Propagation of Systematic Sensor Uncertainty Into the Frequency Domain

The determination of the dynamic characteristics of systems is of great importance in science, engineering, and industry. If the dynamic behavior of a system is known, the optimal design and control can be determined, for instance, through analytical models and their dependence on the design parameters, cf. Pelz et al. [1]. For complex mechanical systems such as hydraulic accumulators or modern turbomachinery, the analytical determination of the transfer behavior may be feasible; however, the identification of the system via experiments is usually preferred. The identification of the time-invariant system, e.g., the identification of the system inherent descriptive linear components, i.e., springs, dampers, and masses, is most commonly performed in the frequency domain [2,3].

In practice, this often but not exclusively involves measuring the response of the dynamic system to harmonic excitations, e.g., the frequency-dependent force of an air spring due to an imposed deflection, cf. Puff and Pelz [4] and Hedrich et al. [5,6]. The measurement signals are subsequently transformed into the frequency domain using fast Fourier transforms (FFT), allowing the frequency-dependent stiffness of the air spring to be determined.

While determining dynamic characteristics in the frequency domain is common practice, a proper quantification of the uncertainty of the identified quantities is often nonexistent. This is mainly due to the nontrivial propagation of the inherent random and systematic uncertainty of the measured signals from the time into the frequency domain or vice versa. Interestingly, however, for one of the two uncertainty types, i.e., the propagation of the stochastic uncertainty, often referred to as type A uncertainty according to the Guide to the expression of uncertainty in measurement (GUM) [7], into the frequency domain or within it, numerous publications are available, cf. Betta et al. [8]; Medina et al. [9]; Medina et al. [10]; Eichstädt and Wilkens [11]; and Moreland et al. [12]. In contrast, the propagation of the systematic measurement uncertainty with an unknown distribution, i.e., the type B uncertainty, has only been addressed by means of elaborate Monte Carlo simulations or propagation laws based on conservative assumptions, cf. Eichstädt et al. [13] and Kuhr [14].

Betta et al. [8] investigate the propagation of stochastic uncertainty into the frequency domain due to quantization errors, sampling frequency, jitter, and finite word length of microprocessors. The work mainly deals with the uncertainty due to the discrete Fourier transform (DFT) itself rather than the propagation of uncertain measurement signals.

Medina et al. [9] gave a general method for uncertainty propagation in parameter identification of frequency response functions. However, the authors did not address type B uncertainty when setting up their covariance matrix. Furthermore, the presented method is only suitable for the identification of linear systems. However, to analyze the uncertainty of the broad range of systems, the propagation of systematic measurement uncertainty into the frequency domain is required.

Medina et al. [10] presented a methodology to propagate experimental uncertainty applied to the identification of a simple two degree-of-freedom mechanical system. Although the suggested method has been developed independently of the types of method to evaluate uncertainty, only type A uncertainty, i.e., stochastic uncertainty, has been investigated.

Eichstädt et al. [13] use Monte Carlo simulations, according to the GUM and its supplements, to calculate dynamic measurement uncertainty. The benefit of this approach is that all types of uncertainties, i.e., type A and B, can be mapped in general. The authors identify a limitation of Monte Carlo simulations in terms of the high demands they place on available memory. In response to this challenge, they present two memory-efficient alternatives. However, the authors still use a workstation computer with 24 cores and 96 GB of RAM, which are still atypical for normal personal computers.

Eichstädt and Wilkens [11] developed a software package to implement closed formulae for the propagation of uncertainty between the time and frequency domain, according to the Joint Committee for Guides in Metrology [7]. The software provides reliable uncertainty propagation for common tasks in dynamic metrology, such as propagation between the frequency and the time domain, between amplitude/phase and real/imaginary part representation, and for estimating the dynamic measured value through deconvolution in the frequency domain. However, only uncertainty of type A is considered.

Moreland et al. [12] measured the static rotordynamic characteristics of liquid annular seals. The uncertainty quantification of the frequency-dependent stiffness and damping values is solely based on the standard deviation of the mean values of the computed complex stiffnesses within the frequency domain. The classical Gaussian uncertainty propagation is used to calculate the measurement uncertainty. It is stated that errors due to instrumentation are largely negligible, and only repetition determines the uncertainty.

Kuhr [14] takes a different approach. The author assumes that the overall measurement uncertainty calculated in the time domain can generally be transferred to the complex amplitude of the Fourier coefficients in the frequency domain. Assuming equal uncertainty of real and imaginary part, a simple relation between time and frequency domain is derived. However, the author explicitly points out that the method represents a conservative estimate of the measurement uncertainty in the frequency domain and is only approximately valid.

All of the above references focus on time-variant systems, i.e., systems without dynamic measurement errors due to the transfer functions of the equipment used. Within the presented paper, we solely focus on time-invariant systems. For further information on the propagation of dynamic measurement errors and their impact on measurement uncertainty, the interested reader is referred to the work of Vasilevskyi [15], Hessling [16,17], and Elster and Link [18,19].

It is evident that most references neglect the propagation of systematic measurement uncertainty with unknown distribution, the underlying reason, except for Monte Carlo simulations, remains unclear. One hypothesis is that the propagation of systematic measurement uncertainties is mainly neglected due to the time-consuming nature of suitable propagation methods, or the extensive memory effort and knowledge requirement of the correlation between the input quantities for covariance analysis. The extensive memory effort is particularly important when dealing with long measurement signals, i.e., where the size of the covariance matrix increases over-proportionally with the samples of a signal. To emphasize this claim, Table 1 gives the size and computation time of the covariance matrix, as well as the overall calculation time.

The propagation from the time into the frequency domain, i.e., the FFT, is carried out by using the advanced measurement uncertainty calculator METAS unclib [20] and the commercial calculation software matlab with 32 digits precision. It should be noted that for a typical long measurement signal, being recorded at a fairly low acquisition rate for dynamic measurements of 6000 samples per second for 5 s, cf. Sec. 4, i.e., the signal length being $105$ samples, the covariance matrix could not be computed using a state-of-the-art workstation, indicated by NaN, due to its requirements of RAM space.

In light of the aforementioned considerations, two fundamental challenges emerge. First, systematic uncertainty is frequently overlooked or subjected to strong assumptions when propagated into the frequency domain, resulting in a significant under- or overestimation of the combined uncertainty of the identified system parameters. Second, time and memory limitations pose significant technical challenges to the adequate quantification of uncertainty. To address those challenges, the paper addresses the following two questions:

• How does the systematic uncertainty affect the magnitude and phase in the frequency domain?

• Are there simple and fast propagation laws for the systematic measurement uncertainty into the frequency domain?

To answer the questions, the main differences between error and uncertainty as well as the difference between the systematic and stochastic uncertainty are briefly recapped. Subsequently, an analytical sensor model for modeling (i) offset, (ii) sensitivity, (iii) linearity, and (iv) hysteresis errors is presented. Combined with the corresponding probability distributions, this allows the description of the systematic uncertainty. The propagation of the four different systematic sensor uncertainties into the frequency domain is investigated by using Monte Carlo simulations. The simulation results are then analyzed to derive simple propagation relations from the time into the frequency domain. It will be shown that the systematic measurement uncertainty is far from being negligible for determining the overall uncertainty of the identified system parameters. Subsequently, the derived simple propagation laws are applied to a complex parameter identification procedure, comparing the derived propagation laws to a previously presented approach.

Prior to analyzing the propagation into the frequency domain, it is first necessary to define and distinguish between the concepts of error and uncertainty.

While an error is a specific deviation, unknown measurement errors are treated statistically using probability distributions. This statistical approach allows the quantification of an uncertainty, providing a framework for analysis through statistical methods. In other words, the uncertainty $δi$ of the measured value $yi$ describes an interval or probability distribution within which the true value is likely to lie in.

In general, there are two distinct types of errors when it comes to the measurement of physical quantities: systematic, and therefore repeatable errors, and random, nonrepeatable errors. The former of the two types, i.e., systematic errors, can be further divided into static and dynamic errors, i.e., errors which are important for time-invariant and time-variant systems. As stated before, time-variant systems are not in the scope of this work.

Regarding the remaining category of nonrepeatable errors, namely, random errors, they can be further subdivided into numerous distinct types, including repeatability, measurement noise, time variance, and quantization errors, cf. Elhoushi et al. [21]. An overview of the different types of errors is provided in Fig. 1. The errors that are the primary focus of this paper are highlighted in gray, i.e., (i) bias errors, (ii) sensitivity errors, (iii) linearity errors, and (iv) hysteresis errors.

For the possibility to take those errors in an experimental setup into account, i.e., addressing the distinct measurement errors as an uncertainty, the use of probability distributions necessitates. For the type A uncertainty defined by GUM [7], the distributions are known and can be treated using statistical methods. In contrast, the probability distributions of systematic measurement errors are unknown. The systematic measurement uncertainty is then defined as type B uncertainty, i.e., an uncertainty which encompasses other uncertainties.

In accordance with the static systematic errors given in Fig. 1, cf. Tränkler and Fischerauer [22], the four main systematic measurement uncertainties regarding the characteristic of a sensor are:

• zero offset, bias, $δbias$⁠,

• sensitivity $δsen$⁠,

• linearity $δlin=fn(x)$⁠, and

• hysteresis $δhys=fn(x,x˙)$⁠.

In this context, bias refers to a shift in the y-axis section of the sensor’s characteristic curve. The uncertainty of sensitivity pertains to the deviation of the sensor’s stated sensitivity from its actual value as specified in the data sheet. Similarly, the uncertainty of linearity represents the deviation between the real value and the measured value due to a nonlinear sensor characteristic. Finally, the uncertainty of hysteresis indicates the maximum deviation of a measured value depending on the direction from which it is approached, cf. Fig. 2.

All of these systematic uncertainties can be reduced by calibration with much more accurate measurement systems. In general, the manufacturer’s specifications are used as the systematic uncertainty in measurements. These are usually given in a sensor’s datasheet. However, for simplicity, some manufacturers give only a general uncertainty band that covers all possible sensor errors. Table 2 gives an overview of the relative and the absolute systematic uncertainties of an exemplary Hottinger Brüel & Kjær force transducer HBK U2B\1KN with a measurement range of $1 kN$ [23].

respectively. To determine the $95%$ confidence interval from this assumption $δ95$⁠, the specified interval of the uniform distribution is reduced by $5%$⁠, i.e., $δ95=0.95/2 [max(C)−min(C)]$⁠. In the following, all uncertainties are given within the $95%$ confidence interval. Therefore, the index 95 is neglected.

Figure 2 gives an overview of the modeled sensor errors. The thin lines represent the ideal sensor, whereas the thick lines give the influence of the error. The dashed lines show the uncertainty band given by the data sheet.

The initial question that arises is how the input parameters of the modeled linearity and hysteresis errors should be distributed to achieve a uniform distribution to fulfill the assumption of the Joint Committee for Guides in Metrology [7] and therefore derive the uncertainty.

To conduct this analysis, an input signal x is used where each point on the sensor characteristic is sampled at the same frequency, i.e., number of occurrences. Sampling a harmonic signal at equidistant intervals results in an arcsine distribution, which is unsuitable here, as it disproportionately sampled the edges of the sensor characteristic. Instead, a triangular signal is selected as a uniformly distributed input.

A full period $T0$ is used so that the characteristic is traversed in both ascending and descending directions. To simplify the analysis, only linearity errors are initially considered.

Subsequently, a Monte Carlo simulation with $N=105$ deviations from the linear characteristic, i.e., Eq. (2), is conducted, and the frequency distribution of the error is analyzed. The distributions of the parameters are chosen to obtain a uniform distribution of the linearity error.

Table 3 shows the initial parameters of the error model with the selected distributions.

In this model, the amplitude $ai$ and phase $φi$ span the full range of possible values, while the frequency $wi$ theoretically has no upper limit. To reflect the decreasing probability of higher frequencies, a half-normal distribution is used, ensuring only positive frequencies are included. The effects of distribution parameters and the number M of spectral components $wi$ on the error are subsequently evaluated.

The uncertainty limits for linearity and hysteresis are set to $δlin=δhys=0.01$⁠, and the errors are added to the ideal sensor signal using Eq. (4) with sensor sensitivity $s=1$⁠.

Figure 3 presents three exemplary resulting error distributions. As the variance of the half-normal distribution for $wi$ increases, the resulting distribution transitions from an arcsine to a normal distribution.

Varying the distributions of $wi$ and $ai$ alone does not yield a uniform error distribution. To address this, an approach based on the Fourier decomposition of a function that is uniformly distributed under equidistant sampling is explored. This approach couples wavelength and amplitude as $ai=fn(wi)$⁠, reducing the model’s variables to two: wavelength $wi$ and phase $φi$⁠.

When this relationship is used, distributions of the error are obtained, cf. Fig. 4. This ensures that the error is distributed uniformly.

For further investigation, the number of spectral components M and the distribution of wavelengths $wi$ have a significant impact on the output distribution. To choose the appropriate parameters, a full factorial simulation design with $M:1…15$ and $wi:|N(0,1…500)|$ is conducted. The maximum of the error distribution with 50 bins is used as the evaluation metric. For a uniform distribution, the maximum is 2%. For all other distributions, the maximum is higher. The closer the determined maximum is to 2%, the closer the error is to a uniform distribution.

Figure 5 shows the results depending on the distribution of wavelengths for a constant number of spectral components.

Generally, higher values of $σ(wi)$ reduce the maximum frequency error, approaching 2% for large $σ(wi)$ values over 500, indicating a uniform distribution.

Furthermore, two curves are noteworthy. If only one spectral component $M=1$ is used, it is not possible to achieve a uniform distribution. Moreover, with spectral components of $M=2$⁠, this is not feasible, despite being closer to a uniform distribution than the previously considered curve. Therefore, $M≥3$ is chosen.

With an increasing number of spectral components, it is observed that the slope of the curves becomes flatter. This means that for $M1>M2$⁠, $σ1(wi)>σ2(wi)$ must also apply to achieve the same results. After this analysis, $M=3$ is chosen. This is also advantageous considering the increasing simulation times and increasing number of spectral components.

Finally, the combined hysteresis and linearity errors are examined. Figure 6 shows the resulting distribution, which is triangular within an error range of −0.02 to 0.02.

This result is expected, as the modeled addition of errors results in a convolution of the two uniform error distributions $Elin∗Ehys$⁠, confirming their statistical independence.

In summary, this analysis determines the input parameter distributions of the modeled linearity and hysteresis errors. For following analysis and the propagation into the frequency domain, an amplitude-to-wavelength dependency $ai=1/wi2$ with parameters $M=3$ and $σ(wi)=500$ is employed.

Here, the $•˜$ indicated frequency domain variables. It should be noted that the uncertainty of the bias as well as the uncertainty of the sensitivity only affect the magnitude of the Fourier coefficients. This becomes evident when taking the linearity of the Fourier transform into account, cf. Brigham [24]. The underlying probability distributions remain unchanged.

It is evident that even when the uncertainties in both bias and sensitivity are determined using the uniform distribution and propagated directly into the frequency domain, cf. Eq. (13), the uncertainties related to the sensor’s linearity and hysteresis, as well as their propagation into the frequency domain, however, are by no means straightforward and sophisticated Monte Carlo simulations are needed.

Monte Carlo simulations have been widely used since the 19th century for various applications, cf. Metropolis et al. [25], Barbu and Zhu [26], etc. According to the GUM [7], Monte Carlo simulations are a suitable method for analyzing the propagation of an uncertainty with a previously unknown distribution. The simulations use numerous calculations to estimate the model output and its distribution with stochastic distributed input parameters [26]. Within this paper, the Monte Carlo simulations are carried out using the commercial software matlab. Each simulation is designed to emulate a real measurement using the aforementioned sensor model. The number of iterations within the Monte Carlo simulation has to be large enough to ensure that the resulting probability distributions no longer change with the number of iterations. This can either be guaranteed by an iterative procedure or by a very high number of iterations. For simple models, the GUM [7] suggests a number of $N=106$ iterations to get accurate results, which will be put to test at the end of the section.

The simulations are carried out in four steps, shown in Fig. 7.

It is assumed that the signal covers the full measurement range of the modeled sensor, i.e., the amplitude $â$ is chosen accordingly. Subsequently, the actual Monte Carlo simulation then starts with N iterations. Second, an error is added to the sensor characteristic curve, creating an erroneous signal. The signal is transferred into the frequency domain in the following step by using matlab’s FFT algorithm. Since the signal length is a multiple of the period, no window function is used. For the statistical analysis, only the coefficient of the frequency of the test signal $Y(Ω)$ is considered. In the final step, the results are evaluated using the standard deviations of the magnitude $σ(|Y(Ω)|)$ and the phase $σ(∠[Y(Ω)])$⁠.

Table 4 gives an overview of the used parameters of the discrete time vector t and the harmonic excitation.

Finally, the suggested number of Monte Carlo iterations N by the Joint Committee for Guides in Metrology [7] has to be confirmed. Figure 8 shows the standard deviation of the magnitude $σ(|Y(Ω)|)$ of the Fourier coefficient depending on the number of Monte Carlo iterations N. Here, the number of iterations varies between $N=103$ and $N=107$⁠.

It exhibits that the evaluation value, i.e., the magnitude of the Fourier coefficient $σ(|Y(Ω)|)$⁠, converges and hardly changes from $N≈106$ iterations onward. Therefore, the number of iterations in every following investigation is set to $N=106$⁠, which is in line with the GUM [7] recommendation. Subsequently, the propagation of the linearity uncertainty and a hysteresis uncertainty is investigated.

With the error model and the Monte Carlo simulations available, the propagation of the linearity and hysteresis uncertainty in the frequency domain can be examined, to derive an equally simple propagation law as for the uncertainty of the bias and sensitivity. To do so, all influencing factors of the Monte Carlo simulations must be examined. These factors include:

• the input parameters and their distributions within the error model,

• the parameters of the test signal used, and

• the boundaries of the uncertainty intervals $δlin$ and $δhys$⁠.

The input parameters of the linearity and hysteresis error model (i) are already set through the investigation of the model itself, cf. Sec. 2.3. The found parameters for the error model meet the assumptions made by the GUM [7].

Initially, the other influencing factors (ii) are varied individually to determine their impact on the results. In the subsequent parameter determination, conservative estimates are aimed for. Finally, the uncertainty intervals (iii) $δlin$ and $δhys$ are examined through a sensitivity analysis. From this analysis, simple and fast propagation rules into the frequency domain can be derived.

must apply. This relationship can also be seen in Fig. 9.

All other parameters of the signal have no influence on the result in the frequency domain. This indicates that it is an adequate representation of systematic errors. Especially, the parameters of the test signal should have no influence on the results. The variations of the remaining parameters can be found in the  Appendix.

It is noteworthy that the sampling rate $fab$ or the length of the signal L do not affect the results. Both quantities increase the number of measurement points when increased. If these were stochastic errors, the uncertainty would decrease as a result, cf. Betta et al. [8] and Eichstädt and Wilkens [11]. However, systematic errors cannot decrease by increasing the sampling rate or the length of the signal, as they are inherent to the measurement system. From this, it can be concluded that the chosen form of modeling concerns solely systematic errors.

After the influences (ii) have been examined, the distributions of the results in frequency domain are shown in Fig. 10. The distributions are oriented toward the magnitude and phase. A cross is formed in the complex number plane. This suggests that magnitude and phase can be considered independently. If they were dependent on each other, an elliptical shape would be expected. The distributions obtained with the chosen parameters do not correspond to a uniform distribution, although this is present in the time domain, cf. Sec. 2.3.

Finally, a sensitivity analysis determines the influence of the uncertainty intervals $δlin$ and $δhys$ on the distribution of magnitude and phase in the frequency domain. From this, the propagation of systematic uncertainty due to linearity and hysteresis errors into the frequency domain is determined. Both boundaries are varied in steps of 0.001 within a range from 0 to 0.01. A full factorial design is used.

Figure 11 shows the results for the uncertainty. Both metrics are viewed against the input parameters, i.e., the uncertainty due to linearity and hysteresis errors. The independence of magnitude and phase is confirmed. The linearity error only leads to a change in the magnitude of the Fourier coefficient, while the hysteresis error only affects the phase. The combination of both errors has no impact on this result. The two errors are independent in the frequency domain, which is not the case in the time domain, see Fig. 6.

Until now, the presented uncertainty quantification is limited to the propagation of the systematic sensor uncertainty of an arbitrary sensor. This section presents the results of the aforementioned methodology applied to a complex system comprising various sensors, including eddy current and hall sensors. Subsequently, the uncertain sensor signals are employed to identify a system of complex system parameters, specifically the stiffness, damping, and inertia coefficients of a liquid-filled annular gap, cf. upper part of Fig. 12, within the frequency domain. Accordingly, the annular gap flow test rig at the Chair of Fluid Systems at Technische Universität Darmstadt, the employed identification procedure, and the previously utilized uncertainty quantification are briefly described. The uncertainty quantification presented in Sec. 3.2 is then employed as a substitute for the former one. Ultimately, the calculated uncertainty of the identified parameter is compared to a previously presented method by Kuhr [14].

Figure 12 shows the annular gap flow test rig for the experimental investigation of the static and dynamic force characteristics of generic fluid-filled annuli, cf. Kuhr et al. [27] and Kuhr [14]. In this setup, either journal bearings or annular seals are investigated not only for their static characteristics, e.g., their induced load carrying capacity or generated pressure drop across the gap, but also for their dynamic behavior, i.e., their stiffness, damping, and inertia properties. The test rig is primarily composed of two active magnetic bearings, which serve the rotor support as well as an intrinsic displacement, excitation, and force measurement system. The displacements of the rotor are measured using two eddy current sensors in an $90 deg$ arrangement at both the entrance and exit of the lubrication gap. Each sensor has a user-defined measuring range of $1 mm$ and an uncertainty in linearity of $δx,lin<±2.4 μm$⁠. As the manufacturer does not specify any further uncertainties regarding bias, sensitivity, and hysteresis, these are assumed to be negligible or included in the linearity uncertainty. The static and dynamic forces generated by the magnetic bearing are obtained by measuring the magnetic flux density $B˜$ in the air gap between the rotor and the active magnetic bearing by using hall sensors on each pole of each electromagnet. The hall sensors have a measuring range of $±2 T$ and an uncertainty in linearity of $δB,lin=±0.1%B$⁠. Similar to the eddy current sensor, the remaining sensor uncertainties in bias, sensitivity, and hysteresis are assumed to be negligible. The force applied per pole is proportional to the magnetic flux density $FH∝B2$⁠. While the identification procedure and uncertainty analysis used in the initial reference are outlined briefly below, a more comprehensive description of the test rig, its calibration, and the actual measurement task can be found in the work of Kuhr et al. [27] and Kuhr [14].

Since the stiffness, damping, and inertia characteristics of annular gaps cannot be directly measured, they must be identified using parameter estimation methods, such as linear and quadratic regressions. The approach detailed by Kuhr [14] employs four linearly independent whirling motions to excite the rotor monofrequently at predefined precessional frequencies $ωi$ and amplitudes $Δe$ and $Δγ$⁠: (i) translational excitation in the direction of rotor rotation; (ii) translational excitation against the direction of rotor rotation; (iii) angular excitation in the direction of rotor rotation; and (iv) angular excitation against the direction of rotor rotation. The induced dynamic forces and moments acting on the rotor $FX$⁠, $FY$ and $MX$⁠, $MY$ along with the rotor motion in the four degrees-of-freedom, $X,Y$ and $αX,βY$⁠, are measured using the aforementioned hall and eddy current sensors. All sensor data are collected simultaneously via a multifunction I/O module operating at a sampling rate of $6000 Hz$ per channel for a duration of 5 s.

Here, $Kij$ are the complex stiffness coefficients depending on the precessional frequency $ω$⁠. $F=DFT(F)$ and $M=DFT(M)$ are the Fourier transformations of the induced forces and moments on the rotor, whereas $Di=DFT(i˜)$ are the Fourier transforms of the translational and angular excitation amplitudes.

The stiffness and inertia coefficients are obtained by applying a quadratic regression if the rotor is excited at different precessional frequencies $ωi$⁠. In contrast, a linear regression gives the damping of the annulus.

To quantify the measurement uncertainty of the identified parameters, Kuhr [14] follows the guidelines set forth in the GUM [7]. However, for the propagation from the time into the frequency domain, i.e., from Eq. (25) to Eq. (26), it is assumed that the total measurement uncertainty of the signal $δ(x)$⁠, i.e., the Euclidean norm of the systematic $δsys(x)$ and stochastic measurement uncertainty $δstat(x)$⁠, can be assigned to each discrete value of the recorded time signal, i.e., each of the 6000 samples. Furthermore, it is assumed that the overall measurement uncertainty can be regarded as an uncertainty in the sensitivity, i.e., the slope of the characteristic, and the real and imaginary part of the Fourier coefficients have the same magnitude of uncertainty after applying the discrete Fourier transform. This leads to a straightforward yet highly conservative propagation law for the propagation of uncertainty from the time domain to the frequency domain. The uncertainty of the stiffness, damping, and inertia coefficients is then calculated following Gaussian uncertainty propagation based on the equations of the identification procedure, i.e., the quadratic and linear regression.

To apply the simple propagation laws presented in this paper, to the complex parameter identification for the stiffness, damping, and inertia coefficients, the assumptions employed by Kuhr are discarded, and the genuine sensor signals are regarded as analogous to the generic signal utilized in the Monte Carlo analysis. Strictly speaking, the simple propagation laws, i.e., Eqs. (18) and (19), are applied directly at the transformation from the time into the frequency domain, i.e., from Eq. (25) to Eq. (26). The subsequent procedure of uncertainty propagation of the transformed signals according to GUM remains unaffected.

Figure 13 depicts the identified coefficients for an annular gap with length $L=92.95 mm$ at a water temperature of $35 °C$⁠, spanning a relative eccentricity range $0.1≤ε:=e˜/h¯˜≤0.8$⁠. Here, $e˜$ represents the eccentricity, and $h¯˜$ denotes the mean gap height of the annulus. The operating conditions of the annulus, i.e., the volume flow through the gap, the angular velocity of the rotor, as well as the velocity of the fluid flow in front of the annulus, are kept constant throughout the measurements. The corresponding dimensionless operating conditions, i.e., the modified Reynolds number, the flow number, and the preswirl ratio, are $Reφ*=0.031$⁠, $ϕ=0.7$⁠, and $Cφ|z=0=0.5$⁠. Here, the makers represent the experimental results including uncertainty bound for a confidence interval of 95%. The solid lines are model predictions of the rotordynamic coefficients obtained by the clearance-averaged pressure model [28]. The left side of the figure shows the mean values of the results, while the figures on the right side compare the calculated measurement uncertainty obtained by the method used by Kuhr [14] and the one presented here.

Considering that 16 stiffness, damping, and inertia coefficients are determined in the original publication, only the change in the uncertainty interval of the direct and cross-coupling stiffness $KYYandKYX$⁠, the direct and cross-coupling damping $CXX$ and $CXY$⁠, as well as the direct and cross-coupling additional mass $MXX$ and $MXY$ is given here.

It is evident that the presented uncertainty quantification has a severe impact on the size of the error bars on the dynamic coefficients. In contrast to the previous method, where the results are clearly visible and comparable to the actual measured value, e.g., for the identification of the direct and cross-coupled added masses $MXX$ and $MXY$⁠, the results presented here are almost indiscernible, obscured by the markers. The new uncertainty quantification is a more realistic one compared to the worst-case scenario used previously. This is mainly due to the reduction of the impact of the systematic uncertainty. Due to the long measurement period of 5 s, the influence of the statistic measurement uncertainty is negligible.

This paper investigates the propagation of four different categories of systematic errors into the frequency domain. Overall, the two research questions:

• What types of systematic error affect magnitude and phase in the frequency domain?

• Are there simple rules for propagating systematic measurement uncertainty into the frequency domain?

Manuel Rexer: Conceptualization, Methodology, Software, Formal analysis, Visualization, Writing—original draft, Writing—review and editing, Validation. Peter F. Pelz: Funding acquisition, Resources. Maximilian M. G. Kuhr: Conceptualization, Investigation, Supervision, Visualization, Writing—original draft, Writing—review and editing.

• Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (Project No. 57157498—SFB805; Funder ID: 10.13039/501100001659).

• Federal Ministry for Economic Affairs and Energy (BMWi) due to an enactment of the German Bundestag (Grant No. 03EE5036B; Funder ID: 10.13039/501100006360).

• Industrial collective research programme supported by the Federal Ministry for Economic Affairs and Energy (BMWi) through the AiF (German Federation of Industrial Research Associations e.V.) due to an enactment of the German Bundestag (IGF No. 21029 N/1; Funder ID: 10.13039/501100002723).

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.