Chapter 4 Proposed Multi-Sensor Power Spectrum Blind Sampling Approach for OMA:
5.2. Error Assessment of the PSBS Approach (Single-Sensor Case)
5.2.1. Structural system and simulation
Consider a viscously-damped linear Multi-Degree-of-Freedom (MDOF) structural system with R modes of vibration, excited by an ideal band-limited white-noise input with zero-mean and
2
w
variance (i.e., OMA assumption), that observes, theoretically, a power spectrum density (PSD) of constant amplitude, 2
( )
w w
G , across all frequencies, ω, in the band of interest. Let
x(t) be the output Gaussian process, representing a real-valued time-domain acceleration response
signal of the MDOF system, which, in the frequency domain, observes a maximum frequency at
ωmax. The input-output PSD relationship is expressed in
2 2 2
( ) ( ) ( ) ( )
x w i w i
G G H H , (5.1)
where H(iω) is the frequency response function of the MDOF system – termed as accelerance or
inertance in the field of modal testing (e.g., Ewins (2000)) for output acceleration response
signals. The above expression can be cast in the form Soong & Grigoriu (1996)
2 2 2 2 2 4 2 2 2 2 2 2 2 2 , 1 ( ) ( ) 4 ( ) [( ) (2 ) ] [( ) (2 ) ] R r s r s r s x rs r s r r r s s s A
G , (5.2)with ωr (ωs) being the resonant frequency and ζr (ζs) the damping ratio of the MDOF structural
system at the r-th (s-th) mode of vibration, while the amplitude Ars is a parameter associated with
the structural modal deflected shapes and the modal participation factors due to the considered stationary input process (see also Soong & Grigoriu (1996)).
A Monte Carlo simulation-based assessment framework is introduced herein, which allows to compare the dynamic properties of MDOF system defined by eq. (5.2) with those estimated by the data-driven PSBS method fused with standard OMA algorithms. The proposed framework is illustrated Figure 5.1, which first defines an analog MDOF structural system with known modal properties (ωr, Ar, ζr) attaining the PSD in eq. (5.2). This represents the “target” PSD which is
sought to be captured by the developed PSBS approach as explained below.
The target PSD is replaced by a surrogate discrete-time auto-regressive moving average (ARMA) filter of order (p, q) subject to clipped white-noise excitation, w[n] (e.g., Spanos &
Mignolet (1989)). Based on the ARMA process, discrete-time Nyquist-sampled signals, xARMA[n], can be generated and treated as NR realisations of an underlying stochastic process representing
the acceleration responses of structural systems with known modal properties (ωr, Ar, ζr). This is
achieved by recursively computing each n sample in xARMA[n] based on past observation and
1 0 [ ] [ ] [ ] p q ARMA k ARMA k x n b x n k c w n
. (5.3)Figure 5.1: Adopted Monte Carlo simulation-based framework to assess the multi-coset sampling device for OMA applications
The coefficients bk, k=(1,2,…,p) and c , (0,1,..., )q of this ARMA filter are derived from the auto/cross-spectrum correlation matching algorithm by Spanos & Zeldin (1998), which is commonly used for spectrum compatible simulation (e.g., Giaralis & Spanos (2009), (2012)). The above coefficients are obtained by solving a (p q) (pq) system of linear equations such that the square modulus of the frequency response function of the ARMA filters closely trace the target PSD of the analog system in eq. (5.2), that is,
2
( ) ( i Ts)
x e
G
H . (5.4)In the above expression, Ts, is the sampling period of the discrete-time process associated with
the Nyquist relation, Ts=π/ωmax, and (ei Ts)
H is the transfer function of the ARMA filter that satisfies the equation (e.g., Spanos (1983); Giaralis & Spanos (2009))
0 1 ( ) 1 s s s q i T i T p ki T k k c e e b e
H . (5.5)For the considered stochastic process, the generated Nyquist-sampled discrete-time structural responses, xARMA[n] (coloured white noise via the ARMA filter) are contaminated further with
are next treated by a single discrete-time model of the multi-coset sampling device depicted in Figure 4.2, to derive M compressed/ sub-Nyquist measurements, yi[k], in eq. (4.5) for i={0,1,…,
1
M }, k={1, 2, …, K}, and MKM . The simulated compressed data are treated by the PSBS strategy in §4, to recover the unknown power spectrum, Gˆx N(2L 1) 1 upon solving the least- squares optimisation problem in eq. (4.26) (i.e., for a=b=1). Thus, an approximation of the target PSD in eq. (5.2) can be obtained and further processed with standard OMA algorithms (e.g., the “peak picking” method detailed in Ewins (2000)) to retrieve the location, amplitude, and width of the recovered spectral peaks, providing approximations of the structural modal parameters,
ˆ ˆ ˆ , ,r Ar r
. The accuracy of these parameters can be assessed with respect to the known modal values, ωr, Ar, ζr, originally defined in the first step of the proposed framework (i.e., in derivingthe target PSD of the MDOF system in eq. (5.2)). The significance of the developed strategy in Figure 5.1 can be appreciated in that parametric dynamic analyses are numerically performed in MDOF systems with low computational cost, bypassing the need for linear response history analyses using standard FE software.
In this study, the simulation-based framework in Figure 5.1 is utilised to approximate a continuous MDOF structural system with R=2 degrees of freedom (i.e., 2DOF system) and a critical damping of ζ1 = ζ2 =5% for both modes of vibration, pertaining to the spectral coefficients A11=0.43, A22=0.5, A12=A21=0.46 in eq. (5.2). Note that the above coefficients can be retrieved
from the PSDs of acceleration responses measured at the quarter-span of a 2DOF dynamically vibrating simply supported beam subjected to a white-noise point force applied at the 3/8 of its length (e.g., Figure 5.2).
Figure 5.2: L-length simply supported beam with two degrees of freedom and the considered location of the excitation and measurement point at the 3L/8 and L/4 respectively
Two different case studies are examined for the above 2DOF with resonances at:
(1) ω1=20 rad/s, ω2=60 rad/s, (case of well-separated modes of vibration); and
For the above systems, their PSDs (target) are first derived from eq. (5.2). It is considered next an ARMA filter of order (200, 20) which is convolved with a clipped white-noise input assuming a sampling period at Ts=0.02s (i.e., the Nyquist frequency is 157.08 rad/s). The
auto/cross-spectrum correlation matching algorithm proposed by Spanos & Zeldin (1998) is then employed to compute the ARMA coefficients using eqs. (5.2), (5.4), and (5.5), and derive discrete-time Nyquist-sampled signals considering NR=10 realisations of the underlying
stochastic process. The generated signals represent acceleration responses of the adopted MDOFs in cases (i) and (ii) under ambient vibrations, to be treated by the proposed PSBS method in §4. For illustration, Figure 5.3 shows the above derived PSDs for the two considered cases, normalised to their peak value to facilitate comparison. It is seen that the PSD curve of the ARMA filter (broken red curves) can efficiently represent the target power spectrum (grey curve) evaluated from eq. (5.2). Figure 5.3 also presents the theoretical PSD (solid blue curve) derived from eq. (4.18) (for a=b=1) together with the autocorrelation function of the ARMA filter,
[ ] E [ ] [ ]
xx x
r p x n x np (see also eq. (4.6)), which is shown to closely trace the target PSD.
Further to the above, Figure 5.4 plots the recovered PSDs at CR=21%, obtained from eq. (4.26) for M=8, N=39 and s=[0,1,3,7,9,14,18,19]T (see also Table 5-2) for the case of well- separated (Figure 5.4(a)) and that of closely-spaced modes of vibration (Figure 5.4(b)). It is readily observed that the PSBS-recovered PSDs are capturing well the salient attributes of the systems frequency response function, such as the location of the two prominent peaks, their widths, and amplitudes, associated with the structural resonant frequencies, the damping ratio, and the modal deflected shapes, respectively, at the pertinent modes of vibration. From a qualitative point of view, the above confirms the efficiency of the proposed PSBS method in retrieving auto-spectral densities from compressed acceleration measurements acquired from a single sensor.
Figure 5.3: Comparison of normalised PSD curves to maximum their amplitude, obtained from the target PSD (analytical expression in eq. (5.2)), the ARMA model, and the theoretical expression in the PSBS method for the two adopted case studies: 2DOF with (a) well-separated and (b) closely-spaced modes of vibration
Figure 5.4: Estimated PSDs from sub-Nyquist multi-coset sampled simulated data (K=1000, M , 8 39
N , L=16) with the multi-coset sampling pattern s=[0,1,3,7,9,14,18,19]T (blue curve)
plotted against the target PSD in eq. (5.2) for the two adopted case studies: (a) 2DOF with well-separated and (b) closely-spaced modes of vibration
An error metric is adopted to quantitatively assess the recovery performance of the PSBS method and ensure reasonable estimates of structural modal properties. In this respect, the root- mean-square error (RMSE)
2 0 ˆ ( ) ( ) 1 n x j x j j G G RMSE n
(5.6)is computed between the power spectral amplitudes derived from the PSBS-recovered PSDs, ˆ ( )x j
G , and those obtained from the target spectrum, Gx(
j), in the frequency band [ω0, ωn].For the two examined 2DOF structural systems, the adopted error metric in eq. (5.6) is calculated for three different frequency bands as reported in Table 5-1, i.e., one wide-band covering the frequency range of interest between [0, 100] rad/s, and two narrow-bands around the resonant frequencies of the adopted structural systems (i.e., ω1, and ω2, respectively). For
illustration, Figure 5.5 and Figure 5.6 plot with a red curve the three different spectral ranges in Table 5-1 for the two 2DOF systems analysed.
Table 5-1: Considered frequency ranges in the PSD estimates for the computation of the RMSE
Case Study 1 Case Study 2
Well-separated modes (ω1=20rad/s,ω2=60rad/s)
Closely-spaced modes (ω1=20rad/s,ω2=25rad/s)
[ω0 - ωn] [ω0 - ωn]
Wide-band [0 - 100] rad/s [0 - 100] rad/s
Narrow-band around ω1 [10 - 30] rad/s [18.5 – 21.5] rad/s
Narrow-band around ω2 [50 - 70] rad/s [24 - 27] rad/s
Figure 5.5: Considered frequency bands in the computation of the RMSE between recovered PSBS-PSD and target PSD for the 1st case study: (a) wide-band; (b) narrow-band around ω
1; and (c)
narrow-band around ω2
Figure 5.6: Considered frequency bands in the computation of the RMSE between recovered PSBS-PSD and target PSD for the 2nd case study: (a) wide-band; (b) narrow-band around ω
1; and (c) narrow-band around ω2