Link 3 (Point D)。 The equations for the output motion with paths 1 and 2 are, respectively,

uuuv        uuuv       uuv    uuuv

equation for the joint force and moment, and it is thus sup- plemental to the equilibrium equation established based on the D’Alembert principle。

For the over constrained mechanism in Fig。 1, the number of force/moment equilibrium equations is 12, but the number of unknown variables (joint forces and moments plus the driv- ing force) is 13。 According to the aforementioned procedure to come up with the deformation compatibility equation, there is one deformation compatibility equation as the output motion of the end-effector has only one (i。e。, the translation along the y-axis)。 As a result, the total number of equations is 13, which is equal to the number of unknown variables, and thus all the joint forces and moments plus the driving force (F in this case) can be found。

3。 Case study

A case study is presented to validate the proposed approach。 This includes the model development, resolution and experi- ment。 The system used for this purpose is a new press ma- chine (Fig。 3)。 This machine is  used  for heavy load (about 600 kN)。   The  size  of  the  press  machine   is  designed     as

2。1 m×1。7 m×3。3 m in length, width and height。

The kinematic diagram of the press machine in Fig。 3(a) is shown in Fig。 3(b)。 The press machine consists of the frame, crank-slide mechanism, tie rods, symmetric elbow bar mecha- nisms, connectors and main slider。

In this mechanism, the upper and lower elbow-bar and con- nector make up a two parallelogram mechanism。 The stiffness of mechanism and determinacy of motion is increased。 The drive force is magnified by elbow-bar mechanism, while most of the vertical loading is absorbed by toggle mechanisms at work。

The  crank is uniform  circular motion  during  the  runtime。

Then the connecting rod l2 allows the guide rail to be traced out by the secondary slider 7 in the vertical direction。 The vertical motion of the secondary slider makes tie rods l31    and

The projection of the above two questions along the y-axis leads to two equations that describe the deformation at point D

l32 bob up and down, so that elbow-bar mechanisms move on sideways。  Finally,  it  achieves  the  reciprocating  motion   of

Fig。 4。 A general planar mechanism。

There are three equilibrium equations to each link based on the D’Alembert principle。 For link i, they are [12]

According to the above theory of dynamic modeling, for the machine of Fig。 4, there are 48 equations (for brevity, details of these equations are not included in this paper) and 49 un- known joint forces and moments plus one driving moment  on

Fig。 3。 The transmission mechanism of YHK-60 type high-speed press machine: (a) The machine transmission mechanism configuration; (b) the machine transmission mechanism schematics。

l1: Crank, l2: Connecting rod, 7: Second slider, O0O1O2 form a crank- slide mechanism, l31and l32 are symmetric tie rod, l41、l42、l43 and l44 are symmetric upper link, l61 and l62 are symmetric connector, l51、 l52、l53  and l54  are symmetric lower link, link l4i、l5i  and l6i  form paral-

lelogram mechanism, 8: Main slider

main slider 8 in the vertical direction。

3。1 Model development

The deformation of frame and transmission mechanism is unavoidable under the action of a force。 There are several assumptions which main factors are considered before the analysis of the press machine [11]。 The first assumption is that the slider, frame, and guider are rigid, while all links are flexi- ble。 The second assumption is that the motor runs at a constant speed。 The third assumption is that the clearance in joint and temperature induced stress is ignored。 The fourth assumption is that the materials of all the links are isotropic and elastic。