Determination of eigenfunction and frequency response function of constrained dynamic system

Structural modification needs to prescribe some structural frequencies and mode shapes. This study derives the eigenfuction and frequency response function matrix of constrained dynamic system based on measured test data. The modified eigenfunction is derived by utilizing the measured modal data of the actual system as constraints to govern a part of the behavior of modified system and minimizing cost functions of the difference between analytical and corrected parameter matrices with them. It is shown that the modified eigenfunction incorporates the modified parameter matrices. The frequency response function matrix modified by measured constraints is also derived by minimizing a cost function of the dynamic strain energy to be expressed by dynamic stiffness matrix and the difference between analytical and measured modal displacements. The validity of the proposed methods is demonstrated in application

Static and dynamic synthesis of partitioned substructures

Substructuring is to subdivide an overall structure into two or more substructures to reduce the model-order of the huge structural system. The problem to synthesize the substructures is established by a mathematical system consisting of equilibrium equations and prescribed compatibility conditions. Considering that the compatibility conditions are constraints, this study derives the analytical methods for describing the responses of constrained static and dynamic systems and provides a structural synthesis method based on the Guyan condensation method and the derived equations. The analysis process is carried out by partitioning into two regions of interior and boundary regions, and giving the compatibility conditions. And the dynamic analysis reduces model-order based on the constraint conditions between modal coordinates by the first several mode shape matrix. The validity of the proposed method is illustrated through the structural synthesis of stable and unstable substructures, and the structural reanalysis to evaluate the structural response for changes in the design without solving the complete set of modified simultaneous equation

FRF based substructuring and decoupling of substructures

This study considers FRF (frequency response function) based substructuring and decoupling of substructures for the dynamic analysis of complicated huge structures utilizing compatibility conditions between adjacent substructures. This work includes: 1) the derivation of updated FRF matrix for dynamic system subjected to frequency or time dependent constraints in the frequency-domain, 2) the synthesis and decoupling of subsystems based on the dual domain approach using compatibility conditions between adjacent subsystems, 3) the evaluation of the validity of the proposed methods through numerical applications. It is expected that the proposed methods will be utilized as the basic formulation in investigating the dynamic characteristics of partitioned or synthesized system

Damage detection of beam structure using response data measured by strain gages

The health state of a structural beam as a flexural member can be evaluated by the curvature or flexural strain. Measured strain data provide more accurate flexural characteristics than the curvature approximation by the central difference method from the displacement mode shape and DFRFs (displacement frequency response functions). The strain sensor may be sensitive to local damage when analyzing the flexural response. This work presents a method to detect damage utilizing only the measured strain data collected from a damage-expected beam structure without intact baseline data. The static method from the measurement strain data and the dynamic method from measured SFRFs (strain frequency response functions) are introduced. It can be observed that damage exists in the region, which represents an abrupt change in the strain response. The validity of the method is illustrated by numerical and experimental applications

Update of FRF matrix and physical parameters of finite element model

Numerically simulated results by finite element model of a dynamic system do rarely coincide with the actual responses of the structure due to the modeling, manufacturing errors or other causes. The parameters of the finite element model should be corrected for subsequent proper analysis. Minimizing a cost function by the difference between the analytical and actual dynamic stiffness matrices along with constrained conditions of measured FRFs (Frequency Response Functions), this work provides the mathematical form on the updated FRF. The updated parameter matrices of the structure are derived from the FRF variation within the prescribed frequency range. It is shown that the exactness of the proposed method depends on the number of the measured data. The validity of the proposed method is illustrated in updating the parameter matrices of a planar truss structure

Damage detection and identification of parameter matrices using residual force vector

Beginning with incomplete mode shape measurement data, this study presents analytical equations to predict the actual stiffness and mass matrices. The measured modal data, including the measurement, manufacturing and modeling errors, should be updated for subsequent analysis. In this study, the incomplete mode shape data are expanded to a full set of degrees-of-freedom (DOFs) based on the generalized inverse method and the concept of residual force vector. The corrected parameter matrices are straightforwardly derived using the estimated mode shape data and the pseudo inverse method. The validity of the proposed method is evaluated based on the number of measured modes in an application, and its limitations are investigated

Damage detection of truss structure based on the variation in axial stress and strain energy predicted from incomplete measurements

This study derives the static equilibrium equation of a damaged system on the basis of stiffness change due to damage as well as the constraint forces at measurements required for obtaining the measured data. Based on the derived equations, this work provides an analytical method to detect damage from the stress and strain energy variations between intact and damaged truss structures. The applicability of the proposed method is evaluated in detecting multiple damages of low rate in the truss structure from measured data contaminated by external noise. It is demonstrated that it is possible to properly detect damage in an isolated substructure by partitioning the damage-expected substructure from an entire structure and using the displacements measured at the boundary of the partitioned subsystem. The partitioning method has the benefits in reducing the computational time and measured data as well as improving the effectiveness of the damage detection process

Identification of parameter matrices using estimated FRF variation

This study presents an analytical method to predict the dynamic parameters of actual structure from measured FRF (Frequency Response Function) data. The inconsistency due to modeling errors between the actual structure and the finite element model exists. The number of measured data is less than the one of a full set of dofs and should be expanded to estimate the parameters. Considering that the stiffness and mass matrices are related with the real part of the expanded FRF data and the damping matrix with the imaginary part, the variation in the parameter matrices is evaluated. A numerical example evaluates the appropriateness of the proposed method

Identification of joint parameters using FRF based decoupling

Structural and mechanical systems are assembled from smaller components using mechanical joints. The mechanical properties of the joints must be modeled in order to perform subsequent design and analysis of the structure or mechanical system. This study provides an analytical method of determining the parameters that describe the behavior of the joints using frequency response function (FRF) data that is measured at joint nodes. The variation in FRFs is derived by utilizing the consistent response conditions at the same joint nodes of the assembly system and the portioned subsystems. The variation reflects the mechanical properties of the joint and is utilized to extract the joint parameters. The validity of the proposed method is illustrated in two numerical applications