DETERMINATION OF SHAFT ROTATION ANGLE FROM ACCELERATIONS OF THE WIRELESS SENSOR BY THE NOVEL NUMERICAL METHOD

2018. Т. 18, No 3. С. 143–149 143 Introduction The scheduled maintenance allows us to prevent breakdowns and unplanned downtime of equipment but limits the flexibility of manufacturing systems. On the other hand, condition monitoring concept enhances the manufacturing systems flexibility but requires a reliable diagnostic information in real-time. Traditionally, accelerometers mounted on a housing of a mechanism are used to receive diagnostic information. Nevertheless, the received data need special techniques of signal processing to diagnosis, for example, low-energy defects [1–3] or machines with variable workloads [4–5]. Unfortunately, these methods have a limitation of machine fluctuation speed or require a high level of computational resources. Modern technologies associated with concept Industrial Internet of Things (IIoT) such as wireless technologies (transfer of power and data) and MEMS-technologies allow a substantial increase in the opportunities of implementing sensors in various fields, such as structural health monitoring [6, 7] and condition monitoring [8, 9]. An example of the wireless technologies for condition monitoring is Wireless Acceleration Sensor (WAS) [10]. The WAS is mounted on a machinery rotating shaft and the WAS measures angular, linear accelerations and angle of the shaft simultaneously. However, the accuracy of angle measurement significantly affects the accuracy of linear accelerations measurement. This paper investigates a problem of calculating the rotation angle of the shaft from the measured angular acceleration of the shaft. The significant feature of the problem is an instability resolve. For that reason, various methods are applied for stabilization of the solution. At present, development of numerical methods for solving inverse problems is of great interest to many researches. Chinchalkar [11] proposed a numerical method for determining the location of a crack in a beam of different depths based on the finite-element method. Also, Zhang [12] used the finiteelement method for detecting delamination in composites. On the other hand, reducing original problem to integral or integro-differential equation is a generally used method for resolving first-order differenDOI: 10.14529/ctcr180315

Introduction tial equation. Jang [13] reduced the differential equation of motion to the nonlinear integral equation and found the regularization method for the integral equation. Also, Parand [14] used Volterra's Population Model which was presented as integro-differential equations. The authors considered two common collocation approaches based on radial basis functions for resolving the equations.
The present study offers a numerical method to resolve differential equations for calculating the rotation angle of the shaft from indirect measurement. The method applies explicit finite-differential scheme. Also, the authors used a regularization technique for stabilization of solution. Additionally, the method is robust to noise.

Statement of the problem
The mathematical model of the wireless sensor, which is reported in [15], is described by the equation system , , a t a t a t -acceleration measured by accelerometers of the wireless sensor,   -angular acceleration of the shaft, x  and z  -linear accelerations of the shaft,  -rotation angle of the shaft, g -gravitational acceleration, r -the distance between the rotational axis of the shaft and the sensitive axis of the accelerometer. Furthermore, the important feature of applying the sensor is incompleteness of initial data which are needed for calculating the instantaneous rotation angle of the shaft. The initial angle of the shaft can be certainly defined from the signals of the wireless sensor. Accordingly, the initial angle is   0 0    . However, the initial angular speed of the shaft is an arbitrary value. As a result, the existing methods for calculating the shaft rotation angle are unstable.
Furthermore, the noise of the measurement is a significant feature of the problem described above.
Therefore, in this study, the authors presumed that instead of the exact    

Numerical method
The section contains the main idea of the numerical method to resolve of the system (1). Direct relationship equation between the unknown function   t   and measured results is shown as [15]  (2) The equations system (2) contains important features. At first, the right hand side of the equation is implicitly depends on the required function   t  . Secondly,   0   is arbitrary value. The first feature leads to essential increase in error of the classical numerical resolve due to small increase in uncertainty of the measured signals. The second feature is the key distinction from the classical numerical methods.
As, due to the fact that classical statement of the problem requires extra condition   0   for a stable solution, the general rule is   . As a result, the features have a significant effect on numerical solution stability.
The authors changed the equation system (1) and rearranged x  and z  to the left side of the equation system. Thus, the changed system is Based on the physical properties of the mechanical systems the authors assumed the functions   Taking into account the properties of       , , t x t z t  the authors proposed the computational scheme based on discrete regularization method (DRM) to calculate the rotation angle. The multidimensional form of DRM was proposed in [16]. The basic principles of the DRM are regularization technique and application of finite-difference equations. The idea of the method is described hereafter.
The finite-difference grid is introduced k k a t a  Using the finite difference analog of the derivatives, the differential equation in (2) was replaced by a finite-difference equation The differential equations from system (3) for   x t  and   z t  are replaced in the same way.
Additional stabilization functionals with parameters 1 2 3 , ,    for a stable solution are introduced into each finite-difference equation. Further, basing on continuity and differentiability of the function , t x t z t     are approximated by linearized analogs. Further, the authors chose the discretization step  and regularization parameters according to the special conditions and calculated the rotation angle 1 .
k  

Computational results
The method described above has been implemented in simulation and applied to the signals of the wireless sensor with random white noise. The parameters and the initial conditions of the sensor are the same as in [15]. We assumed that the shaft moved with the accelerations (5). Also, we added the random white noise to the signals of the accelerometers of the sensor.
The results of the signals processed by the numerical method were compared with the reference angle of the shaft. The reference angle of the shaft was calculated from the angular acceleration by means of numerical integration (Fig. 1). The Fig. 2 shows calculation error of the angle of the shaft by numerical method which did not exceed 0.027 rad.

Conclusion
The calculation of the rotation angle of the shaft from the angular acceleration of the shaft is a co plicated problem. The problem becomes more difficult if the angular acceleration contains noise. The computational results show that the proposed numerical method allows an increase in the accuracy of calculating the shaft rotation angle during the mechan contains high noise. The future studies will analyze work of the numerical method at actual data from the wireless sensor of acceleration.  The calculation of the rotation angle of the shaft from the angular acceleration of the shaft is a co plicated problem. The problem becomes more difficult if the angular acceleration contains noise. The computational results show that the proposed numerical method allows an increase in the accuracy of calculating the shaft rotation angle during the mechanism operation when the angular acceleration contains high noise. The future studies will analyze work of the numerical method at actual data from the wireless sensor of acceleration. The calculation of the rotation angle of the shaft from the angular acceleration of the shaft is a complicated problem. The problem becomes more difficult if the angular acceleration contains noise. The computational results show that the proposed numerical method allows an increase in the accuracy ism operation when the angular acceleration contains high noise. The future studies will analyze work of the numerical method at actual data from