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1、外文资料：Kinematic and dynamic synthesis of a parallel kinematic high speeddrilling machineAbstractTypically, the termhigh speed drilling is related to spindle capability of high cutting speeds. The suggested high speed drilling machine (HSDM) extends this term to include very fast and accurate point-to
2、-point motions. The new HSDM is composed of a planar parallel mechanism with two linear motors as the inputs. The paper is focused on the kinematic and dynamic synthesis of this parallel kinematic machine (PKM). The kinematic synthesis introduces a new methodology of input motion planning for ideal
3、drilling operation and accurate point-to-point positioning. The dynamic synthesis aims at reducing the input power of the PKM using a spring element. Keywords: Parallel kinematic machine; High speed drilling; Kinematic and dynamic synthesis1. IntroductionDuring the recent years, a large variety of P
4、KMs were introduced by research institutes and by industries. Most, but not all, of these machines were based on the well-known Stewart platform 1 configuration. The advantages of these parallel structures are high nominal load to weight ratio, good positional accuracy and a rigid structure 2. The m
5、ain disadvantages of Stewart type PKMs are the small workspace relative to the overall size of the machine and relatively slow operation speed 3,4. Workspace of a machine tool is defined as the volume where the tip of the tool can move and cut material. The design of a planar Stewart platform was me
6、ntioned in 5 as an affordable way of retrofitting non-CNC machines required for plastic moulds machining. The design of the PKM 5 allowed adjustable geometry that could have been optimally reconfigured for any prescribed path. Typically, changing the length of one or more links in a controlled seque
7、nce does the adjustment of PKM geometry.The application of the PKMs with constant-length links for the design of machine tools is less common than the type with varying-length links. An excellent example of a constant-length links type of machine is shown in 6. Renault-Automation Comau has built the
8、 machine named Urane SX. The HSDM described herein utilizes a parallel mechanism with constant-length links.Drilling operations are well introduced in the literature 7. An extensive experimental study of highspeed drilling operations for the automotive industry is reported in 8. Data was collected f
9、romhundreds controlled drilling experiments in order to specify the parameters required for quality drilling. Ideal drilling motions and guidelines for performing high quality drilling were presented in 9 through theoretical and experimental studies. In the synthesis of the suggested PKM, we follow
10、the suggestions in 9.The detailed mechanical structures of the proposed new PKM were introduced in 10,11. One possible configuration of the machine is shown in Fig. 1; it has large workspace, highspeed point-to-point motion and very high drilling speed. The parallel mechanism provides Y, and Z axes
11、motions. The X axis motion is provided by the table. For achieving highspeed performance, two linear motors are used for driving the mechanism and a highspeed spindle is used for drilling. The purpose of this paper is to describe new kinematic and dynamic synthesis methods that are developed for imp
12、roving the performance of the machine. Through input motion planning for drilling and point-to-point positioning, the machining error will be reduced and the quality of the finished holes can be greatly improved. By adding a well-tuned spring element to the PKM, the input power can be minimized so t
13、hat the size the machine and the energy consumption can be reduced. Numerical simulations verify the correctness and effectiveness of the methods presented in this paper.2. Kinematic and dynamic equations of motion of the PKM moduleThe schematic diagram of the PKM module is shown in Fig. 2. In consi
14、stent with the machine tool conventions, the z-axis is along the direction of tool movement. The PKM module has two inputs (two linear motors) indicated as part 1 and part 6, and one output motion of the tool. The positioning and drilling motion of the PKM module in this application is characterized
15、 by (y axis motion for point-to-point positioning) and (z axis motion for drilling). Motion equations for both rigid body and elastic body PKM module are developed. The rigid body equations are used for the synthesis of input motion planning of drilling and input power reduction. The elastic body eq
16、uations are used for residual vibration control after point-to-point positioning of the tool.2.1. Equations of motion of the PKM module with rigid links Using complex-number representation of mechanisms 12, the kinematic equations of the tool unit (indicated as part 3 which includes the platform, th
17、e spindleand the tool) are developed as follows. The displacement of the tool is andwhere b is the distance between point B and point C, r is the length of link AB (the lengths of link AB, CD and CE are equal). The velocity of the tool iswhereThe acceleration of the tool iswhereThe dynamic equations
18、 of the PKM module are developed using Lagranges equation of the second kind 13 as shown in Eq. (7).where T is the total kinetic energy of the system; and are the generalized coordinates and velocities; is the generalized force corresponding to . k is the number of the independent generalized coordi
19、nates of the system. Here, k=2, q1=y1 and q2=y6. After derivation, Eq. (7) can be expressed aswhere n is the number of the moving links; are mass and mass moment of inertia of link i; are the coordinates of the center of mass of link i; hi is the rotation angle of link i in the PKM module. The gener
20、alized force can be determined bywhere V is the potential energy and Fi are the nonpotential forces. For the drilling operation of the PKM module, we havewhere Fcut is the cutting force, F1 and F6 are the input forces exerted on the PKM by the linear motors. Eqs. (1) to (10) form the kinematic and d
21、ynamic equations of the PKM module with rigid links.2.2. Equations of motion of the PKM module with elastic linksThe dynamic differential equations of a compliant mechanism can be derived using the finite element method and take the form ofwhere M, C and K are system mass, damping and stiffness matr
22、ix, respectively; D is the set of generalized coordinates representing the translation and rotation deformations at each element node in global coordinate system; R is the set of generalized external forces corresponding to D; n is the number of the generalized coordinates (elastic degrees of freedo
23、m of the mechanism). In our FEA model, we use frame element shown in Fig. 3 in which EIe is the bending stiffness (E is the modulus of elasticity of the material, Ie is the moment of inertia), q is the material density, le isthe original length of the element. are nodal displacements expressed in lo
24、cal coordinate system(x, y). The mass matrix and stiffness matrix for the frame element will be 66 symmetric matrices which can be derived fromthe kinetic energy and strain energy expressions as Eqs. (12) and (13)where T is the kinetic energy and U is the strain energy of the element; are the linear
25、 1 2 3 4 5 6 and angular deformations of the node at the element local coordinate system. Detailed derivations can be found in 14. Typically, a compliant mechanism is discretized into many elements as in finite element analysis. Each element is associated with a mass and a stiffness matrix. Each ele
26、ment has its own local coordinate system. We combine the element mass and stiffness matrices of all elements and perform coordinate transformations necessary to transform the element local coordinate systemto global coordinate system. This gives the systemmass M and stiffness K matrices. Capturing t
27、he damping characteristics in a compliant systemis not so straightforward. Even though, in many applications, damping may be small but its effect on the systemstability and dynamic response, especially in the resonance region, can be significant. The damping matrix C can be written as a linear combi
28、nation of the mass and stiffness matrices 15 to form the proportional damping C which is expressed aswhere a and b are two positive coefficients which are usually determined by experiment. An alternate method 16 of representing the damping matrix is expressing CasThe element of C is defined as,where
29、 signKij=(Kij/|Kij|), Kij and Mij are the elements of K and M, is 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 numb
30、er 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. Fo
31、r 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 drillingSuppose we know the ideal motion function of the drilling tool. How to determine the input motor motion so that
32、 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 phas
33、e, 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 condition
34、s, 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 afte
35、r 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 correspond
36、ing 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 dr
37、ill 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 angle=53 at the sta
38、rting 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 funct
39、ion 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 a
40、nd 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 m
41、aximum 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 integrat
42、ed into the control algorithmof the PKM.4. Input motion planning for point-to-point 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.
43、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 mod
44、ule used in previous for example. A FEA model is built using ADMAS with frame elements. The positioning motion is the y-axis 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
45、(approximated as 30m/s 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
46、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. Selectin
47、g 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 point-to-point positioning time, the first six coeffcients are resulted:Logically, set the optimization objective aswhere c6 is the independent design variable; is the maximum fluctuat
48、ion of residual vibrations of the tool tip after the point-to-point positioning. Set and start the calculation from c6=0. The optimization results in c6=-10mm/s . Consequently, c5=7.510mm/s , c4 =-1.42510mm/s , c3=8.510mm/s , c2=c1=c0=0. It can be seen that the optimization calculation brought the d
49、esign 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 become 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. 1
