CNC machine tools require higher motion accuracy, and the carrier of fault information is different from general machinery. A large number of diagnostic examples show that analyzing the operating state of the machine tool from the quality of the machined parts is the main idea of ​​the diagnosis machine tool. The spindle is a key component of the CNC machine tool, and its motion accuracy directly affects the machining quality of the workpiece. When the main shaft is unbalanced, the reciprocating inertial force generated causes the bearing bush and the main journal to collide at the thinnest part of the oil film wall, causing motion error of the machine tool spindle. It mainly includes pure radial runout D of the main shaft, pure angle swing and pure axial yaw L. The time series analysis method is based on the pattern recognition theory, and abstracts the dynamic process into a random complex system into a simple physical model to facilitate the monitoring and diagnosis of the actual system operating state. Most of the existing literature is based on one-dimensional time series, but for many practical problems, considering only one factor does not fully reveal the internal laws of the system. In order to further explain the influence relationship of each related physical quantity on the system and improve the fitting precision of the model, a multi-dimensional time series model can be introduced. In this paper, the multi-dimensional time series is applied to the surface roughness Ra monitoring of the workpiece, and the influence relationship between D, L and Ra is analyzed. 1 Timing model selection Common time series models include autoregressive model (AR), moving average model (MA) and autoregressive sliding model (ARMA). The latter two types of models can be approximated by high-order AR models, and the degree of approximation depends on the order of the autoregressive model taken. Since the parameters of the AR model are estimated to be linear regression processes, the calculation is simple and fast, and the actual physical system is often an all-pole system. Therefore, the AR model is widely used, and is especially suitable for mechanical fault diagnosis. The premise for the AR model discussion is to assume that the obtained time series is a sequence of stationary, normally distributed. However, when the machine is in a certain hidden danger and the situation is deteriorating, the data obtained by the sensor will be an unstable time series, which has a certain tendency to develop steadily over time. For non-stationary timing, a smooth time series can be obtained after several orders of differential transformation, and then ARMA modeling is performed. A non-stationary time series ARIMA model is proposed for this purpose. 1 1 AR IMA model The ARIMA model is an extension of stationary time series in non-stationary state. It is closely related to the non-stationary sequence {X t }, and the AR IMA model undergoes finite difference ( Y t = ( 1- B )d X t After B is a post-shift operator, { Y t } is the ARMA sequence. The data suitable for ARIMA model modeling has the characteristics of slowly decaying positive sample autocorrelation coefficient function; after differential transformation, the data suitable for ARMA modeling should have the characteristics of rapidly decreasing sample autocorrelation coefficient function. 1 2 Multidimensional AR model concept From the concept of one-dimensional AR model, a multidimensional AR model can be defined. The multidimensional AR model can fit most of the realities because it can approximate multidimensional ARMA models. There is an m-dimensional AR(p) model: zt = A 1 z t- 1 + A 2 z t- 2 + + A pz t- p + ut(1) where: zt is the m 1 dimensional zero-mean stationary sequence ; A j( j = 1, 2,, p) is a mm matrix, where A p 0; { ut } is an m-dimensional white noise sequence that is independent of each other at different times, ie E < ut > = 0 E < utu T s > = S,t= s 0, ts(2) where: S is a positive order of mm order, and E < utz t- j > = 0 ( j = 1, 2,). Then the condition of multivariable stationary: det pi = 0 A i B i = 0 The root is all outside the unit circle. If { zt } satisfies the equation (1) and the stationarity condition, then { zt } is called an m-dimensional AR(p ) sequence. 2 The establishment of multi-dimensional AR model 2 1 Data acquisition The vibration and corner signals obtained by the sensor are continuous signals, and the establishment of the time series model requires discrete sampling of continuous signals. So first determine the sampling interval. If the sampling interval is unreasonable, aliasing of different frequency harmonics will occur. When the data is too large, the sampled data will lose the original correlation in the observed signal, so that the model is reduced and the signal resolution is reduced. When the time is too small, the high frequency noise is taken as a useful signal, and the model is upgraded. , causing an increase in the amount of calculation. Using the Shannon sampling theorem: When the sampling frequency fs is higher than twice the highest frequency f max of the harmonics, frequency mixing can be avoided. For this reason, 1/2 5f m ax is often used for sampling. 2 2 Parameter Estimation of Multidimensional AR Models The parameter estimation method of the AR model is commonly used in direct estimation methods, including the YuleW alker estimation method and the least squares estimation method. The correlation moment estimation method is the most commonly used method for time series model parameter estimation. It is based on the YW equation. Although the estimation accuracy of this method is not as high as that of the least squares method, its calculation is simple and practical. Especially for time series with normal characteristics, the estimation accuracy can be comparable to the least squares estimation method when the sample data is sufficiently large. This paper uses the YW estimation method. This gives a moment estimate of A j( j = 1, 2,, p ). Theoretically, the YuleW alker moment estimator has many advantages, such as positive definiteness, asymptotic unbiasedness, weak compatibility, and asymptotic normality. The multi-dimensional AR model fixed-order FPE criterion FPE criterion (minimum final forecast error criterion) is determined by the sample-to-model. It is based on the one-step prediction error variance of the model output to determine the model order: the smaller the determinant of the one-step forecast error variance matrix, the more ideal the model fits, and the model order is considered to be the best order. The AR(p) model fitted by m-dimensional observation data of sample length N is shown in equation (1). ^ T i(8) It should be noted that if the value of FPE p(zt) increases monotonically with increasing p, then the model can be judged as a first-order autoregressive model; if the value of FPE p(zt) increases with p If the monotonic decrease, the sample sequence cannot be described by the AR model; if the value of FPE p(zt) increases with the p value, the sample length can be increased and then scaled. In order to investigate whether the law of the m-dimensional sequence can fully describe the system characteristics by its partial components, such as the former q components (q<m), the sub-matrix of the final prediction error variance is further introduced, and its determinant FPE p, q, m ( zt) ^ T i in the upper left corner of the q-order sub-matrix. If m inFPE p, q, m( zt) %m inFPE p, q, q( zt), it is considered that only the former q-dimensional sequence is considered. It is not obvious that the m-dimensional sequence is considered to be significant compared to considering only the former q-dimensional sequence. Benefits, so the secondary factors can be excluded, focusing on the monitoring and diagnosis of the main factors; if m inFPE p, q, m(zt) < m inFPE p, q, q(zt), then the m-dimensional sequence must be thoroughly investigated. Gain a comprehensive understanding of the system. This also provides a concrete method for how to build a multidimensional sequence model. 4 Study of the example This paper establishes the initial four-dimensional AR model based on the surface roughness Ra of a CNC lathe workpiece, the pure radial runout D of the lathe spindle, the pure angle swing and the pure axial 窜. Among them, the maximum working speed of the machine tool design spindle is 15 000r / m in. The number of samples N is 500. The non-stationary time series processing steps are as follows: (1) Observing the first 480 sample sequences of the above four-dimensional non-stationary observations {X t } , {X t } = ( Ra, D, , L)T, where {Ra }, { D } raw data is as shown. After the difference is made to the data, the stationary sequence { Y t }, { Y t } = ( 1- B ) {X t }.{R at } The time series obtained after one difference, the autocorrelation coefficient obtained after one difference is obtained. The figure shows a significant downward trend, indicating that the differentially transformed sequence has met the conditions of ARMA modeling. Then standard normal processing is performed on { Y t }: zt = ( Y t - % y) / y, where % y is the mean of the sequence { Y t } and y is the mean square error of { Y t }. Furthermore, multidimensional AR modeling is performed for { zt }, and Y t =yzt + % y, the inverse transform obtains the time series { Y t }, and finally the inverse transform obtains the sequence {X t }. (2) The parameters of the four-dimensional model AR(p) (where p = 1, 2,) are sequentially estimated by the multi-dimensional YuleWalker algorithm. On this basis, the FPE values ​​of each order model are calculated, and the optimal order of the model is determined. In this paper, the FPE value of the 10th-order AR model is calculated. When the AR(6) is the smallest, the FPE value is the smallest. It can be seen that the optimal order of the four-dimensional AR model is 6, and the four-dimensional AR(6) model is established accordingly. (3) Analyze the degree of influence of each relevant physical quantity on the surface roughness. It can be seen that FPE 6, 1, 2 < FPE 6, 1, 1, the pure radial runout is considered to have a greater influence on the surface roughness of the machined workpiece; FPE 6, 2, 3 < FPE 6, 2, 2 is considered pure Angle swing is also important for the surface roughness of the workpiece; FPE 6, 3, 4 > FPE 6, 3, 3, it can be considered that the motion error caused by the pure axial turbulence of the spindle has no effect on the geometry of the surface of the workpiece. This is also consistent with the actual theory. Based on this conclusion, it can be seen that in the establishment of the multi-dimensional AR model for machine tool spindle diagnosis, only three key factors of Ra, D, need to be considered. (4) Based on the above viewpoint, a { Ra, D,} T three-dimensional AR (6) model is established. According to this model, 20 Ra values ​​after serial number 480 were monitored and predicted, and the results are as follows. From the prediction results, the AR (6) model is used to predict the maximum relative error of 8 7% and the minimum is only 176%, which is basically consistent with the development trend of the measured values. It shows that the multi-dimensional AR model is reasonable for analyzing the failure of the machine tool spindle components. of. 5 Conclusions Through the information carrier of the surface roughness of the workpiece, the influence degree of the motion error of the machine tool spindle is studied. Based on this, a multi-dimensional AR model based on non-stationary sequence is established, and the theoretical basis from the acquisition of the original data to the establishment of the model is given. At the same time, according to the FPE criterion, the main factors causing the system failure are identified, and the interference of the secondary factors is avoided. This simplifies the difficulty and workload of modeling. Through the example calculation, it is proved that the multi-dimensional AR model based on non-stationary sequence meets the requirements of machine tool fault diagnosis, and the prediction accuracy meets the requirements. (Finish) Pa66 Melamine Cyanurate,Melamine Cyanurate Molecular Mas,Melamine Cyanurate Polyethylene,Mca Refine Powder SHANDONG TAIXING ADVANCED MATERIAL CO., LTD. , https://www.fr-chem.com