Question

A four – wheeled trolley car of total 2000 kg running on rails of 1.6 m gauge, rounds a curve of 30 m radius at 54 km/hr. The track is banked at 8”. The wheels have an external diameter of 0.7 m and each pair with axle has a mass of 200 kg. the radius if gyration for each pair is 0.3 m. the height of center of gravity of the car above the wheel base is 1 m. Determine allowing for centrifugal force and gyroscope couple actions. The pressure on each rail. 

Answer 1

Given m = 2000 kg; x = 1.6 m; R = 30 m ; v = 54 km/h = 15 m/s ; θ = 8° d_W = 0.7 m or r_w = 0.35 m; m₁ = 20 kg; k = 0.3 m; h = 1 m

First of all, let us find the reactions R_A and R_B  at the wheel A and B respectively. The various forces acting on the trolley car are shown in fig.

Resolving the force perpendicular to the track,

Now taking moments about B,

We know that angular velocity of wheels,

and angular velocity of precession,

Gyroscopic couple,

Due to this gyroscopic couple , the car will tend to overturn about the outer wheels. Let P be the force at each pair of wheels or rail due to the gyroscopic couple.

 P = C/x = 382/1.6 = 238.75 N

we known that pressure (or total reaction) on the inner rail,

P₁ = R_A - P = 3182 - 238.75 = 2943.25 N

and pressure on the outer rail,

P_o = R_B =+ P = 18336 + 238.75 = 18574.75 N