Dynamic analysis of tippe top on cylinder’s inner surface with and without friction based on routh reduction

M Ariska1*, H Akhsan1 and M Muslim1

1Physics Education Department, Universitas Sriwijaya, Palembang,South Sumatra, Indonesia

*mellyariska@fkip.unsri.ac.id

Abstract. Physics computing can be used to help to solve complex dynamic equations, both translation and rotation. The purpose of this study was to obtain differences in the dynamics of the tippe top with and without friction moving on inner surface of a cylindrical with varying initial state based of Routhian Reduction. The equation of tippe top in flat fields with and without friction has been reduced by the Routhian reduction method with the Poincare equation with computational in the previous research, and computation has also been carried out in the search for numerical solutions to the dynamics of tippe top with friction in the Maple program. In this study the reduction used is a Routhian reduction, so that the equation used in determining the equations of tippe top motion with and without friction that moves in a curved plane in the form of a cylindrical surface with varying initial state based on maple is Poincaré's equation based on Routhian reduction with and without friction. The effect of friction can be seen clearly through the dynamics and graph equations in the return top. This method can reduce the equation of backward motion with and without friction that moves on the surface of the cylinder clearly in the form of a set of differential equations. This research can be continued by solving the dynamic equations of the tippe top in other curved fields such as the torus and ball. The findings of this study are dynamic equations and graphs of friction with and without friction equations that move in curved fields in the inner of surfaces in cylinders with varying initial state based on maple.

1. IntroductionPhysics is one of the components of science [1]. Physics is a science that requires observation and measurements made through experiments [2-4]. One discussion in physics that is quite complex is the problem of kinematics and dynamics. Dynamics movements can be in the form of translational motion and rotational motion. Every object in the universe experiences both translation, rotation and both dynamics [5]. The motion of objects in the translation configuration space as well as rotation is very complicated if analyzed manually. This of course requires computational assistance in completing it. The motion of objects to be discussed in this study is tippe top motion. Given the top motion is an example of the motion of objects that can move in translation or rotation [6]. Especially gasing that can reverse itself, which is called a tippe top. In a study on the research conducted by Ariska (2018), it has succeeded in solving the dynamics of the reverse equation in the flat plane while at the same time describing the equation of motion using computational physics [7]. In this study the researcher will analyze the tippe top motion on the surface of the cylinder with and without friction. The researcher will predict the dynamics of the tippe top in the inner surface of the cylinder can still

be reversed or not.In completing this equation is not an easy thing, because the configuration space that will be passed through the tippe top is a cylinder which is a curved field that has a tube coordinate variable and tippe top’s coordinate that moves using three coordinate systems, so the number of common coordinates to be completed is six coordinates general, namely one translation coordinate and five rotational coordinates. In addition, researchers will also analyze and predict tippe top’s motion with and without friction movement.

In previous studies, there were no clear graphic images of the dynamics of the top tippe moving in the cylinder configuration chamber under various initial conditions when the tippe top was rotated with and without friction. This study provides a clear picture of the difference between tippe top dynamics in the flat plane and on the inside surface of the cylinder through computational physics testing. The results show a graph depicting the dynamics of the top tippe with clear Routhian reduction and detailed analysis in every second with various tilt angles. Previous studies did not provide a clear picture of the dynamics of top tippe detail per second.

This study will review top tippe dynamics using Routhian Reduction with computational physics based on Maple 18. The researcher will review top tippe movements that move both translation and rotation using five general coordinates, namely two general coordinates in translational dynamics and three general coordinates for rotational dynamics. The movement of the top tippe will be changed from the initial conditions in the form of tilt angle when the first top is played with and without friction on the inner surface of the cylinder. The dynamics of the top tippe will be observed and analyzed by testing the motion of the top tippe surface on the tube using Maple 18 based physics computing.

Dynamics analysis of tippe top will be done by computational physics. This research is an implementation of computer courses in physics learning. Based on the research conducted by Bourabee (2004), Bloch (2007) and Blanskentein (2014) who analyzed the dynamics of tippe top in flat fields with various equations namely Eular equations, Lagrange equations and Hamilton equations[8-9]. Previous studies conducted various methods to formulate tippe top dynamics in the flat plane, such as the Eular equation which reviews tippe top dynamics through angular rotation and moment of inertia, Lagrange equation by reviewing the kinetic energy of systems and Hamiltonian equations that analyze system movements through momentum. Previous studies have reviewed Tippe top in flat areas. In this study the authors analyzed the dynamics of the top tippe in the curved field with the Routhian Reduction Method. The formulation of Tippe top dynamics can be done using the Port Controlled Hamiltonia System (PCHS) method, but this method still leaves a Lagrange multiplier. Furthermore, the dynamics can be formulated using another method which is more systematic, namely the Routhian Reduction method [10]. The method illustrates a system that is subject to non-Holonymous constraints and External Style, so that the Lagrange multiplier can be removed from the equation. Before formulating the dynamics of a non-holonmic mechanical system, the researcher will analyze the potential energy that occurs in a system that moves in the cylinder configuration space. Potential energy is the initial key in formulating the dynamics of a system, due to Routhian reduction. The author analyzes two motion systems namely Tippe top dynamics on the inner surface of the tube with and without friction. Of course it will produce two different systems and different equations of motion. This difference can be explained through the similarity of tippe top movements with and without friction which will be explained in the results of this study. With the tippe top with friction paying attention to external forces such as the normal force and the friction force of the system, this can certainly be explained both on the flat plane and on the inside surface of the tube.

The results of the equation obtained from this study are diferential equations which describe the dynamics of tippe top in the flat plane [10]. The results of the analysis obtained are described by the configuration space in the form of translation, namely the flat plane. The researcher will develop the results of the analysis obtained by Bourabee (2004), Bloch (2007) and Blanskentein (2014) by analyzing tippe top motion with a configuration space in the form of a rotation space, namely the inner cylinder surface with and without friction [11-13]. The object used in the system is the top Tippe dynamics that move in the tube, the analysis technique used is to transform coordinates from cartesian coordinates to tube coordinates The following illustration of dynamics is examined.

Figure 1. Tippe top played on the inner surface of the tube viewed from the side

Figure 2. Tippe top played on the inner surface of the tube as viewed from the front

Based on the schematics of figures 2 and 3 schematic analysis can be done in determining the equation of motion of the system with the method of routhian reduction that reviews the system through energy by considering external forces and non-holonomic constraints.The problems to be solved in this study are as follows: How to analyze tippe top motion in surface configuation space in computational based physics cylinders. This research will apply technology to solve the general equation of a system of motion in three-dimensional space. Given the growing development of science and technology in the world of education, the dynamics of objects that have a configuration space that is quite complicated because it consists of translational and rotational movements that are very complicated if completed manually [14].

This research is a solution for lecturers and students to complete complex object dynamics thoroughly and precisely. The limitation of the problem in this study is. This study reviews the dynamics of objects in low energy such as tippe top. The top tippe is only analyzed moving in the arena in the form of a surface in a tube that is considered to be quite large from the size of the top tippe. Tippe top has a ball shape with a small rod as a handle and can be flipped. The dynamics of objects are in 3D space which consists of rotational and translational motion [15].

2. Method

This research is a mathematical theoretical study. The study was conducted with a review of several literature on mechanical systems in the top tippe case that had been developed previously as well as mathematical calculations using computational physics, especially Maple based. This research is a theoretical mathematical study with the utilization of physics computations. Computing was carried out with the help of Maple 18. The research was carried out with a review of several studies regarding mechanical systems in the case of previously developed tippe top and mathematical calculations.

( ∂ T∂ si )−cr

li (q ) s l ∂T∂ sr −

∂T∂ σ i =Si (1)

However, this equation requires quasi velocity be found as a direct time-derivative from the temporary quasi coordinate. Therefore, the Poincare equation used in this study to analyze the dynamics of the tippetopon inener silinder is the Poincare equation which is based on the Routhian reduction [16], which can be written as follows

ddt

∂ R∂ vρ −∑

μ=2

n

∑λ=2

n

cλμρ vμ ∂ R

∂ v λ=0 (2)

The tippe top phenomenon has been known since the 1800s, beginning with the event of a stone-

eyed turning ring, when it was rotated the ring could turn itself. Since 1950, research on this tippe top has only been developed by physics scientists, including Braams, Hugenholtz, and Pliskin. They observed the effect of friction on the tippe top movement. Meanwhile, Synge (1952) examines the role of top tippe mass distribution with axis symmetry [17-18].

Research on modern tippe top originated from Cohen's (1974) study [19], which looked at the effect of friction on top tippe movements through simple simulations. Cohen's research is a reference for some researchers about the dynamics of top tipple including Ciocci, et al. (2012) who developed a mathematical model to describe the dynamics of tippe top that have axis mass distribution [20]. Furthermore, in 1999, Gray and Nickel explained the three motion constants that are important in the tippe top movement, namely the energy constant, Jellett's constant, and the Routh constant. Several years later, research on the tippe top continued, such as by Bou-Rabee et al. (2004) which explained inversion that occurred in the tippe top through modification of the Maxwell-Bloch equation, then continued in 2008, Bou-Rabee, et al. shows the conditions for the existence of heteroclinic orbits in inversion and non-inversion tippe top conditions determined by a complex version of the simple harmonic oscillator equation [21-22].

The equation of the tippe top motion by applying group theory in the form of a rotation group using the Poincare equation in the flat plane has been formulated by Ariska (2018) with computational physics. In addition, previous research on top tippe dynamics was only formulated for tippe top moving in the flat plane [23-24]. Therefore, the authors are interested in continuing the research by formulating tippe top dynamics which are played in curved fields in the form of a cylindrical surface with and without friction. Detailed movement predictions will be analyzed using computational physics.

T=12 (m ( X2+r2 η2+(a θ sin θ cosη)2+2ra θ ηsin θ sin ηcosη )+ I θ2+ I sin 2θ ϕ2+ I 3( β1

I 3 )2

) (3)

Kinetic energy in the tippe top that moves on the surface of the tube due to the presence of these motion constants can be written with and because of this constant cyclic speed can be expressed

ψ=β1

I 3− ϕcos θ (4)

With

U =mg (r (1−cos η )+ (R−acos θ ) cosη+a sin θ cos ϕsin η) (5)

so, the new Lagrangian that is formed is

R=T−U−ψ ∂T∂ψ

¿ 12 (m ( X 2+r2 η2+(a θ sin θ cosη)2+2ra θ η sin θ sin η cosη )+ I θ2+ I sin 2θ ϕ2−

(β1)2

I 3)−mg(r (1−cosη )+( R−acosθ ) cosη+a sin θ cosϕ sin η)+ ϕ β1cos θ

(6)The equation for tippe top motion for coordinates θ,ϕ , ηand y stated by,

ddt ( ∂ R

∂ vθ )−(cϕθψ vϕ β1 )−Xθ R=0 (7)

3. Results and DiscussionThe equation of the tippe top motion that moves on the surface in a tube through the reduction of the Routhian without friction force is explained as follows,The equation for tippe top motion for coordinates θ

ddt ( ∂ R

∂θ )−( (+1 ) ϕsin θ β1 )−∂ R∂θ

=0 (8)

m (ar sin θ η2 cos (2η )+a2 cosη (sin2 θ θ+sin θ cosθ θ2 )+a sin ηcos η ( r sinθ η−2 a sin2θ θ η ))+ I θ+ I sin θ cosθ ϕ2−2 β1sin θ ϕ−mg (a sin θ cos η+acosθ cos ϕsin η )=0

The equation for tippe top motion for coordinates ϕddt ( ∂ R

∂( ϕ))−((−1) θ β1 )−X ϕ R=0 (9)

ϕ=( 1I sin θ )( β1θ−2 I cosθ θ ϕ−

θ β1

sin θ+mgasin ϕ sin η)

The equation for tippe top motion for coordinates ηddt ( ∂ R

∂η )−Xη R=0 (10)

m ¿The equation for tippe top motion for y coordinates

ddt ( ∂ R

∂ X )−X X R=0. (11)

X=0Based on equation (8), (9), (10), (11) can be illustrated a graph that describes the dynamics of tippe

top without friction on the surface in a tube with maple as follows,

rads

Figure 3. a. Graph of angle relations θwith respect to time enlarge b.Graph of angle relations θwith respect to time scaled down

a bFigure 4. a. Graph the relationship of angular velocity θ to time, b. Graph the relationship of

angular velocity ϕ to Time

In Figure (3) and (4), it can be seen that if the inner surface of the tube where the tippe top moves is no friction (slippery), then the turning top will not reverse. Tippe top will only move to rotate (slip) with a constant speed with small precision, so that the turning back will move like a conventional top that slips. This is expressed by the angular velocity θ and the angular velocity ϕ is constant which illustrates that tippe top is in a stable state rotating constant with small precision on the ez. The

a b

backside is stable with angular velocity θ constant value throughout the time the tippe top moves for 12 seconds.

The equation of the tippe top motion that moves on the surface in the tube through reduction of the Routhian with friction force stated as follows,

Motion equation for coordinates θddt ( ∂ R

∂θ )−((+1 ) ϕsin θ β1 )−∂ R∂θ

=Sθ (12)

m (ar sin θ η2 cos (2 η )+a2 cosη (sin2 θ θ+sin θ cosθ θ2 )+a sin ηcos η ( r sinθ η−2 a sin2θ θ η ))+ I θ+ I sin θ cosθ ϕ2−2 β1sin θ ϕ−mg (a sin θ cos η+a cos θ cos ϕsin η )=Sθ

with constraint force in the form of a normal force obtained

|FN|=mg cosη+m z+mr η2=m [ g cos η+a (θ sin θ+θ2 cosθ )+r η2 ] (13)

The equation for tippe top motion for coordinates ϕ

ϕ=( 1I sin θ )( Sϕ

sin θ−2 I cosθ θ ϕ−β1 θ (1−csc θ)+mgasin ϕ sin η)

The equation for tippe top motion for coordinates η

m ¿. (14)

The equation for tippe top motion for y coordinatesm X=SX (15)

A numerical solution of the equation of the tippe top motion played on the surface in a tube for coordinates θ ( t ), ϕ ( t ) , θ ( t ) , ϕ ( t ) , η ( t ) , X ( t) can be described using maple with parameters:I n=I n'=I =45 gr . cm2 , I 3=50 , α=600, mk=1 gr ,mt=3 gr , mb=13 gr , mtotal=17 gr , a=0,6 cm ,R=1,3 cm , D=2,6 cm.Value of initial conditions θ (0 )=0,1 rad , ϕ (0 )=0 , ϕ (0 )=θ (0 )=0and x (0 )=η (0 )=0 , β1=2500 gr . m2 . rad /s and if μ=0,3.

Based on the equation of motion (11), (12), (13), a numerical solution can be described for angular velocity for coordinats θ ( t ), ϕ ( t ) , θ ( t ) , ϕ (t ) , η ( t ) , X ( t) changes with respect to time and angle changes to time as follows,

Figure 5. Tippe top dynamics move in the tube configuration space with a rotational axis a. Graph the relationship of angle θ to Time. b. Graph the relationship of angular velocity θ( t) to Time.

Figure 6. a. Graph the relationship of linear velocity x (t) to Time, b. Graph the relationship of angular velocity η( t)

The graph for the slope angle θ(t) shows the main characteristic of the tippe top reversal. On the graph, it can be seen that after turning back and forth for 22 seconds, then slowly the turning top will experience a reversal, then after θ(t) forming an angle of π, the tippe top will rotate with the stem and rotate stable without precision.

Figure 3,4,5 shows the tippe top motion in rotation, while Figure 6 and 7 show the translational tippe top motion. It is clear that the equation 3,4,5,6 can be used to predict the dynamics of the top tippe in the cylinder configuration space. It can be seen that the top tippe undergoes a reversal process at the 22nd second and stops spinning and reaches stability at the 40th second. The top tippe process occurs in accordance with the predetermined initial requirements. This shows that the differential

a b

a b c

equation that has been derived is true and can be proven computationally using Maple. The purpose of solving the dynamics system by describing the numerical equation of the top tippe is to compare the results of the solution that has been calculated with the nature of the qualitative analysis of the top tippe equation studied in this study.

Based on Figure (3), (4), (5) and (6) it can be seen that the dynamics of the tippe top on the surface in the tube is more random than the tippe top in the flat plane and the reversal process requires a longer time of 22 seconds while in the flat plane 20 seconds [20]. Likewise with the initial conditions of the slope of the rod when the tippe top is rotated also has a difference between the tippe top on the surface in the tube and in the flat plane. Tippe top in the flat plane of the slope of the rod when rotated so that the reverse turning is θ (0) = 0.9 rad. Whereas the slope limit of the tippe top which moves on the surface of the tube is θ (0) = 0.2 rad, if the slope of more than 0.2 rad the tippe top on the inner surface of the tube cannot be reversed.

4. ConclusionBased on the above research it can be concluded that with the Routhian reduction the equation of tippe top’s dynamics that moves on the surface in a cylinder with and without complex friction can be formulated well in the form of a differential equation. The results of the analysis of the previous tippe top dynamics with friction are the tippe top will flip perfectly. The dynamics of Tippe top moves in the tube configuration room can be clearly illustrated through Routhian reduction and the equation has been tested with Physics Computation. Based on computational test results, the top tippe reversal process occurs in accordance with the initial requirements that have been predetermined in this study. This proves that the differential equation that has been derived is true and can be proven computationally using Maple. The completion of the dynamics system by describing the numerical equation of the top tippe on the surface in the tube by comparing the results of the solution that has been calculated with the qualitative analysis properties of the top tippe equation has been tested in this study.

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