Dynamic displacement field reconstruction through a limited set of measurements: Application to plates

0022-460X/$ see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.05.031 n Corresponding author. E-mail addresses: [email protected] (M. Chierichetti), [email protected] (M. Ruzzene). Journal of Sound and Vibration 331 (2012) 4713–4728 Author's personal copy functions into the modal space. Also in this case, the major limitation is the necessity of a large number of basis functions (of the order of 50–60) to represent the forcing function with acceptable accuracy [3,4]. Consequently, FR techniques can be inefficient and not practical. The second class of approaches can be denoted as shape sensing (SS) techniques [5]. The reconstruction of the displacement field at every point of the structure is achieved from a set of discrete strains or displacement measurements. The advantage of the SS approach is that the knowledge of the structural response implies the ability to identify stresses in the structure. This inverse problem is in its own nature ill-posed as it does not necessarily satisfy conditions of existence, uniqueness and stability. Tessler and Spangler [6] present a variational principle to reconstruct the deformed shape in plates and shells. The method has been subsequently developed in a finite element framework in Tessler and Spangler [7]. The technique, called inverse finite element method (iFEM), essentially consists in fitting the measurements through the finite element discretization of the structure and the relative shape functions. In Tessler and Spangler [7], Bogert et al. [8], a cantilevered plate is analyzed, and 28 measurement points are necessary for a plate area of 252 cm. The number of sensors is definitely reduced with respect to FR approaches, but a remarkable number of sensors is still necessary. The possibility of using high-speed cameras with digital image correlation has recently given the possibility of comparing full-field measured data to the numerical response computed through detailed finite element models in order to estimate updates to the physical properties of the model, as for example, Wang et al. [9,10] in which measured full-field mode-shapes are compared to the numerical modes, and Wang et al. [11] that compares full-field strain data to the estimated strain field. The approach presented in this paper belongs to the shape sensing class, and it is based on the technique proposed in Bousman [12] for the analysis of rotor blades. In Bousman [12], the author investigates the possibility of computing the spanwise load distribution from bending moment measurements as a way of obtaining an estimate of blade loading. This procedure is based on a modal expansion of the flap equation of motion and has the advantage that a few modes (10) and in-flight measurements (of the order of 15–20) are required. The approach of [12] is here generalized to analyze different systems other than rotor blades by applying a general FE procedure. In addition, the focus is placed on the accuracy of the reconstructed response rather than the estimation of the load distribution. Description of the results of the proposed approach, here denoted as load confluence algorithm, on a onedimensional structure are found in Chierichetti et al. [13] and for dynamic components (rotor blades) in a rotating environment in McColl et al. [14] and Chierichetti et al. [15]. This approach can also be modified to improve the numerical response by updating the physical properties of the system, as described in [16]. The load confluence algorithm is formulated with a general finite element (FE) framework, and it is based on an iterative procedure which estimates and corrects the externally applied load so that numerical predictions and measured quantities reach an agreement within a specified tolerance. This procedure can accommodate nonlinearities in the structure owing to its iterative nature. The approach in fact applies linearized corrections to a nonlinear model, and it is therefore not limited to linear analysis. Examples of nonlinear analyses are discussed in Chierichetti et al. [15]. Moreover, due to the generality of the FE formulation, the algorithm can include different types of measurements (strains, displacements, velocities, accelerations) with very minor modifications to the algorithm. The iterative approach operates in the modal domain, from which the response of the structure is reconstructed using numerical modes. Reconstruction of modal loads is shown to accurately identify the response field after a few iterations, while the identification of external loads requires a higher computational burden and detailed experimental information. Therefore, the proposed approach appears as promising as it requires a limited number of measurements and of modal information. The paper is organized in three sections, including this introduction. Section 2 describes the general concept which supports the development of the confluence algorithm and provides detailed description of the algorithm, Section 3 describes the experimental set and provides results obtained with the application of the algorithm. 2. Confluence algorithm 2.1. Concept The objective of the proposed algorithm is the development of an effective procedure for the dynamic response mapping using experimental measurements. The motivation comes from the need to monitor the dynamic response of components operating in rotating environments under complex loading as in the case of rotor blades and other rotating components in rotorcrafts [14]. In rotorcraft applications, the main sources of vibrations are air loads, whose accurate prediction remains an outstanding challenge [17]. In contrast, structural dynamic codes have achieved an acceptable level of predictive accuracy, provided accurate air loads are available. This general observation and the application to rotorcraft environment provide the main motivations for the current work and lead to the assumption that the applied loads vary periodically in time. Accordingly, the dynamic response is considered periodic and can be expanded through a Fourier series. Based on the schematic of Fig. 1, the objective is to numerically reconstruct the full-field dynamic response of the structure w(x) from a limited number N of experimental measurements wðxnÞ, n1⁄4 1,: :,N. This is achieved through the modal expansion of the numerical model of the structure and an iterative process. At each iteration, the external loads applied to the model are corrected to obtain the agreement between numerical predictions and experimental measurements at M. Chierichetti, M. Ruzzene / Journal of Sound and Vibration 331 (2012) 4713–4728 4714 Author's personal copy locations xn, n1⁄4 1,: :,N. Iterations are needed in order to account for nonlinearities, model inaccuracies and unmodeled dynamics. The confluence algorithm consists of a numerical model of the structure, which is assumed to be as accurate as possible, a set of experimental measurements at a limited number of locations, and an iterative procedure which estimates the applied loads and maps the measured response over the structure. A schematic representation of the components of the procedure is presented in Fig. 2. The dynamic model of the structure is generally defined by a mass matrix M and a stiffness matrix K, while the set of available experimental measurements is denoted as eEðxÞ, where x is a vector defining the location of the sensors, array eEðxÞ stores the experimental measurements. The following notation is used throughout the paper: bold, lower case define vectors, bold, upper case define matrices. For the configuration depicted in Fig. 1, for example, x1⁄4 1⁄2x1, . . . ,xN T and eEðxÞ 1⁄4 1⁄2wðx1Þ, . . . ,wðxNÞ T . A general formulation allows for the measurement vector eEðxÞ to include displacement, strain, velocity, or acceleration measurements. The procedure is summarized as the following sequence of steps: An initial guess for the load F is applied to the model. Solution of the FE model estimates the dynamic response eMðxÞ at the sensors location x. Numerical and measured responses are compared and a error vector De1⁄4 eE eM is calculated. A load correction DF is calculated based on De through the formulation of the problem in the modal domain. This is performed through the modal mapping procedure described in Section 2.2. The updated load F 1⁄4 FþDF and the corresponding new numerical response is computed, and a new error De is estimated. The process is iterated until De reaches the desired tolerance level. The process is represented schematically in Fig. 3, while Fig. 4 illustrates the improvement of the estimated mapped dynamic response at each iteration. An iterative procedure is particularly important in case of systems including nonlinearities, in the presence of un-modeled dynamics and noisy experimental data, or in situations where the number of modes considered in the modal expansion is lower than the number of experimental measurements, Chierichetti et al. [15]. measurements map x1 x2 W (x2) W (x3) W(x1) X3 Fig. 1. Schematic of the fitting process through discrete measurements in the system. Experimental Measurements CONFLUENCE ALGORITHM map Fig. 2. Components of the confluence algorithm. M. Chierichetti, M. Ruzzene / Journal of Sound and Vibration 331 (2012) 4713–4728 4715 Author's personal copy 2.2. Modal procedure for load estimation Consider a general undamped linear system: M € uðtÞþKuðtÞ 1⁄4 FðtÞ (1) where M and K are the mass and the stiffness matrix of the system, array uðtÞ stores the N degrees of freedom of the solution, and FðtÞ is the array of the generalized applied loads. The modal expansion of the equation requires that uðtÞ 1⁄4 PqðtÞ (2) where qðtÞ are the modal coordinates and matrix P contains the N eigenvectors of the system. The generalized loads are expanded in a similar manner as FðtÞ 1⁄4MPkðtÞ. The governing equations of motion of the system reduce to € qiðtÞþoi qiðtÞ 1⁄4 liðtÞ (3) MAPPING ALGORITHM FULL-FIELD RESPONSE RECONSTRUCTION Yes No C O N V EXPERIMENTS _ em en Δe ΔF F+ΔF F +