Parts of this chapter have been published in:
[GDAM18] S. Gres, M. D¨ohler, P. Andersen, and L. Mevel. Variance computation of the Modal Assurance Criterion. InInternational Conference on Noise and Vibration Engineering - KU Leuven, pages 2939–2950, Belgium, 2018. Katholieke Universiteit, Leuven. Proceedings of the ISMA2018 and the USD2018.
Furthermore, this chapter is currently in preparation for the submission to Me-chanical Systems and Signal Processing journal.
5.6 Dissemination 83
In addition, the algorithm for uncertainty quantification of MAC between different estimates of the mode shapes was implemented in ARTeMIS Modal Pro 6.0 [SVSA18].
CHAPTER 6
Uncertainty quantification of the MAC from a stabilization diagram
The stabilization diagram is a practical tool used in many engineering applications of system identification to handle bias errors in the estimates of modal parameters, considering the order of the system is unknown. There, the estimates of modal param-eters can be clustered into so-called modal alignments by some practical criteria, like the relative difference between the consecutive parameters, and the modal indicators like MAC and MPC. However, no measure yet exists to quantify probability that a modal estimate belongs to a selected modal alignment. In this chapter such statis-tical measure is developed, based on the confidence intervals of the MAC estimates corresponding to the mode shape estimates within tested alignment. That allows to filter the modal alignments from estimates of modes which do not correspond to the same theoretical mode shape, which should result in an improvement of the global estimates of modal parameters.
In this chapter, a framework to quantify uncertainty of the MAC from a stabilization diagram is developed. Moreover, uncertainty computation of a global mode shape estimate and variance computation of the MPC computed from the global mode shapes is also established. These topics are illustrated based on a numerical simulations of a simple chain system and applied to a real-data from the experimental plate.
6.1 Illustrative example
For numerical simulations consider a theoretical 6 DOF chain-like system that, for any consistent set of units, is modeled with a spring stiffnessk1 =k3 =k5 = 100 and k2 =k4 = k6 = 200, mass m1−6 = 1/20 and a proportional modal damping matrix. The system is subjected to white noise signal in all DOFs and sampled with a frequency of 50 Hz for 2000 seconds. The responses are measured at 1, 2 and 5 DOF. Gaussian white noise with 5% of the standard deviation of the output is added to the response at each channel. The computations are performed in a
Monte Carlo setup withm= 1000 simulations. The SSI-UPC and the corresponding variance computation are set up with time lags of 15 and 200 blocks for the covariance computation of the data Hankel matrix. System orders are in the range between 12−40. Six global modes are established, with the respective modal parameters and their variances, in each simulation. For the formation of each modal alignment the following criteria are used
difference in two consecutive natural frequencies≤1%,
difference in two consecutive damping ratios≤10%,
MAC level between two consecutive mode shapes≥99%,
standard deviation of natural frequency computed with the perturbation theory
≤1%.
The results presented first are established based on one realization of the Monte Carlo simulations. Figure 6.1 illustrates the estimates of natural frequency with the corresponding 95% confidence intervals computed for model orders 12−40 together with the estimates of the global natural frequency and its corresponding 95% confidence intervals. Figure 6.2 shows the estimates of the natural frequencies, the damping ratios, the MAC and the MPC from the first modal alignment.
0 2 4 6 8 10 12 14 16 18
natural frequency f (Hz) 15
20 25 30 35 40 45 50
model order n
Estimates f Global estimate f
Figure 6.1: Stabilization diagram for natural frequencies with the corresponding 95%
confidence intervals.
One can see that the natural frequency of each mode can be tracked along estimated model orders and the estimated variances are small, thus not visible in the scale presented on Figure 6.1. From Figure 6.2 it can be observed that the statistical dispersion of the natural frequencies and damping ratios, represented by the horizontal bars, is small and yields less then 0.1 Hz and 0.15% respectively for the estimates of the natural frequency and the damping ratio. The variation of the natural frequencies and damping ratios in the estimates at different modal orders is reflected by the global estimates as described in the scheme in Appendix A.1.2. The MAC values presented on Figure 6.2 are estimated between the mode shape from model order 12 and the remaining mode shapes of the first alignment. Resultant MAC values tend to 1, meaning that the mode shapes used to compute the MAC belong to the same mode.
6.1 Illustrative example 87
1.9 1.95
natural frequency f (Hz) 15
20 25 30 35 40 45 50
model order n
Estimates f Global estimate f
0 2 4
damping ratio (%) 15
20 25 30 35 40 45 50
Estimates d Global estimate d
0.9999 0.99995 1 MAC 15
20 25 30 35 40 45 50
Estimates MAC
0.9998 0.9999 1 MPC 15
20 25 30 35 40 45 50
Estimates MPC
Figure 6.2: Estimates of natural frequencies, damping ratios, MAC and MPC from the first modal alignment.
It also indicates that the estimates of MAC within one modal alignment approach the right boundary of the distribution of the MAC indicator. The quantification of statistical uncertainties for such problems has been investigated for the MPC in Chapter 4 and for the MAC between the estimate of a mode shape and an exact mode shape from a model in Chapter 5. However, a scheme for quantifying the uncertainty of the MAC indicator between two mode shape estimates from the stabilization diagram is missing. For that, three cases of MAC computation can be distinguished, namely
1. gmac( ˆϕa,n,ψˆa,n) where ˆϕa,n and ˆψa,n are estimates of different mode shapes from a modal alignmentaand model ordern,
2. gmac( ˆϕa,global,ψˆa,n) where ˆϕa,globalis an estimate of a global mode shape from a modal alignmenta,
3. gmac( ˆϕa,global,ψˆa,global) where ˆϕa,globaland ˆψa,globalare estimates of a orthogo-nal global mode shapes from two different modal alignments.
Additionally, the global estimates of the MAC and the MPC are functions of the global mode shapes for which the uncertainty scheme is also recalled.
6.1.1 Distribution of MAC from the stabilization diagram
This section illustrates different distributions of MAC between mode shapes corre-sponding to the modes from the stabilization diagram. The distribution of MAC is made available by Monte Carlo simulations of the numerical system described in Section 6.1. The examples that follow are based on the aforementioned MAC computation cases from the section above
1. gmac( ˆϕ1,12,ψˆ1,14) is computed between the mode shapes from first modal align-ment at model orders 12 and 14, namely ˆϕ1,12 and ˆψ1,14. Denote it as the base-case 1.
2. gmac( ˆϕ1,global,ψˆ1,12) is computed between a global mode shape estimate of the first modal alignment and the one at modal order 12, namely ˆϕ1,globaland ˆψ1,12. Denote it as the base-case 2.
3. gmac( ˆϕ1,global,ψˆ3,global) computed between a global mode shape estimate of the first modal alignment and a global mode shape estimate from third alignment, namely ˆϕ1,globaland ˆψ3,global. Denote it as the base-case 3.
These distributions are illustrated on Figure 6.3.
0.99996 0.99997 0.99998 0.99999 1
MAC 0
100 200 300 400 500 600 700
counts
0.999940.999950.999960.999970.999980.99999 1 MAC
0 50 100 150 200 250 300 350
counts
0 2 4 6 8 10 12 14 16 18 20
MAC 10-6
0 50 100 150 200 250 300 350
counts
Figure 6.3: Histogram of the base-case 1 MAC (left). Histogram of the base-case 2 MAC (middle). Histogram of the base-case 3 MAC (right).
For the two first cases the MAC indicator is computed between the mode shapes that correspond to the same theoretical mode, hence its value is asymptotically 1.
For the third case the MAC is computed between mode shapes that are orthogonal, hence the distribution of its values asymptotically tends to zero. All three cases are inadequate to be approximated by a Gaussian distribution, which was shown on Figure 5.10 from Section 5.2.3, since the JacobianJϕg∗mac,ψ∗ used in that approximation is null, as showed in Lemma 5.2. Also, it is clear that the Gaussian approximation should be symmetrical with respect to its limit mean, which is impossible if the limit is on the boundary of the support. Section 6.3 in this chapter focuses on approximating three aforementioned cases of MAC with a quadratic form, which subsequently is approximated by a scaledχ2 distribution. Before, however, a strategy to estimate the uncertainty in the global mode shape estimates is established.