_{1}

It is important to specify the occurrence and cause of failure of machines without stopping the machines because of increased use of various complex industrial systems. In this study, two new diagnosis methods based on the correlation information between sound and vibration emitted from the machine are derived. First, a diagnostic method which can detect the part of machine with fault among the assumed several faults is proposed by measuring simultaneously the time series data on sound and vibration. Next, a diagnosis method based on the estimation of the changing information of correlation between sound and vibration is considered by using prior information in only normal situation. The effectiveness of the proposed theory is experimentally confirmed by applying it to the observed data emitted from a rotational machine driven by an electric motor.

Recently, various industrial systems have increased and become complicated. The estimation and prediction of damage part in rotational machines without stopping these are required for cost reduction and improvement of safety. Most of studies proposed up to now for the diagnosis have analyzed by use of either of sound or vibration emitted from the machine in frequency domain [

In the previous study, a faults diagnosis method was proposed by using correlation information between sound and vibration emitted from a machine in time domain [

In this study, by utilizing correlation information between sound and vibration emitted from the machine more effectively, a new faults diagnosis method is derived from two points of view. More specifically, a faults diagnosis method based on the conditional probability distribution reflected liner and non-liner correlation information of lower or higher order between sound and vibration is proposed. When a specific failure occurs among multiple faults established in advance, a diagnosis method able to detect the machine part with the failure is proposed. By adopting expansion expression of conditional probability distribution based on the multinomial distribution to evaluate several failure situations, a new faults diagnosis method of machine is first proposed. Next, a faults diagnosis method to find change of correlation information between sound and vibration is considered by measuring simultaneously the sound and vibration only in normal situation as prior information. Finally, the effectiveness of proposed theoretical method is experimentally confirmed by applying it to measurement data of sound and vibration emitted from a rotational machine.

First, let us introduce a random variable y with two exclusive values of 0 and 1 corresponding to normal situation without fault of machines and a failure situation with the fault. Furthermore, two kinds of variables on sound and vibration are expressed as

the probability of fault occurrence can be predicted in an expansion expression, as follows:

where,

with

By use of a computer for the observed time series data on sound and vibration, on-line signal processing can easily be carried out.

Since the specific correlation relationship between sound and vibration emitted from an identical machine exists, it is possible to diagnose faults of the machine by detecting the change of correlation characteristic between sound and vibration. As examples of multiple faults of machine,

First, the joint probability distributions

where, correlation information among variables with lower and higher orders is reflected in the expansion coefficient

Thus, from Equation (8), the orthonormal functions can be determined as the Hermite polynomial [

Furthermore, a trinomial distribution is adopted as the standard distribution

Two variables

Normal:

where

Therefore, the arbitrary constants are determined as

The coefficients

Furthermore, in the case of considering more than three kinds of failures, a multinomial distribution can be adopted as the standard distribution of Equation (10). It is possible to derive the fault diagnosis in the same manners as the present study by calculating orthogonal polynomial based on the preciously published method [

The following expression can be obtained from Equations ((5), (6)), using Bayes’ theorem on conditional probability distribution.

Therefore, fault occurrence probability

Since the specific correlation relationship between sound and vibration emitted from an identical machine exists, it is possible to diagnose faults of the machine by detecting the change of correlation characteristic between sound and vibration.

The fault diagnosis methods proposed in 2.1 and 2.2 require prior information in both situations before and after a failure occurs. However, only the prior information before the failure occurs can be really obtained. In this section, a faults diagnosis method is proposed based on only prior information in normal situation. The variables

First, the joint probability distribution

Gaussian distribution is used as the standard distributions

Since the correlation information between

In general, cumulative distribution is more suitable than probability distribution for evaluating the prediction error.

There, by using Equation (18) cumulative distribution

with

Furthermore, the faults diagnosis based on the prediction error when evaluating the probability distribution of vibration from sound data is also possible by exchanging the variable

The proposed method was used to detect faults of a rotational machine by simultaneously observing the sound and vibration emitted from the machine. The correlation relationship between the sound and vibration in the case of failure situations changes from the correlation characteristic in the absence of a fault. Therefore, by detecting information on the change of the correlation characteristic between sound and vibration, it is possible in principle to predict machine faults. The RMS values of the sound pressure level (dB) and the acceleration amplitude (m/s^{2}) emitted from a rotational machine driven by an electric motor were simultaneously measured, as shown in

As an example of fault, three weights were put on the lower part of the bearing and the distortion was made. As the other example of fault, a cogwheel with a small scratch was adopted. The distortion of bearing and the existence of a scratch on a cogwheel were considered for fault 1 and fault 2 as a trial. The 5000 data points were measured for normal, fault 1 and fault 2. The observation data were transformed to the sound pressure level and the acceleration amplitude by use of the following relation.

P: sound pressure [Pa],^{2}].

The scatter diagram between the sound and vibration in three cases before and after occurrence of the fault in the machine is shown in

Furthermore,

First, two different time series data sets (Data Set 1 and Data Set 2) for sound and vibration in three different time intervals were successively measured, in cases with and without fault occurrence. Using the 1500 data points of Data Set 1, which contained both cases of faults occurrence (500 data points for each fault) and fault-free cases (500 data points) as the learning data, the expansion coefficients of Equation (7) were first evaluated. Next, after dividing each data set into 30 sub-data sets consisting of 500 data points with 450 overlapping points, the probability of fault occurrence

The results of prediction of the probability for normal situation, failure situations with the fault 1 and 2 are shown in

The fault diagnosis method with less prior information proposed in 2.3 was applied to real observation data. First, the expansion coefficients in Equation (15) were calculated by using 1000 data points in normal situation. Next, sub-data sets with 1000 data points were made from total 15,000 data points consisting of each 5000 data points in normal, fault 1 and fault 2 situations. Furthermore, probability distribution of vibration (or sound) was predicted by observing the data of sound (or vibration) by using sub-data sets. One of the prediction results is shown in

When fault 1 and fault 2 occur, large prediction errors are obtained for the probability distribution of vibration based on the observation of sound. As shown in

Data point/order | Averaged prediction errors for several percentile levels | Prediction errors for 50 percentile levels | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | |||||||||||

Normal situation | 1-1000 | 0.17078 | 0.17078 | 0.17078 | 0.28333 | 0.05000 | 0.05000 | |||||||||

1001-2000 | 0.27538 | 0.28333 | 0.27538 | 0.36248 | 0.05000 | 0.05000 | ||||||||||

2001-3000 | 0.21148 | 0.21148 | 0.21148 | 0.32702 | 0.15000 | 0.15000 | ||||||||||

3001-4000 | 0.39476 | 0.40311 | 0.39476 | 0.49917 | 0.25000 | 0.25000 | ||||||||||

4001-5000 | 0.15000 | 0.15723 | 0.15723 | 0.25000 | 0.05000 | 0.05000 | ||||||||||

Failure situation | for a bearing | 5001 - 6000 | 0.84080 | 0.84869 | 0.84080 | 0.92391 | 0.65000 | 0.65000 | ||||||||

6001 - 7000 | 1.12854 | 1.13541 | 1.14903 | 1.22395 | 0.95000 | 0.95000 | ||||||||||

7001 - 8000 | 0.73352 | 0.74106 | 0.73352 | 0.81257 | 0.55000 | 0.55000 | ||||||||||

8001 - 9000 | 0.80156 | 0.80709 | 0.80156 | 0.88459 | 0.55000 | 0.55000 | ||||||||||

9001 - 10,000 | 0.85781 | 0.86426 | 0.85781 | 0.93705 | 0.65000 | 0.65000 | ||||||||||

for a cogwheel | 10,001 - 11,000 | 1.52179 | 1.54209 | 1.53632 | 1.60078 | 1.25000 | 1.25000 | |||||||||

11,001 - 12,000 | 2.28236 | 2.30272 | 2.29740 | 2.35189 | 1.55000 | 1.55000 | ||||||||||

12,001 - 13,000 | 1.52834 | 1.53342 | 1.54353 | 1.60009 | 1.05000 | 1.05000 | ||||||||||

13,001 - 14,000 | 2.24456 | 2.26526 | 2.25985 | 2.30513 | 1.75000 | 1.75000 | ||||||||||

14,001 - 15,000 | 1.54281 | 1.54928 | 1.56356 | 1.63240 | 1.25000 | 1.25000 | ||||||||||

This paper has paid attention to the correlation information between sound and vibration emitted from rotational machine, and a method based on the conditional probability distribution with linear or non-linear correlation information of lower order or higher order has been proposed for the diagnosis of multiple faults and fault diagnosis with less prior information. Furthermore, the proposed method has been applied experimentally to observe data emitted from a rotating equipment. The proposed method focuses on the observational data in time domain, and the complicated preprocessing such as frequency analysis is unnecessary. Therefore, the proposed method is suitable for on-line signal processing. More specifically, the proposed fault diagnosis method can specify the fault part of machine by using fault probability. Furthermore, based on the information in only normal situation, it is possible to diagnose the fault in a form of the estimated error of cumulative distribution. Both methods have advantages to diagnose faults numerically. However, the proposed method is still at the early stage of study. Thus, there are a great number of problems in the future. For example, 1) The practical method should be developed at the actual environment existing background noise, 2) The determination method for the most suitable learning period to grasp correlation characteristic has to be proposed, and 3) It is necessary to extend the theory to simultaneous generation of multiple faults.

This work was supported by JSPS KAKENHI Grant Number 24760322.

Hisako Orimoto, (2016) Statistical Fault Diagnosis Methods by Using Higher-Order Correlation Information between Sound and Vibration. Intelligent Information Management,08,87-97. doi: 10.4236/iim.2016.84007