Nonlinear dynamics of the rattleback: a nonholonomic model

Chaotic dynamics in the problem of the fall of a screw-shaped body in a fluid

“…В качестве секущей в фазовом пространстве была выбрана поверхность p 3 = 0. При расчете бифуркационных диаграмм применяют схему с наследованием начальных условий для ускорения сходимости [33], однако численные расчеты показали, что при од-них и тех же значениях параметров сосуществуют различные аттракторы. В связи с этим при построении бифуркационной диаграммы осуществлялось сканирование с различными начальными условиями ε ∈ [0, π/2].…”

Understanding reversals of a rattleback

“…This system has two first integrals F 1 = γ, γ , F 2 = 1 2 M, ω − mg a, γ , which are used for checking accuracy of calculations. Since the direction of the unit vector γ at the contact point A is connected to the vector a pointing from the geometrical center of a whole tri-axial ellipsoid F(a) = , allows for expressing a in terms of γ [9,7]. Our vector a = (a 1 , a 2 , a 3 ) pointing from CM toward the contact point A, after taking into account displacement of the mass center for the halv-ellipsoid due to adding two mass points m p , has coordinates…”

“…(3.2). If the angular momentum of the body with respect to the contact point is M = Iω + ma × (ω × a) , then these equations can be written in a more compact formṀ = M × ω + mȧ × (ω × a) + mga × γ,γ = ω × γ (see [9,7]). This system has two first integrals F 1 = γ, γ , F 2 = 1 2 M, ω − mg a, γ , which are used for checking accuracy of calculations.…”

“…In a more general framework of nonholonomic systems on a Lie algebra change of stability of stationary motions and reversion of direction of motion has been discussed in [14]. Further references about rattleback dynamics may be found in review papers [9,7].…”

Rolling resistance modelling in the Celtic stone dynamics

“…The first scientific work on rattleback was published at the end of the 19th century [17] and it has been an object of interest till today [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], both from the point of view of dynamics and of modelling. During modelling and analysis of the Celtic stone, rolling without sliding is often assumed, leading to non-holonomic constraints [17, 21, 22, 24-26, 29, 31].…”