The Mathematical Model for the Tippe Top Inversion

Hindawi Advances in Mathematical Physics Volume 2021, Article ID 5552369, 11 pages https://doi.org/10.1155/2021/5552369

Research Article The Mathematical Model for the Tippe Top Inversion

R. Usubamatov ,1 M. Bergander ,2 and S. Kapayeva 3

1Kyrgyz State Technical University, Bishkek, Kyrgyzstan 2Magnetic Development, Inc., Madison, USA 3Serikbayev East Kazakhstan Technical University, Kazakhstan

Correspondence should be addressed to R. Usubamatov; [email protected]

Received 24 February 2021; Revised 26 March 2021; Accepted 19 April 2021; Published 3 May 2021

Academic Editor: Ivan Giorgio

Copyright © 2021 R. Usubamatov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The dynamics of rotating objects is an area of classical mechanics that has many unsolved problems. Among these problems are the gyroscopic effects manifested by the spinning objects of different forms. One of them is the Tippe top designed as the truncated sphere which is fitted with a short, cylindrical rod for rotation. The unexplainable gyroscopic effect of the Tippe top is manifested by its inversion towards the support surface. Researchers tried to describe this gyroscopic effect for two centuries, but all modelings were on the level of assumptions. It is natural because the Tippe top has a more complex design than the simple spinning disc, which gyroscopic effects did not have an analytical solution until recent time. The latest research, in the area of gyroscopic effects, reveals the action of the system of several interrelated inertial torques on any spinning object. The gyroscopic inertial torques are generated by their rotating mass. These inertial torques and the variable ratio of the angular velocities of the spinning object around axes of rotations constitute the fundamental principles of gyroscope theory. These physical principles of dynamics of rotating objects enable to description and compute of any gyroscopic effects and also the Tippe top inversion.

1. Introduction the top served as an amusing toy for a population that was more attractive than an ordinary disc-type top. The Tippe The dynamics of classical mechanics contains the section of top for the short time got popularity all over the world. gyroscope theory that considers different types of spinning Nevertheless, the number of publications dedicated to the objects [1–3]. One of them is well-known from ancient times physics of Tippe top is less than for a disc-type top. The com- aspinningtopanditsmodification the Tippe top. This top plexity of the Tippe top’s motions and the unexplainable is designed with a truncated sphere that is fitted by a short, gyroscopic effect of inversion interested only persistent cylindrical rod for rotation. The Tippy top center mass is researchers. Analysis of these publications and methods for located below its center of the sphere. The gyroscopic effect solutions demonstrate the application of the same mathe- of the spinning Tippe top is manifested when its spherical sur- matical tools and more intuitive assumptions as for the spin- face contacts the support surface and by the following inver- ning top [16]. Their main principles for describing the Tippe sion and spinning on the rod [4–6]. Following spinning of top inversion were the angular momentum, kinetic energy, the Tippe top is the same as a regular design of the top. This friction force, and its weight. In reality, the physics of gyro- gyroscopic effect of the spinning Tippe top manifests other scopic effects are many times harder. This is the reason that designs of the rotating objects as hardboiled eggs, an American the Tippe top’s gyroscopic effect does not have the accepted football, round smooth ovals, etc. [7–9]. All similar spinning physics and recognized mathematical model. objects with the property of inversion attracted physicists In engineering, the visual example of the manifestation of and mathematicians [10–12]. They have been studied exten- the spinning object inversions is presented at conditions of sively the gyroscopic effect of Tippe top from the beginning the weightless at the orbital flight. This example and other non- of the nineteenth century [13–15]. The inversion property of discovered ones are waiting for investigations of their physics. 2 Advances in Mathematical Physics

All similar examples with the spinning objects are formulating around axes and is well described in the publication [17, by the defined principles for gyroscopic effects that can explain 18]. The dependency of angular velocities of the rotation and describe the physics of acting forces and inversion by the around axes of the spinning sphere is defined by the principle rules of classical mechanics [17]. of the kinetic energy conservation law [17]. Schematic of the This work proposes the analytical circumscribing of the torques acting on the Tippe top (Figure 1) is used for the for- physics for Tippe top’s inversion based on the latest defined mulation of the equation of the kinetic energy balance for the principles of gyroscope theory [17, 18]. inertial torques acting around axes of rotation. These inertial torques express the equality of the kinetic energy of the spin- 2. Methodology ning objects (Chapter 4, Section 4.1.2, of [17]): 3 2 3 The mathematical model for motions of the Tippe top is for- − ðÞπ − 2 π + ðÞπ − 2 Jωω fi ff 8 8 x mulated by the de ned principles for gyroscopic e ects that 3 3 can explain and describe the physics of acting forces and − π − 2 2π +1 ωω = π − 2 2π +1 ωω 1 inversion by the rules of classical mechanics [17]. The system 8 ðÞ J y 8 ðÞ J x ð Þ of the inertial torques generated by the centrifugal, common 3 3 inertial, Coriolis forces, the change in the angular momen- − π − 2 2π + π − 2 ωω : 8 ðÞ8 ðÞJ y tum of the spinning object, and the ratio of the angular velocities of the top around axes are applied for the mathematical fi model for its motions. For the analysis, the top axis is located Simpli cation of Equation (1) yields the dependency of vertically at the movable system of 3D coordinates Σoxyz angular velocities of the rotation around axes for the spin- which center disposes at the center mass of the Tippe top. ning sphere: "# The location of the center mass m of the top is below the cen- 2 ter of the truncated sphere on the distance r. The top is spin- 6ðÞπ − 2 π +3ðÞπ − 2 +8 ω = − ω , ð2Þ ning with the angular velocity ω in the counterclockwise y 8 − 3ðÞπ − 2 x direction around axis oy that passed the center mass. The combined action of external, inertial torques, forces, and where the sign (-) shows the direction of the action of inertial motions around three axes for the running inclined Tippe fi torques that can be omitted for the following considerations top as the truncated sphere with the tted rod is represented (Figure 1). in Figure 1. The mathematical models for the expressions of the iner- At starting condition, the top spins on the spherical surface tial torques for the spinning sphere are described in the work of the radius R and contacts the support surface at the point P. [18], and the dependency of angular velocities of the rotation The spinning axis of the top passes the center mass m and dis- around axes (Equation (2)) is presented in Table 1. poses perpendicular to the support surface. Any inclination of J is the variable mass moment of inertia of the Tippe top the top on the angle β or the roughness of the contact surfaces around the axis of rotation; Ti is the variable inertial torques gives the shift of the point P of the top to the side. At this situ- that depend on the geometry of the top and its axis inclination ation, the spinning axis passes the center mass and does not on the angle β; ω is the angular velocity of the spinning top; and pass the center of the sphere and the contact point P.Thesize ω i is the angular velocity of the rotation of the top around axis i. of the shift of the contact point P depends on the displacement The action of the inertial torques and the motions of the r of the center mass from the center of the sphere (Figure 1). At spinning Tippe top around axes are presented by the follow- this condition, the sphere slides on the support surface. ing components of gyroscopic effects (Figure 1): The point P moves by the circular line around the axis oz ’ fi of the top s rotation. The sliding motion is ful lled at the con- (i) The frictional torque T which action is the counter- dition of the action of the kinetic frictional force F = mgf (g fx ffi clockwise direction around axis ox produces the resis- is the gravity acceleration; f is the coe cient of dry sliding = − − − − − tance torque Trx Tctx Tcrx Tiny Tamy Trlx, kinetic friction). The frictional force F acts around two axes = cos β ox and oz and produces the frictional torque T = mgf h where Trlx mgf rR is the rolling frictional tor- fx que, and other torques are as specified in Table 1. The cos β (h = R – r sin β is the distance from the center mass resulting torque around axis ox is the sum of fric- to the support surface) and T = mgf r cos β, respectively. fy tional and resistance torques T = T − T turns The frictional torque T turns the angular momentum vec- x fx rx fx the spinning top with the angular velocity ω γ x tor H of the spinning top on the small-angle x, Hx around axis ox at the plane yoz in the counterclockwise direction (Figure 1). (ii) The torque Tx produces the resulting precession The vector of the change in the angular momentum ΔHx pre- torque around axis oy = + − − − − sents the precession torque that turns the angular momentum Ty Tinx Tamx Tcty Tcry Trly Tmgy, where γ T = mgf rR cos β is the rolling frictional torque, vector H of the spinning top on the small-angle y around axis rly = cos β oy at the plane xoz in the counterclockwise direction. Tmgy mgr is the torque produced by the The action of the external torque on the spinning object top weight due to the inclination of an axis on generates the system of interrelated inertial torques that the angle β, and other torques are as specified in depends on the ratio of the angular velocities of its rotation Table 1 Advances in Mathematical Physics 3

Z Y Z �Hx l �c �H c y H m H Y x � O X x y Y H Yy y Ty

Tfx Tctx Tcrx Tamy Trlx Tx �x � m RrO h X

T rly P F Tcry Tmgy Tfx T amx � Tinx

Figure 1: Schematic of the torques acting on the Tippe top spinning on the sphere and its motions.

Table 1: Equations of the inertial torques acting on the spinning Tippe top.

Type of the torque generated by Equation (kg·m2/s2)

Centrifugal forces 3 2 T = Tin = ðÞπ − 2 πJωω Inertial forces ct 8 x 3 = π − 2 ωω Coriolis forces Tcr 8 ðÞJ i Change in angular momentum T = Jωω am i 3 3 = + = π − 2 2π + π − 2 ωω Resistance torque Tr Tct Tcr Tr 8 ðÞ8 ðÞJ x 3 = + = π − 2 2π +1 ωω Precession torque Tp Tin Tam Tp 8 ðÞ J x

The dependency"# of angular velocities of the rotation around axes oy and ox, rad/s 6 π − 2 2π +3 π − 2 +8 ω = ðÞ ðÞω y 8 − 3ðÞπ − 2 x

ﬃ (iii) The inertial torques Tctx and Tinx at expressions of the rod and high coe cient of sliding friction can Tx and Ty are self-compensated as far as there is change in the direction of spinning. This condition one expression and one sign (Table 1) (Chapter 4, occurs at the time of the contact of the rod with the Section 4.1.2 of [17]) support surface. The change in the direction of the top spinning is explained by the principle of the (iv) The action of the inertial torques on the spinning top kinetic energy conservation law and by the principle leads to its rotation around axes ox and oy with the of the angular impulse and momentum ω ω interrelated angular velocities x and y,respectively. The following turn of the spinning top around two (v) The spinning top maintains its vertical rotation in so axes brings to contact of its rod with a support sur- far as the point P of its contact with the support surface, inversion, and its vertical spinning on the rod. face that is movable because the top rolls on the The light model of the top with a big diameter of spherical surface under the action of the inertial 4 Advances in Mathematical Physics

torques. This rolling leads to the turn of the top’s axes ox and oy, respectively (Table 1); η is the coefficient axis and to change the mass moment of the inertia of the change in the inertial torque due to the action of of the Tippe top around axes of rotation. The top’s the frictional torques on the top; Jy and Jx are the mass moment inertia is different for its vertical and hori- moments of inertia of the top around axes oy and ox, zontal location due to its nonsymmetrical design rel- respectively; other components are as specified above. ative to its axis of spinning The gyroscope parameters of Equations (3)–(5) and their mathematical processing were described in the publication (vi) The action of the inertial torques is interrelated and (Chapter 6, Section 6.2 of [17]). The frictional torque acting looped by the ratio of the angular velocities of the around axis oy on the spinning top (Figure 1) reduces the top around axes of rotation. The action of the fric- value of the precession torques acting around axis ox that is tional force F activates all inertial torques generated presented by the correction coefficient η. The coefficient η by the rotating mass of the top is expressed by the ratio of the difference between the precession torques ðT + T Þ and the external torques T and This detailed presentation of the action of frictional inx amx rly and inertial torques on the Tippe top enables us to Tmgy of axis ox, to the precession torques. Substituting equa- describe and explain the physics of its motions and inver- tions of Table 1 into the ratio of these torques and transfor- ffi η sion. Analysis of the action of the inertial torques on the mation yields the expression of the correction coe cient . spinning top demonstrates its inversion is the result of the action of two precession torques around two axes. A T + T − T − T T + T η = inx amx rly mgy =1− rly mgy mathematical model for the Tippe top motions is formu- + + lated by the equations of the inertial torques and the ratio Tinx Tamx Tin:x Tam:x 6 + cos β ð Þ of the angular velocities of the top around of axes of rota- =1− ÂÃðÞmgf rR mgr , 3/8 π − 2 2π +1 ωω tion (Table 1). The physics of inversion is considered for ðÞðÞ J x two conditions of the Tippe top spinning, i.e., on the spherical part and on the rod. The mathematical models for the Tippe top motions where J is the mass moment of inertia; f r is the rolling fric- around two axes are expressed by the following Euler’sdiffer- tional coefficient; β is the angle of the top inclination, ω is ential equations (Chapter 6, Section 6.2 of [17]): the angular velocity of the spinning top, and other components are as specified above. dω x = − − − − η, ð3Þ For the solution of Equation (4), the precession torques Jx T fy Trlx Tctx Tcrx Tam:y = + dt Tpx Tin:x Tam:x (Table 1) should be expressed via the fric- ω ω tional load torque T fy and the angular velocity x by x. The value of the precession torque of Equation (4) should be cor- dω rected by the principle of the conservation of the kinetic y = + − − − , ð4Þ Jy Tinx Tamx Tmgy Tcry Trly energy. Since the angular velocity of the gyroscope around dt axis oy is bigger than around axis ox (Equation (5)), then the value of the precession torque should be proportionally "# 2 increased on the ratio of the angular velocities (Equation 6ðÞπ − 2 π +3ðÞπ − 2 +8 ω = ω , ð5Þ (3)) and decreased on the ratio n of the precession and resis- y 8 − 3ðÞπ − 2 x tance torques. This transformation expresses the interdepen- dency of load and inertial torques acting around two axes of ω ω where x and z are the angular velocity of the top the spinning object (Chapter 5, Section 5.2, of [17]). The around axes and , respectively; is the time; , ω ω ox oz t Tctx expression of x is replaced by y (Equation (5)) into the pre- , , , Tcrx Tcry Tiny Tamx, and Tamy are the variable inertial tor- cession torque of Equation (4) and increased on the ratio of ques generated by the centrifugal, Coriolis, inertial forces the angular velocities; then, the ratio of the precession and and the change in the angular momentum acting around resistance torques is represented by the following:

ÂÃÀÁ2 ÂÃ2 ÂÃÀÁ2 6 π − 2 π +3 π − 2 +8 /8− 3 π − 2 3/8 π − 2 π +1 Jωω /6π − 2 π +3 π − 2 +8 /8− 3 π − 2 = ðÞ ðÞ ðÞðÞðÞðÞ y ðÞ ðÞ ðÞðÞ n 3/8 π − 2 ωω ðÞðÞJ y 8 = ðÞπ − 2 π + : 3ðÞπ − 2 ð7Þ Advances in Mathematical Physics 5

ω fi "# The expression of the angular velocity x is de ned from fi 6 π − 2 2π +3 π − 2 +8 the right side of Equation (1). Substituting de ned parame- − ðÞ ðÞ ters (Table 1, Equation (6)) into the right side of Equation 8 − 3ðÞπ − 2 (4)), and transformation yields the following: "#ð8Þ + cos β 1 − ðÞmgf rR mgr ωω =0: Á ÂÃ2 J 3/8 π − 2 π +1 Jωω x 3 3 ðÞðÞ x cos β − sin β − π − 2 2π + π − 2 ωω mgf ðÞR r 8 ðÞ8 ðÞJ x Simplification of Equation (8) brings the following:

( ) ÂÃÀÁÀÁ2 ÀÁ2 f cos β R − r sin β +6π − 2 π +3 π − 2 +8 /8− 3 π − 2 / 3/8 π − 2 π +1 f R + r cos β ω = ðÞðÞ ðÞÀÁðÞðÞðÞðÞ ðÞr mg : 9 x 2 2 ð Þ ðÞ3/8 ðÞπ − 2 π +ðÞ 3/8 ðÞπ − 2 +6ðÞπ − 2 π +3ðÞπ − 2 +8 /8ðÞ− 3ðÞπ − 2 Jω

ﬁ Then, the modi ed formula of the precession torque Tp:x is presented by the following expression:

ÂÃÀÁ2 T =ðÞ 3/8 ðÞπ − 2 π +1 /ðÞðÞπ − 2 π +8/3ðÞðÞðÞπ − 2 Jω p 8 9 > > > ÂÃÀÁÀÁÀÁ> <> 2 2 => f cos β R − r sin β +6π − 2 π +3 π − 2 +8 /8− 3 π − 2 / 3/8 π − 2 π +1 f R + r cos β ðÞðÞ ðÞÀÁÀÁðÞðÞðÞðÞ ðÞr mg > 3/8 π − 2 2π + 3/8 π − 2 +6π − 2 22 π +3 π − 2 +8 /8− 3 π − 2 > Jω 10 > ðÞðÞðÞðÞ ðÞ ðÞ ðÞðÞ > ð Þ :> ;> ( ÂÃ ÂÃÀÁ ) 3/8 π − 2 2π +1 cos β − sin β +6π − 2 2 π +3 π − 2 +8 /8− 3 π − 2 + cos β = ðÞðÞ f ðÞR r ðÞ ðÞ ðÞðÞðÞf rR r : ÂÃ2 ÀÁÀÁ2 mg ½ðÞπ − 2 π +8/3ðÞðÞðÞπ − 2 ðÞ3/8 ðÞπ − 2 π +ðÞ 3/8 ðÞπ − 2 +6ðÞπ − 2 π +3ðÞπ − 2 +8 /8ðÞ− 3ðÞπ − 2

Substituting Equations (6) and (5) into Equation (3) and Equations (7) and (10) into Equation (4) yields the following equations:

"#"# 2 dω 3 2 3 6ðÞπ − 2 π +3ðÞπ − 2 +8 ðÞmgf R + mgr cos β J x = mgf cos βðÞR − r sin β − ðÞπ − 2 π + ðÞπ − 2 Jωω − mgf R cos β − 1 − ÂÃr Jωω , x 8 8 x r 8 − 3 π − 2 3/8 π − 2 2π +1 ωω x dt ðÞ ðÞðÞ J x ð11Þ ( ) ÂÃ2 ÀÁÀÁ2 dω 3/8 π − 2 π +1 f cos β R − r sin β +6π − 2 π +3 π − 2 +8 /8− 3 π − 2 f R + r cos β 3 y = ðÞðÞ ÂÃðÞðÞÀÁÀÁðÞ ðÞðÞðÞr − π − 2 ωω − cos β − cos β, Jy 2 2 mg ðÞJ y mgf R mgr dt ½ðÞπ − 2 π +8/3ðÞðÞðÞπ − 2 ðÞ3/8 ðÞπ − 2 π +ðÞ 3/8 ðÞπ − 2 +6ðÞπ − 2 π +3ðÞπ − 2 +8 /8ðÞ− 3ðÞπ − 2 8 r ð12Þ

where the components of the external resistance torques Simpliﬁcations and transformations of Equations (11) = β + cos β Try mgf rRcos mgr are removed from Equation and (12) bring the following: (12) as far as Equation (9) contains the correction factor η; all other components are as speciﬁed above.

"#"# ω + cos β 3 3 6 π − 2 2π +3 π − 2 +8 d x = cos β − sin β + ðÞf rR r − − π − 2 2π + π − 2 + ðÞ ðÞ ωω , 13 Jx f ðÞR r 2 f R mg ðÞ ðÞ J x ð Þ dt ðÞ3/8 ðÞπ − 2 π +1 r 8 8 8 − 3ðÞπ − 2 ( ) ÂÃ2 ÀÁÀÁ2 dω 3/8 π − 2 π +1 f cos β R − r sin β +6π − 2 π +3 π − 2 +8 /8− 3 π − 2 f R + r cos β 3 y = ðÞðÞ ÂÃðÞðÞÀÁÀÁðÞ ðÞðÞðÞr − π − 2 ωω : 14 Jy 2 2 mg ðÞJ y ð Þ dt ½ðÞπ − 2 π +8/3ðÞðÞðÞπ − 2 ðÞ3/8 ðÞπ − 2 π +ðÞ 3/8 ðÞπ − 2 +6ðÞπ − 2 π +3ðÞπ − 2 +8 /8ðÞ− 3ðÞπ − 2 8 6 Advances in Mathematical Physics

Z dω 3 3 �H y = π − 2 2π +1 ωω − π − 2 ωω �H y Jy 8 ðÞ J x 8 ðÞJ y y H dt ð16Þ Y Hx x �y Y − cos β, Yy mglc Hy Ty

where the spinning of the top in the same direction, Ji,is 3 Tfx Tctx Tcrx Tamy Tx �x the top’s mass moment of inertia around the support ; � = cos β sin β m Tmgf mglc is the frictional torque (for the top O = cos β X rotating on the rod); Tmgy mglc is the torque generated by the action of the top weight; lc is the length of the leg; other expressions are as specified in Equations (3) and (4). ffi η fi FP The correction coe cient is de ned similarity as for Tcry Tfz Equation (6) and presented by the following expression: T mgy � Tamx mgl cos β Tinx η =1− ÂÃc : ð17Þ 3/8 π − 2 2π +1 ωω ðÞðÞ J x

Figure = + 2: Schematic of the torques acting on the Tippe top spinning The precession torques Tp:x Tin:x Tam:x of Equation on the rod and its motions. (16) is expressed via the load torque T and the angular ω ω velocity x by x. The solution is the same as presented Equations (13) and (14) are used for direct computing of for Equation (4). The ratio of the precession and resistance the angular velocities ω and ω of the Tippe top around axes torques is represented by Equation (7). The expression of x y ω fi ox and oy, respectively. The value of the mass moments of the angular velocity x is de ned from the right side of Equation (15). Substituting defined parameters (Table 1, inertia J and Ji of the top around axes i of rotation is a variable that depends on the angle β of the top inclination. The Equation (17)) into the right side of Equation (15)) and action of the resulting torque around axis ox and oy turns transformation yields the following: the top until the contact of the rod with the support surface. 3 3 The following turn of the top disconnects the spherical part cos β sin β − π − 2 2π + π − 2 ωω with the support surface and the top starts to rotate on the mgf lc 8 ðÞ8 ðÞJ x rod. The top rotates around the center mass, and its rod "# 6 π − 2 2π +3 π − 2 +8 moves by the spiral trajectory on the support surface. The − ðÞ ðÞ action of the frictional torque F on the rod and following 8 − 3ðÞπ − 2 "# inertial torques generated is almost the same as in Figure 1. mgl cos β The rolling friction force does not consider due to the small 1 − ÂÃc Jωω =0: 3/8 π − 2 2π +1 ωω x diameter of the tip of the rod. The torques acting on the ðÞðÞ J x top spinning around the vertical axis oz and its motion are ð18Þ presented in Figure 2. The method and processing of the mathematical models for the Tippe top rotation on the rod around axes are the Simplification of Equation (18) brings the following: same as for Equations (3)–(5) and presented by the following "# ÀÁÀÁ2 equations: fl cos β sin β + l cos β / 3/8 π − 2 π +1 ω = c ðÞÀÁcÀÁðÞðÞ mg : x 2 2 ðÞ3/8 ðÞπ − 2 π +ðÞ 3/8 ðÞπ − 2 +6ðÞπ − 2 π +3ðÞπ − 2 +8 /8ðÞ− 3ðÞπ − 2 Jω 19 dω 3 2 3 ð Þ J x = mgf l cos β sin β − ðÞπ − 2 π + ðÞπ − 2 Jωω x dt c 8 8 x fi − ωω η, Then, the modi ed expression of the precession torque J y ðT : = Tin: + T : Þ is defined by the same method as for 15 p x x am x ð Þ Equation (10) and presented by the following expression:

( ) ÂÃ2 3/8 π − 2 π +1 f sin β +1 l cos β = ÂÃðÞðÞ ðÞÀÁÀÁc : Tp:x 2 2 mg ½ðÞπ − 2 π +8/3ðÞðÞðÞπ − 2 ðÞ3/8 ðÞπ − 2 π +ðÞ 3/8 ðÞπ − 2 +6ðÞπ − 2 π +3ðÞπ − 2 +8 /8ðÞ− 3ðÞπ − 2 ð20Þ Advances in Mathematical Physics 7

Table 2: Technical data of the Tippe top.

Mass of Tippe top (kg) 0.070 Mass of the rod (kg) 0.005 Mass moment of inertia around the vertical axis (J kgm2) β ° Angle c (Figure 1) (degrees) 53.130102

a Vertical location of the rod Linear dimensions (mm) ÀÁ ÀÁ ÀÁ2 r = 2 2 + 1 d =1:734 × 10−5 b J 3 msR 2 mr 2 m R R =20:00; a =50:00; b =30:00; r =10:00; d =5:00.

lc k

Linear dimensions (mm) HorizontalÀÁ ÀÁ location of the rod = 2 2 + 1 2 =1:8×10−5 J 3 msR 12 mra m k =10:00; =40:00 lc .

Substituting Equation (19) and η into Equation (15) and Equation (19) into Equation (16) and transformation yields the following equations:

"#"# ω cos β 3 3 6 π − 2 2π +3 π − 2 +8 d x = cos β sin β + lc − π − 2 2π + π − 2 + ðÞ ðÞ ωω , 21 Jx flc 2 mg ðÞ ðÞ J x ð Þ dt ðÞ3/8 ðÞπ − 2 π +1 8 8 8 − 3ðÞπ − 2 ( ) ÂÃ2 dω 3/8 π − 2 π +1 f sin β +1 l cos β 3 y = − ÂÃðÞðÞ ðÞÀÁÀÁc − π − 2 ωω , Jy 2 2 mg ðÞJ y dt ½ðÞπ − 2 π +ðÞ 8/3ðÞπ − 2 ðÞ3/8 ðÞπ − 2 π +ðÞ 3/8 ðÞπ − 2 +6ðÞπ − 2 π +3ðÞπ − 2 +8 /8ðÞ− 3ðÞπ − 2 8 ð22Þ

where all components are as specified above. face, its spherical part disconnects from it, the top inverses, = cos β Equation (22) does not contain torques T fy mglc and it turns to vertical. The technical data of the Tippe top as far as the precession torque modified with the correction is presented in Table 2. The starting condition accepted the factor η contains this torque. vertical location of the top axle. The coefficients of the dry kinetic sliding and rolling friction of the top and table sur- =0:42 =0:001 faces are accepted at f and f r , respectively 3. Case Study and Working Example [19]. The angular velocity of the spinning top is accepted at 2000 rpm. The angle of the contact of the rod with the table Analysis of the mathematical model for motions of the steel fi β = Tippe top on the steel support surface with the action of is de ned from the geometry of the top that is c 53:130102° the frictional and inertial torques demonstrates the turn of (Figure 1). For simplicity of computing accepted, its rod’s axle from the vertical around axes ox and oy. This the mass moment of inertia for the sphere does not change top’s turn is considered for the two conditions. The first con- the property of the top for inversion. dition is the top spins on its spherical part and turns until the contact of the rod with the support surface. The second con- 3.1. The Numerical Solution of the Case Study. The axis of the dition is the rod of the spinning top contacts the support sur- spinning of the Tippe top remains vertical, but the rod 8 Advances in Mathematical Physics inclines then contacts the support surface, inverses, and then (1) The spinning Tippe top motions on the spherical sur- rotates on the rod. The rotation of the top’s about horizontal face around vertical axis oy (Figure 1). The angular , changes in the values of the mass moment of inertia J Ji and velocity of the top around axes of rotation is com- inertial torques Ti. For simplicity of the solution and to avoid puted for the angle of the rod contact with the sup- cumbersome computing, the values of J and Ji for the inter- port surface. The turn of the top from vertical to the mediate inclination of the top on the angle β are accepted angle of the contact of the rod with the surface is β nearest to the vertical or horizontal location. This simplifica- = 143:130102°. The top parameters defined above tion does not bring a big difference in the results of comput- and presented in Table 2 are substituted into Equa- ing because the values of J and Ji for vertical and horizontal tions (13) and (14) that yield the following equations locations are almost the same (Table 2). of the top motion around axis oy and ox

2 ° ° 3 0:42 cos ðÞ−53:130102 ×0fg:02 − 0:01 × sin ðÞ−53:130102 − 6 7 "# 6 ° 7 2 −5 dω 0:001 × 0:02ðÞ−53, 130102 + 3 2 3 6ðÞπ − 2 π +3ðÞπ − 2 +8 −5 2π 1:734 × 10 x = 6 7 ×0:07 × 9:81 − ðÞπ − 2 π + ðÞπ − 2 + ×1:734 × 10 × 2000 × × ω , 6 ° 7 8 8 8 − 3 π − 2 60 x dt 4 ðÞ0:001 × 0:02 + 0:01 ðÞ−53, 130102 5 ðÞ 2 ðÞ3/8 ðÞπ − 2 π +1 8 ÂÃ2 ° ° 9 > ðÞ3/8 ðÞπ − 2 π +1 ×0:42 × cos ðÞ−53:130102 ×0ð :02 − 0:01 × sin ðÞ−53:130102 + > > ÂÃÀÁ > <> 2 ° => −5 dω 6ðÞπ − 2 π +3ðÞπ − 2 +8 /8ðÞ− 3ðÞπ − 2 ðÞ0:001 × 0:02 + 0:01 cos ðÞ−53:130102 3 −5 2π 1:734 × 10 y = ÂÃÀÁÀÁ×0:05 × 9:81 − ðÞπ − 2 ×1:734 × 10 × 2000 × × ω : dt > π − 2 π +8/3π − 2 3/8 π − 2 2π + 3/8 π − 2 +6π − 2 2 π +3 π − 2 +8 /8− 3 π − 2 > 8 60 y >½ðÞðÞðÞðÞðÞðÞðÞðÞ ðÞ ðÞ ðÞðÞ> :> ;>

ð23Þ

Following simpliﬁcation and transformation yield the The right integral is simple, and integrals have the following equations as follows: solution:

ω dω ω x 1:734 × 10−5 x =0:006465 − 0:035698ω , ð24Þ −ln d x − ω = 2058:7082 t , 30 x 0:181102 − ω x tj0 ð Þ dt x 0

ω ω −5 d y −4 −ln 0:494579 − ω − ω y =89:653979 t , 31 1:734 × 10 =7:688728 × 10 − 0:0015546ω : y y 0 tj0 ð Þ dt y ð25Þ that gave rise to the following:

Separating variables and transformation for these diﬀer- ω −2058:7082 1 − x = e t, ð32Þ ential equations gives the following: 0:181102 ω ω −89:653979 d x = 2058:7082 , 26 1 − y = e t: 33 0:181102 − ω dt ð Þ 0:494579 − ω ð Þ x y

dω The right side of Equations (32) and (32) has the small value of y =89:653979 : 27 the high order that can be neglected. Solving these equations 0:494579 − ω dt ð Þ y yields the angular velocity for the top around axis oy and ox as the result of the action of the frictional torque. Transformation and presentation of the obtained equations ﬁ by the integral form with de ned limits yield the following: ω =0:181102 rad/s = 10:376413°/ , 34 x s ð Þ ð ð ω t x 1 ° ω = 2058:7082 , ð28Þ ω =0:494579 − ω rad/s = 28:337299 / : 35 0:181102 − ω d x dt y y s ð Þ 0 x 0 Analysis of the angular velocities of the top around axes ox ðω ð y 1 t ω =89:653979 : 29 and oy demonstrates the turn down around axis oy faster than 0:494579 − ω − ω d y dt ð Þ around axis ox, and the rod contacts with the support surface. 0 y y 0 Following the action of the inertial forces on the Tippe top lifts The left integral of equationsÐ is transformed and repre- upitscentermassanddisconnectsthesphericalpartfromthe sented by the tabulated integral dx/ða − xÞ = −ln ja − xj + C. support surface. Advances in Mathematical Physics 9

(2) The spinning Tippe top motions on the rod around Equations (21) and (22) that yield the following vertical axes ox and oy (Figure 2). The top parameters equations: deﬁned above and in Table 2 are substituted into

"#"# ω 0:04 cos −53:130102° 3 3 6 π − 2 2π +3 π − 2 +8 2π 1:734 × 10−5 d x =0:42 × 0:04 cos −53:130102° × sin 53:130102° + ðÞ×0:07 × 9:81 − π − 2 2π + π − 2 + ðÞ ðÞ ×1:734 × 10−5 × 2000 × × ω , ðÞ 2 ðÞ ðÞ x dt ðÞ3/8 ðÞπ − 2 π +1 8 8 8 − 3ðÞπ − 2 60 ( ) ÂÃ2 dω 3/8 π − 2 π +1 0:42 × sin − 53:130102° +1 ×0:04 cos −53:130102° 3 2π 1:734 × 10−5 y = ðÞðÞÂÃðÞÀÁÀÁðÞ ×0:07 × 9:81 − π − 2 ×1:734 × 10−5 × 2000 × × ω : 2 2 ðÞ y dt ½ðÞπ − 2 π +8/3ðÞðÞðÞπ − 2 ðÞ3/8 ðÞπ − 2 π +ðÞ 3/8 ðÞπ − 2 +6ðÞπ − 2 π +3ðÞπ − 2 +8 /8ðÞ− 3ðÞπ − 2 8 60 ð36Þ

Following simplification and transformation is the same as axes. The Tippe top turns in the counterclockwise direction presented above, and all comments for mathematical pro- around axes ox and oy (Equations (34), (35), (47), and cessing are omitted: (48)). The action of the inertial forces on the spinning Tippe top that has the only contact of the rod with the support sur- dω 1:734 × 10−5 x =0:012037 − 0:035698ω , ð37Þ face lifts up its center mass and the top goes to the vertical. dt x Following spinning of the Tippe top on the rod is the same as for ordinary top. dω 1:734 × 10−5 y =9:589250 × 10−4 − 0:0015546 ω , ð38Þ The time of the spinning Tippe top inversion is com- dt y puted for two conditions, i.e., for the turn from vertical to ω the angle of the contact of the rod with the support surface d x = 2058:7082 , 39 and for the following turn on the rod to vertical. Equations 0:337184 − ω dt ð Þ x (35) and (48) demonstrate the angular velocities of the Tippe dω top rotation are changed by the cosine law. For solution is y =89:653979 , 40 formulated the functional dependency of the variable angular 0:616830 − ω dt ð Þ y ω velocity from the angle of the to turn that is as follows: i ðω ð = a cos β where a is coefficient that defined for the two con- x 1 t i i ω = 2058:7082 , ð41Þ ditions, i.e., for the contact of the rod of the support surface 0:337184 − ω d x dt 0 x 0 and after disconnecting the spherical surface of the top of ðω ð y 1 t the support surface. ω =89:653979 , 42 28:337299°/ = cos −53:130102 = 0:616830 − ω d y dt ð Þ s ac ð Þ where ac 0 y 0 47:228831°/ 90°ω =0 s and for x . ω 35:341755°/ = cos −53:130102 = −ln 0:337184 − ω x = 2058:7082 t , 43 s ad ð Þ where ad jjx 0 t 0 ð Þ 58:902925°/ −90° ω =0 s and for ð Þ x .The time of the turn of ω β −ln 0:616830 − ω y =89:653979t t , ð44Þ the top on the angle can be computed by the equation: t y 0 0 = β /ω β fi i i,where i is the xed angle of the top turn down that ° ° ω −2058:7082 is β = 143:130102 and for the turn up β =90 − 1 − x = e t, ð45Þ d u 0:337184 53:130102° =36:869898°.The first derivative of variable ω angular velocity yields the following: 1 − y = −89:653979t, 46 0:616830 e ð Þ dω = −a sin βdβ: ð49Þ ω =0:337184 rad/s = 19:319220°/ , 47 i i x s ð Þ ω =0:616830 rad/s = 35:341755°/ : 48 Integral presentation of Equation (49) brings the following: y s ð Þ ð ð ω e Equations (34), (35), (47), and (48) demonstrate the top i ω = − sin β β, 50 d i ai d ð Þ rotates faster around axis oy. At the time of the contact and 0 s disconnecting of the rod with the support surface, the angu- ff ω =28:337299°/ lar velocities are di erent that are y s (Equa- where the limits of the integral s are the start angle and e is ω =35:341755°/ tion (35)) and y s (Equation (48)), the end angle of the turn of the top. respectively. This difference is explained by the change in Solution of the integral Equation (50) is presented by the the radius of the frictional torque that is l = R + r sin β and following expression: l = ðR + k + rÞ sin β and by the removing of the rolling fric- ω e = cos β e, 51 tional torque, respectively. The increase of the radius ijs ai js ð Þ increases the value of the frictional torque and hence increases the angular velocities of the top rotation around where all parameters are as specified above 10 Advances in Mathematical Physics

Substituting the initial data presented above for the top unexplainable gyroscopic motions are solved by the contem- turn down and up, and solution yields the following results: porary principles of classical mechanics, and the Tippe top ω ω ω d =47:228831° cos β −53:130102° ω d = dj0 j90° and uj0 inversion is a good example of this one. ° −90° 58:902925 cos βj−53:130102° that gave rise to the integrated angular velocities: Nomenclature

ω =47:228831 cos −53:130102° − cos 90° =28:337299°/ , a, b, c, d, h, k, r, l , R: Geometrical sizes of the top d ½ðÞ s c 52 components ð Þ m: Mass of the top g: Gravity acceleration ω =58:902925° cos −90° − cos −53:130102° =35:341755°/ : u ½ðÞ ðÞ s f : Kinetic coefficient of sliding friction ffi ð53Þ f r: Coe cient of rolling friction i: Index for axis ox or oy The time of the to turn down and up is defined by the J: Mass moment of inertia of the top = β /ω expression ti i i. Ji: Mass moment of inertia of the top around axis i 143:130102 = =5:050 , Ti: Torque i td s 28:337299 T : : Torque of the change in the angular ð54Þ am i 36:869898 momentum acting around axis i t = =1:043s: , , : u 35:341755 Tcti Tcr:i Tin i: Torque generated by centrifugal, Cor- iolis, and common inertial forces, The total time of the Tippe top inversion is as follows: respectively, and acting around axis i Tp:i: Precession torque acting around axis i = + =5:050 + 1:043 = 6:093 : 55 t td tu s ð Þ Tr:i : Resistance torque acting around axis i t: Time 4. Results and Discussion β: Angle of inclination of the rotor’s axle γ: Angle of minor inclination of the The mathematical model for the motions of the Tippe top is rotor’s axle derived on the principles of the theory of gyroscopic effects ω: Angular velocity of the rotor ω for rotating objects [17]. The inversion process of the Tippe i: Angular velocity of precession around top is the result of the action of the system of interrelated axis i. inertial torques generated by the rotating mass. The intuitive statement of the researchers that that frictional force acting Data Availability on the Tippe top resulting in its inversion is not correct [5– 16]. The frictional force acting on the spinning top activates The equation of inertial torque data used to support the find- the inertial torques of the Tippe top. The Tippe top manifests ings of this study has been deposited in the journal AIP two precession rotations around two axes which are mani- Advances that is in the process. fested by its inversion. The mathematical models for the Tippe top motions were used for computing the time of Conflicts of Interest inversion from vertical to the upside-down vertical location. Practical observation of the similar Tippe top designs con- The authors declare that they have no conflicts of interest. firms the obtained results. The time of inversion will be less for the inclined axis of the Tippe top at the starting condition Acknowledgments and for the rational size of the center mass displacement. This solution is analytical validation of the physics of the The work was supported by the Kyrgyz State Technical Uni- spinning Tippe top inversion. versity after I. Razzakov.

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