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.
Task
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,
b) the transverse force in the crosshead guidance (crosshead bearing) and
c) the torque in the crankshaft.
![Crank drive for a certain angle of rotation](https://pickedshares.com/wp-content/uploads/2021/01/tm1-12-1-1024x458.png)
Solution
The following video is in german language.
Sketch for the solution
![Forces on the crank drive for a certain angle of rotation](https://pickedshares.com/wp-content/uploads/2021/01/tm1-12-2.png)
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.