Exploring possible chaotic and stable motions of astrojax pendulum

Abstract
This work studies the movement of a toy astrojax pendulum in space and what types of movement it can describe. Chaotic and stable motions were used to describe the entire system. The work was carried out to provide for more research on the topic and develop the understanding of a system as complex as astrojax. In the past, this topic has been understudied; thus, this research can be partially considered the first work in this area, though it has big assumptions.
Introduction
An interesting toy problem is an Astrojax pendulum. Astrojax is, in fact, a double spherical pendulum, in which the first swing is allowed to slide freely along the cable. Astrojax players can produce a lot of exciting and unexpected movements. After all, even without external coercion from the outside player, the movement of two beans exhibits confusing and chaotic behavior.
Oscillatory mechanical systems are a subset of dynamical systems that can describe the evolution of states for almost all physical phenomena. Fluctuations of various types are found in real mechanical systems, almost all of them are thermodynamically irreversible (due to damping, friction, energy loss for heat, etc.). This study investigates a set of coupled chaotic oscillations formed by an astrojax pendulum. Pendulums are one of the most fundamental physical systems studied, and although they can be ordinary, they are ideal only in the simplest cases. While one pendulum one can simply solve the equations of motion, especially in the limit of small amplitudes, a \”double\” swingarm has a grip that gives go to chaotic behavior. But an astrojax pendulum can create both chaotic and stable movements. The astrojax system will be described by a system of differential equations and then described using a python program. Then an experiment will be carried out and a comparison between the results of the simulation and the experiment will be carried out. Thus, it can be understood how much theory can describe the movements of the astrojax pendulum.
In the past, the question of the movement of the astrojax pendulum has been raised in two works. In the Astrojax pendulum and the n-body problem on the sphere: A study in reduction, VARIATIONAL integration, and pattern Evocation, the system is described by Lagrangian mechanics, but it is not taken into account that the support can move. Also, the experiment does not fully show the similarity with the theory and what kinds of trajectories a toy can create. In the study of the astrojax pendulum, the system is described by Newtonian mechanics, but it is also not visible what types of trajectories the toy describes and how, in principle, it behaves. It also does not describe how a toy with a movable fulcrum will behave. In general, all the articles written in the past years consider a special case of a toy in which only the upper ball moves and does not show how the system can behave under different initial conditions. This work will study the astrojax pendulum in general, describing movements with a movable fulcrum and a moving upper ball. We will also consider all types of trajectories that can form and how the system will behave as a whole.

Theory


Figure 1: The Astrojax Pendulum
A force is applied to the end of the rope to make the balls move and create different trajectories. This toy is called an astrojax pendulum. Due to the different movements of the end of the thread, the system can create two types of movement: chaotic and periodic.
Chaotic movement is a movement that a mathematical model cannot accurately predict and that a person cannot intuitively imagine. On the other hand, stable motion is a motion that can be accurately predicted by a mathematical model and describes trajectories that are intuitive to humans, such as a circle or an ellipse. Stable movement and chaotic movement are shown in Figure 2 and Figure 3, respectively.

Figure 2 describes the stable motion where the green circle represents the trajectory of the end of the rope which rotates by the motor, blue and yellow describe trajectories of balls

Figure 3 describes the chaotic trajectories of the second ball in x y system
In the first part of the theory, a model for orbital motion will be presented. In the second part of the theory, the chaotic movement of an astrojax pendulum will be presented.

Orbital Motion

When the top end of the rope rotates, it is noticeable that the toy itself behaves stably and predictably. Balls create orbital trajectories that can be predicted by a mathematical model.

Figure 4: describes the entire pendulum system. R is the radius of motor trajectory, is the angular velocity of trajectory, – length between end and first ball, – length between the first and second ball
In Figure 4, a diagram of the entire system is shown. When the end of the rope begins to rotate in a circle of a certain radius with a certain angular velocity, the lower balls also begin to rotate in a circle. The system has no choice regarding rotation because of the angular momentum. According to the law of conservation of momentum, the angular momentum created by the rotation of the end must remain unchanged. Therefore, the balls begin to spin.
To describe the movement of the balls, it must be clarified why first the ball will not move up and down. This is possible only when the centripetal force is greater than or equal to the difference between the mass of the ball and the friction of the thread on the ball. In this model, the tension of the thread above and below the ball will play an insignificant role due to the centripetal force.
(1)

Figure 5
Figure 5 schematically shows the forces acting on the ball: downward gravity, friction in the direction of the rope, centripetal force, and support reaction force generated by the tension on the rope. As already mentioned, if the projection of the centripetal force is greater than the difference between the ball\’s mass and friction, then the ball will be at rest and the motion will be stable. However, options for exclusion are not excluded when the ball starts moving downward, and this will lead the system into chaos.
The force of air resistance should also be taken into account. Since the balls rotate in the air, the drag force can be written as the formula below. Let’s write down the dependence of distance, time, and momentum on time
(2)
(3)
(4)
Now, using Newton\’s second law in impulse form, we write down the formula for the drag force, inserting the result from equation (3) into (4)
(5)
It is now noticeable that the force of air resistance depends on the surface area of ​​the surface of the ball, speed, angle of attack, and air density. To simplify the model, the angle of attack will be 90 degrees. Later in the experiment, it will be possible to say exactly how accurately the angle of attack was chosen.
Now, to calculate the area, integrate over the phi angle as shown in the picture

Figure 6: integrating over the whole sphere from 0 to pi/2
(6)
(7)
now substitute for dA the equation (5) and integrate from 0 to
(8)
(9)
Since this is the force of air resistance, it is necessary to enter a correction factor that is equal to the Reynolds number

(10)
(11)
where ρ is the density of fluid;
υ is the velocity of the fluid;
l is the diameter of the pipe;
µ is the dynamic viscosity of the fluid.
Now It is possible to write Newton\’s second law to describe this system in three planes: x, y, and z. Friction forces, centripetal force, resistance force were taken into account. Since the changes occur along with the x and z coordinates, it is necessary to add the ratio of the coordinate with the angular velocity and radius of the circle that describes the motor
As shown in the picture below. The phi and theta angles refer to 1 ball, and the alpha and beta angles refer to the second ball.

Figure 7: describes a stable system with and theta angles
Below is the equation for the x coordinate
(12)
(13)
As can be seen from the equation, the resistance force is directed opposite to the motion of the ball. Now let\’s write down the equation in the y coordinate. It will have downward gravity.
(14)
(15)
And the equations for the z coordinate
(16)
(17)
It is noticeable that the tension force differs from above and below one of the balls due to the bending of the thread inside the ball during rotation. The tensile and centripetal forces themselves form the rotational movements, while the resistance force prevents this. Phi and theta were used for Ball 1, while alpha and beta were used for Ball 2. However, since the tension of the threads depends on both balls in the equations of motion regarding Ball 1, the angles alpha and beta are also present.
Now the equations of motion for the second ball are written down. It is noticeable that the trajectory of the ball, as in reality, is affected by the movement of the first ball.
(18)
(19)
(20)
(21)
(22)
(23)
Since these systems of equations are quite complex, a program was written for solving and visualizing. As these are differential equations, the initial conditions were set coinciding with the experiment. At the very beginning, all angles were 0 degrees. After the end began to rotate, the angles began to change.
Some preliminary conclusions can be drawn so far. The angular velocity remains constant throughout the flight. The starting radius of one ball will decrease over time, while the radius of the two balls will increase. It is also possible to conclude that, with sufficiently large omega balls, a different trajectory will occur due to the tension of the rope, the ball shedding, and the possible small vibrations

Chaotic Motion

To describe chaotic movements, one must understand that Ball 1 will move along the rope. Let us describe its movement.

Figure 8
As seen in Figure 8, different tensions will form at the ends of the ball due to the second ball and the bending of the thread inside the ball. To describe the tensions being formed, it is necessary to enter the theta and phi angles describing the bends at the upper and lower ends. There is also a nominal thread tension, which is constant. Given that the angles change, one can integrate over the angle to find the dependence of the tension on the angle. The formulas are given below.
(24)
(25)
As it can be seen, the pattern is inversely exponential, and it is possible that, at high angles, the tension will drop. This can just cause the ball to fall.
The reaction force of the support also needs to be considered. This force is the tension force on the thread. This is exactly how the ball interacts with the thread. To introduce the reaction force of the support, one also must enter the alpha angle describing the bend of the thread at the top of the ball.

Figure 9: here introducing angle alpha, which describes the movement of rope inside the ball
After the introduction of the angle alpha, the equation for the reaction force of the support can be written.
(26)
However, for a more accurate description, it is necessary to take into account the friction force of the thread on the ball.

Figure 10:here introducing friction between ball and rope
As seen in this figure 10, the friction force is directed along the line of the thread at the top of the ball and plays a role in keeping the ball in a state of equilibrium or creating small vibrations. Now that the frictional force is considered, one can rewrite the equations and enter the reaction force of the support created by this friction.
(27)
(28)
(29)
(30)
Now, considering friction and the fact that the tension of the threads is different at the ends, one can apply the Kabstan equation. A revised version of this equation was used for this case. There are no major differences.
(31)
There is a dependence of the tension of the threads, which refers to the movement of the ball itself upon rotation. Considering the force of air resistance derived in the last part, the equations of motion describing this system can be written.

Figure 11: full chaotic system with forces applied to each ball and the rope
As seen in this figure 11, the rope length differential was introduced because the ball moves along it. The forces acting on the balls were described. These are resistance forces, designated F and F1, acting against the movement of the ball. The forces of gravity of the balls, m1g, and m2g, are directed downward. The tension forces, T1 and T2, are directed along the threads and centripetal forces. Also, the forces created by vertical oscillatory movements still act as y = Acoswt.
These are the equations of motion. The basis equation was Newton\’s second law.
(32)
(33)
(34)
(35)
(36)
(37)
Since this is a system of differential equations, the only way to solve this numerical solution is by programming the system. Therefore, this system was solved numerically.
Now it is necessary to describe the dependence of the coordinates of the balls in the x and y plane for a more accurate description of the model and understand the system of motions and the initial conditions for solving differential equations. Without these data, the solution of differential equations is not possible.

Figure 12
As seen in this figure 12, the coordinates themselves were calculated taking into account the angles that are already in theory and the oscillations of the end to produce an accurate description of the movement.
(38)
(39)
(40)
(41)
Now knowing the coordinates, one can calculate the Lyapunov exponent. Its meaning and the interpretation of the results will be presented in the Experiment and Discussion section. It is possible to assume that the system, which looks chaotic, can be partially predicted and everything can converge at one point.

Method

A physical model of the astrojax pendulum toy was investigated for chaotic movement and stable movement. To confirm the theory, one must study the trajectory of the balls. Later, comparing the trajectory of the balls, it can be concluded that the theory is reliable.
To more accurately study the behavior of the system, the constants must be changed, namely the ratio of the masses of the balls, the initial distance between them, the speed of rotation or vibrations at the end of the thread, and the thread itself. This will change the coefficient of thread tension.
For the experiment, a system of rope with two balls was used. One is attached to the end and the second can freely move along the rope attached to the mechanism of rotation or create vibrations along a rolled line. On the side and below, there are fast cameras that record the movements of the balls and the tracker program, which draws the trajectory.

Figure 13: Full system of motor, DC power supply, and computer with program tracker that tracks system
The picture above shows the setup for the experiment. Above, there is a motor that rotates. The necessary mechanism is already attached to it to create rotation or oscillation. The laboratory power supply unit supplies the motor and, by changing the voltage, changes the rotation speed and the computer that the cameras, which record all movement, are connected to.

Figure 14
Figure 14 shows a blade attached to a motor. The thread is attached to the blade to create rotation and rotates along a certain radius at a certain angular velocity. The torque of the motor is transmitted to the blade. The thread is fixed at a certain level to rotate with a certain radius, thereby imitating the movements of a human hand.
While a different mechanism is used for vibrations, it is also attached to the motor. It is designed so that the torque of the wheel is transmitted at the linear moment of the part. Thanks to such a simple and well-known mechanism, it is possible to create vertical oscillations with the help of a motor. To measure the period of oscillation, tests were initially carried out and a table recording voltages, corresponding frequencies, and periods of oscillation was created. The mechanism diagram is presented below.

Figure 15: the diagram of the system that creates vertical oscillations
Two cameras and a tracker program on a computer were used to outline the trajectories to observe the balls. Below is a picture of the tracking of the balls.

Figure 16: the program Tracker tracks the first ball

Figure 16: the program Tracker tracks the second ball
For theoretical graphs and trajectories of the balls, a program was written that visualizes the trajectories. Later, the data collected by the tracker program was interpreted into a schematic and applied to the main drawing. The program displays the coordinates, which are already applied to the theoretical data for analysis.

Results


Figure 17:describes the result of the simulation of the stable motion where the green circle represents the trajectory of the end of the rope which rotates by the motor, blue and yellow describe the trajectories of the balls.

Figure 18:describes the result of the simulation of the stable which created flower trajectory

Figure 19:describes the result of the simulation of the stable which created flower trajectory

Figure 20: describes the correlation between theoretical and experimental results on flower trajectory

Figure 21: describes the correlation between theoretical and experimental results on flower trajectory

Figure 22: describes the correlation between theoretical points and experimental on the chaotic motion of the second ball. Here masses of the ball equal 1kg, rope length =2m, w=4.7rad/s and A=0.3m

Figure 23: describes simulation of chaotic movement of second ball in x y plane with initial conditions : mass=1kg, rope length=1m,w=2rad/s,A=0.1m

Figure 24: describes the correlation between theoretical and experimental trajectories in stable motion

Figure 25: describes the ratio of the rising angular velocity of the motor and the angular velocity of the balls and the experimental points of the angular velocities

Figure 26: describes the ratio of the angular velocity of the motor and the angular velocity of the balls and the experimental points of the angular velocities


Figure 27:describe Lyapunov exponent of the system on a long time

Figure 28: describes the Lyapunov exponent of the system on lower times

Figure 29:describes correlation between theoretical and experimental points of Lyapunov exponent for the whole system

Figure 30: describes the trajectory of the first ball in chaotic movement
In Figure 17, it is noted that the balls describe a circular or more ellipsoidal trajectory at an angular velocity of less than 11 rad / s. Meanwhile, in Figure 18 and Figure 19, at angular velocities above 11 rad / s, the balls begin to describe a trajectory in the form of a flower. The first and second balls in Figure 30 and Figure 22 tend towards chaotic movements. Their drawings are quite similar. In Figure 27, the Lyapunov exponent initially holds at -7.70 for up to 6–7 seconds; however, it later begins to grow throughout the chart up to -4.25. The same trend is visible in Figure 28.

Discussion

Based on the graphs, it can be concluded that the theory agrees with the experiment with little error. In Graph 25, it is noticeable how the points lie almost on a straight line, which means that the angular velocity of the end of the thread coincides with the angular velocity of the balls. One of the hypotheses was confirmed. In Graph 26, it is also noticeable that the experimental points fall neatly on the theoretical line. Another conclusion that was drawn from the experiment is that at high omegas, the system becomes critical and no longer describes circular paths, and at even larger ones, the upper ball begins to fall on the lower one and the system stops moving.
As mentioned previously and predicted by the theory, at high angular velocities, the centripetal force becomes less than the force of gravity and the ball falls onto the second ball. This was also confirmed experimentally.
Looking at Graph 24, it is also noticeable that the theoretical and experimental trajectories coincide in pattern, but do not coincide in height. In the future, the initial conditions will strongly influence the trajectory of movement.
In figure 22 it can be seen that on 4.7 rad / s the lower ball will have a stable trajectory and it can be compared with theoretical points as seen in the figure. As a result, the points are noticeable enough to converge
The charts themselves coincide in pattern in Figure 20 and Figure 21, but the error is sufficient. Considering that the movement here is already partially chaotic, the match can be considered sufficient. Another conclusion that can be made is that the initial conditions have the most influence, which is not surprising. Euler\’s method for solving differential equations is very dependent on the initial conditions. Therefore, both in theory and in experiments, everything depends on the initial conditions. Indirect factors may also affect the trajectory.
In Graph 27 and Graph 28, the Lyapunov exponent grows slowly and has negative values. This means that the points should converge, which is seen on the charts. However, the fact that the exponent grows indicates that the system becomes more chaotic over time, which is also observed in the trajectory graphs. At the very beginning, the exponent does not change. The only thing that can be said is that at the beginning, there are the factors caused by the ball\’s movement and other indirect factors.
In Graph 29, in the time from 0 to 40 seconds, the theoretical and experimental points of the Lyapunov exponent converge with small deviations, but already from 40 seconds, discrepancies occur. The only thing that can assume is that at large the system becomes so chaotic that the theoretical values ​​have large deviations from the experimental ones.

Conclusion

In this paper, chaotic and stable movements were considered. Later they were described by differential equations in 3 planes. Differential equations were solved using a python program. The results of the study showed that theory quite accurately coincided with the experiment. In some situations, the initial conditions influenced the course of the experiment, but these were not significant deviations. All hypotheses were confirmed and a new flower trajectory was revealed at w 11.34rad / s flower trajectory. The Lyapunov exponent was also calculated and compared with experimental data, from which it can be concluded that the theory for chaotic motion can accurately describe the system\’s behavior.
In the future,ít will be possible to describe all variants of the system using Lagrangian mechanics without dividing them into types of motion. It is also possible to study the conditions of motion at very high angular velocities and describe the motion during horizontal and other types of oscillatory movements.
In general, this kind of movement should be studied to better understand chaotic movements and, in general, the movements of various pendulums. This knowledge can help make the toys themselves better.
References
Karsai A, Harrington S, Campbell C. [Pdf] the astrojax pendulum: Semantic scholar [Internet]. undefined. 2014 [cited 2021Aug6]. Available from: https://www.semanticscholar.org/paper/The-Astrojax-Pendulum-Karsai-Harrington/1d65c87ac85170ae87652acd02da036a1750d7f1
Toit PD. [PDF] the Astrojax pendulum and the n-body problem on the sphere: A study in reduction, VARIATIONAL integration, and pattern Evocation .: Semantic Scholar [Internet]. undefined. 2005 [cited 2021Aug6]. Available from: https://www.semanticscholar.org/paper/The-Astrojax-Pendulum-and-the-N-Body-Problem-on-the-Toit/533cf17f06055a76a5ffb2001557712f4f8c3707
Landau LD, Lifshit︠s︡ E. M., Sykes JB, Bell JS. Mechanics. Oxford: Pergamon Press; 1976.

Leave a Comment

Your email address will not be published. Required fields are marked *