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.chuibin.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  lABu  lAB ( u) , (5)

where v is the angular velocity of the unit vector u and it can be written as (ddt)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)