Roller coaster dynamics -- from point particles to a continuum model using Lagrange density

Abstract

Analyzing the motion of a roller coaster allows for an instructive introduction of various theoretical concepts in a concrete and enjoyable context. We start by modeling the roller coaster train as a point particle. We then develop more realistic models for the train and finally we show how to introduce a continuum limit in a simple way. These studies instructively illustrate the relationships between different formalisms (Newtonian mechanics, Lagrangian mechanics of the first and second kind, as well as continuous Lagrangian mechanics using a Lagrangian density). We derive the equations of motion in all considered models and calculate the forces acting on the track and on the passengers. Numerical results are also provided and discussed.

Figures (9)

• Figure 1: (a) Energy distribution and (b) normal force on a roller coaster train modeled by a point mass on a Gaussian track. The dotted green line denotes the track shape. Parameters chosen in this example are a height of $H = 30m$ and a half width at half maximum of $10m$, i.e., $a = \sqrt{\ln 2}/10m$. The initial conditions are $x_0 = -4/a$, $z_0 = h(x_0)$, and $\dot{x}_0 = 1.018 \times \dot{x}_{0,\mathrm{min}}$. Since the initial velocity is only slightly above the minimum value, the velocity at the peak of the Gaussian almost vanishes.
• Figure 2: (a) The profile of a step-like track. (b) The modulus of the normal force, track height and tangent angle for the step-like track. The two kinks in the modulus of the normal force result from a change of its direction. The normal vector is defined to point always upward away from the track, which is identical to the direction of the normal force at most points on the track. Between the two kinks, the normal force points downward toward the track. The initial velocity is $9.0\,\mathrm{m/s}$.
• Figure 3: Roller coaster train -- "hilltop phase" at three successive times. The thick dark blue arrow marks the position at which the passengers in the last wagon feel an airtime, i.e., they are pressed to the track by the security bar in the wagon and would lose contact with the seat without it. The lengths of the light blue arrows (denoted by $v$) indicate the velocity of the wagon at the hilltop. Arrows at the outer wagons indicate the direction of the tangential component of the gravitational force on these wagons. Green arrows with label f represent tangential components in forward direction, whereas red arrows with label b show tangential forces in backward direction.
• Figure 4: Roller coaster train -- "valley phase". The thick dark blue arrow marks the position at which the passengers of the middle wagon feel the largest normal force. The lengths of the light blue arrows indicate the velocity of the wagon in the valley. Arrows at the outer wagons have the same meaning as in Fig. \ref{['fig:train_hill']}.
• Figure 5: Distance $d$ between the wagons 1 and 2 as well as 10 and 11 as a function of time. Oscillations initiated by the springs are clearly visible. Each wagons has a mass of $m = 500kg$. Their equilibrium distance is $s_0 = 1m$, and the spring constant is $k=100kN \per m$. The track has the Gaussian shape \ref{['eq:gaussian']} with a height of $H = 20m$ and a width parameter of $a = \sqrt{\ln 2}/10m$.