Resonance Frequencies for 8202A10
6.3 Results
6.3.1 Numerical Validation of the Jitter Analysis
We present here the main results of the study in terms of quantification of the tacho signal jittering effects. The jitter variables are modelled as a centered uniform distribution in the interval[/2,/2]. Such a choice is motivated if we think of the prevalent error in pulse arrival time being due to the discretization of the keyphasor signal used in the tacho device for the pulse time of arrival determination. Figure6.3(similarly to [7] shows a keyphasor signal sampled with a sampling frequency fKP. The rate of samplingfKP at which the keyphasor is sampled determines the resolution of the keyphasor pulse arrival times. A pulse will be detected when, e.g., the rising edge of the signal surpass a threshold. We model the uncertainty due to such resolution as a uniform random variable, hence we assume D 1/fKP in the distribution of the jitter variables.
Such an assumption is reasonable when the main uncertainty is coming from the discretization of the keyphasor signal.
In order to validate the statistical analysis of Sect.6.2.3, the results in terms of statistical moments and distributions are compared to numerical simulations drawn from the proposed model in a Montecarlo-fashion. The remainder of the section is organized as follows: first, the expected value and variance computed forgkiare validated against numerical simulation. Then, the Laguerre-expansion approximation to the distribution of the absolute value is compared to the numerically simulated distribution. The same comparison is finally made for the phase error distribution. All the simulations are run in MATLAB® environment.
6.3.1.1 Expected Value and Variance of the Functionsgki
The free parameters used in the simulation and in the analytical expressions are given in Table6.1. The high value for the number of revolutionsNis fed to the simulation to obtain an estimate of the expected value and the variance of the simulated quantity. Figure6.4shows the analytical mean value in closed form and approximated through Eq. (6.30) as compared with the outcome of numerical simulations from the model. In the upper part of the plot it can be seen that the mean values (exact, approximated, simulated) ofg1iare virtually superposed. Concerning the expected value ofg2i, the scattering of the simulated values around the predicted value is increasingly higher with increased variance of the function (see Fig.6.5), suggesting that the zero-mean mismatch is mostly related to an insufficient number of averages. Figure6.5shows the estimations of the
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0 0.1 0.2 0.3 0.4 0.5
Time [s]
0.6 0.7 0.8 0.9 1
Analog signal Discrete samples Threshold
Amplitude
Fig. 6.3 Example of discrete sampling of an analog keyphasor signal
Table 6.1 parameters used for analytical expected value and variance validation
Parameter Value
N 1,000,000
T 1
f* 1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0.9 0.92 0.94 0.96 0.98 1
E[g1i]
Montecarlo simulation Exact analytical Approximated analytical
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
−1
−0.5 0 0.5
1x 10−3
Δ E[g2i]
Fig. 6.4 Mean values, comparison of analytical expressions and numerical simulations varying jitter entity. Upper plot:g1iexpectancy. Lower plot:g2iexpectancy
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0
1 2 3 4 5 6x 10−3
Var[g1i]
Montecarlo simulation Approximated analytical
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Δ Var[g2i]
Fig. 6.5 Variance values, comparison of analytical expressions and numerical simulations varying jitter entity.Upper plot:g1ivariance.Lower plot:g2ivariance
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01 0 0.01 0.02
Δ E[g1i] relative error [%]
Exact analytical Approximated analytical
Fig. 6.6 Mean values forg1i, comparison of analytical expressions and numerical simulations varying jitter entity. Relative error in percentage points between simulation (reference) and analytical solutions
variance respectively forg1iandg2i. It can be seen that the second-order approximation for the variance of g1i is increasingly worse with increasing entity of the jitter. This is consistent with the hypothesis done in the truncated expansion and suggest to add more terms for improving the analytical approximation in the high error region. Figures6.6and6.7provide a zoom on the accuracy of the approximations for the expectance ofg1iand the variance ofg2i, that could not be seen clearly in the previous figures. The values are reported in terms of relative error and show for Fig.6.6consistently decreasing accuracy for the approximated solution as the jitter entity grows, and oscillating precision of the analytical solution which can be explained as for the case of the expectancy ofg2i in Fig.6.4. Figure6.7shows consistently increasing (though relatively small) approximation error still attributable to the neglected terms in the power series expansion. Note that the parameter controlling the extent of the jitter,can be interpreted in terms of relative importance of the uncertainty with respect to the considered frequency component’s period as already explained in the theoretical part.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4
Δ Var[g2i] relative error [%]
Approximated analytical
Fig. 6.7 Variance values forg2i, comparison of analytical expressions and numerical simulations varying jitter entity. Relative error in percentage points between simulation (reference) and analytical solutions
Table 6.2 Parameters used for simulating the distribution of the absolute value
Parameter Value
Number of iterations 100,000
N 200
T 1
f* 1
1e-4*2000
6.3.1.2 Approximation of the Absolute Value Distribution
Although extensive simulations were performed to validate the model, for the sake of brevity here only one comparison is reported. The used parameters are reported in Table6.2. From the developed model, it is known that the distribution of the squared absolute value will be affected by the number of revolutions averaged. The more revolutions will be averaged, the less will be the variance. Seemingly, it is expected to have higher attenuation and higher variance when the jitter is of a higher entity. For the analyzed case, the number of averagesN was set to a typical value used in helicopter vibration diagnostic systems. The value of the jitter parameter is approximatively one that could have for the first harmonic of the gearmesh frequency (i.e. one prominent harmonic in geared transmission typical signature, computed as shaft rotational speed times number of teeth of a connected gear) when sampling withfKPD10 kHz. Figure6.8shows the predicted probability density function usingdD20 terms in the expansion, compared to the simulated one for the squared absolute value. Seemingly, Fig.6.9shows a comparison between the estimated distribution of the variable Rin and the simulated one for the same set of parameters. Fair agreement between prediction and simulations is observed. It can be concluded that the analytical approximations offer a viable way of predicting the attenuation and phase errors due to the jittering error in pulse arrival times according to the proposed model. Note that since the jitter effects are proportional to frequency, an attenuation of around 10% is reached for the 2 kHz harmonic only by considering the uncertainty due to the sampling of the keyphasor signal. Moreover, the values present themselves a certain variability. The approximated analytical and simulated expected value and variance for the considered simulation are given in Table6.3.