Kite Flyer

Silvio Zuin – silviozuin1@gmail.com

Introduction

The purpose of this work is to simulate the Kite Flyer, a rotating amusement park ride.

This ride is particular because it’s moved by two different rotational motors. The axis of the two motors are not parallel because during the first phase of the motion the axis of the second motor inclinates with respect to the axis of the first motor, which remains always perpendicular to the ground. The inclination of the second motor axis is obtained with a curved guide and a pair of pistons.

Objectives

A lot of informations can be obtained by the simulation of the system, but the ones I wish to analyze are:

  • the forces and the torques that the motors need to guarantee to perform the wished motion;
  • the reactive forces and torques between the base frame of the ride and the connected rotating body;
  • the accelerations of the passengers of the ride.

Modelling

The ride has been modelled as in the following figure:

kite_flyer_modellazione

Fig. 1 Modelling.

In order to clarify the ride mechanism, the latter is reported in the following figure:

kite_flyer_schema

Fig. 2 Ride mechanism.

The bodies are:

  1. base frame (maize);
  2. column (red);
  3. pair of piston rods (yellow);
  4. pair of cylinder barrels (blue);
  5. fixed centre (green);
  6. rotating centre (cyan);
  7. twelve independent gondolas (orange).

The inertias of the gondolas have been set to be those of the case in which all the 24 seats are occupied by people with a mass of 75 kg each.

The joints are:

  1. fixed joint between the ground and the base frame;
  2. revolute joint between the base frame and the column;
  3. Hooke joint between the column and the piston rods;
  4. cylindrical joint between the piston rods and the cylinder barrels;
  5. Hooke joint between the cylinder barrels and the fixed centre;
  6. a general constraint between a lower point of the fixed centre and the column, so that the former has to lay on the latter mean line;
  7. a general constraint between an upper point of the fixed centre and the column, so that the former has to lay on the latter mean line;
  8. an inplane joint between a central point of the fixed centre and the column, so that the former has to lay on the symmetry plane of the latter;
  9. a revolute joint between the fixed centre and the rotating centre;
  10. a revolute joint between each gondola and the rotating centre.

The two pistons are treated as a single piston because:

  • otherwise the would have been a redundance in their action;
  • they share the same oil circuit and as a consequence the total load that they have to carry is always equally split.

The number of degrees of freedom locked by the joints are:

  1. 6
  2. 5
  3. 4
  4. 4
  5. 4
  6. 2
  7. 2
  8. 1
  9. 5
  10. 5 (x12)

The remaining degrees of freedom, obtained by applying the Grubler Equation, are:

DOF=18×6-1×6-1×5-1×4-1×4-1×4-1×2-1×2-1×1-1×5-12×5=15.

Twelve are associated with the oscillation of the gondolas with respect to the rotating centre, one is associated with the rotation between the base frame and the column, one is associated with the rotation between the fixed centre and the rotating centre and the last is associated with the motion of the pistons.

The degrees of freedom associated with the oscillation of the gondolas with respect to the rotating centre are damped by twelve linear dampers. The purpose of these dampers is to avoid infinite oscillation of the gondolas.

The degree of freedom associated with the rotation between the base frame and the column is controlled by a rotational motor in joint 2 that respects the following velocity law (counter-clockwise rotation):

kite_flyer_rotazione_base_colonna

Fig. 3 Velocity law of the motor in joint 2.

The degree of freedom associated with the rotation between the fixed centre and the rotating centre is controlled by a rotational motor in joint 9 that respects the following velocity law (clockwise rotation):

kite_flyer_rotazione_centro_fisso_centro_girevole

Fig. 4 Velocity law of the motor in joint 9.

The degree of freedom associated with the motion of the pistons is controlled by a translational motor in joint 4 that respects the following displacement law:

kite_flyer_traslazione_stantuffi_cilindri

Fig. 5 Displacement law of the motor in joint 4.

Simulation and results

The model has been simulated for 190 seconds with 10000 steps using a SI2 solver.

Forces and torques applied by the motors

The torque applied by the motor in joint 2 to body 2 is reported in the following figure:

kite_flyer_M_motore_base_colonna

Fig. 6 Torque applied by the motor in joint 2 to body 2.

The torque applied by the motor in joint 9 to body 6 is reported in the following figure:

kite_flyer_M_motore_centro_fisso_centro_girevole

Fig. 7 Torque applied by the motor in joint 9 to body 6.

The two torques are compared together in the following figure:

kite_flyer_confronto_M

Fig. 8 Comparison of the torques applied by the two rotational motors.

The two torques are approximately the same and are typical of a clockwise movement. This is due to the fact that that the majority of the rotational inertia is given by the rotating centre and the gondolas, which are above the motor in joint 9, while the rotational inertia of the bodies between the two motors is very low. As a consequence, the majority of the torque applied by the motor in joint 2 is needed to counter the torque applied by the motor in joint 9 to the bodies between the two motors. A slight difference is noticed at the end of the acceleration and at the beginning of the deceleration because in these phases the axis of the motor in joint 9 is inclined with respect to the axis of the motor in joint 2.

The force applied by the motor in joint 4 to body 4 is reported in the following figure:

kite_flyer_F_motore_stantuffi_cilindri

Fig. 9 Force applied by the motor in joint 4 to body 4.

It’s easy to observe that the majority of the force is needed to balance the weight of the sliding bodies. The force applied in the middle of the motion is lower than the force applied at the beginning and at the end mainly because when the sliding bodies are inclined a part of the weight is held by the column. The four peaks occur when the joints 6 and 7 are over the curved part of the column mean line.

Reactive forces and torques between the base frame and the column

The reactive forces between the base frame and the column will be given in the coordinates of a moving frame attached to column, with the z axis perpendicular to the ground and the y axis perpendicular to the symmetry plane of the column, as in the following figure:

kite_flyer_frame

Fig. 10 Frame used to obtain the components of the reactive forces and torques between the base frame and the column.

Furthermore, the reactions have to be considered as applied by the base frame to the column.

The force along the x axis is:

kite_flyer_Fx_base_colonna

Fig. 11 Reactive force between the base frame and the column along the x axis.

The force along the y axis is:

kite_flyer_Fy_base_colonna

Fig. 12 Reactive force between the base frame and the column along the y axis.

The components of the force along the x and y axis are equal to zero at the beginning and at the end because there are no inertial forces and the weight is applied along the z axis. During the motion components along the x and y axis arise because of the dynamic effects.

The force along the z axis is:

kite_flyer_Fz_base_colonna

Fig. 13 Reactive force between the base frame and the column along the z axis.

The majority of the force along the z axis is needed to balance the weight of the rotating bodies. The force is slightly different in some phases of the motion because of the dynamic effects of which the most important is the force needed accelerate the sliding bodies along the z axis.

The torque along the x axis is:

kite_flyer_Mx_base_colonna

Fig. 14 Reactive torque between the base frame and the column along the x axis.

The torque at the beginning and at the end is different from zero because it balances the moment of the weight caused by the fact that the center of mass of the fixed centre is not located in the symmetry plane of the column. During the motion the torque is different because of the dynamic effects.

The torque along the y axis is:

kite_flyer_My_base_colonna

Fig. 15 Reactive torque between the base frame and the column along the y axis.

The torque at the beginning and at the end is different from zero because it balances the moment of the weight caused by the fact the center of mass of the rotating bodies has a negative component along the x axis. During the motion the torque is different because of the dynamic effects. Among the causes of the big negative value of the torque in the middle there are:

  • the moment of the weight caused by the fact that in this phase the center of mass of the rotating bodies has a considerable positive component along the x axis;
  • the moment of the centrifugal forces along the x axis;
  • the gyroscopic torque caused by the fact that the rotation axis of the motor in joint 9 is rotating around the axis of the motor in joint 2.

The two positive peaks occur because when the joint 7 is over the curved part of the column mean line the fixed centre has a big angular acceleration along the y axis, as shown in the following figure:

kite_flyer_alfay_centro_fisso

Fig. 16 Angular acceleration of the fixed centre along the y axis.

Acceleration of the passengers

To approximate the acceleration of the passengers the following marker attached to a gondola has been considered:

kite_flyer_passeggeri

Fig. 17 Marker used to calculate the acceleration of the passengers.

The magnitude of the acceleration of this marker is:

kite_flyer_accelerazione_passeggeri

Fig. 18 Magnitude of the acceleration of the marker.

During the middle of the motion the magnitude of the acceleration of the marker oscillates between 6.5 and 8 m/s^2 with a frequency equal to the frequency of rotation of the motor in joint 9. A steady state is therefore reached in the middle of the motion.

Conclusions

The Kite Flyer ride has been modelled with the hypothesis of fully occupied seats and simulated.

The forces and the torques applied by the motors, the reactive forces and torques between the base frame and the column and the acceleration of the passengers have been obtained and commented.

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