Optimal sensor placement for detection of non-linear structural behavior

Transcription

1 Optimal sensor placement for detection of non-linear structural behavior R. Castro-Triguero, M.I. Friswell 2, R. Gallego Sevilla 3 Department of Mechanics, University of Cordoba, Campus de Rabanales, 07 Cordoba, Spain mecatrr@uco.es 2 College of Engineering, Swansea University, Swansea SA2 8PP, UK 3 Department of Structural Mechanics and Hydraulic Engineering, University of Granada, 807 Granada, Spain Abstract Most Structural Health Monitoring (SHM) techniques are based on the variation of modal properties of structures. Modal analysis is established under the assumption that the structure behaviour is linear. However, the presence of non-linearities (local or global, smooth or strong) can affect the dynamic properties of the structure. Most of techniques to detect non-linear behaviour in structural dynamics are based on frequency domain data. Fewer techniques only deal with the use of time responses. The main aim of the present research is to improve the detection of non-linear behaviour by the use of efficient sensor placement methodologies. In addition, a minimal number of sensors and information is considered. Optimal sensor placement algorithms have been extensively employed in modal analysis which have been re-oriented to the case of non-linear behaviour. In particular, the Principal Component Analysis (PCA) technique is applied to the time series of a cantilever beam with an attached non-linear spring at its free end. Two cases are considered, where the beam is excited by harmonic force and by a constant amplitude random force. Several amplitude levels are considered where non-linear behaviour is expected. Numerical results for different number of sensors and configurations are presented. Introduction The optimal placement of a limited number of sensors in a structure is usually performed to obtain accurate experimental real-time data. In particular, sensor placement is considered one of the most important pre-test steps in experimental/operational modal analysis [, 2, 3, ]. Structural Health Monitoring (SHM) techniques also incorporate optimal sensor placement (OSP) methodologies for different purposes: system identification, structural damage identification, and finite element updating [, 6, 7, 8, 9]. There has been little discussion of OSP methods to measure the response of a non-linear system, partly because the most measurements are taken on non-linear structures with a small number of degrees of freedom, or the response is considered as a perturbation to the linear response. Ritto [0] recently considered the application of the effective independence method to the proper orthogonal modes of a non-linear structure; this paper investigates the properties of this approach. 3033

2 303 PROCEEDINGS OF ISMA20 INCLUDING USD20 2 Proper Orthogonal Decomposition (POD) Suppose that the response of a structure is measured at n degrees of freedom and at m samples. Thusx i (t j ) represents the measurement of the i th degree of freedom at the j th time instant. This time response history is assembled into a matrix given by X = x (t )... x (t n )..... x n (t )... x n (t m ) The singular value decomposition (SVD) of the matrix X is then calculated as. () X = USV (2) where U and V are orthogonal matrices of dimensions n n and m m respectively, and S is an n m matrix where only the diagonal terms are non-zero. The columns of U are called the Proper Orthogonal Modes (POMs) and the diagonal elements of S are the corresponding Proper Orthogonal Values (POVs) [, 2]. Since we mostly interested in the POMs and POVs, these are more conveniently calculated by an eigenvalue decomposition of the symmetric matrix XX = US 2 U. (3) Typically the majority of the response is within a subspace of low dimension, and this is clearly shown by the POVs, which have high values for those POMs that contribute significantly to the response. Thus the POVs are ordered and a threshold decided, for example based on the percentage of energy retained in the response [2]. Thus, the POMs are split into two sets, given by U = [ ] U r U e () where the subscript r denotes those POMs retained and the subscript e denotes those POMs neglected. 2. Quantifying subspace overlap The retained POMs define the subspace of the significant response of the structure, and the question is then how closely two response subspaces overlap. Several measures to quantify the degree of overlap of two subspaces will now be given, where the subspaces are defined by the orthogonal basis matrices U and U Angle between subspaces The principle angle, θ, between the subspaces is calculated from the matrix C = U U 2. () Since the matrices are orthogonal, the singular values of C are less than, and hence the principal angles, θ i, are defined by cosθ i = σ i (6) where σ i is the i th singular value. The subspace angle [3] is then θ = max(θ i ). (7)

3 NON-LINEARITIES: IDENTIFICATION AND MODELLING Residual projection error The subspaces U andu 2 were obtained from the sampled time response matrices X andx 2 respectively. The choice of the subspace was based on capturing most of the corresponding response. Hot et al. [2] suggested an alternative measure, where the amount of the responsex 2 captured by subspace U is determined. This is based on the projection matrix U U, and hence the response X 2 within the subspace defined by U is ˆX 2 = U U X 2 (8) and hence the residual is r = ˆX2 X 2 = [ U U I] X 2. (9) 3 Understanding the spatial response of a non-linear system The response of a linear systems is generally best understood using the mode shapes of the structure; the excitation applied to a structure is transformed into the excitation of individual modes, and the response is obtained as a summation of contributions of these modes. Usually the number of degrees of freedom is very high, and hence only a small subset of lower frequency modes are of interest. In non-linear systems there are difficulties with this approach, specifically the absence of a set of constant mode shape vectors, and the existence of multiple solutions for a given excitation. However, for weak non-linearities the response can be interpreted in the modal space of the linear system, with the non-linearities introducing interactions between a reduced set of the linear modes. In this way the non-linear response is reduced to the modal participation factors of the linear structure. Alternatively the modal transformation may be viewed as a projection of the non-linear response onto the subspace spanned by the mode shapes of the linear system. Using the POD is a similar approach, where the response is projected onto a different subspace based on the POMs. There are several issues with this approach that will now be discussed in more detail. An important question for the reduced order model is the extent to which the dynamics of the full system is captured. In terms of the projection onto a subspace, the question is the amplitude of the residual of the dynamic response of the full system that is outside the subspace. Often the number of linear modes included in the reduction transformation is higher for a non-linear structure than a linear structure; this is a reasonable approach to allow for increased spatial complexity in the non-linear response. However, it should be emphasised that the linear modes may not be the optimum basis for the response, and it may be that the POMs are better able to capture the response. If the model reduction captures most of the dynamics, then clearly the modal interactions of interest will occur within the projected subspace. A non-linear structure will often produce harmonics of the excitation frequencies, and hence will transfer energy from low frequencies to high frequencies. In linear systems high frequencies always means higher modes and a more complex spatial response. This is not necessarily the case in non-linear system; depending on the form of the non-linearity, the excitation and the particular solution obtained, the change in the spatial response may look similar to a range of linear modes. One difficulty with the POD is that it requires the response to a particular excitation from a particular initial condition to obtain the POMs. This response should be representative of the response of the structure of interest during the analysis. For example, if the structure is able to exhibit multiple solutions, and only one solution is used to obtain the POMs, then there is no guarantee that the reduced model based on these POMs will correctly capture the other solutions. 3. A cantilever beam example This example is motivated by that used by Hot et al. [2] although used here to emphasise different points. The cantilever beam, shown in Figure, has length m, width mm and height mm, and is made of

4 3036 PROCEEDINGS OF ISMA20 INCLUDING USD20 Force Non-linear Spring Figure : The cantilever beam example with a cubic spring at the free end, and modelled using 30 finite elements steel (Young s modulus20 GPa, mass density7800 kg/m 3 ). At the free end is a spring with a cubic stiffness characteristic with coefficient0 MN/m 3. Band-limited ( to20 Hz) white noise is applied at the free end of the beam, with RMS amplitudes of 0.00 N and 2. N. The beam is modelled with 30 Euler-Bernoulli beam elements, giving the first five linear natural frequencies as.73, 73., 20.9, 03. and Hz. More elements are used than Hot et al. [2] (hence the slightly different natural frequency estimates) to give smoother plots for the mode shapes and POMs, although the beam model is reduced to the first6linear modes for the time simulations, and modal damping with damping ratio 0.% has been used. The response of the beam to the random forcing is simulated for 00 s and the first 20% is discarded. This response is then used to estimate the frequency response function at the tip of the beam and the result is shown in Figure 2 for the two excitation levels considered. The responses are very close, although the first resonance frequency increases slightly as the force level increases. Also at higher excitation levels there is a response at the third harmonic of the first resonance. Figure 3 shows the mode shapes of the linear system and the POMs at the two RMS force levels, namely 0.00 N and2. N. The first three shapes are all very similar, which is perhaps not surprising as the bandwidth of excitation covers the first three linear modes. The subspace angles between the linear modes and the POMs are 0.6 and 0.2 for the 0.00 N and 2. N excitation respectively. At the low levels of forcing, the beam is behaves linearly, but the higher linear modes are not excited. Hence the fourth POM does not correspond to the fourth mode shape. As the excitation level increased, more energy is transferred to frequencies outside the excitation bandwidth, and this means that higher POMs look very similar to the corresponding higher mode shapes, although they are not exactly the same. To investigate a stronger nonlinear dynamic response the simulations are repeated with a swept sine excitation. The amplitude of the force is kept constant and the excitation frequency is swept from Hz to 20 Hz in 00 s. The simulations are repeated for excitation amplitudes of 0.00 N, 2. N and 20.0 N. Figure shows the time responses of the tip displacement for these cases. The response for 0.00 N excitation is essentially a linear response. The response at 2. N excitation shows that the non-linear spring has stiffened the first mode significantly and hence the first resonance occurs at a higher frequency (i.e. a later time in the frequency sweep). The response for 20.0 N excitation shows jumps in the response, particularly in the second mode. Figure shows the POMs for these cases; for 0.00 N and 2. N excitation the POMs are very close to the linear modes, whereas the first POM for 20.0 N excitation clearly shows that the non-linear spring is constraining the tip displacement. Although the time response clearly shows a jump at the second resonance, the second POM is very similar to the second linear mode. This highlights that complexity in terms of the non-linear dynamics and the time response, does not necessarily lead to complexity in the spatial response. The subspace angles between the first three linear modes and the first three POMs for the swept sine case are.2,.8 and.89 for excitation levels of 0.00 N, 2. N and 20.0 N respectively. It is clear that although the first POM is significantly different for the excitation level of 20.0 N, the response subspaces are very close. The response is significantly non-linear for the swept sine even at an excitation level of 2. N. Suppose the excitation frequency is swept from Hz to 30 Hz in 200 s. The response at the fundamental excitation

5 NON-LINEARITIES: IDENTIFICATION AND MODELLING N RMS 2.N RMS FRF amplitude (m/n) Frequency (Hz) Figure 2: The FRF at the beam tip due to random excitation at two force levels frequency is then obtained by fitting short windows of data consisting of four periods of the excitation. The resulting frequency response functions (FRFs) are shown in Figure 6 for excitation amplitudes of 0.00 N and 2. N. Clearly the response at the lower force level shows a linear response whereas the response at the higher force level shows a typical stiffening response with a jump in amplitude. Even at these slow swept rates the transients are very clear just after the resonances. Optimal sensor placement Classical OSP approaches are sub-optimal methods. Kammer [] proposed an iterative method, the Effective Independence method (EFI), based on the maximisation of the determinant of the Fisher Information matrix (FIM). This matrix is defined as the product of the mode shape matrix and its transpose. The number of sensors is reduced in an iterative way by deleting the degrees-of-freedom (DOFs) successively from the mode shape matrix. This idea can also be extended to biaxial and triaxial accelerometers []. The FIM can also be weighted by the use of the mass matrix from a finite element model [6]. Energy matrix rank optimization (EMRO) techniques are similar to the EFI method but in this case the strain energy of the structure is maximised rather than the determinant of the FIM [7]. A similar method based on the kinetic energy is obtained by using the mass matrix instead of the stiffness matrix [8]. Information entropy has also been used as the performance measure of a sensor configuration. A reduction in the information entropy is directly related to a measure of uncertainty. Papadimitriou and Beck [9] and Papadimitriou [20] employed entropy-based optimal sensor location, measuring the uncertainty in the structural system parameters, for structural model updating and this method is known as parametric identification.. The Effective Independence Method The EFI method, developed by Kammer [], is one of the most popular methods to optimally locate sensors in structural dynamic tests. The starting point of the EFI method is the full modal matrix from a finite element model of the structure to be studied. The finite element model could be built with any type of elements, but not all DOFs under consideration can be measured. Some of these DOFs are internal, while others are rotations, neither of which can be measured conveniently. Hence, the rows corresponding to DOFs that cannot be measured are deleted from the full modal matrix. Furthermore, system identification methodologies

6 3038 PROCEEDINGS OF ISMA20 INCLUDING USD20 Linear Modes POMs 0.00N Force POMs 2.N Force Figure 3: The first five linear modes and POMs due to random excitation at two force levels Response (µm) N Force Time (s) Response (mm) N Force Time (s) Response (mm) N Force Time (s) Figure : The time response at the beam tip due to swept sine excitation at three force levels

7 NON-LINEARITIES: IDENTIFICATION AND MODELLING 3039 POMs 0.00N Force POMs 2.N Force POMs 20N Force Figure : The first five POMs due to swept sine excitation at three force levels FRF amplitude (m/n) N 2.N Frequency (Hz) Figure 6: The FRFs for the first resonance for swept sine excitation at 0.00 N and 2. N amplitudes

8 300 PROCEEDINGS OF ISMA20 INCLUDING USD20 from sensor data records can only extract some of the mode shapes of the structure under study. Therefore, a limited number of target modes must be selected in order to find the best sensor configuration. Consequently, only some rows (potential DOFs locations) and columns (target modes) of the full modal matrix are retained. The EFI sensor placement algorithm is based on the Fisher Information Matrix (FIM), which is defined as FIM = Φ Φ (0) where Φ is the mode shape matrix. The FIM matrix is symmetric and positive semi-definite. Furthermore, if the mode shape vectors are linearly independent, then the FIM matrix is full rank, i.e. the rank is equal to the number of target mode shapes. The main aim of the EFI method is to select the best DOF configuration (where sensors are to be placed) that maximizes the FIM determinant. These DOFs are selected in an iterative way, thereby producing a sub-optimal solution to the problem. The selection procedure is based on the orthogonal projection matrix E, defined as E = ΦFIM Φ. () The matrix E is an idempotent matrix whose rank is equal to the sum of the diagonal terms. Hence, the i th diagonal element of matrix E, denoted by E ii, represents the fractional contribution of the i th DOF to the rank of E. The DOF with the lowest value of E ii is deleted as a candidate sensor location and the corresponding row removed from the modal matrix. The contribution of E ii to the rank of E is often cited as the reason that this procedure retains the linear independence of the mode shapes. However, it is easily shown [9] that det(fim) = det(fim 0 )( E ii ) (2) where FIM 0 is the original FIM and FIM is the FIM after the removal of the i th sensor. Thus, every time one row is deleted from the mode shape matrix, Φ, the determinant of the FIM decreases, and the objective of the selection method is to maintain a high value for this determinant. Many authors (for example, [2]) have highlighted that the high determinant ensures the variance of the estimated response is low, and this is used as the criteria for sensor location selection..2 The EFI method for non-linear structures For non-linear structures at low levels of excitation, the response is often well represented by the the sum of modal responses that are excited. In this case selecting sensor locations based on the linear modes represents a good solution. However, when the excitation is increased so that the non-linearities are excited then the linear modes may not represent the best basis for the response subspace, and hence an alternatives basis could be used. The POMs represent a suitable basis for the response subspace, and Ritto [0] suggested their use in place of the mode shapes to choose sensor locations. This approach relies on two significant and linked requirements. First, the example excitations and associated responses used to calculate the POMs must be representative of the dynamics of the system. This means that the dynamics of the structure can be projected onto the subspace spanned by the POMs with only a small residual. Second, the number of POMs used to choice the sensor locations is not fixed by the frequency range of interest, but must be able to capture the majority of the response. Numerical example A numerical example is used to demonstrate the influence of sensor position for non-linear detection. In this case, the cantilever beam response to a white noise force is studied. In this section only one non-linear detection method (residual projection error) is considered. In order to calculate the residual projection error, the time responses are first divided into windows. Thus, values of r, corresponding to each part of

9 NON-LINEARITIES: IDENTIFICATION AND MODELLING 30 the signal, are calculated for the 2. N excitation level. The sensor configuration is studied based on the following assumptions: only vertical displacement degrees of freedom can be measured and only the first five mode shapes and POMs are considered. Under these assumptions two different numbers of sensors were chosen to illustrate the effect on the residual projection methodology: all DOFs were chosen as measurement sensors, or a reduced number of sensors chosen. The literature strongly recommends choosing the number of sensors to be equal or lower than the number of mode shapes or POMs considered. Thus, a total of five sensors are considered in the reduced cases. 6 x 0 7. residual error part number 2 x 0 6 (a) All degrees of freedom selected. residual error part number (b) Five degrees of freedom selected: -*- EFI with POMs, -o- EFI with mode shapes Figure 7: Evolution of the residual error, r, with a non-linear cubic stiffness for the case of white noise force with an RMS amplitude of 2. N For the first case, Figure 7(a) shows the residual projection error evolution when all vertical DOFs are chosen at the same time. The evolution of the residual error through every window of the response may be observed. Thus, windows 9 and give the maximum residual error. In the second case, a total of five sensors were studied with the EFI methodology applied to both mode shapes and POMs. The sensor configuration selection is an iterative process, as described in the EFI methodology section, and the POMs may be considered in place of the mode shapes. The previous sections showed that the differences between both mode shapes and the POMs were not very high. However, the EFI methodology chooses different sensor configurations for both cases. Only one sensor location is shared in both configurations. The others locations are very close, but still different. Figure 7(b) shows two important results, the residual projection error decreases with the reduced number of DOFs measured, and the sensor configuration using POMs improves the non-linear detection. In both cases windows 9 and still remain as maxima; in the case of the POMs configuration they are global maxima, in contrast to the case of the mode shapes configuration.

10 302 PROCEEDINGS OF ISMA20 INCLUDING USD20 6 Conclusions Non-linear detection methodologies have been introduced. Relationships between the Proper Orthogonal modes and non-linear detection have been discussed. A cantilever beam with a nonlinear spring has served to understand the spatial response of a typical nonlinear system. It was demonstrated that these structures try to minimise energy and hence will tend to deform with simple shapes if possible. For this reason the lower modes can form a good basis for a range of deformations. A sensor configuration methodology was then introduced to improve the detection of non-linear behaviour. The Effective Independence method was employed with Proper Orthogonal modes and compared with the results obtained from the mode shapes. The outcomes of these results show the potential of the proposed methodology and will be the subject of further investigation. Acknowledgements The authors acknowledge the support of the Engineering and Physical Sciences Research Council through grant number EP/K References [] C.R. Pickrel, A practical approach to modal pretest design, Mechanical Systems and Signal Processing, Vol. 3 (999), pp [2] J.E.T. Penny, M.I. Friswell, S.D. Garvey, Automatic choice of measurement locations for dynamic testing, AIAA Journal, Vol. 32 (99), pp [3] M. Papadopoulos, E. Garcia, Sensor placement methodologies for dynamic testing, AIAA Journal, Vol. 36 (998), pp [] S.D. Garvey, M.I. Friswell, J.E.T. Penny, Evaluation of a method for automatic selection of measurement locations based on subspace-matching, Proceedings of IMAC XIV, Dearborn, Michigan, 2- February 996, pp [] L. He, J. Lian, B. Ma, H. Wang, Optimal multiaxial sensor placement for modal identification of large structures, Structural Control and Health Monitoring, Vol. 2, No. (20), pp [6] K. D Souza, B. Epureanu, Sensor placement for damage detection in nonlinear systems using system augmentations, 8th AIAA/ASME/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2007), Paper No. AIAA [7] M. Reynier, H.A. Kandil, Sensors location for updating problems, Mechanical Systems and Signal Processing, Vol. 3 (999), pp [8] R. Castro-Triguero, S.M. Murugan, R. Gallego, M.I. Friswell, Robust sensor placement under parametric uncertainty, Mechanical Systems and Signal Processing, Vol., No. -2 (203), pp [9] R. Castro-Triguero, E.I. Saavedra Flores, F.A. Diaz de la O, M.I. Friswell, R. Gallego, Optimal sensor placement in timber structures by means of a multi-scale approach with material uncertainty, Structural Control and Health Monitoring, (20). [0] T. G. Ritto, Choice of measurement locations of nonlinear structures using proper orthogonal modes and effective independence distribution vector, Shock and Vibration, Vol. 20 (20), article ID

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