4 Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed 31 4.3 The Cubic Polynomial f(u) The classical EOM is of the form: u u0 du f u =t,wheret is time (s) [6] and is typically shown on the literature as a derivation from the conservation of energy equation. The roots of this cubic polynomial f (u) furnish the angles at which ˙θ changes in sign; in other words, the extreme values of the nutation (θ) trajectory. The next section shows the importance of selecting the most suitable value of the root u3 to facilitate the creation of a closed-form solution. The author of this paper used a novel approach to formulate f (u) as shown in the following equations. The general expression shown in (4.4) could be further reduced by making the angular momentums equal (i.e., p=pφ = pψ and then substituting in (4): ˙θ ˙θ0=0 ˙θd˙θ = θ θ0 p 2 I1 2 (1−cosθ) sinθ (cosθ −1) 1−cos2θ + Mgl I1 sinθ dθ (4.5) The parameter qis defined as follows: p I1 2 =q Mgl I1 , (4.6) and (4.5) is solved as 1 2 ˙θ 2 = θ θ0 q Mgl I1 1 sinθ (cosθ −1) (1+cosθ) + Mgl I1 sinθ dθ (4.7) ˙θ 2 = 2Mgl I1 θ θ0 q (cosθ −1) sinθ (1+cosθ) + sinθ dθ (4.8) Resolving (4.8) in terms of a new variable u= cos θ is convenient, as shown on the next line: ˙θ 2 sin2 θ = ˙u 2 = β sin 2 θ θ θ0 q (cosθ −1) sinθ (1+cosθ) + sinθ dθ, (4.9) where β = 2Mgl I1 and the expressions (4.10), (4.11), and (4.12) are used: sin2 θ = 1−cos 2 θ = 1−u 2 (4.10) θ θ0 q (cosθ −1) sinθ (1+cosθ) = −q (1+cosθ) + q (1+cosθ0) (4.11) θ θ0 sinθ dθ =−cosθ +cosθ0 (4.12)
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