Modal Truncation Error Research Articles

A complete SRSS format for the response spectrum method of high-cycle fatigue life assessment considering modal truncation error correction

In this paper, a more accurate and efficient format of the fatigue response spectrum method is further proposed considering modal truncation error. Firstly, the quasi-static response of higher-order modes is taken into account to modify the modal truncation error. Then, an analytical description of the cross-correlation between the modal responses as well as quasi-static response is developed by two coefficients associated with the natural frequency and modal damping ratio. Finally, the expected damage rate is transformed into a summation of modal damage responses strictly, resulting in a dual improvement in accuracy and efficiency. In numerical examples, the correctness and efficiency are verified and the influences of load spectrum and material fatigue properties on the modal truncation error are further investigated.

A generalized elastic coordinate method for unconstrained structural dynamics

A generalized elastic coordinate method (GECM) is proposed for the vibrational (elastic) response analysis of unconstrained structural dynamics. Firstly, the generalized rigid body and elastic coordinates are defined to describe the overall motion of a discrete unconstrained structure. Secondly, the corresponding equations of motion for generalized elastic coordinates are established with a tiny computational cost, and the stiffness matrix is still sparse. Finally, the elastic deformation in physical space can be exactly caught by the proposed equations of motion after determining the rigid body modes and the coordinate system of the rigid body motion. Different from directly solving the original semi-positive definite system in physical space, in which the overall displacement can only obtain, and the direct modal projection, the modal truncation error problem can be effectively overcome based on the established positive definite system when the modal order reduction is carried out. In light of this, the modal truncation error correction of the elastic displacement for the transient time domain and steady-state frequency domain of unconstrained structures is directly realized, summed the above as GECM, and the validity and practicability are illustrated through the numerical examples.

Improved mode acceleration-based vibroacoustic coupling analysis of functionally graded shell under random excitation

This paper focuses on the vibroacoustic response of functionally graded material (FGM) shell structures under stationary random excitations. An eight-node finite element model of FGM shells is developed, which is valid for both flat and curved shells of FGM. Then, the finite element formulation of the acoustic-structural coupled system for FGM shells under random excitation is established. It is shown that the commonly used combined method of the pseudo excitation method (PEM) and mode displacement method (MDM) in previous work is highly efficient for stochastic dynamics problems, but its accuracy is reduced by the modal truncation error. To remedy this, the combined method of PEM and the conventional mode acceleration method (MAM) is an effective alternative, but not applicable to the coupled acoustic-structural system due to its singular stiffness matrix. To circumvent this, an efficient and accurate method integrating the PEM and the improved MAM (IMAM) is proposed for the vibroacoustic coupling analysis of FGM shell structures under stationary random excitation. Numerical examples are given to verify the effectiveness of the proposed method and to explore the random vibroacoustic properties of the FGM shells with different power-law distributions.

Maximum drift estimation based on only one accelerometer for damaged shear structures with unknown parameters

This paper presents a method to estimate the maximum inter-story drift and relative displacement time history of multi-degree-of-freedom (MDOF) shear structures underground-motion excitation by using only one accelerometer installed on a floor when the structures are damaged during the event and hence modal parameters change and become unavailable. First, the absolute acceleration and relative displacement are formulated in modal coordinates and a state-space expression is derived. Then, a scheme to determine the unknown natural frequencies of the damaged structure is devised with the help of the genetic algorithm (GA) and a reasonably chosen fitness function, and the mode shapes are updated by solving a set of nonlinear equations with respect to each group of frequency variables in the GA. The applicability of this method was studied in numerical simulations that examined the effect of the extent of damage to the structure and the selection of the lower bound of the GA variables. Further simulations were conducted to investigate the modal truncation error and robustness against measurement noise of the proposed approach. Finally, the method was validated in a shake table experiment. The results indicate that the time history of the relative displacement and maximum inter-story drift can be accurately estimated in the case of a significant reduction in structural stiffness and a large search range of GA variables. The results also show the method is robust against noise and performs well even when only the lower modes are included in the model.

An improved hybrid FE-SEA model using modal analysis for the mid-frequency vibro-acoustic problems

Based on the hybrid finite element (FE)-statistical energy analysis (SEA) method, an improved hybrid model with less time-consuming is proposed for the mid-frequency vibro-acoustic problems. Mode shapes are used to reduce the order of the deterministic acoustic component modeled by the finite element method in this paper. The governing equation of the residual response, which is derived to correct the modal truncation error, can be reduced by constructing a set of basis vectors. The model order reduction (MOR) equation of the acoustic component is formulated through a combination of the truncated mode shapes equation and the reduced residual response equation. The dimension of the overall equation of motion is efficiently decreased. The statistical structural component is modeled by SEA, and the responses of a mid-frequency vibro-acoustic system can be solved after introducing the diffuse field reciprocity relationship. Based on the efficient correction of the modal truncation error, the proposed hybrid MOR-SEA method promotes the reduction of order using mode shapes in mid-frequency vibro-acoustic systems. Compared with the traditional hybrid FE-SEA method, the present method has an evident advantage in computational efficiency, as shown in two numerical examples.

Full-field strain estimation of subsystems within time-varying and nonlinear systems using modal expansion

Virtual sensing allows for the estimation of stress and/or strain response in unmeasured locations of a system. Often, these virtual sensing techniques assume a linear and time-invariant system with proportional damping. In this article, one virtual sensing technique - the modal expansion - is proven applicable for stress/strain estimation of subsystems within time-varying and nonlinear systems with general viscous damping where the time-varying and nonlinear effects act externally on the subsystems. This technique uses the mode shapes of the subsystem to expand the response by a subspace projection. It is proven that the mode shapes of the underlying undamped and linear system form a basis for the response of the time-varying and nonlinear systems with general viscous damping. Therefore, a truncation of the mode shapes results in modal truncation errors that depend on the span of the applied mode shapes. Thus using an appropriate set of undamped and linear mode shapes of the subsystem, the modal expansion allows for estimation of the stress/strain response for “linear” subsystems within time-varying and nonlinear systems with general viscous damping. This concept is proven both numerically and experimentally.

Towards robust statistical damage localization via model-based sensitivity clustering

Damage diagnosis is a fundamental task for structural health monitoring (SHM). With the statistical sensitivity-based damage localization approach, a residual vector is computed from vibration measurements in the reference and the damaged state. The residual is analyzed statistically in hypothesis tests with respect to change directions defined by the sensitivities of the structural parameters associated to elements of a finite element (FE) model of the investigated structure. If the test for a parameter reacts, then the respective element of the structure is indicated as damaged. This approach offers a very generic and theoretically sound framework to analyze parametric changes in systems, and takes into account the intrinsic statistical uncertainty related to measurement data. Depending on the definition of the residual and of the parameterization, the approach offers a simple computation of the test statistics directly from the measurement data in the damaged system, without the need of system identification. Since an FE model is used, the approach is applicable on arbitrary structures, while no model updating is required and therefore the requirements on the FE model accuracy are less strict. While the theoretical framework has been developed previously, it lacked robustness so far for an application on real structures. The purpose of this paper is the development of this framework into a working damage localization method that is applicable on real data from complex structures. To achieve this goal, robust hypothesis tests are used, the sensitivity computation of the residual is revisited for more precision thanks to reduced modal truncation errors, and an adequate clustering approach is proposed for the case of a high-dimensional FE parameterization for complex structures. Furthermore, several robustness properties of the method are proven. Finally, an application of this framework is shown for the first time on experimental data for damage localization, namely in an ambient vibration test of a 3D steel frame at the University of British Columbia.

A power series expansion method based on frequency response function matrix for sensitivity analysis of viscously damped system

The modal method represents the derivative of the eigenvector as a superposition of all modes. Although this method is relatively effective, it has a problem of modal truncation error. In this paper, based on the two kinds of calculation methods of frequency response function matrix, the transformation relationship between system matrices and eigensolutions is proposed. Meanwhile the transition from the high-order modes to the low-order modes is emphasized. Then the conversion relationship is applied to reduce the modal truncation error, and finally a more accurate method for sensitivity analysis is derived.

A free interface component mode synthesis method for viscoelastically damped systems

The analysis of large, complex systems can be simplified and more computationally efficient if smaller, simpler subcomponents can be treated and assembled. Component Mode Synthesis (CMS) methods allow one to reduce the order of the model at the subcomponent level. Although CMS methods are widely applied in the field of structure dynamics for undamped or viscously damped systems, their counterparts for viscoelastically damped systems seem not well-developed. Motivated by such points, this paper develops a model-order-reduction approach for viscoelastically damped systems based on a free interface CMS method. Due to the fact that the nonviscous modes and higher-order elastic modes are difficult to be obtained accurately for viscoelastic systems, the mode truncation error is often introduced. To reduce the mode truncation error problem, two residual attachment modes of viscoelastically damped systems are proposed in terms of the lower modes and system matrices by using the first one or two terms of Neumann expansion. Finally, three numerical studies are provided to illustrate the accuracy and effectiveness of the derived methods. It is shown that the proposed methods can receive good results and reduce the modal truncation error significantly over the conventional method.

Improved Modal Truncation Error in the Directly Analytical Method for Damage Identification in Civil Engineering

Improved Modal Truncation Error in the Directly Analytical Method for Damage Identification in Civil Engineering

Operational modal analysis with uncertainty quantification for SDDLV-based damage localization

The Stochastic Dynamic Damage Locating Vector (SDDLV) approach is a vibration-based damage localization method based on a finite element model of a structure in a reference state and output-only measurements in both reference and damaged states. A stress field is computed for loads in the null space of a surrogate of the change in the transfer matrix at the sensor positions, where the null space is obtained based on the identified modal parameters in both structural states. Then, the damage location is related to positions where the stress is close to zero. The localization results of this generic approach are perturbed by mainly two sources: modal truncation (not all modes of the structure are available) and modal parameter identification errors (estimation is subject to statistical uncertainties). In this paper, we show how damage localization with the SDDLV approach is improved by taking into account the estimation uncertainties of the underlying identified modal parameters.

Efficient and accurate calculation of sensitivity of damped eigensystems

The exact methods (the algebraic method and Nelson’s method) to calculate eigensensitivities need a matrix decomposition for each eigensensitivity, and therefore are time-consuming. The modal method may be the most efficient method for eigensensitivity analysis, but suffers from modal truncation error. A method is presented by exactly expressing the modal truncation error as a sum of available modes and system matrices. The proposed method maintains original space without having to use the state-space form. It is shown that the proposed method yields good trade-off between accuracy and computational complexity in terms of the theoretical analysis and numerical studies.

Eliminating the modal truncation problem encountered in frequency responses of viscoelastic systems

Increasing the number of degrees of freedom used in finite element analysis for mechanical and structural systems with viscoelastic damping, the need to consider the modal truncation problem of viscoelastic systems is more than ever before. The higher modes may be unnecessary to obtain in dynamic analysis for engineering applications. For viscoelastic systems, the modal truncation problem may be more frequently encountered since the nonviscous modes are difficult or even impossible to be found accurately even if a small-scaled problem is considered for some eigensolution methods. This study aims at eliminating the influence of the higher modes on the frequency responses of viscoelastically damped systems. A method is presented by making the equilibrium equations of motion into a subspace equation spanned in terms of the columns of a projection basis obtained by considering the use of the contribution of the lower modes and the first two terms of the Neumann expansion of the contribution of the unavailable modes. Finally, three example studies are provided to illustrate the effectiveness of the derived results. It is shown that the proposed method can reduce the modal truncation error significantly.

Closed-form modal flexibility sensitivity and its application to structural damage detection without modal truncation error

The closed form of the sensitivity of modal flexibility for a real undamped system is derived based on the algebraic eigen sensitivity method. The formula is then adopted to detect the damage location and severity. The merit of the proposed approach over many other exiting modal flexibility sensitivity-based schemes is that it eliminates the adverse effect of the truncating higher-order modes. Moreover, the proposed approach could identify results with sufficient accuracy by employing one mode or multiple modes. To tackle the operational mode shape normalization problem ascribed to the unknown ambient excitation, the scaling factors are employed. A number of assumed damage scenarios in beams has been used to demonstrate the applicability of the proposed damage detection method. It is demonstrated that the proposed method is not sensitive to noise, and it can effectively identify single and multiple damage.

Study on the Method of the Damage Identification for Frame Structure

First order iterative algorithm, mixed iterative algorithm, structural damage identification using static and dynamic data were put forward. The first and second order sensitivity matrixes of modal parameters that respect to the damage member were derived, and the modal truncation error which produced during the derivation of modal mode sensitivity was improved. The first and second order sensitivity equations were established respectively based on the principle of Taylor series expansion. And the solving method of these sensitivity equations was studied. Mixed iterative algorithm took up the second order nonlinear analytical solution as the first substituting value, and then the first substituting value was modified based on the Taylor series bias error using the solution of the first order sensitivity equation. It showed that the mixed iterative algorithm in this paper had a better convergence and a faster iteration speed because the higher precision second order nonlinear analytical solution was adopted. Because the method using static and dynamic data combined the static information and dynamic information of the structure, it could react the inside information of the structure more comprehensively, the result of damage identification was more accurate and it would be adapted more widely.

Reconstruction of the interior sound pressure of a room using the probabilistic approach

In this paper, a probabilistic approach is introduced and used to find the optimal values assigned to the uncertain parameters of a room acoustic model, which is used for the reconstruction of the interior sound pressure distribution. The acoustic model selected here to examine the capacity of the probabilistic approach is a rectangular room with two air leakages that is subject to an external uniform noise. The model is set up using the modal analysis method. The values of the uncertain parameters of the model are identified using the time domain sound pressure responses at selected measurement points. In the simulation results, the comparisons between the proposed approach and the typical least error square method show that the former clearly assigns optimal values to the uncertain model parameters for the best prediction, while the latter generates a set of values for which the prediction errors are very similar and close to the minimum (in other words, it is difficult to identify the optimal values using the least error square method). Moreover, the optimal values for the uncertain parameters can be found individually, unlike those from the least error square method. It is also found that the measurement points should not be located at the nodal points of the dominant acoustic modes; otherwise, the identification process becomes unidentifiable. For cases in which large modal truncation errors (or modeling errors) exist, the identification process also becomes unidentifiable.

Vehicle axle loads identification using finite element method

Existing methods on moving force identification are usually based on modal decomposition and are subject to modal truncation error in the dynamic responses. A new moving load identification method is presented basing on finite element method and condensation technique. The measured displacements are formulated as the shape functions of the finite elements of the structure which is modeled as a straight beam. The measured responses can be limited to a small number of master degrees-of-freedom of the structural system. Numerical simulations and experimental results demonstrate the efficiency and accuracy of the method to identify a system of general moving loads or interaction forces between the vehicle and the bridge deck. The number of master degrees-of-freedom of the system should be selected smaller than or equal to the number of measured points, and the identified results are relatively not sensitive to the sampling frequency, velocity of vehicle, measurement noise level and road surface roughness when a minimum of eight beam-column finite elements are used to model the bridge with measured information from three measuring points.

Quasi-static data recovery for dynamic analyses of structural systems

Data recovery is a post-processing procedure employed in dynamic analyses to recover the physical response from the modal responses obtained using the mode-displacement method. An enhanced data recovery process can improve accuracy by compensating for the modal truncation errors. The mode-acceleration method is a well-known data recovery technique. However, its domain of application is restricted to problems having a low-frequency band of interest. In this paper, new data recovery methods, the quasi-static data recovery methods, are presented. These new methods are based upon a quasi-static compensation technique that is applicable for any banded frequency domain of interest, with both high- and low-frequency modes truncated. Numerical examples are given to demonstrate the improvements on accuracy and applicability of the proposed method compared with traditional data recovery.

Dynamic analysis of a heavy vehicle using a condensation technique

For conducting dynamic design studies on commercial heavy vehicles, finite element analysis is usually required. Existing dynamic analysis techniques involve significant computation time. In order to reduce the computation time, a dynamic condensation method is proposed in this paper. The condensation technique involves sub-dividing the system under consideration into master and slave sub-systems; usually the number of master degrees-of-freedom (DOF) is significantly smaller than the slave sub-system DOF. A heavy vehicle ride comfort analysis is carried out using the proposed condensation technique and the results are shown to be as accurate as the complete model simulation. A saving in computation time is also demonstrated. However, there are issues such as modal truncation errors that occur when using this technique. The paper briefly addresses guidelines to overcome such issues.

HYBRID EXPANSION METHOD FOR FREQUENCY RESPONSES AND THEIR SENSITIVITIES, PART II: VISCOUSLY DAMPED SYSTEMS

Frequency responses and their sensitivities have been broadly applied to finite element model updating, structural damage detection, dynamic optimization, vibration control and so on. In this paper, the modal acceleration method for the frequency responses and the double-modal acceleration method for their sensitivities, which have been discussed in the previous paper for undamped systems, are extended to viscously damped systems. The two methods are based on the hybrid expansion, power series expansion and modal superposition, of the inverse of the complex dynamic system matrix. Three steps are required to calculate the sensitivities using the proposed method. Firstly, frequency responses of a system excited by external forces are calculated by using the modal acceleration. Pseudo-force vector is then computed from the production of the sensitivity matrix and the frequency response vector. Finally, a second modal acceleration is applied to obtain the general frequency responses under the pseudo-forces, that is, the sensitivities. Two modal truncation schemes, middle-high-modal and low-high-modal truncation schemes, are presented according to the values of the exciting frequencies. The modal truncation errors of the modal acceleration method for frequency responses and the double-modal acceleration method for the sensitivities are also given to show the convergence of the proposed methods. Although the frequency responses and their sensitivities are discussed in this paper, the proposed methods are also valid for the frequency response functions, responses in time domain and their sensitivities. The results of a floating raft isolation system show that the proposed modal acceleration methods are efficient, especially for the sensitivity analysis. The modal truncation errors of the frequency responses and their sensitivities will reduce quickly when the two-modal acceleration methods are adopted.