外文翻譯----高速鉆床動(dòng)力學(xué)分析（英文）

1、International Journal of Machine Tools received in revised form 20 February 2004; accepted 22 April 2004AbstractTypically, the term ‘‘high speed drilling’’ is related to spindle capability of high cutting speeds. The su

2、ggested high speed drill- ing machine (HSDM) extends this term to include very fast and accurate point-to-point motions. The new HSDM is composed of a planar parallel mechanism with two linear motors as the inputs. The p

3、aper 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 drilling operation and accurate poi

4、nt-to-point positioning. The dynamic synthesis aims at reducing the input power of the PKM using a spring element. # 2004 Elsevier Ltd. All rights reserved.Keywords: Parallel kinematic machine; High speed drilling; Kinem

5、atic and dynamic synthesis1. IntroductionDuring the recent years, a large variety of PKMs were introduced by research institutes and by indus- tries. Most, but not all, of these machines were based on the well-known Stew

6、art 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 main disadvantages of Stewart type PKMs are the s

7、mall workspace relative to the overall size of the machine and relatively slow oper- ation 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

8、 of a planar Stewart platform was mentioned 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 o

9、ptimally reconfigured for any prescribed path. Typically, chan- ging the length of one or more links in a controlled sequence does the adjustment of PKM geometry.The application of the PKMs with ‘‘constant-length links’’

10、 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 mac

11、hine named ‘‘Urane SX’’. The HSDM described herein utilizes a parallel mechanism with con- stant-length links. Drilling operations are well introduced in the litera- ture [7]. An extensive experimental study of high-spee

12、d drilling operations for the automotive industry is reported in [8]. Data was collected from hundreds con- trolled drilling experiments in order to specify the para- meters required for quality drilling. Ideal drilling

13、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 the suggestions in [9]. The detailed mechan

14、ical 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, high-speed point-to-point motion and very high drilling speed. The

15、 parallel mechanism pro- vides Y, and Z axes motions. The X axis motion is pro- vided by the table. For achieving high-speed? Corresponding author. Tel.: +1-734-647-7325; fax: +1-734-615- 0312. E-mail address: lizhe@umic

16、h.edu (Z. Li).0890-6955/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2004.04.007where n is the number of the moving links; (mi,Igi) are mass and mass moment of inertia of link

17、i; (ygi,zgi) are the coordinates of the center of mass of link i; hi is the rotation angle of link i in the PKM module. The gen- eralized force Qj can be determined byQj ¼ ? @V@qj þ X ni¼1 F0 i@ri @qj 

18、0;9Þwhere V is the potential energy and F0i are the non- potential forces. For the drilling operation of the PKM module, we haveQ1 Q2? ?¼?gP 5i¼2 mi@_ zgi @ _ y1 ? Fcut@_ zg3 @ _ y1 þ F1?gP 5i¼2

19、mi@_ zgi @ _ y6 ? Fcut@_ zg3 @ _ y6 þ F68 > > >> > :9 > > > => > > ;ð10Þwhere Fcut is the cutting force, F1 and F6 are the input forces exerted on the PKM by the linear mot

20、ors. Eqs. (1) to (10) form the kinematic and dynamic equa- tions 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

21、 can be derived using the finite element method and take the form of½M?n?nf€ Dgn?1 þ ½C?n?nf _ Dgn?1 þ ½K?n?nfDgn?1 ¼ fRgn?1 ð11Þwhere [M], [C] and [K] are system mass, damping and

22、 stiffness matrix, respectively; {D} is the set of general- ized coordinates representing the translation and rotation deformations at each element node in global coordinate system; {R} is the set of generalized exter- n

23、al forces corresponding to {D}; n is the number of the generalized coordinates (elastic degrees of freedom of the mechanism). In our FEA model, we use frame element shown in Fig. 3 in which EIe is the bending stiffness (

24、E is the modulus of elasticity of the material, Ie is the moment of inertia), q is the material density, leis the original length of the element. di (i ¼ 1, 2,. . .,6) are nodal displacements expressed in local coor

25、dinate system (x, y). The mass matrix and stiffness matrix for the frame element will be 6 ? 6 symmetric matrices which can be derived from the kinetic energy and strain energy expressions as Eqs. (12) and (13)ddt@T@ _ d

26、? ?? @T@d? ?¼ ½m?ef€ dg ð12Þ@U@d? ?¼ ½k?efdg ð13Þwhere T is the kinetic energy and U is the strain energyof the element; fdg ¼ ½d1 d2 d3 d4 d5 d6?T, are the linear and an

27、gular deformations of the node at the element local coordinate system. Detailed derivations can be found in [14]. Typically, a compliant mechanism is dis- cretized into many elements as in finite element analy- sis. Each

28、 element is associated with a mass and a stiffness matrix. Each element has its own local coordi- nate system. We combine the element mass and stiff- ness matrices of all elements and perform coordinate transformations n

29、ecessary to transform the element local coordinate system to global coordinate system. This gives the system mass [M] and stiffness [K] matri- ces. Capturing the damping characteristics in a com- pliant system is not so

30、straightforward. Even though, in many applications, damping may be small but its effect on the system stability and dynamic response, especially in the resonance region, can be significant. The damping matrix [C] can be

31、written as a linear com- bination of the mass and stiffness matrices [15] to form the proportional damping [C] which is expressed as½C? ¼ a½M? þ b½K? ð14Þwhere a and b are two positive

32、coefficients which are usually determined by experiment. An alternate method [16] of representing the damping matrix is expressing [C] as½C? ¼ ½M?½C0? ð15ÞThe element of [C0] is defined as C

33、0 ij ¼ 2fðsignKijÞðKij=MijÞ1 2, where signKij ¼ ðKij= Kij ? ? ? ?Þ, Kij and Mij arethe elements of [K] and [M], f is the damping ratio of the material. The generalized force in a f

34、rame element is defined asRe i ¼ X mj¼1 Fxj@xj @di þ Fyj@yj @di þ Mhj@hj @di? ?ði ¼ 1;2;...;6Þð16Þwhere Fj and Mj are the jth external force and moment including the inertia f