3。3 Deformation compatibility equation
Based on the material mechanics, the deformation of com- ponent can be expressed with constraint force, material prop- erties and structural attributes。 So the relation between con- straint force of components and position error of end-effector (main slider 8) can be established。 Because the number of position errors of end-effector is independent of whether the over-constrained is introduced in mechanism, and the number of dynamic equations obtained is equal with that of position errors of the end-effector。
During the run time of the mechanism, the position of pivot points O51, O61, O52 and O62 of statically indeterminate sub- mechanism is determined by kinematic constraint of the origi- nal mechanism。 Therefore, the constraint forces F51, F52, F53 and F54 are considered as the input driver of the system。 The link produces micro-deformation, in terms of the assumption in Sec。 3。1 (the main slider 8 is assumed as a rigid body), the slider produces only displacement and deflection angle with- out deformation under the state of the workload。
The base coordinate system O-xy which is attached to the frame is established; the x axis is horizontal, the y axis is verti- cal, and they are all through the pivot joint O0 on the basis of the characteristics of the mechanism。 A moving coordinate system T xyis fixed on the center of mass of the main slider 8, which is parallel to the coordinate system O-xy。 The posi- tion vector diagram of a link is shown in Fig。 6。
The rotation matrix of coordinate system T xywith re- spect to the base coordinate system O-xy can be expressed as
RO I 。 ` (12)
In terms of Sec。 2 and Fig。 6, the closed-loop constraint equation associated with the ith kinematic chain (connectivity path) can be written as
r RT Ai aie1 bie2 Liui , (13)
where r is the position vector of the origin T with respect to
the coordinate system O-xy, which is defined as r (x, y, 0)T ,
Ai is the position vector of the pivot joint Ai with respect to the
coordinate system T xywhich is defined as A (x , y , 0)T ,
ai is the x coordinate of Bj with respect to the coordinate sys- tem O-xy, bi is the y coordinate of Bj with respect to the coor- dinate system O-xy, L is the length of link AB , and e = (1,0)T, e = (0,1)T。
Taking the derivative of Eq。 (13) with respect to time, the equations can be written as
Fig。 7。 Force analysis of under link。
According to the material mechanics, the axial deforma- tion li of link li can be express as
where Fi is axial force acting on link, li is the effective length of link, Ei is elastic modulus, and Si is section area of link。
In terms of Eq。 (19), the axial deformation li of the link can
The link produces elastic deformation at runtime under the action of the workload, which will make the end-effector (main slider 8) produce the position errorsx andy in the
be derived by unknown force Fi。 Then, the relation between position error of main slider 8 and force Fi can be established。
Thus, the axial deformation li of the kinematic chain can be expressed as
direction of the x axis and the y axis and the deflection angle
error, which are vertical to the moving plane。 Then the Eq。 (14) can be changed towherer (x,y)T 。
Taking the dot product with ui on both sides of Eq。 (15) leads to 压力机的动力学建模英文文献和中文翻译(6):http://www.youerw.com/fanyi/lunwen_87320.html