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# Forces on a crank drive

This exercise is about a statics investigation of the forces on a crank drive with a crosshead guidance for a specified rotation angle. The kinematics of a crank drive is considered here.

The force F acts on the piston of a steam engine. The following parameters have to be calculated:

a) the force in the driving rod,

c) the torque in the crankshaft.

## Solution

The following video is in german language.

### Sketch for the solution Forces on the crank drive for a certain angle of rotation $\require{cancel}$ $\newcommand{\myvec}{{\begin{pmatrix}#1\end{pmatrix}}}$

The angle α between the force F and the driving rod results from

$\tag{1} \alpha = arctan \left( \frac{r}{l} \right)$

Between the force F and the force FS in the driving rod exists the following relationship

$\tag{2} F = F_S \cdot \cos \alpha$

$\tag{3} F_S = \frac{F}{\cos \alpha}$

The cosine of the arctangent α leads to

$\tag{4} \cos \left( arctan \left( \frac{r}{l} \right) \right) = \frac{1}{\sqrt{\left( \frac{r}{l}\right)^2 + 1}}$

So this is the force in the driving rod:

$\tag{5} F_S = F \cdot \sqrt{\left( \frac{r}{l}\right)^2 + 1}$

The transversal force in the crosshead guiding results from

$\tag{6} F_Q = F_S \cdot \sin \alpha$

The sine of the arctangent α leads to

$\tag{7} \sin \left( arctan \left( \frac{r}{l} \right) \right) = \frac{\frac{r}{l}}{\sqrt{\left( \frac{r}{l}\right)^2 + 1}}$

So the transversal force is

$\tag{8} F_Q = F \cdot \bcancel{\sqrt{\left( \frac{r}{l}\right)^2 + 1}} \cdot \frac{\frac{r}{l}}{\bcancel{\sqrt{\left( \frac{r}{l}\right)^2 + 1}}}$

$\tag{9} F_Q = F \cdot \frac{r}{l}$

The torque at the crankshaft is

$\tag{10} M_K = F_S \cdot r$

$\tag{11} M_K = F \cdot r \cdot \sqrt{\left( \frac{r}{l}\right)^2 + 1}$

This exercise is part of the collection Engineering Mechanics 1 - Statics.