Analysis of the Accelerometer Input–Output Energy Distribution Based on the Upper Bound of Absolute Dynamic Error

: This paper presents an analysis of the accelerometer input–output energy distribution based on the upper bound of the absolute dynamic error (UBADE). This analysis corresponds to the input and output accelerometer signals, determined previously by mathematical modeling. Obtained results may provide the basis for verifying the correctness of the algorithms intended for the determination of the UBADE.


Introduction
In the field of theory and practice related to accelerometers, as for the other types of measurement sensors, there are known test procedures used for calibration [1][2][3][4][5]. This type of measurement experiment usually consists of determining the time or frequency responses of such sensors [1,2,4]. On the basis of these responses, the selected parameters of the sensors are determined [2], which are then compared with their equivalents declared by the manufacturer in the corresponding datasheet. For accelerometers, the calibration process usually determines voltage sensitivities [2,4]. This parameter is determined based on a measurement of one frequency response (amplitude) or the simultaneous measurement of two frequency responses (amplitude and phase) [6][7][8][9].
Accelerometers are widely used in the power industry to continuously monitoring the operation of the kinetic system components of turbines used in wind and hydro plants [10][11][12] as well as in the case of drilling platforms [13,14]. Such monitoring is aimed at current determination of the technical condition of such components and early diagnosis of all kinds of potential failures. It is very important due to the significant costs associated with the removal of the effects of failure of turbine components used in various types of power plants [15]. Therefore, it seems justified to select an appropriate type of accelerometer with a specific accuracy of the processing of measurement signals. Such accuracy can be achieved through the use of the appropriate procedures dedicated to determining of the errors of accelerometer processing [16][17][18][19], which can only ensure their mutual comparability.
For accelerometers with the same operating range but from different manufacturers, there are no clear criteria for mutual comparisons of the accuracy of measurement signal processing [4,5]. This is due to the obvious fact that for all instruments intended for measuring quantities that vary over time (dynamic), there is no accuracy class, such as in the case of instruments intended for measuring constant quantities (static). However, mutual comparability among accelerometers can be achieved by using procedures for determining the upper bound of the dynamic error (UBDE) [16][17][18][19]. Taking into account that the value of the dynamic error [20][21][22][23][24][25] and the dynamic properties of the accelerometer are also influenced by the shape of the input signal [16,17], to determine the UBDE, it is necessary to which is obtained based on amplitude response, where 0 = 2π 0 and c = 2π c while 0 and denote natural undamped frequency [Hz] and damping factor (dimensionless), respectively. Both the cut-off frequency and tolerance ∆ are provided by the accelerometer datasheet. By expanding Equation (1), we obtain Based on Equation (2), we can easily obtain 0 = 0 2π ⁄ when the parameters ∆ , , and c are given. Figure 2 shows the block diagram of the methodology for determining the UBADE [16,17]. The basis for determining the UBADE is the mathematical model of an accelerometer. This mathematical model is represented by the accelerometer transfer function a ( ) [30,34,39]. The parameters included in this function are obtained as the results of the accelerometer's parametric identification [6−9]. The percentage tolerance ∆S of the accelerometer voltage sensitivity S V/ ms −2 for frequency f [Hz] equal to the cut-off frequency f c [Hz] is represented by the following formula: which is obtained based on amplitude response, where ω 0 = 2π f 0 and ω c = 2π f c while f 0 and β denote natural undamped frequency [Hz] and damping factor (dimensionless), respectively. Both the cut-off frequency and tolerance ∆S are provided by the accelerometer datasheet. By expanding Equation (1), we obtain Based on Equation (2), we can easily obtain f 0 = ω 0 /2π when the parameters ∆S, β, and ω c are given. Figure 2 shows the block diagram of the methodology for determining the UBADE [16,17].

Theoretical Background
The mechanical construction of the accelerometer is shown in Figure 1 [40,41]. Formulas referring to the mathematical model of accelerometer are described in the previous authors papers [30,31,34,39]. The percentage tolerance ∆ of the accelerometer voltage sensitivity [V/(ms −2 )] for frequency [Hz] equal to the cut-off frequency c [Hz] is represented by the following formula: which is obtained based on amplitude response, where 0 = 2π 0 and c = 2π c while 0 and denote natural undamped frequency [Hz] and damping factor (dimensionless), respectively. Both the cut-off frequency and tolerance ∆ are provided by the accelerometer datasheet. By expanding Equation (1) Based on Equation (2), we can easily obtain 0 = 0 2π ⁄ when the parameters ∆ , , and c are given. Figure 2 shows the block diagram of the methodology for determining the UBADE [16,17]. The basis for determining the UBADE is the mathematical model of an accelerometer. This mathematical model is represented by the accelerometer transfer function a ( ) [30,34,39]. The parameters included in this function are obtained as the results of the accelerometer's parametric identification [6−9]. The basis for determining the UBADE is the mathematical model of an accelerometer. This mathematical model is represented by the accelerometer transfer function K a (s) [30,34,39]. The parameters included in this function are obtained as the results of the accelerometer's parametric identification [6][7][8][9].
For the modeling purposes in this paper, the accelerometer is assumed to be a low pass system. Additionally, the values of the accelerometer parameters were assumed in advance for the purposes Energies 2020, 13, 5816 4 of 14 of the analysis. Several sets of accelerometer parameters were analyzed. Of course, the procedure presented here can also be used for real data obtained based on parametric identification.
The value of UBADE is determined based on [16]. A high-order low-pass analog filter with a cut-off frequency corresponding to the frequency band of the accelerometer is used as a reference [42,43]. Use of this reference ensures that the analysis of the UBADE concerns only the frequency range of the accelerometer. For the purposes of this paper, an analog Butterworth filter of the 10th order with the transfer function denoted as K r (s) was adopted.
The procedure for determining the UBADE was carried out based on the mathematical models K a (s) and K r (s). The procedure for the UBADE determination is discussed in detail in Section 4.
This procedure obtains the two-constrained shapes of signal x u (t) that produce the UBADE. These constraints apply simultaneously to the magnitude A and the rate of charge ϑ of signal x u (t). The second constraint is derived from where T is the time of accelerometer testing, and h a (t) is the impulse response of the accelerometer, which is determined based on the relation, where L −1 denotes the inverse Laplace transform [16,17]. Figure 3 shows an example signal x u (t) with two constraints and five time switchings denoted by t s1, t s2, . . . , t s5. The magnitude constraints are denoted by +A (positive constraint) and -A (negative constraint), while the rate of change constraints are denoted by +ϑ (signal increase) and +ϑ (signal decrease) [16,17]. For the modeling purposes in this paper, the accelerometer is assumed to be a low pass system. Additionally, the values of the accelerometer parameters were assumed in advance for the purposes of the analysis. Several sets of accelerometer parameters were analyzed. Of course, the procedure presented here can also be used for real data obtained based on parametric identification.
The value of UBADE is determined based on [16]. A high-order low-pass analog filter with a cut-off frequency corresponding to the frequency band of the accelerometer is used as a reference [42,43]. Use of this reference ensures that the analysis of the UBADE concerns only the frequency range of the accelerometer. For the purposes of this paper, an analog Butterworth filter of the 10th order with the transfer function denoted as r ( ) was adopted.
The procedure for determining the UBADE was carried out based on the mathematical models a ( ) and r ( ). The procedure for the UBADE determination is discussed in detail in Section 4. This procedure obtains the two-constrained shapes of signal u ( ) that produce the UBADE. These constraints apply simultaneously to the magnitude and the rate of charge of signal u ( ). The second constraint is derived from where is the time of accelerometer testing, and ℎ a ( ) is the impulse response of the accelerometer, which is determined based on the relation, where ℒ −1 denotes the inverse Laplace transform [16,17]. Figure 3 shows an example signal u ( ) with two constraints and five time switchings denoted by s1, s2, … , s5. The magnitude constraints are denoted by + (positive constraint) and -(negative constraint), while the rate of change constraints are denoted by + (signal increase) and + (signal decrease) [16,17]. Signal u ( ) producing the UBADE has properties such that any signal ( ) contained in its constraints can only produce the lower value of ADE.
The function ( ) presents the ADE for ∈ [0, ]. The maximum value of the function ( ) can be obtained for time = . Then, this function is denoted as u ( ). Figure 4 shows an example function u ( ). The UBADE is obtained for = . Signal x u (t) producing the UBADE has properties such that any signal x(t) contained in its constraints can only produce the lower value of ADE.
The function d(t) presents the ADE for t ∈ [0, T]. The maximum value of the function d(t) can be obtained for time t = T. Then, this function is denoted as d u (t). Figure 4 shows an example function d u (t). The UBADE is obtained for t = T.
The value of UBADE determined for accelerometers with the same operating range (frequency band) but produced by different manufacturers may provide the basis for their mutual comparability in terms of signal processing accuracy. Thus, an analogy can be found here with the accuracy class of the measuring instruments intended for measuring the constant quantities over time. It should be clearly emphasized that there is no accuracy class for instruments (sensors) intended for measuring the dynamic quantities (variables over time) [5,16]. Hence, the analysis presented in this paper is justified. The value of UBADE determined for accelerometers with the same operating range (frequency band) but produced by different manufacturers may provide the basis for their mutual comparability in terms of signal processing accuracy. Thus, an analogy can be found here with the accuracy class of the measuring instruments intended for measuring the constant quantities over time. It should be clearly emphasized that there is no accuracy class for instruments (sensors) intended for measuring the dynamic quantities (variables over time) [5,16]. Hence, the analysis presented in this paper is justified.

Procedure for Determining the UBADE
When the input signal is constrained in both its magnitude and rate of change , the signal u ( ) producing the UBADE ( u ) is determined indirectly, as follows: where ∆ is the quantization step of time , which is the duration time of accelerometer testing [16,17,28]. Based on the signal u ( ) and the impulse response ℎ( ), we can easily determine the error u by using the following formula [16,17]: The impulse response ℎ( ) is obtained as the difference where ℎ r ( ) is the impulse response of the reference. This response is calculated based on the following formula Equations (16) and (17) in their digital form are [17] and where = ∆ .
The function ( ) is obtained in several steps used for transforming the impulse response ℎ( ), as follows [16,17,28]: 1. Determine the pulse function 1 ( ) whose duration is equal to 2 , according to the following equation:

Procedure for Determining the UBADE
When the input signal is constrained in both its magnitude A and rate of change ϑ, the signal x u (t) producing the UBADE (D u ) is determined indirectly, as follows: where ∆ is the quantization step of time T, which is the duration time of accelerometer testing [16,17,28]. Based on the signal x u (t) and the impulse response h(t), we can easily determine the error D u by using the following formula [16,17]: The impulse response h(t) is obtained as the difference where h r (t) is the impulse response of the reference. This response is calculated based on the following formula Equations (16) and (17) in their digital form are [17] x and where N = T ∆ . The function ϕ(t) is obtained in several steps used for transforming the impulse response h(t), as follows [16,17,28]: Determine the pulse function ϕ 1 (t) whose duration is equal to 2A ϑ , according to the following equation: An example function ϕ 1 (t) is shown in Figure 5. This function was obtained for the following accelerometer parameters: S = 1 V/ ms −2 , f 0 = 4581 Hz, β = 0.02, and f 0 = 1 kHz.
An example function 1 ( ) is shown in Figure 5. This function was obtained for the following accelerometer parameters: = 1 V/(ms −2 ), 0 = 4581 Hz, = 0.02, and 0 = 1 kHz. This is the typical shape of this function for any accelerometer analysis. Only the pulse width of 2 is changed. Example figures for the subsequent stages of the procedure for determining the upper bound of the absolute dynamic error were also obtained for the above accelerometer parameters.  (12) Figure 6 shows the shape of the function 2 ( ) , which is typical for any accelerometer parameters.  This is the typical shape of this function for any accelerometer analysis. Only the pulse width of 2A ϑ is changed. Example figures for the subsequent stages of the procedure for determining the upper bound of the absolute dynamic error were also obtained for the above accelerometer parameters.

2.
Calculate the convolution integral ϕ 2 (t) based on the function ϕ 1 (t) and the impulse response h(t): Figure 6 shows the shape of the function ϕ 2 (t), which is typical for any accelerometer parameters.
An example function 1 ( ) is shown in Figure 5. This function was obtained for the following accelerometer parameters: = 1 V/(ms −2 ), 0 = 4581 Hz, = 0.02, and 0 = 1 kHz. This is the typical shape of this function for any accelerometer analysis. Only the pulse width of 2 is changed. Example figures for the subsequent stages of the procedure for determining the upper bound of the absolute dynamic error were also obtained for the above accelerometer parameters.  (12) Figure 6 shows the shape of the function 2 ( ) , which is typical for any accelerometer parameters. (13) Figure 6. Convolution integral ϕ 2 (t).

3.
Determine the rectangular function based on the following formula ( Figure 7): Energies 2020, 13, x FOR PEER REVIEW 7 of 15

6.
Determine the signal x u (t) with two constraints by substituting ϕ 5 (t) for ϕ(t) in Equation (5) (Figure 10). This signal produces the upper bound of the absolute dynamic error. 6. Determine the signal u ( ) with two constraints by substituting 5 ( ) for ( ) in Equation (5) (Figure 10). This signal produces the upper bound of the absolute dynamic error.
Finally, based on Equation (16), we can easily obtain the upper bound of the absolute error by using the following formula: which corresponds to Equations (6) and (10).

Analysis of the Accelerometer Input-Output Energy Distribution
Next, we present the procedure for analyzing the input-output energy distribution for the three selected sets of parameters included in the mathematical model of the accelerometer. This analysis relates to the input signal u ( ) and the related output signal u ( ) and allows an assesment of the frequency components included in these signals. Thanks to this analysis, it may be possible to assess the correctness of the implementation of procedures dedicated to UBADE determination.
The analysis of the input-output energy distribution uses a scalogram that employs both the wavelet method and MATLAB software (R22a, MathWorks, Natick, MA, USA). The scalogram procedure is based on the following formula: Figure 10. Signal x u (t) with two constraints.

7.
Determine the absolute error as the time function: Finally, based on Equation (16), we can easily obtain the upper bound of the absolute error by using the following formula: which corresponds to Equations (6) and (10).

Analysis of the Accelerometer Input-Output Energy Distribution
Next, we present the procedure for analyzing the input-output energy distribution for the three selected sets of parameters included in the mathematical model of the accelerometer. This analysis relates to the input signal x u (t) and the related output signal d u (t) and allows an assesment of the frequency components included in these signals. Thanks to this analysis, it may be possible to assess the correctness of the implementation of procedures dedicated to UBADE determination.
The analysis of the input-output energy distribution uses a scalogram that employs both the wavelet method and MATLAB software (R22a, MathWorks, Natick, MA, USA). The scalogram procedure is based on the following formula: where Ψ * is the complex conjugate of the wavelet used for the analysis [30,[44][45][46], for which a Morlet wavelet (Version, Manufacturer/Company, City, State Abbr. if USA or Canada, Country) was used. Three sets of accelerometer parameter values for the analysis are tabulated in Table 1. These values were selected from among many other sets as they provide the most characteristic results for the analysis. For all checked cases, a voltage sensitivity S equal to 1 V/ ms −2 was assumed. The tests were carried out for a constant value ( f c = 1 kHz) of the accelerometer operating band. It was also assumed that the magnitude constraint A is equal to the accelerometer' voltage sensitivity (A = ±1 V). The value of frequency f 0 was calculated with Equation (2), while the rate of change ϑ constraint was calculated via Equation (3). The time T of accelerometer testing corresponds to a steady-state of the impulse response h(t).  Figure 11 shows the signals x u (t) and d u (t) obtained for the accelerometer parameters included in the first row of Table 1. The value of UBADE (D u ) is equal to 27.15 Vs.
the analysis. For all checked cases, a voltage sensitivity equal to 1 V/(ms −2 ) was assumed. The tests were carried out for a constant value ( c = 1 kHz) of the accelerometer operating band. It was also assumed that the magnitude constraint is equal to the accelerometer' voltage sensitivity ( = ±1 V). The value of frequency 0 was calculated with Equation (2), while the rate of change constraint was calculated via Equation (3). The time of accelerometer testing corresponds to a steady-state of the impulse response ℎ( ).  Figure 11 shows the signals u ( ) and u ( ) obtained for the accelerometer parameters included in the first row of Table 1 Figure 11. Figure 13 shows the signals u ( ) and u ( ) obtained for the accelerometer parameters from the second row in Table 1 ( = 0.026). The value of UBADE ( u ) is equal to 21.16 Vs. The signal u ( ), as in the previous case, has a trapezoidal shape. The signal u ( ) is similar in shape to the signal determined for = 0.026, but its phase is inverted.  Figure 13 shows the signals x u (t) and d u (t) obtained for the accelerometer parameters from the second row in Table 1 (β = 0.026). The value of UBADE (D u ) is equal to 21.16 Vs. The signal x u (t), as in the previous case, has a trapezoidal shape. The signal d u (t) is similar in shape to the signal determined for β = 0.026, but its phase is inverted. Figure 12. Scalogram and normalized amplitudes vs. frequency for signals u ( ) (top plots) and u ( ) (bottom plots) shown in Figure 11. Figure 13 shows the signals u ( ) and u ( ) obtained for the accelerometer parameters from the second row in Table 1 ( = 0.026). The value of UBADE ( u ) is equal to 21.16 Vs. The signal u ( ), as in the previous case, has a trapezoidal shape. The signal u ( ) is similar in shape to the signal determined for = 0.026, but its phase is inverted.      Figure 15 shows the signals x u (t) and d u (t) for β = 0.031. The value of UBADE (D u ) is equal to 17.60 Vs. In this case, the signal x u (t) has a trapezoidal or triangular shape. The signal's triangular shape means that it has not reached the magnitude constraint. Figure 14. Scalogram and normalized amplitudes vs. frequency for signals u ( ) (top plots) and u ( ) (bottom plots) shown in Figure 13. Figure 15 shows the signals u ( ) and u ( ) for = 0.031. The value of UBADE ( u ) is equal to 17.60 Vs. In this case, the signal u ( ) has a trapezoidal or triangular shape. The signal's triangular shape means that it has not reached the magnitude constraint.    Additional tests were performed for a damping factor β equal to 0.1, 0.2, 0.3, and 0.4. With an increase of this factor, analysis of the signal x u (t) shows a decrease in the normalized amplitude for the frequency f 0 . The normalized amplitude for β = 0.4 is equal to 0.33. The normalized amplitude for frequency f c has a constant value equal to 1. Considering the frequency f c , for all the above-mentioned values of the factor β, the uniform energy distribution for time t in the range of 0-7 s is visible. On the other hand, for frequency f 0 , the energy distribution is narrowed to time t in the range of 9-10 s. Analysis of the signal d u (t) for the damping factor β equal to 0.1, 0.2, 0.3, and 0.4 shows that with an increase in the value of this factor, there is an increase in the normalized amplitude of frequency f c . For β = 0.4, the value of this amplitude is 0.65, while the energy distribution for frequency f c is uniform for time t with a range of 0-7 s. On the other hand, for frequency f 0 , just as for signal x u (t), the energy distribution is narrowed to time t within the range of 9-10 s.

Conclusions
The procedure presented in this paper, which is intended for analyzing the input-output energy distribution of accelerometers, can verify the correctness of the algorithm for determining the upper bounds of the absolute dynamic error (UBADE) in any mathematical or computational program (e.g., MATLAB and MathCad). This verification consists of determining the dominant frequencies in the accelerometer input and output signals obtained by modeling, as well as an analysis of the energy distribution of these signals.
Based on the conducted research, we conclude that the input signal with two constraints, which was obtained by modeling, contains two dominant frequency components. These components correspond to the undamped natural and cut-off frequencies, which are associated with the accelerometer model. Analysis of the accelerometer output signal, which was likewise determined by modeling, also showed the dominance of the above frequencies.
Graphical representation of both the normalized amplitude and the energy distribution for the accelerometer input and output signals can be used to test the implemented algorithms intended for UBADE determination. The need to carry out of such verification is because UBADE may provide the basis for assessing the accuracy of signal processing by a given accelerometer or determining the comparability of the accuracy of accelerometers produced by different manufacturers. Importantly, for accelerometers, as for many other sensors that process time-varying measured signals, there is no clear criterion to assess their accuracy.
The procedure presented in the paper allows verification of the implementation correctness of the procedure for UBADE determination. Therefore this procedure can ensure an error-free determination of the accuracy of an accelerometers through their appropriate selection, and thus the improvement of the process of monitoring the operation of many devices used in the power engineering systems, e.g., monitoring of the turbines applied in the power plants by accurate measurements of their vibrations. Funding: This research was financially supported by the National Science Centre Poland as part of the scientific activity no. 2017/01/X/ST7/00394.

Conflicts of Interest:
The authors declare no conflict of interest.