DIAGRAMA DEL MODELO DE GESTIÓN DE PROCESO DE NEGOCIO

The results of MC simulations on the power reflection coefficient are given in this section. Conclusions below are all drawn regarding this set of MC simulations. Figure 3.4 shows the MC simulations for the case with noise level σ₁=σ₂ =5%. The x-axis

b

k Δ is proportional to the non-dimensional frequency ξ. In this case, k_bΔ =29.412ξ when the transducer spacing Δ =0.05 m. MC simulations are performed at 100 frequencies at equally spaced wavenumbers in a range of 0< Δ <k_b 3.5. At each frequency, 10,000 calculations are performed with random perturbations on “measured” transducer outputs. The mean value of MC simulation and the true value of the power reflection coefficient are also shown in the figure. The deviation and fluctuation of the simulated points can be seen from the “spread” of the data. For frequencies such that

0.28

b

k Δ > , the transducers are more than one wavelength from the discontinuity and the nearfields can be neglected.

When ρ=0, the energy will be totally transmitted through the discontinuity. Referring to equation (2.52), 0r_PP = , i.e. μϑξ3+2ϑξ2−2μ=0. Substituting the values of μ and

beam discontinuity density, ρ _modulus, Young’s _E width×_{b h}thickness, _× mass,m moment of _{inertia, J}

7800 9

194 10× 0.050 0.006× 0.300 4

ϑ into the this equation, it gives ξ =0.042, thus k_bΔ =1.235. Here it should be noticed that ξ =k_bκ, where κ =h/ 12. When τ =0 and t_PP =0, i.e. ϑξ3−μξ + =4 0. In this case 0.086ξ = and k_bΔ =2.529 . The energy will be totally reflected from the discontinuity.

Figure 3.5(a) shows the mean value of the estimated power reflection coefficient ρˆ, obtained from MC simulations with a transducer spacing Δ =0.05m and transducer noise standard deviation levels σ₁=σ₂ =5%. Also shown is the noise free value, the approximate solution given by perturbation (equation (3.19)), and the resulting upper bound given by equation (3.20). The MC simulation is close to the approximate solution over the whole frequency range, and the upper bound of this is fairly conservative. The power reflection coefficient is estimated well except near zero frequency and k_bΔ =π

where bias in the estimate is most apparent. Near these frequencies the sensor spacing is nearly 0 and 1 times half a wavelength respectively. This causes sink_bΔ ≈0 in equations (3.19) and (3.20).

Figure 3.4 Monte Carlo simulations of the power reflection coefficient:σ₁=σ₂ =5%, 1.20

a= m and Δ =0.05m: , ρˆ; , E

[ ]

[ ]

Figure 3.5(b) gives the corresponding estimates for the variance of ρˆ. The approximate solution given by equation (3.21) is again in close agreement with the MC simulations. The upper bound given by equation (3.22) appears fairly conservative except near zero

frequency and k_bΔ =π . This is due in part to fluctuations in the approximate solution for ρˆ arising from the term _e−2ik ab _{in equation (3.21). The exponent,}

b

k a can become very large, especially at high frequencies, so the variance changes rapidly with frequency. The estimate of the variance is smallest at about k_bΔ =1.2, where the power reflection coefficient is the smallest. This is due to the definition of the noise model.

10-3 10-2 10-1 100 101 ρ (a) 0 0.5 1 1.5 2 2.5 3 3.5 10-5 10-3 10-1 101 σ 2 ρ k bΔ, rad (b)

Figure 3.5 First order approximations and MC simulations of (a) the mean value and (b) the variance of ρˆ: Δ =0.05m and σ₁ =σ₂ =5%: , ρ(noise free); , perturbation solutions; • , Monte Carlo simulations; , upper bound of perturbation

solutions.

The effect of noise level was similarly investigated using the approximate solution of equation (3.22). Figure 3.6 contrasts the effect of transducer noise levels of 1% and 5% for a fixed sensor spacing of Δ =0.05m. The results are expressed in terms of the standard deviation normalised by the true value in order to assess the extent to which the wave decomposition process amplifies noise on the transducers. The first peak at about 1.2k_bΔ = is inevitable since there is no true reflection at this frequency to measure, and the second peak corresponds to a half-wavelength transducer spacing. At best (i.e. at frequencies for which the transducer spacing is a quarter wavelength) the noise on the power reflection coefficient is twice as large as that of the individual sensor measurements. The imperative for high fidelity measurements is clear if high precision of the reflection coefficients is desired.

0 0.5 1 1.5 2 2.5 3 3.5 10-2 10-1 100 101 102 103 k bΔ, rad σ ρ / ρ

Figure 3.6 Closed form solutions for the upper bound normalised standard deviation of ˆ

ρ: , σ₁=σ₂ =5%; , σ₁=σ₂ =1%.

If the mass or moment of inertia changes, usually the frequencies where ρ=0 and 1

ρ = will change correspondingly. If the discontinuity is a spring-like discontinuity, the trend of ρ is different from that of the mass and moment of inertia discontinuity. However, the noise on the power reflection coefficient keeps a similar trend (see Figure 3.4), i.e. when ρ is small, the noise is small; when ρ is large, the noise becomes large. Also at frequencies where k_bΔ =nπ , the variance of ρ is very large because of the ill- condition of the wave amplitude decomposition method.

3.6.3 Statistical Distribution of the Simulated Noisy Power Reflection