【文章內(nèi)容簡介】 ball array （ R） Thermal errors at different z coordinates 1. z = 50 2. z = 150 3. z = 250 4. z = 350 plot s the timehistory of thermal drift Δ z at different z coordinates under a test . It shows that the resultant thermal drift s are obvious positiondependent . The thermal drift s at z 1 ,z 2 , z 3 , z 4 are coincident initially but separate gradually as time passes and temperature increases. The reason is that , initially most of thermal drift s result f rom the positionindependent thermal growth of the spindle housing which would rise fast and go to thermalequilibrium quickly pared to other machine ponent s with longer thermaltimeconstant s. However , as time passes , those positiondependent thermal errors such as the lead screw and the column cont ribute to the resultant thermal drift s of the tool more and more. As a result , the thermal drifts at different z coordinates have different magnitude and thermal characteristics. However , the thermal errors at different coodinates vary with z coordinate continuously. 2 AR MODEL FOR THERMAL ERROR Precise prediction of thermal errors is an important step for accurate error pensation. Since the knowledge of the machine structure , the heat source and the boundary condition are insufficient , a precise quantitative prediction based on theoretical heat transfer analysis is quite difficult . On the other hand , empiricalbased error models using regression analysis and neural works have been demonst rated to predict thermal errors with satisfactory accuracy in much application. Thermal errors are caused by various heat sources. Only the influence of the heat caused by the fiction of spindle which is the most significant heat source is considered. The influence of external heat source on machining accuracy can be diminished by environment temperature control. From the obtained data , it is found that thermal errors vary continuously with time. The value of error at one moment is influenced by that of the previous moment and the rotation speed of spindle. So a model representing the behavior of the thermal errors as written is the form where Δ z ( t) ——— Thermal error at time t k , m ——— Order of the model ai , bi ——— Coefficient of the model n ( t i) ——— Spindle rotation speed at time t i The order k and m are determined by the final predictionerror criterion. The coefficients ai and bi are estimated by artificial neural work technique. A neural work is a multiple nonlinear regression equation in which the coefficient s are called weight s and are t rained with an iterative technique called back propagation. It is less sensitive than other modeling technique to individual input failure due to thresholding of the signals by the sigmoid functions at each node. The neural work for this problem is shown in . ( k = 1 , m = 0) . The number of hidded nodes is determined by a trialand error procedure. Using the data obtained (thermal errors and correspondence speed) , four models for the errors at z 1 , z 2 , z 3 and z 4 are established. Thermal errors at positions other than z 1 , z 2 , z 3 , z 4 are calculated by an interpolating function. So the errors at any z coordinates can be obtained. In order to verify the prediction accuracy of the model , a number of new operation conditions are used. Fig14 shows an example of predicted result o