At a point in time t, the configuration of the mechanism is considered as “frozen” to be a “structure”。 Dynamic force analysis of the running mechanism then becomes the problem of dynamic force analysis for that structure。 This structure is indeterminate in terms of joint force and moment。 Naturally, the deformation compatibility principle is applied to get the supplemental equation。 Deformation compatibility means that the deformation of the members in the structure is subjected to the same geometrical constraint as the one that constrains the structure。
A general deformable model of the link is shown in Fig。 2, where let us focus on the link AB。 At time t, we have the fol- lowing vector equation:
workload (P)。 The redundancy of the constraint is under the
where rA is the vector from O to A , rB
is the vector from O to
is the vector from A to B。
along the y-axis, namely, taking the dot product with uAD on
Then Eq。 (1) can be further written as
rB rA lABu , (2)
where lAB iusuuvthe length of link AB, and u is the unit vector of the vector AB with respect to the coordinate system O-xy that
can be written asu (u ,u , 0)T (cos,sin, 0)T , (3)
where α is the angle between link AB and x axis of coordinate system O-xy。
Derivative of Eq。 (2) with respect to time leads to
both sides of Eqs。 (8) and (9)。 Then the deformation compati- bility equation is obtained by letting them be equal:
[lAB uAB lAB (AB )vAB ] uAD =C uAD 。 (10)来,自,优.尔:论;文*网www.youerw.com +QQ752018766-
+[lCB uCB lCB (CB )vCB ] uAD
All the deformation terms in Eq。 (10) can be represented by the joint force and moment (which are unknown) based on the well-known linear beam theory with the known forces and moments applied on the beam。 Particularly, resultant (internal) forces and moments can be found and expressed by the exter- nal forces and moments plus the joint forces and moments, and then the deformation at any end point can be found and expressed by the joint forces and moments。 Therefore, the
deformation compatibility equation is in fact the constraint
rB rA lABu lAB ( u) , (5)
where v is the angular velocity of the unit vector u and it can be written as (ddt)k , where the k is the unit vector perpen- dicular to the paper plane。
We can write Eq。 (5) into a difference form, namely,
where v ( v k u ) is a unit vector perpendicular to the unit vector u counter-clockwise and in the paper plane。
Therefore, at time t t , B goes to B*, and the position of
B* is represented by
Likewise, the position of A* can also be gained。
Eq。 (6) is more general in that they do have the bending term, that is l()v 。
The deformation compatibility equation can then be created based on the end-effector motion; particularly, the differential motion of the end-effector (both translation and rotation in general) should be the same, calculated from different link connectivity paths。 For the over-constrained mechanism as shown in Fig。 1, the end-effector is point D (particularly, the y-axis translation at point D) and the input is on link 4 via point C (particularly, the x-axis translation at point C)。 There are two link connectivity paths from point A to point D。 Path 1: Link 2 (Point A) Link 4 (Point C) Link 1 (Point B) Link 3 (Point D)。 Path 2: Link 2 (Point A) Link 1 (Point B) 压力机的动力学建模英文文献和中文翻译(3):http://www.youerw.com/fanyi/lunwen_87320.html