Because φ1,1 and φ2,1 are assumed to be the same, Eqs。 (6a) and (6b) are the same。 This implies that the similarity ratios n1 and n2 are the same。
ð2Þ i ¼ k
If φ1,k =φ2,k, then Δ O p1,k p1,k+1 and Δ O p2,k p2,k+1 are similar。 The similarity ratio is nkrk ¼ nk。
Fig。 6。 Diagram for the proof of Proposition 1。
The tangent values of angles φ1,k and φ2,k are described as follows:
Because φ1,k and φ2,k are assumed to be the same, Eqs。 (7a) and (7b) are the same。 This implies that the similarity ratios nk
and nk+1 are the same。
By mathematical induction, the similarity ratio of step k (nk) is the same as that of step 1 (n1)。 This means that if two plane figures have the same first-order derivative in all discrete segments, they are mathematically similar。 This can also be applied to a continuous system by decreasing the interval angle ξk。
3。2。 Geometrical meaning of second-order derivative (change in angle of slope) of coupler point
In the previous section, the sign of the second-order derivative was also used to distinguish the coupler points。 Thus, we pro- pose the following geometrical interpretation of the derivative:
Proposition 2。 If two plane figures have the same first and second-order derivative profiles, they are the same figure。
Proof Of Proposition 2。 The second-order derivative value can be obtained as follows:
If the second-order derivative values of each plane figure are the same, Eq。 (8a) is equal to Eq。 (8b)。
2 1 ∂2 C
9Þ
∂Cx 2 big ¼ nk ∂Cx 2 small ð
Eq。 (9) shows that the similarity ratio nk has to be one。 In other words, the two plane figures are exactly the same。
3。3。 New approach for four-bar linkage design
To obtain an optimal trajectory numerically without the possibility of generating an unintended trajectory shape, the following two-step optimization problem was developed:
First step of optimization:
min RMSð1st deriv:G — 1st deriv:D Þ
Xlink ∈R5
Subject to: Desired shape (distinguished by geometrical features)。
T1 , T2 , T3 N 0 (Grashof condition)。
Second step of optimization:
min RMS 2nd deriv:G 2nd deriv:D
Xsize ∈R
Subject to: Xlink ¼ Xlinkm:
where:
RMS(x): Root mean square value of x
n th derive。G: n th-order derivative equation of generated trajectory
n th derive。D: n th-order derivative equation of desired trajectory
Xlink ¼ 。 l2 l3 l4 lcx lcy 。=l1
Xlinkm: Optimal value of first step of optimization
Xsize ¼ l1
Because of Proposition 1, the shape of the trajectory is determined in the first step, in which the RMS value of the first-order derivative error between the generated trajectory and target is minimized。 After that, due to Proposition 2, the size of the trajec- tory is determined by the second step of optimization, in which the RMS value of the second-order derivative error between the generated trajectory and target is minimized。 The reason for piding the optimization process into two steps is that the design variables can be separated into two groups: shape-dependent and size dependent variables。 The shape-dependent variable is Xlink, which contains the rations of the link lengths to l1。 The only size-dependent variable is l1, which is the standard of Xlink。 曲柄摇杆机构英文文献和中文翻译(6):http://www.youerw.com/fanyi/lunwen_82614.html