3.1. Determining the flexible combination constraintsThe dynamic characteristics of joints can be intro-duced into the dynamic analysis of a whole machine-tool structure by the flexible combination constraints,which can be illustrated with Fig. 1. Elements 1 and 2of Fig. 1 are subsystems of a machine-tool structure sys-tem. The second end of element 1 is linked to the firstend of element 2 by a flexible joint. Each end of theelements has six freedom degrees (three translational andthree rotational). A is the junction point between the twolinked elements.Because the element 1 and 2 are connected at junctionpoint A, we have the restraints of force as follows,{F12}   TRV·{F21} (1)where F21,F12 are the amplitude vectors of force forelement 1 at its second end and element 2 at its first endrespectively; TRV is the coordinate transform matrix.Fig. 1. Synthesis of two elements. At junction point A, the dynamic equation can be writ-ten as,{[KJ]   iw[CJ]}·(TR·{X21}   TV·{X12})   {F12} (2)where X21,X12 are the amplitude vectors of displacementsfor element 1 at its second end and element 2 at its firstend, respectively; w is the excitation frequency; TRTV arethe coordinate transform matrices.From eq. (2), the restraint conditions of the jointbetween force and displacement can be obtained as fol-lows,[QM]·{X1 X2}T  {F12} (3)where [QM] is the coefficient matrix including CJ, KJand w; X1,X2 are the amplitude vectors of displacementfor element 1 and 2 at their ends, respectively.3.2. Determining the rigidity combination constraintsWhen stiffness of a joint between two elements is suf-ficiently high, it is called the rigidity combination con-straint, which is expressed as,{X12}   TRV·{X21} (4)3.3. Establishing dynamic equations of a wholemachine-tool structureBased on the physical and geometrical parameters ofthe two elements, the dynamic equations of the twoelements can be established by receptance method [9]as follows,{X1 X2}T   R1 00 R2 ·{F1 F2}T(5)where F1,F2 are the amplitude vectors of forces forelement 1 and 2 at their ends, respectively; R1,R2 are thereceptance matrices of element 1 and 2, respectively,which determined by their physical and geometricalparameters.If eq. (5) is substituted into eq. (3), we have[QMM]·{F11 F21 F12 F22}T  {0} (6)From eqs (6) and (1), we can obtain the following equ-ation,{F12}   [FM]·{F11 F22}T(7)If eqs (7) and (1) are substituted in eq. (5), we canobtain a dynamic equation of a new sub-system syn-thesised by these two elements as,{X11 X22}T  [R12]·{F11 F22}T(8)Repeating the above steps from Eqs (1–8) and syn-thesising the new sub-system with the next element oneby one until all the elements are synthesised into onesystem, then the dynamic equation of a whole machine-tool structure can be obtained as follows,{X}   [RR]·{F} (9)where X and F are the amplitude vectors of displacementand force at the element ends of interest in a wholemachine-tool structure, respectively; [RR] is thereceptance matrix of the whole machine-tool structure.From the above procedures we can see that thereceptance matrix [RR] contains the information of fre-quency characteristics. Each excitation frequency atwhich there is a receptance matrix [RR]. The matrix[RR] comprises the direct- and cross-receptances withrespect to six coordinates (three translational and threerotational). The computation is processed at a large num-ber of discrete frequencies, which are specified as inputdata. When the computed receptance becomes large andchanges its sign from positive to the negative, the closeproximity to a resonant frequency is indicated. Resonantfrequencies up to the Nth order can be identified in thismanner. The matrix [RR] can be calculated in the fre-quency range of interest by the frequency scanningmethod. Therefore the natural frequencies of a wholemachine-tool structure can be obtained by this way at itsdesign stage, if the synthesis of the structure are known.4. Analysis of dynamic characteristics of aguideway jointAlthough the dynamic equations of a whole machine-tool structure have been established by the above pro-cedures, the values of stiffness and damping of joints ineq. (2) have still not been determined. As an example,the dynamic characteristics of a guideway joint are ana-lysed as the following procedures. 机床结构动态性能预测英文文献及中文翻译(2):http://www.youerw.com/fanyi/lunwen_52473.html