Velocity, acceleration, jerk, snap and vibration: forces in our bodies during a roller coaster ride

Equation (1) can be used to express the speed v at the bottom of the valley in figure 1 in terms of the speed v₀ at the top and the elevation difference ΔH. For the estimates, we neglect energy losses and also contributions due to the finite length of the roller coaster train [8, 9]. Using v₀ > 0, this gives an estimate of the upwards centripetal acceleration a_c at the bottom of the valley in terms of the radius of curvature, R:

$$\begin{equation} a_c = \frac{v^2}{R} > \frac{2\ \mathrm{g} \Delta H}{R} \approx 3\ \mathrm{g} \end{equation} \tag{ 2 }$$

where we have used the photo to obtain an approximation of the ratio ΔH/R ≈ 3/2, as indicated in figure 3.

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The normal force needs to compensate for the force of gravity and provide the force required for the centripetal acceleration. The force from the ride on your body should then be close to N = 4 mg, which is also found in the accelerometer data [3, 6].

An interesting aspect of the forces is that they do not depend on the scale, only the ratio between elevations and radii (as long as we neglect energy losses). Double height would lead to speeds that are higher by a factor of $\sqrt{2}$ and a time exposure to the forces for a period that is longer by the same factor.

Figure 3(b) shows the accelerometer data from the end of the launch, over the top and into the drop and the valley of the Helix roller coaster. The maximum force on the rider is found to be close to the 4 mg predicted from the visual analysis above. The graph also shows small oscillations in the data, in particular after rapid changes—large jerks—e.g. when the train leave the linear launch section of the track and enters the curve to move uphill. This can be attributed to 'ringing', and depends on the resonance frequency of the sensor within the measurement device.

Every roller coaster you have ever ridden upon has a pre-ride safety briefing. It is likely to include instructions to keep your heads back or keep your heads against the headrest. Why is this important? Your head has mass and inertia. If there is a gap between your head and the head-rest during a launch your head will impact the head-rest and give you an unpleasant knock on the back of your head. With extreme rides you may experience a mild traumatic brain injury (m-TBI), unless the head is supported. Mild TBIs were found to have the potential to lead to long-lasting cognitive and motor deficits in mice, increasing the risk of future behavioural, neurological, and other disorders [10]. The risk of experiencing more severe injuries was found to increase with each repeated m-TBI. After a severe bump you may experience a headache, dizziness, nausea, or a combination of these symptoms, or simply a stiff neck. Sudden changes in forces can excite vibrations in a range of frequencies, discussed in the next section.

Our bodies have two direct vibration sensory mechanisms, both of which are applicable to roller coasters. The first mechanism is our balance organs within our ears while the second mechanism is the small mechano-receptors distributed around our body [14]. Vibration may also directly influence the central nervous system by altering levels of arousal and providing surplus information for the brain to deal with. If this is the case then it would be expected that cognitive skills, such as concentration and memory would be affected. Although the effect of vibration on the human body seems to be poorly understood, it has been hypothesised that there are two distinct regimes which can affect our performance. These are 0.1 to 1 Hz in which kinetosis (motion sickness) is important, and 1 to 10 Hz in which resonances of the body dominate. In addition to effects on comfort during car rides [11], research has also investigated the effect on performance, particularly involving motor co-ordination skills and visual performance. The research in the area of human vibration [15–17] has been primarily for the military where the exposure duration is considerably longer than the duration of a typical roller coaster ride. Nevertheless, one could assume with reasonable confidence that staying below exposure limits from these studies would provide safe exposure limits for use by roller coaster designers.

Rapid changes in forces lead to rapid changes in acceleration. However, these rapid changes need to be transmitted between the components of the body, that do not all accelerate together, since our bodies are not rigid. Consider for example a football player hit in the face by a ball. If the player and the ball had been rigid bodies the forces would theoretically be infinite. Slow motion video footage shows the face is not rigid and work ($\mathbf{F}\cdot \mathbf{s}$) is performed on the player and the ball as they both deform. Even with the damping provided by the elasticity of the ball and the soft facial tissue, the force on the player will be high but of a relatively short duration, giving large values for acceleration, jerk and snap. High speed video will also reveal that after separation both the face and the ball will oscillate at their respective natural frequency, albeit highly damped.

Jerk and snap can lead to breakage. Let us illustrate with the example of the traditional 'inertia ball' demonstration, where a massive ball is suspended on a string and an identical string is attached to the bottom of the ball. A slow pull will lead to the top string breaking, whereas a rapid jerk causes the bottom string to break, since the ball would need a much larger force to achieve the acceleration of the bottom string. This concept can be transferred to that of a ligament that connects two bones, or a muscle to a bone. If the ligament is lightly jerked, it pulls whatever bone it is attached to along with it. If a ligament is jerked hard (or one of the segments it is attached to is accelerated quickly), its tensile strength may not be able to handle the change in force and it may tear. Further, consider that repeated jerks are analogous to repetitive micro-trauma; multiple mild or medium jerks may lead to fatigue in the ligament and eventual failure.

Examining the actual jerk, as opposed to previous studies of linear and angular accelerations, may provide a better indicator of repetitive micro-trauma severity.

A very large jerk involves a rapid change in force and acceleration which may be approximated by a step function. It can excite resonances in a broad range of frequencies, and the oscillations can continue long after the application of the force. (Try lifting a slinky and then hold your hand still and observe the 'ringing' as the oscillations continue.) As the train travels along the track it can excite the track and this multi-resonant vibration can be felt by queuing riders as the vibration travels ahead of the train at more than 5 000 m s⁻¹ and exciting the supporting structure and the wall panels lining the queue. Some roller coaster designers use this noise and vibration to good effect as it increases the arousal level of the queuing patrons, whereas in other situations the tracks are filled with sand to reduce the noise level.

Figure 7 shows a sequence of screen shots from a video of the first drop of the dive coaster Valkyria. The valley following the drop is underground and thus invisible, unless you are on the ride. However, an accelerometer taken along on the ride can reveal the forces on the body.

Figures 1(b) and 8(b) show the accelerometer data in the long drops and valleys of the Helix and Valkyria roller coasters (in both cases, only the component orthogonal to the track is shown). We can note that the force on the body at the bottom of the valley is close to 4 mg, for both coasters.

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We can also note that the force during the dive in Valkyria does not quite correspond to a free fall ('zero g'), but has a negative component, {indicating that the coaster seat restraint pushes you toward the track. During the dive, this force is horizontal and does not influence the downward acceleration. The negative value can be attributed to the shape of the track being a more narrow parabola than would correspond to an object leaving the top of the track in pure free fall (figure 8(a)). You may not notice this force as you ride, unless you have been made aware of it. The slight pressure from the restraint system is more likely to be perceived as just providing comfort during a scary drop.

As we compare the forces in the large valleys of Helix and Valkyria, shown in figures 3(b) and 8(b), we can also note that forces from the ride between 3.5 mg and 4 mg are experienced for considerably longer times on Valkyria than on Helix, although both are well below the recommended 6 second limit for 4 mg force, discussed in section 4.

As the drop shown in figure 8 turns into a valley, the forces on the body change rapidly, from slightly negative G's to nearly $4\ \mathrm{g}$. Figure 9 shows the time derivative of the radial G force—the jerk—which reaches about half of the permitted 15 g s⁻¹. It can be noted that this transition feels very smooth when you ride it.

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The jerk, in turn, also grows rapidly. A numerical derivation of the jerk hints to a smooth shape, although the effect of the sensor noise becomes more prominent with every differentiation. Instead we indicate in figure 9 the maximum size of the snap as dashed lines: The derivative of the jerk reaches a magnitude of about 22 g s⁻² in the beginning of the valley and then rapidly drops to −32 g s⁻² as the G force reaches it is maximum and the jerk drops to zero.

Roller coaster tracks involve changes between different radii of curvature. In older roller coasters, individual track segments had different radii of curvature, whereas modern coasters involve smooth transitions, which can be obtained using clothoids and space curves [19].

What shape of the track could lead to the time dependence of the force, shown in figure 8? An extended period of nearly constant G force is possible (see e.g. [20, 21]). However, the transition from a nearly linear vertical track requires a transition curve, such as the clothoid, which is based on a Cornu (Euler) spiral (known from single-slit diffraction in optics). The Cornu spiral has the property that the radius of curvature is inversely proportional to the distance, s, from a starting point (the 'center' of the spiral, which extends both ways from this point) and is discussed in more detail in the appendix.

After the dive into the valley, the track turns into a loop, as seen in figure 10. The top segment is possibly with a circular arc, but following a valley with constant G force may also be a segment constructed to give a constant centripetal acceleration [20]. To compare the track shapes with the accelerometer data, the shapes needs to be combined with a time dependence of the motion and forces. The radial or vertical component of $\mathbf{G}$ can be expressed as

$$\begin{equation} G_z = v^2/r + g \cos \theta \end{equation} \tag{ 3 }$$

where v = ds/dt and $dv/dt = -g \sin \theta$.

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In an attempt to reproduce the accelerometer graph of the data collected on the ride, shown in figure 8(b), we constructed a track shape from a combination of rules together with the resulting G forces. Although the result, shown in figure 11 is less than perfect, the main features of the accelerometer data are reproduced. The real track is shown in figure 10.

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Clothoids are more complex: They are part of Cornu spirals (figure A1), which have the property that the radius of curvature is inversely proportional to the distance, s, from a starting point (the 'center' of the spiral, which extends both ways from that point), i.e. r = 1/ks. This rule can be implemented in a code by expressing the change in angle along the curve as d θ/ds = ks, as discussed in [20]. Figure A1 shows a generic Cornu spiral. No real track includes segments with repeated smaller and smaller radii with larger and larger G forces for every full turn.

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For a clothoid segment of the track, the angular velocity can be expressed as $d\theta/dt = v/r = vks$. The change in radius can be expressed in terms of the speed, distance and the constant k, giving $dr/dt = (dr/ds) (ds/dt) = (-1/ks^2)v$. Using these expressions, equation (A2) can be rewritten as

$$\begin{equation} \frac{dG_z}{dt} = \left( -3vks g\sin\theta - (vks)^2 \frac{(-v)}{ks^2}\right ). \end{equation} \tag{ A3 }$$

giving

$$\begin{equation} \frac{dG_z}{dt} = \left( kv^3 - 3vks g\sin\theta)\right) \end{equation} \tag{ A4 }$$

An important observation is that all these expressions for dG_z /dt can be differentiated - and differentiated repeatedly—without generating discontinuities. This is in contrast to shapes where the radius of curvature changes abruptly (as e.g. in classical Schwarzkopf coasters, where every track segment was bent to a well-defined radius of curvature [19]). However, even with smooth changes in the radius of curvature, abrupt transitions between different mathematical expressions for the shapes of the track segments lead to discontinuous time derivatives of the G force, as can be observed in figures 8 and 11. In real life the changes are damped and the jerk is not discontinuous, but the sudden change in jerk still leads to very large snap—and, of course, large higher derivatives such as the crackle and pop.