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Nonlinear and linearised behaviour of the Levitron®

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Abstract

The paper presents a magneto-rotordynamic model that allows to obtain the nonlinear equations of motion for all rigid body modes of the Levitron®. These equations describe the complete dynamic behaviour of the spinning top and highlight the presence of stability fields related to its spin speed and vertical position of levitation. The developed model takes into account the effect of the aerodynamic drag torque on the sides of the spinning top too. An experimental investigation has been developed to experimentally identify the complex dynamic behaviour of the spinning top; a dedicated test bench and some tests are discussed. By means of the numerical integration of the equations of motion, the spatial trajectories of the spinning top have been computed and validated by comparison with the experimental test results.

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Abbreviations

L :

Lagrangian function of a dynamic system

q i , \( \dot{q}_{i} \) :

i-th generalized coordinate displacement and velocity

Q i :

Generalized force applied to the i-th generalized coordinate

U :

Potential energy

\( {\varvec{\mathcal{M}}} \), \( {\mathbf{B}} \) :

Residual magnetization vector and magnetic flux density vector

F, T :

Magnetic force and torque vectors

∇, ×:

Rotor operator, external product

M, I, G :

Linear mass, inertial and gyroscopic matrices

C tr , C rot :

Linear translational and rotational damping matrices

K tr , K rot , K c :

Linear translational, rotational and coupling stiffness matrices

I nl , R nl , S nl , T nl , Q nl :

Nonlinear matrices and vectors

B x , B y , B z , J B :

Components of the magnetic flux density vector and Jacobian matrix

x, y, z, ψ, χ, φ :

Displacement and rotational degrees of freedom of the spinning top

\( \bar{x},\,\,\,\bar{y},\,\,\,\bar{z},\,\,\,\bar{\psi },\,\,\,\bar{\chi },\,\,\bar{\varphi } \) :

Displacement and rotational degrees of freedom of the spinning top referred to its equilibrium position

m, I P , I T :

Mass, polar and transversal inertia of the spinning top

k tr , k rot , k c :

Stiffness terms of translational, rotational and coupling behaviour

c tr , c rot :

Damping terms of translational and rotational behaviour

A 0, A 1, A 2, B 0, D 0, D 1 :

Taylor coefficients

B r , M :

Residual magnetic flux density of the base magnet and magnetization of the spin volume

V, S, A :

Respectively magnetic volume, static moment and area of the spinning top

g :

Constant of gravity

l g :

Distance between centre of gravity and magnetic volume centre of the spinning top

e, i, π:

Respectively Nepero number, imaginary operator, pi-greco

q :

Planar complex coordinate vector

r, η :

Planar complex translational and rotational coordinates of the spinning top referred to its equilibrium position

M pl , G pl , C pl , K pl :

Linear mass, gyroscopic, damping and stiffness matrices for the planar behaviour

s, σ, λ :

Eigenvalue, decay rate and natural frequency

M a :

Aerodynamic drag torque

C m :

Non-dimensional coefficient of the aerodynamic drag torque

R e :

Reynolds number

r o  = 15 mm:

Outer diameter of the spinning top

ω :

Angular speed of the spinning top

ρ a  = 1.225 kg/m3 :

Air density

ρ :

Material density

μ a  = 1.81 × 10−5 kg/m/s:

Air dynamic viscosity (normal condition of 20 °C)

c v  = 1.07 × 10−8 Nms/rad:

Equivalent viscous damping coefficient

x eq  = 29.6 mm:

Reference vertical equilibrium levitating position of the spinning top

ω min , ω max :

Minimum and maximum angular speed stability limits

ω x :

Fundamental harmonic of the vertical oscillatory behaviour

x min , x max :

Minimum and maximum vertical stability limits

t, T :

Time and simulation time

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Authors and Affiliations

  1. Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy

    Elvio Bonisoli & Cristiana Delprete

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  1. Elvio Bonisoli
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  2. Cristiana Delprete
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Correspondence to Elvio Bonisoli.

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Bonisoli, E., Delprete, C. Nonlinear and linearised behaviour of the Levitron® . Meccanica 51, 763–784 (2016). https://doi.org/10.1007/s11012-015-0238-5

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