INTERPRETACIÓN DE DATOS DE LA ENCUESTA

The output of this function is the drag force acting on the chassis, bogies, and wheels. The script in this function is based on the theory explained in Appendix G. The functions GondolaDragForce and

ChassisDragForce can both be found in the block named Air Drag Force in Figure A.15.

1 function Drag Force Chassis = ChassisDragForce(velocity)

2

3 %% Input

4

5 % Specify the air density (kg/mˆ3):

6 rho = 1.2;

7 % Presumed drag coefficient of chassis (equal to drag coefficient of a cube):

8 Cd = 0.8;

9

10 %% Chassis Drag Force Calculation

11

12 % Frontal surface of chassis:

13 S front = 0.386; %mˆ2

14 % Drag Force of the main chassis beam and bogies:

C. Track Generator

C.1

NoLimits Track Data to Rail Coordinates

This appendix describes the conversion of NoLimits track data to the Cartesian coordinates that correspond to the left and right rail of the track. The professional version of NoLimits 2 allows the user to export and save track data in CSV-format. The user can specify the distance between the nodes along the track curvature, and each line in the CSV-file corresponds with a node on the centre spline of the track. The centre spline can either be chosen equal to the editor spline or the centre of the rails, where the editor spline is unequal to the centre of rails for most track types due to the addition of for instance a backbone that strengthens the track. However, on the spinning rollercoaster the editor spline and centre of rails coincide with each other thanks to the simple track geometry.

The CSV-file that is exported from NoLimits consists of thirteen columns. The first column specifies the node number, while the next three columns specify the Cartesian coordinates of the nodes with respect to a global reference frame. The remaining nine columns specify thex-,y-, andz-components of respectively the frontFˆ, leftLˆ, and upper vectorUˆ that start at the corresponding centre node, as depicted in Figure C.1a. Each of these three vectors are unit vectors that are decomposed in Cartesian components with respect to a global reference frame. Together the Fˆ,Lˆ, andUˆ vectors can be regarded as a local reference frame, since the three vectors are always mutually perpendicular. The origin of this local reference frame is located at point P in Figure C.1b.

(a) Specification of the vectors as exported by NoLimits (b) The frames and parameters used during the analysis Figure C.1: The vectors and frames used for converting the NoLimits track data to Cartesian coordinates of the left and right rail

Prior to calculating the coordinates of the left and right rail, the data should first be converted from millimeters to meters. Any track width can be specified by the user, but in case of the spinning rollercoaster the width of the track is set equal to 0.9 meters. In Figure C.1b,r_{P A}P refers to the distance between P and either the left or the right rail, formulated with respect to the local reference frame. The value forrP_{P A}equals 0.45 meters in case of the left rail, and -0.45 meters for the right rail. The direction ofFˆ andUˆ can solely be determined based on the direction ofLˆ, thanks to the orthogonality between the three vectors. Therefore, it suffices to only use Lˆ for determining the rail coordinates. The position of a certain rail coordinate can be computed with the expression in Equation C.1.

rO_OA=r_OPO +ROPr_{P A}P (C.1)

frame in point O. The distancerO

OP is equal to the position of the centre node, and hence the value for this

parameter is known. The rotation matrixROP between the global reference frame at point O and the local reference frame at point P is depicted in blue in Figure C.1b. A rotation matrix between any two frames is defined as the matrix that relates the orientation of the axes of both frames. Taking into account thatLˆ is a vector with unit length, the rotation matrix is determined betweenLˆ and a local vector [1,0,0] for each node along the track. This could also be regarded as the specification of a unit vector along thex-axis of the local reference frame at point P, after which a multiplication with the rotation matrix leads to a decomposition of this vector in Cartesian coordinates with respect to the global reference frame at point O. The rotation between the two vectors can be calculated using the built-in functionvrrotvecinMatlab. The output is a four-element rotation row vector, where the first three elements specify the rotation axis and the last element defines the angle of rotation. The rotation row vector can subsequently be represented in matrix-form by using the Matlab-function vrrotvec2mat, which results in the rotation matrix ROP with three rows and three columns. The left and right rail coordinates can subsequently be determined with respect to the global reference frame by multiplyingROP withrP_{P A}, after whichrO_OP is added to the result. Please note that r_{P A}P

is defined as [0.45,0,0] or [-0.45,0,0] for respectively the left and right rail, whose direction is in agreement with the local vector [1,0,0] that was previously defined.

Performing the computation for each node results in a list of xyz-coordinates for the left and right rail. Plotting these coordinates along with the coordinates of the centre rail results in the plot shown in Figure C.2. It should be noted that the global reference frames in NoLimits and Simulink are orientated differently with respect to each other. Hence, the left and right rail coordinates should first be transformed from one coordinate system to the other before the rail data can be used in Simulink. The user can choose a convenient location for the global reference frame in Simulink, and subsequently subtract the coordinates of the chosen location from the rail coordinates. This procedure can for instance be used to assign the station as the start position of the rollercoaster train. If the default zero-length initial train position vector is used in Simulink, the train always starts at the origin of the global reference frame which coincides with the station. Once the transformation has been completed and the global reference frame origin has been specified, the rail coordinates are saved in two separate text files e.g. LeftRail.txt and RightRail.txt. These files specify the xyz-coordinates of the left and right rail, which are used by Simulink to define two cubic interpolation splines that form the track. A copy of theMatlab-script that has been described in this appendix can be found in Section C.2.