liu.seSök publikationer i DiVA
Ändra sökning
RefereraExporteraLänk till posten
Permanent länk

Direktlänk
Referera
Referensformat
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Annat format
Fler format
Språk
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Annat språk
Fler språk
Utmatningsformat
  • html
  • text
  • asciidoc
  • rtf
Tippe Top Equations and Equations for the Related Mechanical Systems
Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
2012 (Engelska)Ingår i: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 8, nr 19Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

The equations of motion for the rolling and gliding Tippe Top (TT) are nonintegrable and difficult to analyze. The only existing arguments about TT inversion are based on analysis of stability of asymptotic solutions and the LaSalle type theorem. They do not explain the dynamics of inversion. To approach this problem we review and analyze here the equations of motion for the rolling and gliding TT in three equivalent forms, each one providing different bits of information about motion of TT. They lead to the main equation for the TT, which describes well the oscillatory character of motion of the symmetry axis 3ˆ during the inversion. We show also that the equations of motion of TT give rise to equations of motion for two other simpler mechanical systems: the gliding heavy symmetric top and the gliding eccentric cylinder. These systems can be of aid in understanding the dynamics of the inverting TT.

Ort, förlag, år, upplaga, sidor
Kyiv: National Academy of Science of Ukraine , 2012. Vol. 8, nr 19
Nyckelord [en]
tippe top; rigid body, nonholonomic mechanics, integrals of motion, stability, gliding friction
Nationell ämneskategori
Matematisk analys
Identifikatorer
URN: urn:nbn:se:liu:diva-76924DOI: 10.3842/SIGMA.2012.019ISI: 000303833100001OAI: oai:DiVA.org:liu-76924DiVA, id: diva2:523761
Tillgänglig från: 2012-04-26 Skapad: 2012-04-26 Senast uppdaterad: 2017-12-07
Ingår i avhandling
1. Analysis of Dynamics of the Tippe Top
Öppna denna publikation i ny flik eller fönster >>Analysis of Dynamics of the Tippe Top
2013 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

The Tippe Top is a toy that has the form of a truncated sphere with a small peg. When spun on its spherical part on a flat supporting surface it will start to turn upside down to spin on its peg. This counterintuitive phenomenon, called inversion, has been studied for some time, but obtaining a complete description of the dynamics of inversion has proven to be a difficult problem. This is because even the most simplified model for the rolling and gliding Tippe Top is a non-integrable, nonlinear dynamical system with at least 6 degrees of freedom. The existing results are based on numerical simulations of the equations of motion or an asymptotic analysis showing that the inverted position is the only asymptotically attractive and stable position for the Tippe Top under certain conditions. The question of describing dynamics of inverting solutions remained rather intact.

In this thesis we develop methods for analysing equations of motion of the Tippe Top and present conditions for oscillatory behaviour of inverting solutions.

Our approach is based on an integrated form of Tippe Top equations that leads to the Main Equation for the Tippe Top (METT) describing the time evolution of the inclination angle $\theta(t)$ for the symmetry axis of the Tippe Top.

In particular we show that we can take values for physical parameters such that the potential function $V(\cos\theta,D,\lambda)$ in METT becomes a rational function of $\cos\theta$, which is easier to analyse. We estimate quantities characterizing an inverting Tippe Top, such as the period of oscillation for $\theta(t)$ as it moves from a neighborhood of $\theta=0$ to a neighborhood of $\theta=\pi$ during inversion. Results of numerical simulations for realistic values of physical parameters confirm the conclusions of the mathematical analysis performed in this thesis.  

Ort, förlag, år, upplaga, sidor
Linköping: Linköping University Electronic Press, 2013. s. 24
Serie
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1500
Nationell ämneskategori
Matematisk analys
Identifikatorer
urn:nbn:se:liu:diva-88316 (URN)978-91-7519-692-3 (ISBN)
Disputation
2013-02-26, BL32 Nobel, Hus B, Campus Valla, Linköping University, Linköping, 10:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2013-02-01 Skapad: 2013-01-31 Senast uppdaterad: 2019-12-03Bibliografiskt granskad

Open Access i DiVA

Fulltext saknas i DiVA

Övriga länkar

Förlagets fulltext

Person

Rutstam, Nils

Sök vidare i DiVA

Av författaren/redaktören
Rutstam, Nils
Av organisationen
Tillämpad matematikTekniska högskolan
I samma tidskrift
SIGMA. Symmetry, Integrability and Geometry
Matematisk analys

Sök vidare utanför DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetricpoäng

doi
urn-nbn
Totalt: 191 träffar
RefereraExporteraLänk till posten
Permanent länk

Direktlänk
Referera
Referensformat
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Annat format
Fler format
Språk
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Annat språk
Fler språk
Utmatningsformat
  • html
  • text
  • asciidoc
  • rtf