updated on June, 2020

## Introduction

All over the world, rollercoasters offer to users an experience of pure adrenaline and extremely high accelerations, probably experienced elsewhere only by astronauts and Formula 1 pilots.

Although they are meant for joy and entertainment, the design is very crucial and regulated. Understanding the interaction between components and humans can help create a more thrilling and safer ride.

In this personal project I tried to find and carry out some fundamentals of the modelling and dynamic analsis of a roller coaster. A unique simpliefied roller coaster was designed, simulating a ride on a particular track and investigating the its velocities and accelerations and the forces on the wheels. This work does not pretend to be exhaustive about the design of a roller coaster, but provides a start point for more detailed analysis and some results that can be used to deal with a structural analysis of the wheel carriers and the chassis of the cart.

## Objectives

From an engineer’s perspective, roller coasters are the attemp to push a ground vehicle to its speed limit and, for this reason, require several integrated dynamic and structural analysis in order to ensure both amusement and safety of passengers.

The dynamics of a rail vehicle moving on a three-dimensional track, such as a roller coaster, is very difficult to handle and, often, only a multibody simulation can predict the temporal evolution of such a complex system.

The main goal of this work was to model a three-dimensional typical path that combine inclines, loops and curves with some banking, to model the cart with its suspensions and kinematic constraints between wheels and rail and to measure velocities, accelerations and forces along the path.

A first dynamic simulation has been performed without any type of loss of energy due to friction, rolling or resistance or aerodynamic drag, so that the simulation results could be compared with those of analytical calculations and the model could be validated.

A second dynamic simulation, instead, has taken into account rolling resistance and drag force too, to give more realistic results.

## The modelling problem

### Carts and Wheelsets

The carts and trains of a roller coaster can vary based on the type and theming. Typically the body of the cars of a roller coaster are made of fiberglass materials. This material is easy to mold, lightweight and durable. The profile of the body will contribute to the effects of drag on the cart’s movement throughout the course. Within the fiberglass shell, the car usually contains the seating and restraints for passengers.

Another important feature of the train is its wheels. The wheels are the main interaction and connection between the train and the track. When choosing the wheels for a ride various aspects must be considered, such as low rolling resistance, high load endurance, smooth ride performance, and high durability. Nowadays, the wheels are manufactured of polymers such as polyurethane to allow for a smoother ride and less ware on the track.

Modern roller coasters typically have three types of wheels: road wheels, side friction wheels, and up-stop wheels (Figure 3.2). Road wheels are the wheels that ride along the top of the track, the wheels usually considered when imagining a moving object. Next, the side friction wheels are perpendicular to the road wheels. They travel along the side of the rails allowing the train to maneuver through turns without sliding off the track. Lastly, trains have up-stop wheels located below the track. These prevent the train from rising off the track and derailing [3].

### The multibody model

The multibody model has been entirely built up within MSC Adams View.

The track has been modelled in Adams defining some reference points in space and then connecting them with lines, circular arcs and splines. The track tries to reproduce a part of the path presented in [4].

As you can see in the Figure 4.1, the track starts with an horizontal part (A-B), then continues with the initial drop (B-C), which provides at its end the velocity needed to overcome the loop (D-E-F). The loop is built with a spline, so it isn’t perfectly circular: its average radius is about 30 m. After this, a straight path characterized by a torsion of the rail (D-G), the roller coaster enters in a curve with constant 45° banking (G-H-I). Finally, in I-J the banking gradually returns to zero.

For both right and left side of the rail, the various arcs have been connected with the boolean “chain tool” of Adams, in order to delete the discontinuities between them.

The coordinates in the table below are referred to the right side rail. The left side rail is defined accordingly, with a distance between the two sides of 1 meter.

x (m) | y (m) | z (m) | |

A | -2 | 0 | 0 |

B | 50 | 0 | 0 |

C | 147 | 0 | -80 |

D | 190 | 0 | -80 |

E | 190 | 1.5 | -20 |

F | 190 | 3 | -80 |

G | 250 | 3 | -80 |

H | 350 | 103 | -80 |

I | 250 | 203 | -80 |

J | 180 | 203 | -80 |

For our purpose, the roller coaster has been simpliefied with a single cart consisting of two rigid bodies, Front Body and Rear Body, connected by a revolute joint oriented in the direction of travel, that leaves free the relative rolling between them. A torsional spring-damper (K_{j} = 16.9 Nm/°, C_{j} = 4.2 Nm*s/° )has been applied on the revolute joint in order to realign them after every relative rotation.

The wheels don’t turn around their own axis, so that the contact point of each wheel remains the same throughout the simulation. This wheels rotation locking prevents taking gyroscopic effects into account, but since the wheels have a small radius (0.1 m), these effects have little influence on the dynamics of the roller coaster. The wheel have been connected to the carbodies through four translational joints, that leave free only their lateral displacements.

The wheelbase is 1 meter.

The mass and inertia properties realated to the CM marker assigned to each body are reported in the table below:

ID | Rigid bodies | Mass (kg) | Ixx (kg*m^{2}) |
Iyy (kg*m^{2}) |
Izz (kg*m^{2}) |

1 | Carbody Front | 180 | 30 | 75 | 75 |

2 | Carbody Rear | 110 | 20 | 48 | 48 |

3 | Front DX wheel | 20 | 0.1 | 0.05 | 0.05 |

4 | Front SX wheel | 20 | 0.1 | 0.05 | 0.05 |

5 | Rear DX wheel | 20 | 0.1 | 0.05 | 0.05 |

6 | Rear SX wheel | 20 | 0.1 | 0.05 | 0.05 |

The mass of each wheel refers to the mass of a whole wheelset, composed by 5/6 wheels and the boogie (carrier), tipically present in real roller coasters and here simplified with a single wheel.

As mentioned above, five joints have been used to connect the bodies of the cart.

In addition to these ones, four point-curve constraints have been defined to constrain the rollercoaster to move along the track. These are kinematic contacts, so they do not take into account the rolling of the wheels, the variability of the contact points and the deformability of the rail and the wheel during the contact, so they only provide a first attempt modelling, which can be used for a preliminary sizing of the wheel bogies and the track supports.

A single point-curve constraint (PTCV) defines a connection between a marker attached to the moving body and a floating marker on the curve, so it leaves 4 DOF: the translation along the spatial curve and the three rotations in space. The x-axis of the floating marker is tangent to the curve at the contact point, its y-axis points outward from the curve’s center of curvature at the contact point and its z-axis is along the binormal. For more detail, see [5, 6].

The table below summarizes the type of joints and the bodies connected:

ID | Type of joint | Body i | Body j |

1 | Revolute | Carbody Rear | Carbody Front |

2 | Translational | Carbody Rear | Rear DX wheel |

3 | Translational | Carbody Rear | Rear SX wheel |

4 | Translational | Carbody Front | Front DX wheel |

5 | Translational | Carbody Front | Front SX wheel |

6 | PTCV | Rail DX | Rear DX wheel |

7 | PTCV | Rail SX | Rear SX wheel |

8 | PTCV | Rail DX | Front DX wheel |

9 | PTCV | Rail SX | Front SX wheel |

The Grubler check should give the following output:

*3 Gruebler Count (approximate degrees of freedom)*

*6 Moving Parts (not including ground)*

*1 Revolute Joints*

*4 Translational Joints*

*4 Point_Curves*

*3 Degrees of Freedom for .RollerCoaster_ptcv*

*There are no redundant constraint equations.*

Finally, independent suspensions have been added to the roller coaster, by inserting spring-dampers on the translational joint of each wheel, so as to allow some lateral movement in the event of variations in track width. The stiffnesses and damping coefficients have been obtained from a single-wheel model, imposing that the eigenfrequency in lateral direction was f_{n} = 11 Hz [4]:

where K^{eq} is the equivalent stiffness of the system, m is the sprung mass and f_{n} is the eigenfrequency of the system. In the suspenson model of each mid-car (carbody Front and carbody Rear), there are two springs acting in parallel.

From these expressions results:

The characteristic of the lateral dampers is estimated assuming that the roller coaster vehicle has a damping factor ζ equal to 0.9 [4]. Under this assumption, the damping coefficient of each mid-car can be written as:

From these expressions results:

In Figure 4.4 you can see the roller coaster assembly in a wireframe view.

An initial velocity of 7 m/s has been applied to the roller coaster, in order to be able to travel the first horizontal section of the path.

## Simulations and analysis of results

### Simulations

A preliminar static simulation has been performed to check the static vertical forces on the contact points.

The resulting force output is 3619.2 N (Figure 5.1) and is about equal to the static weight, so it is correct.

Then, a dynamic simulation has been performed first without any loss of energy and then adding the drag force and the rolling resistance. In the Figure 5.2 is reported the plot of the magnitude of the velocity in the different cases: the blue line is the velocity related to the case without losses, while the dashed lines are related to the drag and drag plus rolling resistance cases.

In order to validate the model, you can compare the velocities resulting from the simulation with the ones obtained from the conservation of energy:

The velocity at the end of the slope (point C) is equal to 40.2 m/s (145 km/h), while the one at the top of the loop (point E) is 21.0 m/s (75.6 km/h), so the results are correct and the model verified.

At this stage, a drag force has been added as a single component force applied on the CM (the center of pressure, if known, would be more correct) of the Carbody Front, pointing backwards towards the travel direction. Drag is the resistance felt caused by the density of the fluid which something is traveling through, the profile of the object and the velocity. It’s magnitude is given by the following expression:

where ρ is the air density, here assumed constant and equal to 1.2 kg/m^{3}, v is the travel velocity, c_{D} is the drag coefficient, here assumed equal to 0.7 such as for convertible cars with open top [7], and A is the cross section area, here assumed equal to 1 m^{2}.

Also a rolling resistance force has been applied for each wheel on its CM, with a magnitude given by:

where F_{z} is the vertical load, f_{v} is the coefficient of rolling friction, about equal to 0.030 inches (0.000762 m) for polyurethane wheels on steel rail [8], and R is the wheel radius, equal to 0.1 m.

The dynamic simulation settings were:

*SIMULATE/DYNAMIC, DURATION=42, DTOUT=5.0E-02*

with the integrator GSTIFF I3.

### Results

As shown in the Figure 5.2, the maximum velocity is about 38 m/s (137 km/h) at the end of the initial drop, while the minimum velocity is about 4 m/s (14 km/h) at the top of the loop.

The forces on the contact points, i.e. the reactions on the wheels, are plotted in the following figures:

A peak of force is generated on the wheel at the beginning of the curve, due to the banking of the roller coaster imposed by the track.

As you can see, the entity of the forces generated on the wheels is very high, so every single component of the roller coaster has to be design with the maximum attention. The forces here calculated, because the used constraints are stronger than the real ones, could be considered in advantage of safety and used for a preliminary sizing and verification both of the roller coaster vehicle and for the track. For example, the maximum force can be applied on the FE model of the rail exactly at the beginning of the curve in order to compute maximum displacements and stresses on that point and consequently sizing sections, joints and supports.

The maximum longitudinal acceleration is about equal to 1g and is reached at three quarters of the loop, when the vehicle is pointing downward.

The maximum lateral acceleration is about equal to 0.78g and is reached at the beginning of the curve, when the vehicle has reached the maximum banking.

The maximum vertical acceleration is about equal to 5.79g and is reached at one quarter of the loop, when the vehicle is pointing downward, while the minimum is equal to -1.46g and is reached at the end of the slope.

## Conclusion

A simplified model of a roller coaster moving along a track with typical monoeuvers has been built and the results of the velocities, accelerations and forces on wheels have been reported.

Of course, several improvements can be made, to make the model more realistic and accurate and to be able to focus attention also on the structural aspects of the vehicle and the track.

By way of example and not limited to, some additional things are reported below to improve the model accuracy:

- to use force contacts solid-solid, instead of point-curve (kinematic) constraints; in this way, you can use the entire wheelsets of the vehicle (Figures 7.1 and 7.2) and study the interaction between wheels and rail, implementing proper friction models of railway derivation, considering the rolling of the wheels with their gyroscopic effects and possible rail irregularities;

- to make the wheel boogies flexible, to compute strain, stresses and vibration modes;
- to insert some bushings to connect the wheels to the boogie, in place of ideal revolute joints, to emulate the compliance and friction of the wheel hubs and the bearings;
- to define the track with the curvilinear abscissa through a ribbon frame (Figure 7.3), in a proper pre-processing program or in a spreadsheet, and then interpolate the points with cubic splines in the MBS, to ensure at least the continuity of the first two derivatives of the position, to have more freedom in the geometry construction.

## References

[1] https://www.bolliger-mabillard.com

[3] Measuring the wheel-rail forces of a roller coaster, A. Simonis, C. Schindler, JSSS, september 2018

[4] Modelling tracks for roller coaster dynamics, J. Pombo, J. Ambrósio, Int. J. Vehicle Design, Vol. 45, No. 4, 2007

[5] MSC Adams View User’s Guide

[6] Point-to-Curve Constraints and other Contact Elements, Proceedings 7th Modelica Conference, Como, Italy, Sep. 20-22, 2009

[7] https://www.engineeringtoolbox.com

[9] https://www.gupta-verlag.com

[10] https://www.coaster101.com

[11] Design Analysis of Roller Coasters, Kristen Hunt, Master Thesis, Worcester Polytechnic Institute, 2018