Strain sensors optimal placement for vibration-based structural health monitoring. T. H. Loutas*, A. Bourikas Dpt of Mechanical Engineering & Aeronautics, University of Patras, Greece
[email protected]
ABSTRACT The optimal sensor placement of engineering structures problem with an emphasis on in-plane dynamic strain measurements and to the direction of modal identification as well as vibration-based damage detection for structural health monitoring purposes is revisited in this paper. The approach utilized is based on the maximization of a norm of the Fisher Information Matrix built with numerically obtained mode shapes of the structure and at the same time prohibit the sensorization of neighbor degrees of freedom as well as those carrying similar information, in order to obtain a satisfactory coverage. A new convergence criterion of the Fisher Information Matrix (FIM) norm is proposed in order to deal with the issue of choosing an appropriate sensor redundancy threshold, a concept recently introduced but not further investigated concerning its choice. The numerical model of a composite sandwich panel serves as a representative aerospace structure upon which our investigations are based.
INTRODUCTION A great deal of effort has been devoted the last two decades to the optimal sensorization of engineering structures with a predefined (or not) number of sensors. The practical interest is huge and the relevant literature has flourished giving a number of very interesting contributions that address the optimal sensor placement (OSP) problem. Among the many sensor types and relevant tests, we focus on vibration testing which can be utilized towards either system identification, finite element model updating and/or quality control and damage detection. OSP is formally an optimization problem having been described by various objective functions. One of the first and most well established approaches was the maximization of a norm of the Fisher Information Matrix (FIM). Fisher information is a concept from mathematical statistics which measures the amount of information that an observable random variable conditional on unknown parameter(s). Kammer [1] was among the first researchers to utilize the maximization of the trace and determinant of a FIM built with mode shapes obtained from Finite Element Modeling. He called his method the Effective Independence method for modal identification and showed its superiority over the Kinetic Energy approach. Udwadia in [2], focused on OSP for parameter identification and gave a more general framework relying on the maximization of the trace of the FIM of any of the targeted parameter(s). _____________
Theodoros Loutas & Alexandros Bourikas, Applied Mechanics Laboratory, Department of Mechanical Engineering & Aeronautics, University of Patras, 26500 Rio, Greece .
Li et al. [3] discussed and compared the modal kinetic energy (MKE) and effective independence (EI) methods, whilst Meo and Zumpano [4] investigated six different OSP techniques for the sensorization of a bridge and utilized the mean square error (MSE) between experimental and numerical mode shapes as a criterion for the efficiency of each OSP approach. They concluded that the EI method and its variation EI-DPR coupled with the driving point residue (DPR) coefficient give the least total MSE for the three first mode shapes of the bridge. Papadimitriou et al. [5] and Papadimitriou [6] presented a rigorous probabilistic formulation of the optimal sensor placement problem for structural identification. They utilized the information entropy as the objective function of the OSP task searching for the sensor configuration that minimizes it. They adopt a Bayesian framework and rationale, conditioning all the associated variables on measured data which are most probably non-existent in the design stage prior to the sensorization of a structure. They tackle this problem by making an assumption of a large number of measured data and an asymptotic approximation of information entropy. Papadimitriou and Lombaert in [7], categorize OSP problems in two categories: a) OSP for parameter identification and model updating and b) OSP for modal identification. Not surprisingly, the OSP problems of the second category end up to an information entropy proportional to the determinant of the Fisher information matrix, resulting in an identical OSP as in the EI method, if equal variances of the prediction error are assumed for each measured DOF. Yuen et al. [8], discuss scenarios where changes due to damage induce larger uncertainties in the parameters to be identified, demonstrating that the locations of the candidate sensors do not actually change. This is in contrast to our findings reported in section 4, where sensor location changes arise even for very small induced damage. Li and Der Kiureghian in a recent study [9], develop a novel probabilistic framework for the OSP problem having as objective the maximization of the expected utility function from information theory. It is generic enough to accommodate any utility function, the authors though admit that there is no way to say which proposed configuration is the best unless specific evaluation criteria are introduced and posterior experimental evidence under various sensor configurations is obtained. Li et al. [10], propose a load-dependent OSP strategy developing the respective framework theoretically and testing their approach experimentally. Their approach though, relies on structural responses/measurements which are usually not available in the design phase of an OSP of an engineering structure problem. Moreover, it can only provide with OSP configurations under a priori known load histories which may or may not be available for a given structure. Stephan in [11], following the work of Kammer [1], also builds his FIM with mode shapes obtained from Finite Element models and introduces for the first time a redundancy criterion to avoid the unnecessary placement of sensors in neighborhood degrees of freedom (DOF) and thus obtain a wider coverage of the structure. He ends up with more expanded sensor configurations compared to the EI method which is desirable for subsequent experimental mode shape determination and damage detection. Papadimitriou and Lombaert in [7] also discuss a similar issue on spatial correlation of candidate sensors and introduce an exponential correlation function to account for it. In the present work, we develop an OSP framework for inplane dynamic strain sensors with a view on Fiber Bragg Grating (FBG) sensors
utilization. A lot of focus has been given the last 15 years to the utilization of FBG sensors for static as well as dynamic strain measurements for SHM purposes in various industries (aeronautics, space, wind energy, etc.). Our focus is on the pre-service planning of FBG strain sensor placement having no prior experimental data other than the numerical model of the structure. To this direction, we extend the framework set by Stephan in [11], utilizing strain mode shapes from two directions simultaneously instead of accelerations or displacements- and proposing a simple convergence criterion to decide on the redundancy threshold he introduced. Moreover, we discuss the effect of damage on the initially nominated optimal sensor placement, an issue marginally investigated in the relevant literature. Through numerical simulations with Finite Element Analysis (FEA) on a honeycomb sandwich composite panel, it can be shown that even very minor damage in the form of skin to core debonding can change locally the mode shapes and shift the sensors from their initially assigned locations.
METHODOLOGY - SENSOR PLACEMENT ALGORITHM The governing differential equation of motion for a linear time-invariant dynamic system under external excitation F(t) is expressed as: 𝑴𝒙̈ (𝑡) + 𝑪𝒙̇ (𝑡) + 𝑲𝒙(𝑡) = 𝑭(𝑡)
(1)
Where x(t) ∈ ℝ𝑁𝑑 , M, C, K ∈ ℝ𝑁𝑑 ×𝑁𝑑 the total nodal displacement, mass, damping and stiffness matrices of the structure. 𝑁𝑑 is the number of all degrees of freedom (depends on the discretization of the structure). After the displacement modal decomposition and since we focus on strain sensors it turns out 𝜺(𝑡) = 𝜳𝒒(𝑡). 𝜳 are the strain mode shapes and 𝒒(𝑡) are the modal co-ordinates. Hence the equation can take the form: 𝜳𝒙 𝒒𝒙 (𝑡) 𝜺𝒙 (𝑡) [𝜺𝒚 (𝑡)] = [ 𝜳𝒚 𝒒𝒚 (𝑡)] 𝜳𝒛 𝒒𝒛 (𝑡) 𝜺𝒛 (𝑡)
(2)
From the numerical model, the mode shape matrices 𝜳𝒙 and 𝜳𝒚 can be obtained since the focus is on planar strain measurements on flat composite panels. In an engineering structure we focus on the first few 𝑁𝑚 modes and thus the size of the matrices is 𝑁𝑑 × 𝑁𝑚 . The optimization principle used for the present approach in OSP is the maximization of a norm of the Fisher Information Matrix of the modal coordinates 𝑰(𝒒(𝑡)) which according to the Cramer-Rao bound minimizes the variance ̂) of an unbiased estimator 𝒒 ̂ of 𝒒(𝑡). This leads to a total FIM dependent only 𝑉𝑎𝑟(𝒒 on strain mode shapes 𝜳. We consider and propose the theoretical total FIM combining information 𝜳𝒙 and 𝜳𝒚 (possible sensorized directions) and hence: 𝑵
𝑵
𝒅𝒚 𝑇 𝑇 𝒅𝒙 𝑰𝑡𝑜𝑡𝑎𝑙 = 𝑰(𝒒𝒙 ) + 𝐼(𝒒𝒚 )~ ∑𝒊=𝟏 𝛹𝑖𝑥 𝛹𝑖𝑥 + ∑𝒊=𝟏 𝛹𝑖𝑦 𝛹𝑖𝑦
(3)
Where 𝑁𝑑𝑥 , 𝑁𝑑𝑦 are the number of available DOF in each direction respectivly. In a free-free condition 𝑁𝑑𝑥 = 𝑁𝑑𝑦 . Equation (3) actually states that as more degrees of
freedom are sensorized more information is added to the respective FIM. The rationale is to determine which from the available DOF contribute the most to the maximization of some norm of the total FIM taking into account spatial and information redundancy. The algorithm decides which DOF (in x or y direction) contributes more to the total FIM and puts the sensor on it. Stephan in [11] introduced redundancy as a normalized Euclidean distance of the elementary FIMs of two sensors measuring at DOF i and j as: ‖𝑰 −𝑰 ‖
𝑅𝑖𝑗 = ‖𝑰𝑖 +𝑰𝑗‖ 𝑖
𝑗
(4)
Setting a threshold Ts in redundancy (i.e. 𝑅𝑖𝑗 < 𝑇𝑠 ), below which the candidate DOF shall be excluded, more sparse sensor configurations can be obtained. The issue with the redundancy threshold is that it is not a priori known or determined and different choices of 𝑅𝑖𝑗 can give quite different sensor configurations. In eq. (5), we introduce a numerical convergence criterion to deal with this shortcoming, ‖𝑰𝑁𝑠 ‖−‖𝑰𝑁𝑠 −1 ‖ ‖𝑰𝑁𝑠 ‖
≥ε%
(5)
Where 𝐈Ns the FIM of the Nsth sensorized DOF. Regarding the chosen matrix norm utilized in the calculations we follow the argumentation in Stephan [11] and utilize the spectral radius ρ(I) of a given FIM which represents a lower bound for any norm of the FIM. The rationale behind eq. (5) is to secure that each subsequently selected DOF to be sensorized, will increase at least by ε % the spectral radius of the FIM. Figure 1 summarizes the rationale of the developed sensor expansion algorithm. Initially all available DOF are sorted according to the norm of the elementary FIM 𝐈𝐤𝐞𝐥𝐞𝐦 of the kth DOF. The highest is kept and sensorized, mathematically: bdof=argmax ρ(𝐈𝐤𝐞𝐥𝐞𝐦 )
(6)
𝑘
The redundancy criterion of eq. (4) comes to eliminate the DOF with redundant information from 𝜳𝒙 and 𝜳𝒚 in every subsequent iteration. The best DOF (bdof) from the available DOF which contributes most to the norm of total FIM is searched out and added to the previous subtotal FIM. 𝐞𝐥𝐞𝐦 𝐈𝐤+𝟏 = 𝐈𝐤 + 𝐈𝐛𝐝𝐨𝐟
(7)
When all DOF are searched, all sensors are put on the structure and convergence has not been achieved, the process restarts at an increased Ts by a step of 0.01. The procedure ends when the last available sensor is put and simultaneously the convergence criterion we introduced is satisfied. The introduction of the convergence criterion of eq. (5) secures a selection of the smallest redundancy threshold which ensures the placement of all the available number of sensors Ns on the structure. CASE STUDY – OSP OF A COMPOSITE SANDWICH PANEL
The structure under examination is a honeycomb sandwich panel. The skin’s layup is [0/-60]s. The panel’s dimensions are 1050 mm x 700 mm x 35.42mm. Start ➢ ➢ ➢ ➢ ➢
set a number of available sensors Ns create the mode shape matrices [𝜳𝒙 ]Ndx×Ndx and [𝜳𝒚 ]Ndy×Ndy from the FE model choice of Nm modes of interest calculate and sort the elementary 𝐈𝐤𝐞𝐥𝐞𝐦 for all k=Ndx+Ndy DOF bdof (1)=argmax ρ(𝐈𝐤𝐞𝐥𝐞𝐦 )
➢
Initiate: Ts =0, I0 = 0̃, 𝑖 = 1
𝑘
while Ts 0 and i ≤ Ns
Else
For every remaining candidate DOF k in x and y direction compute: ➢ information matrix: 𝐈𝒌 ➢ spectral radius: ρ(𝐈𝒌 )
𝒆𝒍𝒆𝒎 ➢ keep bdof=argmax ρ(𝐈𝒌 ) and build the elementary matrix 𝐈𝒃𝒅𝒐𝒇 𝑘
𝒆𝒍𝒆𝒎 ➢ update information matrix: 𝐈𝒊+𝟏 =𝐈𝒊 +𝐈𝒃𝒅𝒐𝒇 ➢ Eliminate this DOF from 𝜳𝒙 or 𝜳𝒚
➢ ➢
Compute the redundancy R according to eq. (5) between bdof and every remaining DOF of Ndx and Ndy Eliminate every DOF of Ndx and Ndy with R
