An example of correlation matrix based mode shape expansion in OMA

Transcription

1 An example of correlation matrix based mode shape expansion in OMA Rune Brincker 1 Edilson Alexandre Camargo 2 Anders Skafte 1 1 : Department of Engineering, Aarhus University, Aarhus, Denmark 2 : Institute of Aeronautics and Space - IAE, São José dos Campos, Brazil Abstract In cases of reducing the number of modes in an operating response for instance when using band pass filtering often time domain identification techniques suffer from a tendency to over fitting. As a consequence, it might be useful to decrease the number of measured DOF s to moderate the number of modes in the modal model. In these cases the dimension of the identified mode shapes is diminished accordingly and therefore it is preferable to have a way to expand back to the full set of DOF s so that the estimated mode shapes can be animated in detail. In the present paper it is considered to use the correlation matrix for the filtered response including all measured DOF s as a basis for the mode shape expansion. This matrix contains the normal modes that can easily be extracted from the column space of the correlation matrix. In this investigation the main focus has not been on the reduction problem, so engineering judgment has been used to secure a reasonable choice of reduced channels. The technique is illustrated on an OMA case where the modes of the tail part of a Panther helicopter is estimated during different flight conditions. Keywords: OMA, DOF reduction, Expansion, Helicopter, Time domain 1. Introduction Nearly all classical modal analysis techniques in the time domain can be used for OMA using the correlation functions as input for the techniques instead of impulse response functions. This is true for instance for techniques like the Ibrahim time domain (ITD) technique, [2-5], the poly reference (PR) technique [6-7] and the Eigen realization algorithm (ERA) technique [8-10] that has been used in the present analysis. These techniques typically have a lower limit to (or a fixed size like the classical ITD) to the number of modes included in the model. This lower limit is normally equal to the number of measured degrees of freedom (DOF s). For instance in the PR technique, normally the lowest model order to be considered corresponds to 2 auto regressive matrices, defining the lowest number of modes in the model to be equal to the number of measured DOF s. This is also the case for the classical ITD technique. It is well known that it is normally an advantage to use an oversized model including a reasonable number of noise modes in order to obtain good modal estimates. However, it is also well known, that if too many noise modes are being used, problems with over fitting might arise, and the modal results can be affected in a negative way. The over fitting problem often becomes severe in cases with a high amount of noise in the signals. In such case it might be useful to consider a reduced number of measured DOF s in the identification, so that the number of noise modes can be reduced accordingly. If the modes are identified in a reduced number of DOF s, then the estimated mode shapes are only known in these DOF s, and therefore, after the identification in the reduced set, it is useful to expand the mode shapes to the full set of DOF s. In the following we will consider the case where the random response to be used for OMA is band pass filtered in order to reduce the number of modes present in the random response. This means that the number of modes is often quite small such as 2-4 modes. Further we will consider the possibility to perform the mode shape expansion based on the correlation matrix of the band pass filtered response.

2 First we will present the basic theory for why it makes sense to use the correlation matrix for expansion, the main reasons being that in case of band pass filtered signals containing only a limited number of modes and in case of OMA tests with a reasonable channel count, the correlation matrix provides a very good way of checking the number of modes in the band pass filtered signal and for expansion. The resulting modes will be normal modes (real mode shapes) so that direct comparison can be made with a finite element model. It is clear that going from a high number of measured DOF s to a small number of DOF s for identification might also be dangerous, because in such case there is a risk of excluding important modal information if the number of identification DOF s are not chosen reasonably. In this investigation however the main focus is not been on the reduction problem, so engineering judgment has been used to secure a reasonable choice of reduced channels. In the following we will illustrate how the classical techniques mentioned above combined with the proposed reduction/expansion technique perform on a difficult OMA case with a relative high channel count and high amount of noise. The considered case is OMA of the tail part of a Panther helicopter estimated during different flight conditions. 2. Correlation matrix theory Let us consider a zero mean random response y (t) with N d number of measured DOF s, i.e. the response vector y (t) is a column vector with N d number of components. In this case the response correlation matrix is Nd Nd and is defined by T (1) C E y ( t) y ( t) y For a measured time series, the matrix will normally be calculated in terms of time averaging (2) 1 T C y y( t) y T ( t) dt T0 0 0 where T 0 is the length of the considered random response. However, if the response is given as a spectral density function, we can use that the correlation function matrix is defined as T (3) R ( ) E y ( t) y ( t ) So that the correlation matrix can be found as (4) Cy R y (0) y Since the correlation function matrix is the inverse Fourier transform of the of spectral density matrix we can obtain the correlation matrix from (5) C y G y ( ) d Thus, the correlation matrix can be found as the integral of the spectral density matrix. It should be noted that in case we are dealing with sampled data, the integration should be performed only over the Nyquist band. Further it should be noted that since the imaginary part of a spectral density function is an odd function of frequency, the integral will always be real as it is given by the definition in Eq. (1). Assuming that the measured response is a linear combination of N normal modes we have

3 (6) y( t) a1q1 ( t) a2q2( t) an qn ( t) Aq( t) where A is the mode shape matrix A a1, a2,, a N q t ) q ( t), q ( t),, q T N ( t). Using this is Eq. (1) we have ( 1 2 T T T (7) Cy AE q( t) q ( t) A and q (t) is the modal coordinate vector ACqA where C q is the correlation matrix of the modal coordinates. 3. Expansion from a smaller set of DOF s We shall assume that the external loading on the considered structure is not of a kind where C q has a rank that is smaller than the number of modes, in fact we shall consider a situation where the modal coordinates are approximately stochastically independent. Further we shall assume that the number of modes N in the response is significantly smaller than the number N d of measured DOF s. In this case, when taking a singular value decomposition of the response correlation matrix (8) T Cy USU any mode shape a participating in the response given by Eq. (6) is in the subspace U and the rank of C y is equal to the number of modes N in the response. We then have the linear relation between the modal matrix A and the subspace U (9) A UT where the transformation matrix T is DOF s, N N r N N. If we have reduced the response to consider only N r number of measured d, then we have the similar equation in the reduced set of DOF s (10) Ar UrT If we have obtained the experimental modes in the reduced set of DOF s, we can find the transformation matrix from Eq. (10). If the reduced number of DOF s Nr is larger than the number of modes, then the equation can be solved approximately (11) Tˆ U r Ar where U r is the pseudo inverse of the matrix U r. In this case we solve the equation in an over determined way and the estimate Tˆ is then less sensitive to the noise on the mode shape estimates. In cases where the reduced set of DOF s is equal to the number modes (usually a model without noise modes), Eq. (10) is a normal equation that can be solved taking the inverse of the matrix U r that is now square (and full rank) (12) Tˆ 1 U r Ar In this case it should be considered if the noise on the mode shape estimate might significantly influence the transformation matrix estimate Tˆ.

4 When we have an estimate of the transformation matrix we can expand the mode shapes to full size using Eq. (9) to get an estimate of the mode shapes in all the measured DOF s (13) Aˆ UTˆ 4. Panther helicopter test case The test case is an investigation of the tail part of the Panther helicopter shown in Figure 1.a. The helicopter was tested in different flight conditions where some results for two of the flight conditions are reported here. A more comprehensive reporting of the results of all investigated flight conditions can be found in Camargo and Brincker [11]. The measurement of the flight condition analyzed in this article, were performed with the aircraft flying in a fixed altitude of 5000 feet with cruising speed under constant weather conditions and payload (Figure 1a). The altitude of the flying, at cruising speed, is taking place according to the simple principle that the pitch of the rotor determines the forward speed, and power determines the altitude of the aircraft. As a consequence, in the cruising speed condition considered here, the flight condition is characterized by constant rotor pitch angle and constant power consumption of the engine. The experimental setup was configured with 40 measurement channels, divided into three measurement units with 17 channels each, using 16 unidirectional accelerometers 4515 type (8 in X axis and 8 in Y axis) and 10 tri-axial accelerometer 4520 type. Of the 10 tri-axial accelerometers four where using all three channels, and 2 where using only two of the three channels in the OMA. The placement of the sensors is shown in Figure 1b. The data acquisition was performed using the software called LabShop from Brüel & Kjaer (B&K) and the analysis were done, in time domain, using the OMA MatLab toolbox related to the OMA book by Brincker and Ventura [1]. (a) (b) Figure 1: (a) The Panther helicopter at cruiser speed, (b) The sensor locations, left: all 40 measured DOF s, right: The four DOF s used in the OMA (reduced set). The measurements of the full set of 40 DOFs were performed for 60 seconds at an acquisition rate of 2,000 samples per second. Analysis were performed using just 4 DOFs, located at the edge of the horizontal stabilizers (HS), see Figure 1b right, decimating the data to a Nyquist frequency of 80 Hz with sampling interval of seconds. This resulted in a spectral density estimate with a resolution of 1024 lines giving a frequency step of 0.04 Hz as shown in Figure 2. A main goal of the test was to identify the first four modes of the tail part of the helicopter, that is the XX mode, that is the first tail torsion mode, the first HS bending mode, the first tail bending mode, and finally the first HS torsional mode. The natural frequencies of these modes were known to be around 12,8 Hz, 14,8 Hz, 17,7 Hz and 19,5 Hz respectively.

5 In order to isolate these four modes, the data were band-pass filtered using the fftfilt.m function in the OMA toolbox for each mode with the center frequency of 12,8 Hz, 14,8 Hz, 17,7 Hz and 19,5 Hz respectively for each mode, a flat band around the center frequency with a band width of 1 Hz and roll-off bands on each side with a width of 1 Hz. Figure 2 shows the highlighted peaks of the four modes with the small cross marking the central frequency. The first harmonic of the main rotor of the helicopter at 23,6 Hz is also indicated in Figure 2. Figure 2. FDD plot showing the singular values of the spectral density matrix of the cruiser speed measurements. The highlighted peaks indicate the first four modes of the tail part and the first harmonic of the rotor is also indicated in the plot. The numbers of block rows on the AR/PR and ITD were chosen to na = 2. For the ERA, the numbers of block rows of the Hankel matrices used was equal to s = 2, and the realization was not reduced so that the returned number of modes were the same for the three techniques. In all three techniques, the correlation matrix was used as multiple input, and 100 discrete time lags were used in the identification. Using all 40 measurement channels the techniques return in principle 40 modes of which some might be non-physical due to non-positive damping, so typically modes were returned. Since we have band pass filtered the signal to contain only one single mode, among the returned modes we need to identify only the single mode that we are looking for. In the following we will illustrate the identification principle on the first of the four modes indicated in Figure 2, this mode has a natural frequency around 12.8 Hz. The modal participation for each mode was estimated as described in Brincker and Ventura [1], and used to identify the physical mode following the principle that noise modes should have a small modal participation and a physical mode should have a strong modal participation. Some of the returned modes are shown in Table 1 for the case where we are using all 40 DOF s. From the results in Table 1 we see that the mode with the highest participation using the PR technique is mode 13 with a modal participation of 22.9 %. Table 1: OMA of the first mode (12.8 Hz) using 40 DOF s PR Mode 1 Mode 2 Mode 3 Mode 4 Mode 11 Mode 13 Mode 15 Mode 27 Frequency, [Hz] Damping, [%] Contribution, [%] ITD Mode 1 Mode 2 Mode 3 Mode 4 Mode 7 Mode 9 Mode 11 Mode 19 Frequency, [Hz] Damping, [%] Contribution, [%] ERA Mode 1 Mode 2 Mode 3 Mode 4 Mode 11 Mode 13 Mode 15 Mode 27 Frequency, [Hz] Damping, [%] Contribution, [%]

6 It seems like a reasonable result since the corresponding natural frequency of Hz is also close to the expected value. However, we see that the modal participation is also high for adjacent modes with frequencies quite close to the target frequency. A similar picture is seen to be present for identification using ITD and ERA identification. As a result in this case it is difficult to use the modal participation clearly to distinguish between noise modes and physical modes Table 2: OMA of the first mode (12.8 Hz) using 4 DOF s PR Mode 1 Mode 2 Mode 3 Mode 4 Frequency, [Hz] Damping, [%] Contribution, [%] ITD Mode 1 Mode 2 Mode 3 Mode 4 Frequency, [Hz] Damping, [%] Contribution, [%] ERA Mode 1 Mode 2 Mode 3 Mode 4 Frequency, [Hz] Damping, [%] Contribution, [%] Reducing the number of DOFs to the 4 sensor signals shown in Figure 1.b, the identification techniques return only 4 modes. Table 2 shows the similar results for the first mode. We see that in this case mode 2 has a modal participation that is significantly larger than all other modal participation factors, and thus, using the reduced set of DOF s we can much easier make a decision of which mode to choose as the physical one. The distribution of the modal participating factors of the modes is shown in Figure 3 and as it appears, all three techniques have a high modal participation of 2-3 modes making it difficult to distinguish noise modes from the physical mode. The plot to the left shows the similar distribution of the modal participation factors in the case of using the reduced set of DOF s and as it appears, in this case only one mode has a modal participation factor that is significantly larger than the other ones. Figure 3. The plots show the modal participation factors of the three different techniques PR (blue), ITD (red) and ERA (gray). 5. OMA results using expansion The top plots of Figure 4 shows the results when only the four DOF s are being used, and as it appears from the plots, using only 4 DOFs the mode shapes are not well defined in a graphical representation. The mode shapes were then expanded using Equation (11/13) and the results shown the bottom plots of Figure 4 clearly shows the advantage of using the mode shape in full resolution.

7 4 DOFs 40 DOFs First tail torsion mode (12.8 Hz) First HS bending mode (15Hz) First tail bending mode (17.8Hz) First HS torsional mode (19.5Hz) Figure 4: Mode shapes of 4 DOFs and after expand the modes for 40 DOFs. Top plot: Mode shape plots using only 4 DOFs. Middle plot: Full mode shapes for flight condition cruising speed, bottom plot: Full mode shapes for the flight condition maximum speed. 6. Conclusions It is shown that the correlation matrix can be used of expansion of mode shapes and it has been illustrated how the expansion can be applied in cases with high channels counts and where over fitting might occur when performing identification using classical time domain techniques. The use of some classical time domain techniques has been illustrated using OMA on a Panther helicopter under different flight conditions. Acknowledgements The financial support received from CAPES, through the Science without Borders Program, for this research, is gratefully acknowledged. References [1] Brincker, R. and Ventura, C.E.: Introduction to operational modal anlaysis. Wiley [2] Ibrahim, S.R. and Milkulcik, E.C.: A method for direct identification of vibration parameters from the free response. The Shock and Vibration Bulletin, 47, p , [3] Ibrahim, S.R.: Random decrement technique for modal identification of structures. J. of Spacecraft and Rockets, 14, p , [4] Ibrahim, S.R.: Modal confidence factor in vibration testing. J. of Spacecraft and Rockets, 15, p , [5] Fukuzono, K.: Investigation of multiple-reference Ibrahim time domain modal parameter estimation technique. M.Sc. Thesis, Department of Mechanical and Industrial Engineering, University of Cincinnati, [6] Vold, H., Kundrat, J., Rocklin, G.T. and Russell, R.:A multi-input modal estimation algorithm for mini-computers. SAE Paper NUmber , 1082.

8 [7] Vold, H. and Rocklin, G.T.: The numerical implementation of a multi-input modal estimation method for mini-computers. In Proc. of the International Modal Analysis Conference (IMAC), p , [8] Juang, J.N. and Pappa, R.S.: An eigen system realization algorithm for modal parameter identification and modal reduction. J. Guidance, V. 8, No. 5, p , [9] Pappa, R.S., Elliott, K.B. and Schenk, A.: Consistent-mode indicator for the eigensystem realization algorithm. J. Guidance, Control and Dynamics, V. 16, No. 5, p , [10] Pappa, R.S.: Eigensystem realization algorithm, user s guide for VAX/VMS computers. NASA technical memorandum , [11] Camargo, E.A. and Brincker, R.: Operational modal analysis of a Panther helicopter during different flight conditions. To be submitted to XX.