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Determination of the reaction forces

Task

A railway wagon stands on a sloping track on a (frictionless) buffer stop. Which reaction forces occur on the wheels and on the buffer stop?

Railway wagon on a sloping track at the buffer stop
Railway wagon on a sloping track at the buffer stop

Solution

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Reaktionskräfte ermitteln - Technische Mechanik 1, Übung 17

Solution sketch

Reaction forces on the track and buffer stop
Reaction forces on the track and buffer stop

Solution approach

\[ \require{cancel} \] \[ \newcommand{\myvec}[1]{{\begin{pmatrix}#1\end{pmatrix}}} \]

The coordinate system is rotated by the angle α to solve the problem. Left turning moments are positive.

The equilibrium of forces in the x-direction is

\[\tag{1} \sum F_x = 0 = F_3 -- G \cdot \sin \alpha \]

\[\tag{2} F_3 = G \cdot \sin \alpha \]

The equilibrium of forces in the y-direction is

\[\tag{3} \sum F_y = 0 = F_1 -- G \cdot \cos \alpha + F_2 \]

Due to the geometric specifications, the equilibrium of moments can only be considered around G and leads to

\[\tag{4} \sum M(G) = 0 = F_3 \cdot h -- F_1 \cdot l + F_2 \cdot l \]

\[\tag{5} 0 = G \cdot \sin \alpha \cdot h -- F_1 \cdot l + F_2 \cdot l \]

\[\tag{6} F_2 = \frac{F_1 \cdot l -- G \cdot \sin \alpha \cdot h}{l} \]

From equation 3 follows

\[\tag{7} 0 = F_1 -- G \cdot \cos \alpha + \frac{F_1 \cdot l -- G \cdot \sin \alpha \cdot h}{l} \]

\[\tag{8} G \cdot \cos \alpha = F_1 + \frac{F_1 \ \cdot \bcancel{l}}{\bcancel{l}} -- \frac{G \cdot \sin \alpha \cdot h}{l} \]

\[\tag{9} F_1 = \frac{G}{2} \cdot \left( \cos \alpha + \frac{\sin \alpha \cdot h}{l} \right) \]

F2 follows from equation 6

\[\tag{10} F_2 = \frac{\frac{G}{2} \cdot \left( \cos \alpha + \frac{\sin \alpha \cdot h}{l} \right) \cdot l -- G \cdot \sin \alpha \cdot h}{l} \]

\[\tag{11} F_2 = \frac{ \frac{G}{2} \cdot \cos \alpha \cdot l + \frac{G}{2} \cdot \frac{\sin \alpha \cdot h}{\bcancel{l}} \cdot \bcancel{l} -- G \cdot \sin \alpha \cdot h}{l} \]

\[\tag{12} F_2 = \frac{ G \cdot \cos \alpha \cdot l -- G \cdot \sin \alpha \cdot h}{2 \cdot l} \]