The classification in this paper has two advantages compared to exist classification [19]。 First, because four categories are easy to distinguish from each other, a designer can easily choose a category that involves the shape of desired trajectory by visual in- spection。 Second, the geometrical properties used for the classification are simple to calculate。 This means that the classification process in computer simulation is so fast that if this classification is used for the design of four-bar linkage, whole design time can be reduced。
3。 New approach for four-bar linkage design
The first and second-order derivatives of the coupler points include the geometrical features used to distinguish the coupler curves, which we prove in this section。 Furthermore, the new methodology based on these features is presented。
Table 1
Geometrical properties of each shape type。
Type I II III IV
Num。 of Infinite points 2 2 2 2
Zero-slope point ratio 1:1 1:1 or 1:3 1:3 2:2
Sign of second derivative at zero slope point Upper - - or + + +/-
Lower + + + -/+
Radius of curvature Finite Infinite Finite Finite
Table 2Classification results with methods 1 & 2。
First method (visual inspection) Second method (using geometrical features)
Type I 4081 (0。5922) 6757 (0。6647)
Type II 742 (0。1077) 600 (0。0590)
Type III 1485 (0。2155) 2001 (0。1969)
Type IV 574 (0。0836) 804 (0。0791)
Exception 7 (0。0010) 3 (0。0003)
Total 6889 (1。0000) 10,165 (1。0000)
3。1。 Geometrical meaning of first-order derivative (slope) of coupler point
The slope is frequently used to distinguish coupler curves。 Based on this, we propose the following geometrical interpretation of the first-order derivative of the coupler point:
Proposition 1。 If two plane figures have the same first-order derivative profile, they can be called mathematically similar。
Proof of Proposition 1。 In the discrete system and based on Fig。 6,
If φ1,1 =φ2,1, then Δ O p1,1 p1,2 and Δ O p2,1 p2,2 are similar。 The similarity ratio is n1 r1 ¼ n1。
The tangent values of angles φ1,1 and φ2,1 are described as follows: 曲柄摇杆机构英文文献和中文翻译(5):http://www.youerw.com/fanyi/lunwen_82614.html