206 C. Papadimitriou 23.2 Background Consider a structural model used to predict the temporal variability of the response vector z t;ϕ,u ∈ Rnz (e.g. accelerations, velocities, displacements, strains, stresses) at nz degrees of freedom (DOF) of an underlining structure given the value of a structural model parameter set ϕ (e.g. stiffness, mass and damping related parameters) and the excitation vector u(t) ∈ Rnu. Let y(t) ∈ Rny be response time history data collected from sensors. These data depend on the sensor configuration vector δ containing the location and measurement direction of sensors placed in a structure. The data may consist of either acceleration, displacement and strain measurements. In what follows, a linear model of the structure is assumed. Also it will be assumed that the excitation response time histories u(t) are not available. Given output-only data, there are a number of methods to reconstruct the response of the structure at unmeasured locations. Two such cases are next considered for reconstructing estimates of response at output QoI. In the first case the response reconstruction is obtained using modal expansion techniques, while in the second case state estimation or input-state estimation is performed using available filtering techniques [1–6]. Due to system linearity, the estimate of the response z(t) at time t given the data is derived to be a multi-variable Gaussian vector with mean that depends on the measurement output time histories and covariance matrix that depends on the structure of the linear model, its parameters ϕ, and state, output and input error covariance matrices. This Gaussian distribution of the output vector QoI z(t) is denoted herein as N(z; z(t), P (t)), where the mean estimate z(t) ≡ z t;ϕ,D,δ depends on the measured data, the model parameter set and the sensor configuration, while the covariance P(t) ≡ P t;δ,ϕ of the error in the estimate depends on the sensor configuration and the parameter set, but is independent of the measured data D. For practical convenience and without loss of generality, stationarity conditions are assumed, where the covariance matrix of the error in the output response estimate or the error in the input-output response estimate is independent of t. We next explore further the two cases for which the multi-variable Gaussian distribution for the response quantity arises. Using modal expansion for linear systems, the measured time histories, restricted for demonstration purposes to displacements at N0 DOF, are given with respect to the modal coordinates as y(t) = δ,ϕ ξ(t) +e(t) (23.1) where δ,ϕ ∈ RN0×m is the modeshape matrix corresponding to mcontributing modes, ξ ∈ Rm is the vector of modal coordinates, and e(t) is a zero-mean measurement error with covariance matrix Qe. For estimation purposes the following condition should be met N0 ≥min order for the system to be identifiable. Displacement and strain predictions at output locations or DOF are given by the prediction equation z(t) = ϕ ξ(t) +ε(t) (23.2) where ϕ ∈Rnz×m are the corresponding displacement or strain modeshapes that relate modal coordinates to predicted displacement or strain quantities, andε(t) is a zero-mean prediction error with covariance matrixQε. The modeshape matrices δ,ϕ and ϕ are available by analyzing the model (e.g. finite element model) of the structure. Assuming Gaussian prediction errors e(t) andε(t) and using Bayesian inference to estimate the parameters ξ(t) and propagate to output quantities z(t), it is straightforward to show that the output z(t) is Gaussian with mean that depends on the data and covariance matrix Pz δ,ϕ given by Pz δ,ϕ = ϕ T δ,ϕ Qe δ,ϕ δ,ϕ T ϕ +Qε (23.3) which is independent of the data. Also, the covariance matrix does not depend on the timet provided that the error covariance matrices are assumed to be independent of time. In the second case, introducing the state vector consisting of displacement and velocities at all DOF, the equation of motion is re-formulated in the state-space continuous form ˙x(t) =A(ϕ)x(t) +B(ϕ)u(t) +es(t) (23.4)
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