.

This study presents a new approach for expanding mode shapes by introducing a perturbed force vector and utilizing a regularization algorithm in order to include the modelling errors in the analytical model and reduce the influence of noise in measured modal data. The proposed approach takes the perturbed force vector containing modelling errors as basic parameters, which can be obtained from modal data measurements, and then applies it to predicting the unmeasured part of the expanded mode shapes. In order to give stable solutions for the perturbed force vector from the noisy measurements, a regularization algorithm based on the Tikhonov solution incorporating the generalized cross-validation (GCV) method is employed to reduce the effect of measurement noise. An investigation with respect to several evaluation criteria is conducted to compare the proposed approach with the commonly used existing expansion methods and assess the robustness and reliability of the proposed approach. The results from two numerical examples show that the proposed approach dealing with both modelling errors and measurement uncertainties produces by far the most accurate and reliable mode shape expansion results, in particular in severe adverse situations.

Section snippets

General expansion methods

For a dynamic structural system with global stiffness matrix K and mass matrix M, the characteristic equation of an n DOFs dynamic system can be expressed as(K-ωi2M)φi=0where ωi and φi are the ith natural frequency and the corresponding mode shape, respectively. In general, the mode shapes obtained from experiments only exist at the DOFs associated with test points and need to be expanded over the full set of analytical DOFs for structural dynamic applications such as correlation studies, model

Model uncertainty and perturbed force

In structural dynamic applications, the finite element model usually has uncertainties in modelling the associated actual tested structural dynamic system. The model uncertainties are mainly related to the unknown perturbations of stiffness (ΔK) and mass (ΔM) between the analytical model and the tested system. The global stiffness matrix (K˜) and mass matrix (M˜) of the tested dynamic structure then can be expressed asK˜=K+ΔKM˜=M+ΔMThe characteristic equation for the experimental structural

Expansion to include regularization method

Due to the inevitable noise in modal data measurements, the solution of the perturbed force vector obtained from the Moore–Penrose pseudoinverse in Eq. (22) may not be stable. In order to reduce the influence of noise in modal data measurements on the performance of mode shape expansion, a regularization method is now employed to obtain reasonable solutions for the perturbed force vector. Consider the linear system in Eq. (20) for solving the perturbed force vector ri (the same process can be

Evaluation of expanded mode shape

In order to assess the performance of the proposed approach, it is assumed that the full set of actual mode shapes of the tested structure φ˜i* as well as its mass M˜ and stiffness K˜ are available. These are purely used here as a reference for performance evaluation. Several evaluation criteria are introduced to compare the proposed approach with the existing expansion methods and assess the mode shape expansion predictions by the proposed approach. The Modal Assurance Criterion (MAC) factors

Numerical example

Two numerical examples, a plane frame model and a thin plate model, are employed to demonstrate the accuracy and effectiveness of the proposed mode shape expansion technique. The results obtained from the proposed approach for the two different types of structure models are then compared with those from commonly used existing expansion methods outlined in Section 2 with respect to the performance evaluation criteria defined in the preceding section. Several simulated scenarios are considered to

Conclusions

A new approach is proposed for effectively expanding mode shapes of a complex dynamic system with limited modal data measurements, significant modelling errors and severe measurement noise. The newly developed expansion approach, based on the introduced perturbed force vector, includes the effect of discrepancies between the finite element model and the actual tested dynamic structure. The perturbed force vector containing modelling errors is adopted as the basic parameter that can be obtained

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