机械钻床外文翻译论文中英文(2)

where a and b are two positive coefficients which are usually determined by experiment. An alternate method [16] of representing the damping matrix is expressing [C]as

The element of [C’] is defined as,where signKij=(Kij/|Kij|), Kij and Mij are the elements of [K] and [M], ζis the damping ratio of the material.

The generalized force in a frame element is defined as

where Fj and Mj are the jth external force and moment including the inertia force and moment on the element acting at (xj ,yj), and m is the number of the externalforces acting on the element. The element generalized forces

,are then combined to formthe

systemgeneralized force {R}. The second order ordinary differential equations of motion of the system, Eq. (11), can be directly integrated with a numerical method such as Runge-Kutta method. For the PKM we studied, each link was discreted as 15 frame elements. Both Matlab and ADAMS software are used for programming and solving these equations.

3. Input motion planning for drilling

Suppose we know the ideal motion function of the drilling tool. How to determine the input motor motion so that the ideal tool motion can be realized is critical for high quality drillings. The created explicit input motion function also provides the necessary information for machine controls. According to the study done in [9], the drilling process can be divided into three phases: entrance phase, middle phase, and exit phase. In order to increase the productivity and quality of the drilling, many operation constraints such as minimum tool life constraint, hole location error constraint, exit burr constraint, drill torsion breakage constraint, etc. must be considered and satisfied. Under these conditions, the feed velocity of 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 25.4 mm (1 in) deep hole with Tc=0.4s, 0.3s for drilling, 0.1s for retracting. Set T1=T3 0.06s, T4=T6=0.03s. Under these con-ditions, 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=53° 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=143.48mm/s, vC=165.77mm/s, vE=-557.36mm/s, vF=-499.44mm/s. When plotting the velocity curve with Eq. (18), no visual difference can be found with the curve shown in Fig. 5. Eq. (18) is composed 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 1.5%. 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 point-to-point positioning

In 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 commonly 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 y-axis motion, which is

realized 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/s2 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 0.1 s. Assuming the material damping ratio as 0.01, 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 …… 此处隐藏：3783字，全部文档内容请下载后查看。喜欢就下载吧 ……