The notation for dynamic analysis is illustrated in Fig。 4: Fpi and Mpi are driving force and moment acted on link i, respec- tively。 Fi and Mi are resultant external force and moment acted on the centroid of link i, respectively。 Fji is the joint force act- ed on the joint point Oi of link i and link j, particularly the link j imposing to link i。

link l1。 Therefore, one deformation compatibility equation (supplementary equation) is needed to solve the dynamic model。

3。2 The resolution of mechanism

To the problem of over-constraint, a method which removes the redundant constraint in statically determinate mechanism must be found。 Some scholars, based on the equivalent princi- ple of kinematic characteristics, have simplified it into a non- over-constrained mechanism, in which one link of the paral- lelogram mechanism is removed or two linkages are merged into one, so the over-constrained problem is solved。 But the dynamic characteristics of simplified mechanism and that of original mechanism might not be identical。 The reliability of analysis result awaits proof [13-15]。

Aiming at the problem, in this paper, several passive joints are virtually cut to form a statically determinate sub- mechanism and a statically indeterminate sub-mechanism of original mechanism on the basis of the specific characteristics of the mechanism。 In the sub-mechanism, the joints are re- placed with negative constraint force。 The disassembly princi- ple of the mechanism is that the static characteristics are equivalent between sub-mechanism and its corresponding part in original mechanism。 This process corresponds to removing the over-constraint in the system。 From microcosmic point of view, the geometric relation that the deformation of compo- nent   should   keep   consonance   of   all   the   components is

Fig。 6。 The position vector diagram of a link。

formation of the sub-mechanism is analyzed to find a supple- mentary equation。

Fig。 5。 The resolution of mechanism: (a) The statically determinate sub-mechanism; (b) the statically indeterminate sub-mechanism。

searched。 The deformation compatibility equation can then be established。 Finally, the statically indeterminate problem (over-constraint) is solved combining the  dynamic equations by D’Alembert principle。

The 16 moving elements are pided into two independent sub-mechanisms to obtain a dynamic model that can be solved analytically。 In the process of resolution of the mechanism, one is a statically determinate sub-mechanism and the other is a statically indeterminate sub-mechanism that must be satis- fied。 The result of resolution is shown in Fig。 5。

The constraint forces F51, F52, F53 and F54 of original mech- anism are regarded as known external loads, and the sub- mechanism is composed of frame and 11 moving elements but excluding under link l5i and main slider 8, as shown in Fig。 5(a)。 The DOF (Degree of freedom) of the sub-mechanism is 1 (15 revolute joints and 1 moving pair), then the determinate base is obtained。 The 33 dynamic equilibrium equations can be listed out by D’Alembert principle。 Then the relation between the constraint force of the kinematic pair, driving moment and F51, F52, F53 and F54 of determinate base can be found。

The constraint forces F51, F52, F53 and F54 of the original mechanism are regarded as input driver, and the sub- mechanism is composed of five moving elements, that is un- der link l5i and main slider 8, as shown in Fig。 5(b)。 The DOF of sub-mechanism is -1 (8 revolute joints), then the statically indeterminate sub-mechanism is obtained。 The 15 dynamic equilibrium equations can be listed out by D’Alembert princi- ple, but the number of unknown constraint forces is 16, and the number of statically indeterminateness is 1。 Then the    de-