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4.1. The dynamic fundamental characteristicparameters of joint surfaces at unit areaThe dynamic characteristics of joints in a machine-tool structure are determined by many factors, such assize of a joint, excitation frequencies, the displacementamplitude of joints, distributed pressure on the joint sur-face, the types of joints, material of joints, lubricativestate on the joint surfaces, machining method of the jointsurface, etc. In this paper, these factors are categorisedinto two types. The first type, including the size of ajoint, excitation frequency, distributed pressure on thejoint surface, the displacement amplitude of joints andthe types of joints etc., is taken into account through thedynamic analysis of joints; and the other type is takeninto account as the dynamic fundamental characteristicparameters of joint surfaces at unit area obtained throughexperimental method.The dynamic fundamental characteristic parameters of joint surfaces at unit area, namely the stiffness anddamping at unit joint area, are obtained by experimentsas follows [4,10],Normal dynamic stiffness knd an·pbnn ·wgn·XhnnTangential dynamic stiffness ktd at·pbtn ·wgt·XhttNormal damping cn anc·pbncn ·wgnc·XhncnTangential damping ct atc·pbtcn ·wgtc·Xhtct(10)where a,b,g,h are the dynamic fundamental character-istic coefficients determined by the second type of fac-tors such as machining method of joint surfaces, lubri-cative state on the joint surface, material of joints, etc.;pn is the normal pressure at unit joint area; Xn,Xt are thedisplacement amplitudes of joints in normal and tangen-tial directions on the joint surface respectively.4.2. Analysis of dynamic characteristics of a guidewayjointGuideway joints are very common in a machine-toolstructure. This section mainly investigates the analysisof dynamic characteristics of a guideway joint. A guide-way joint in a machine-tool structure is usually com-posed of several plane joint interfaces. Usually the press-ure on the unit joint area is not high. Therefore theanalysis of dynamic characteristics of a joint developedin this paper is based on an assumption that plane jointinterfaces remain plane during deformation of the joint.Fig. 2 shows a guideway joint with several plane jointinterfaces depicted by solid bold lines. O is the globalcoordinate system of the guideway joint. Oiis the localcoordinate system of the ith plane joint interface of theguideway joint. The amplitude vector of displacementsof the guideway joint is expressed as,{XJ} {XJ1 XJ2 % XJ6}T(11)where XJ1,XJ2,XJ3 are the three amplitudes of trans-lational displacement; XJ4,XJ5,XJ6 are the three ampli-tudes of rotational displacement.The amplitude vector of displacements of the ith jointsurface in the local coordinate system Oiis expressed as,{XiJ} {XiJ1 XiJ2 % XiJ6}T(12)Based on the above assumption, we can obtain thefollowing equation,{XiJ} [Tiw] 1·{XJ} (13)where [Tiw] is the coordinate transform matrix; i =1,2,...,N; N is the total number of plane joint interfacesin the guideway joint.Fig. 3 shows a plane joint interface in the local coordi-nate system Oi. Based on the above assumption, the dis-placement amplitudes at point B(x1,x2,x3) on the ith jointinterface are expressed as follows, Xi1 XiJ1 XiJ5·x3 XiJ6·x2Xi2 XiJ2 XiJ4·x3 XiJ6·x1Xi3 XiJ3 XiJ4·x2 XiJ5·x1(14)From eq. (10), the dynamic stiffness and damping perunit joint area at point B(x1,x2,x3) on the joint surfaceare as follows,Dynamic stiffnesskn an·pbnn ·wgn·|X2|hnkt at·pbtn ·wgt·|X1|htk3 at·pbtn ·wgt·|X3|ht (15)Dampingcn anc·pbncn ·wgnc·|X2|hncct atc·pbtcn ·wgtc·|X1|htcc3 atc·pbtcn ·wgtc·|X3|htc (16)The unit forces at point B(x1,x2,x3) on the joint surfaceare as follows, Fn (kn iw·cn)·Xi2Ft (kt iw·ct)·Xi1F3 (k3 iw·c3)·Xi3(17)The forces on the ith joint surface are obtained byintegration of the unit forces over the area of the contactsurface as follows,FiJ1 sFtdsFiJ2 sFndsFiJ3 sF3dsFiJ4 s( Fn·x3 F3·x2)dsFiJ5 s(Ft·x3 F3·x1)dsFiJ6 s(Fn·x1 Ft·x2)ds(18)According to the definition of dynamic stiffness, thecomplex stiffness equation for the ith joint surface canbe expressed as{FiJ} [KiJ]·{XiJ} (19)The total force on all of joint surfaces of the guidewayis obtained as,{FJ} ( Ni 1[Tiw]·[KiJ]·[Tiw] 1)·{XJ} (20)The complex stiffness matrix of the guideway joint isobtained from eq. (20) as,[KCJ] ni 1[Tiw]·[KiJ]·[Tiw] 1(21)where KCJ = KJ + iwCJ.From the above analysis, it can be seen that the com-plex stiffness matrix of the guideway joint contains allthe factors that affect the dynamic characteristics of thejoint, and it has an evident physical meaning. Thereforethe proposed analysis is reasonable.The dynamic fundamental characteristic parameters ofjoint surfaces at unit area have nothing to do with thesize and type of joints. These parameters of joint sur-faces are common for the dynamic analysis of joints,when joints are variable in a machine-tool structure.