3. Mechanical Advantage

Derivation

The mechanical advantage of a system is defined by:

$MA = \begin{vmatrix} \frac{\omega_{in}}{\omega_{out}} \end{vmatrix} \begin{vmatrix} \frac{r_{in}}{r_{out}} \end{vmatrix}$

(Eq. 11)

For my case, the input radius and angular velocity comes from the crank itself, not link 2. The output angular velocity can either be ω₄ or ω₅. The output radius is the same for link 4 and link 5.

$MA = \begin{vmatrix} \frac{\omega_{crank}}{\omega_{out}} \end{vmatrix} \begin{vmatrix} \frac{r_{crank}}{r_{out}} \end{vmatrix}$

(Eq. 12)

I have solved for the angular velocity ratio for | ω₂ / ω_out | in the previous section. We can thereby solve for the equivalent angular velocity ratio for | ω_crank / ω_out |:

$\begin{vmatrix} \frac{\omega_{crank}}{\omega_{out}} \end{vmatrix} = \begin{vmatrix} \frac{\omega_{crank}}{\omega_{2}} \end{vmatrix} \begin{vmatrix} \frac{\omega_{2}}{\omega_{out}} \end{vmatrix}$

(Eq. 13)

$\begin{vmatrix} \frac{\omega_{crank}}{\omega_{2}} \end{vmatrix} = \frac{N_{2}}{N_{crank}}$

(Eq. 14)

Substituting equation 13 and equation 14 to equation 12 gives:

$MA = \begin{vmatrix} \frac{N_{2}}{N_{crank}} \end{vmatrix} \begin{vmatrix} \frac{\omega_{2}}{\omega_{out}} \end{vmatrix} \begin{vmatrix} \frac{r_{crank}}{r_{out}} \end{vmatrix}$

(Eq. 15)

Finally the total mechanical advantage is the summation of both end effector forces:

$MA_{total} = MA_{proximal} + MA_{distal}$

(Eq. 16)

Mechanical Advantage Plot

I input the following parameters from table 2:

Parameter	Value
r_in	1.25
r_out	1.875
N₂	40
N_crank	12

Robot Mechanism Design

3. Mechanical Advantage

Derivation

Mechanical Advantage Plot

Table 2, mechanical advantage parameters

Figure 8, mechanical advantage plots