高速鉆床動(dòng)力學(xué)分析翻譯中英文(編輯修改稿)
2024-09-14 17:19 本頁(yè)面
【文章內(nèi)容簡(jiǎn)介】 the tool should be slow at the entrance phase to reduce the hole location errors. The tool velocity should also be slow at the exit phase to reduce the exit burr. At the middle phase, the tool drilling velocity should be fast and kept constant. The retraction of the tool after finishing the drilling should be done as quickly as possible to increase the productivity. Based on these considerations, we assume that the ideal drilling and retracting velocities of the tool are given by Eq. (17).where vT1 is the maximum drilling velocity, T1, T2,and T3 are the times corresponding to the entrance phase, the middle phase and the exit phase. vT2 is the maximum retracting velocity. T4, T5, and T6 are corresponding to accelerating, constant velocity, and decelerating times for retracting operation. is the cycle time for a single drilling. As a numerical example, suppose we drill a mm (1 in) deep hole with Tc=, for drilling, for retracting. Set T1=T3 , T4=T6=. Under these conditions, vT1=106(mm/s), vT2=363(mm/s). The graphical expression of the ideal tool motion is shown in Fig. 4. If the link length in PKM r=500 mm, the angleβ=53176。 at the starting point of drilling, the corresponding input motor velocity relative to the idealtool motion is shown in Fig. 5. Generally, the curve fitting method can be used to create the input motion function. But according to the shape of the curve shown in Fig. 5, we create the linear motor velocity function manually section by section as shown in Eq. (18).where vB=, vC=, vE=, vF=(18), no visual difference can be found with the curve shown in Fig. 5. Eq. (18) is posed of six parts with four cycloidal functions and two linear functions. If we control the two linear motors to have the same motion as described in Eq. (18), the drilling and retracting velocity of the tool will be almost the same as shown in Fig. 4. The absolute errors between the ideal and real tool velocity are shown in Fig. 6, in which the maximum error is less than 8 mm/s, the relative error is less than %. At the start and the end positions of the drilling, the errors are zero. These small absolute and relative errors illustrate the created input motion and are quite acceptable. The derived function is simple enough to be integrated into the control algorithmof the PKM.4. Input motion planning for pointtopoint positioningIn order to achieve fast and accurate positioning operation in the whole drilling process, the input motion should be appropriately planned so that the residual vibration of the tool tip can be minimized. Conventionally the constant acceleration motion function is monly used for driving the axes motions in machine tools. Although this kind of motion function is simple to be controlled, it may excite the elastic vibration of the systemdue to the sudden changes in acceleration. Take the same PKM module used in previous for example. A FEA model is built using ADMAS with frame elements. The positioning motion is the yaxis motion, which isrealized by the two linear motors moving in the same direction. Suppose the positioning distance between the two holes is 75mm, the constant acceleration is 3g(approximated as 30m/s178。 here). The input motion of the linear motors with constant acceleration and deceleration is shown in Fig. 7, in which the maximum velocity is 1500 mm/s, the positioning time is s. Assuming the material damping ratio as , the residual vibration of the tool tip is shown in Fig. 8. In order to reduce the residual vibration and make the positioning motion smoother, a six order polynomial input motion function is built as Eq. (19)where the coeffcients ci are the design variables which have to be determined by minimizing the residual vibration of the tool tip. Selecting the boundary conditions as that when t=0, sin=0, vin=0, ain=0。and when t=Tp, sin=h, vin=0, ain=0, where Tp is the pointtopoint positioning time, the first six coeffcients are resulted:Logically, set the optimization objective aswhere c6 is the independent design variable。 is the maximum fluctuation of residual vibrations of the tool tip after the pointtopoint positioning. Set and start the calculation from c6=0. The optimization results in c6=10mm/s . Consequently, c5=10mm/s , c4 =10mm/s , c3=10mm/s , c2=c1=c0=0. It can be seen that the optimization calculation brought the design variable c6 to the boundary. If further loosing the limit for c6, the objective will continue reduce in value, but the maximum value of acceleration of the input motion will bee too big. The optimal input motions after the optimization are shown in Fig. 9. The corresponding residual vibration of the tool tip is shown in Fig. 10. It is seen from paring Fig. 8 and Fig. 10 that the amplitude and tool tip residual vibration was reduced by 30 times after optimization. Smaller residual vibration will be very useful for increasing the positioning accuracy. It should be mentioned that only link elasticity is included in above calculation. The residual vibration af
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