Keywords

1 Introduction

As a consequence of non-perfect conversion of electrical or chemical (fuel) energy to desired work, any rotary machinery emits sound and vibration, and as a consequence due to the heat energy dissipation increases temperature. All of these physical phenomena can be quantitatively measured and converted into meaningful, physical quantities. The temperature is the easiest indicator to interpret, because it is a simple scalar value. Since typically, permissible operating temperatures of some mechanical elements are given by a manufacturer, it is a convenient “overall” diagnostic indicator. From a definition, vibrations refer to oscillatory movement, so the primary vibrations are displacement values. However, due to some scientific versus economical trade-off, typical vibrations are measured as “acceleration of motion”, and are given as a fraction of [g] units. Still, the most widely used indicator, namely the “velocity RMS” is given as the “velocity of vibrations”. The last signal, the sound, is an acoustic wave, which is this part of vibrations, which is not counterbalanced by machine foundations. Although sound bandwidth is theoretically unlimited (just like in case of any other value), it is accepted that the sound is constrained to a human hearing sense, which is about 20 Hz–15 kHz (numbers might vary).

In the current paper, the widely used vibrations signals are taken into account as a source of information of machine technical condition. The Health Indicators (HI), also called “signatures”, “features”, or diagnostic estimators are scalar results [1] of some processing of signal, e.g. statistical, filter-based or various customized [2].

From practical point of view, processing of vibration signals is like playing hide-and-seek. Regardless the used processing technique, the idea is always the same, i.e. to define such HI that would:

  • detect a fault with a highest rate of reliability,

  • detect a fault the earliest,

  • identify the faulty component most accurately,

  • approximate the Remaining Useful Life (RUL) most accurately.

From signal processing point-of-view, the requirement No. 1 calls for baseband signal analysis, where all signal components are present, i.e. no part is filtered out. On the other hand, requirements No. 2 and No. 3 aim in tracking narrowband, phase-locked frequency components or narrowband envelope characteristic components. In case of wireless hardware, which typically does not support phase markers, the scope of offered HIs includes broadband time domain estimators, narrowband frequency estimators, envelope estimators, and velocity-based estimators. The paper proposes some novel methods of reduction of number of calculations in the process of calculation of these HIs [3], which is irrelative for power-supplied data acquisition units, but is a true added value for wireless equipment, where power consumption is of upmost importance [4, 5]. In case of the wireless condition monitoring which are battery powered it is crucial to have the shortest calculations possible for the extension of the battery life-time.

Chapter 2 covers the classical way of calculating of the selected HIs. Chapter 3 discusses fast calculations algorithms that could be implemented into the wireless Condition Monitoring System (CMS) in terms of energy optimisation. Finally, the last chapter summarizes this paper.

2 Health Indicators

This section describes the most commonly used HIs in wireless condition monitoring.

2.1 Peak-to-Peak

The peak-to-peak value \(x_{peak}\) of the time sequence is simply a difference between the maximum and minimum values encountered in the given signal [6].

$$\begin{aligned} x_{max}=\max x \end{aligned}$$
(1)
$$\begin{aligned} x_{min}=\min x \end{aligned}$$
(2)
$$\begin{aligned} x_{peak}=x_{max}-x_{min} \end{aligned}$$
(3)

2.2 Root Mean Square Value

The Root Mean Square (RMS) value \(x_{RMS}\) stands for a “root mean square” value. The name of the indicator explain the process of its calculation (reading from left to right). For a discrete signal, RMS is given as [6]

$$\begin{aligned} x_{RMS}=\sqrt{\frac{1}{n}\sum _{i=1}^{n}x_i^2} \end{aligned}$$
(4)

The basic idea of using the RMS value as a diagnostic criterion is the fact, that any sort of failure generates additional vibrations, which increase the total energy of the system. The RMS value is an indicator of the average energy of the signal; thus, it may be used as a failure detection indication.

2.3 Crest Factor

The crest factor is simply the ratio of the peak value of the signal to the RMS value given as [6]

$$\begin{aligned} C=\frac{x_{peak}}{x_{RMS}} \end{aligned}$$
(5)

It gives the idea how much of impacting is occurring in the vibration signal. Impacting is associated with the roller bearing and gear tooth failures.

2.4 Kurtosis

The kurtosis of the signal is defined as the measure of the “tailedness” of the signal. It is calculated according to the following equation

$$\begin{aligned} K=\frac{1}{n}\frac{\mu _4}{\sigma ^4} \end{aligned}$$
(6)

where \(\mu _4\) is fourth moment about the mean and \(\sigma \) is the standard deviation.

2.5 Velocity Root Mean Square Value

The VRMS value \(x_{VRMS}\) stands for a “velocity root-mean-square”. The name of the indicator explains the process of its calculation (read from right to left) of the velocity signal. Most commonly used sensor for condition monitoring is accelerometer. Therefore it is expected at first to integrate the signal from acceleration to velocity

$$\begin{aligned} x_{vel_0}=\frac{x_{acc_0}}{F_s} \end{aligned}$$
(7)
$$\begin{aligned} x_{vel_n}=x_{vel_{n-1}}+\frac{x_{acc_n}}{F_s} \end{aligned}$$
(8)

Following step is the band-pass filtration in range of \(f=10\div 1000\,\mathrm{Hz}\), as this is required by most of the vibrodiagnostic standards. This filtration is easily achieved by the Fast Fourier Transform (FFT) combined with the selection of frequency bins that corresponds to the given frequency range and Inverse Fast Fourier Transform (IFFT).

$$\begin{aligned} X(\omega )=\int _{-\infty }^{\infty }x(t)e^{-2\pi j\omega t}dt \end{aligned}$$
(9)

where \(\omega \) is the frequency variable, j is a complex value operator. For the filtration a new variable should be considered as

$$\begin{aligned} Y(\omega )= {\left\{ \begin{array}{ll} 2X(\omega ) &{} \quad \text {, if 10 }\le \omega \le \text { 1000}\\ 0 &{} \quad \text {, otherwise}\\ \end{array}\right. } \end{aligned}$$
(10)

where \(Y(\omega )\) is filtered signal to range of \(f=10\div 1000\,\mathrm{Hz}\) given in frequency domain, multiplication by 2 is required to compensate for the negative frequencies. Following step considers the IFFT operation given as

$$\begin{aligned} y(t)=\int _{-\infty }^{\infty }Y(\omega )e^{2\pi j\omega t}d\omega \end{aligned}$$
(11)

where y(t) is filtered signal given in time domain. Last step considers the calculation of RMS of obtained signal as

$$\begin{aligned} V_{RMS}=\sqrt{\frac{1}{n}\sum _{i=1}^{n}y_i^2} \end{aligned}$$
(12)

2.6 “Band Limited” Energy

Spectrum analysis is performed using a FFT. It is assumed that spectrum will be grouped into several “Band Limited” Energy (BLE), i.e. energy will be integrated over given frequency bands. Different frequency bands will be related to different phenomena’s, i.e. low frequency bands are related to, e.g. misalignment and/or unbalance, medium frequency bands are related to higher orders of operation, e.g. x2, x3, x4 etc., high frequency bands are related to, e.g. gear meshing frequencies.

At first FFT is calculated in the same manner as in Eq. (9). Energy content is integrated over given frequency ranges, for \(BEC_1\) it is given as

$$\begin{aligned} BEC_1=\sum _{\omega =0}^{\omega =30}2X(\omega ) \end{aligned}$$
(13)

2.7 Envelope Root Mean Square

Envelope RMS calculation requires at first computation of FFT given as

$$\begin{aligned} X(\omega )=\int _{-\infty }^{\infty }x(t)e^{-2\pi j\omega t}dt \end{aligned}$$
(14)

Following, filtration of appropriate band of interest should be done as

$$\begin{aligned} Y(\omega )= {\left\{ \begin{array}{ll} 2X(\omega ) &{} \quad \text {, if 4000 }\le \omega \le \text { 10000}\\ 0 &{} \quad \text {, otherwise}\\ \end{array}\right. } \end{aligned}$$
(15)

Once the signal is filtrated, IFFT should be calculated following the formula given as

$$\begin{aligned} y(t)=\int _{-\infty }^{\infty }Y(\omega )e^{2\pi j\omega t}d\omega \end{aligned}$$
(16)

For the resulting function envelope should be calculated as

$$\begin{aligned} y_{env}(t)=|y(t)| \end{aligned}$$
(17)

Finally, envelope RMS can be calculated as

$$\begin{aligned} E_{RMS}=\sqrt{\frac{1}{n}\sum _{i=1}^{n}y_{env_i}^2} \end{aligned}$$
(18)

3 Fast Calculation Algorithms

This section describes fast calculation algorithms that could be implemented in wireless condition monitoring for energy efficient calculations.

3.1 Multiplication Order Considerations

It is in common sense to recalculate the measured discretized values to the physical values at the beginning of the entire signal processing, to remain in the physical world units as

$$\begin{aligned} X_{PU}=(X_{DV}-X_{SO})*A_{DtEC}*B_{EtPC} \end{aligned}$$
(19)

where \(X_{PU}\) are the values of measured signal in Physical Units (PU), \(X_{DV}\) are the values of measured signal in Discretized Values (DV), \(X_{SO}\) is the Systems Offset (SO), that has to be subtracted to obtain measured signal with negative values, \(A_{DtEC}\) is the Digital to Electric Coefficient (DtEC), that has to be used to obtain values in electrical units, \(B_{EtPC}\) is the Electric to Physical Coefficient (EtPC), that have to be used to obtain values in physical units.

Apart of above mention recalculations, it is always a good practice to remove the DC constant from the signal for the purpose of the further signal processing, as time-domain based methods are sensitive to DC offsets

$$\begin{aligned} X_{mean}=\frac{1}{n}\sum _{i=1}^{n}X_{PU_i} \end{aligned}$$
(20)
$$\begin{aligned} X_{PU0M}=X_{PU}-X_{mean} \end{aligned}$$
(21)

where \(X_{PU0M}\) is the finally obtained measured signal in Physical Units with 0 Mean (PU0M), which should be used in further calculations.

All of the abovementioned calculations are very simple and fast, but they do consume power. In order to obtained the finally measured signal PU0M, a list of operations that have to be conducted for every measured sample (we assume the n samples).

  • Subtract the System Offset (n subtraction operations)

  • Multiply by the Digital to Electric Coefficient (n multiplication operations)

  • Multiply by the Electric to Physical Coefficient (n multiplication operations)

  • Calculate the mean value (n addition operations)

  • Subtract the mean value (n subtraction operations)

In total 5n operations have to be conducted, before actual signal processing. The proposition is to conduct signal processing on values with minimal preprocessing, and apply the appropriate coefficients after the signal processing. The proposed minimal preprocessing is as follows

$$\begin{aligned} X_{mean}=\frac{1}{n}\sum _{i=1}^{n}X_{DV_i} \end{aligned}$$
(22)
$$\begin{aligned} X_{DV0M}=X_{DV}-X_{mean} \end{aligned}$$
(23)

where \(X_{DV0M}\) is the measured signal in Discretized Values with 0 Mean (DV0M). The list of operations that have to be conducted for every measured sample (we assume the n samples)

  • Calculate the mean value (n addition operations)

  • Subtract the mean value (n subtraction operations)

In total 2n operations. After the signal processing correction coefficients have to be applied to a trend value, so in fact to a single sample. So it could be assumed that this results in reduction from 5n operations to 2n operations in preprocessing stage—reduction of 60% of operations.

After careful analysis it has been found that the application of the scaling coefficients should be done for the following trends:

  • Peak-to-peak,

  • RMS,

  • Velocity RMS,

  • BEC.

There is no need to apply the scaling coefficients for the following trends:

  • Kurtosis,

  • Crest factor.

3.2 Application of Parseval’s Theorem for Envelope RMS Calculation

Classical calculation of envelope RMS assumes one FFT and one IFFT as shown in Sect. 2.7, which are very time consuming. Alternative way of calculation of envelope RMS could be achieved by application of Parseval’s theorem given as

$$\begin{aligned} \sum _{i=1}^{n}|x_i|^2=\frac{1}{n}\sum _{k=1}^{n}|X_k|^2 \end{aligned}$$
(24)

where \(x_i\) is envelope signal given in discrete time domain, \(X_k\) is FFT of a envelope signal. Considering the equation for the RMS as

$$\begin{aligned} x_{RMS}=\sqrt{\frac{1}{n}\sum _{i=1}^{n}x_i^2} \end{aligned}$$
(25)

Having this assumption made, RMS can be calculated directly from the frequency domain as

$$\begin{aligned} x_{RMS}=\sqrt{\frac{1}{n^2}\sum _{i=1}^{n}|X_k|^2} \end{aligned}$$
(26)

In this manner calculation of envelope RMS could be done without a need to go back to the time domain via IFFT. Such a simplification can save almost 50% of calculation time of envelope RMS, as the FFT and IFFT are the most time consuming operations in this scheme.

3.3 Velocity RMS Calculation

Classical calculation of velocity RMS is done at first by numerical integration as shown in Sect. 2.5. This integration requires n operations. An alternative version assumes the integration in frequency domain as follows

$$\begin{aligned} X_{vel}(\omega )=\frac{X_{acc}(\omega )}{2\pi \omega } \end{aligned}$$
(27)

In this manner a fraction of operations are needed to be done. Nowadays CMS analyse bandwidth of \(B_w=10\,\mathrm{kHz}\), this means that the sampling frequency is at least \(F_s=20\,\mathrm{kHz}\). Since there is a need for the analysis of the frequency range \(10\div 1000\,\mathrm{Hz}\), this means a rough reduction of required operations for the numerical integration of 20 times.

4 Conclusions

In this article fast calculation algorithms for calculation of classical Health Indicators have been presented. It has been shown that such operations could save more than 50% of calculation time of certain operations, without any impact on the obtained results by performing calculations in different domains and applying constant coefficients on estimated indicators rather than on to raw data.