38 H. Laos – The angular momentump=pφ =pψ is equal. – The last root of the cubic polynomial has a value of 1 (u3 =1), and a link between the parameters q and u0 is created (4.14): u3 = q (1+u0) − 1 =1 – Therefore, all the parameters p, q, and u0 are interconnected with each other (4.6): p I1 2 =q Mgl I1 . – Closed-form solutions are defined for all the positive values of the parameter h, where h defines the value of the initial angle θ0 for the calculations according to (4.18): u0 =f(h) = h−1 h , for h≥1. The results shown in this paper are only defined for the positive value of ˙u(or ˙u(+)). Therefore, the solutions only cover a region of the total trajectory of the nutation angle θ. The percentual error (%) is higher at lowθ values. For practical purposes, this could be considered an ideal situation. What started as research for closed-form solutions to check the numerical results for the EOMs of the heavy symmetrical top with one point fixed has evolved into developing a full-procedure to produce closed-form solutions for a subset of EOMs that are able to fulfill a series of conditions. A practical application of these results may be to incorporate formulas like (4.37) and (4.39) into the controls of gyroscopic systems. References 1. Euler, L.: Du mouvement de rotation des corps solides autour d’un axe variable. Histoire de l’Academie Royale des Sciences. 14, 154–193 (1765) 2. Routh, E.J.: The Advanced Part of A Treatise on the Dynamics of a System of Rigid Bodies. MacMillan, London (1884) 3. Klein, F., Sommerfeld, A.: The Theory of the Top, vol. II. Birkhauser, Boston (2010) 4. Lagrange, J.L.: Mécanique Analytique, vol. 2. Mme Vve Courcier, Paris (1815) 5. Marion, J.B., Thornton, S.T.: Classical Dynamics of Particles and Systems, 4th edn. Saunders College Publishing)., Chapter 11 (1995) 6. Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd edn. Pearson, ISBN 9780201657029 (2001) 7. Cline, D.: Variational Principles in Classical Mechanics, vol. 206, 2nd edn. University of Rochester (2019) 8. Udwadia, F.E., Han, B.: Synchronization of multiple chaotic gyroscopes using the fundamental equation of mechanics. ASME J. Appl. Mech. 75, 1–10 (2008) 9. MacMillan, W.D.: Dynamics of Rigid Bodies, vol. 245. Dover Publications, New York (1960) 10. Fetter, A.L., Waleka, J.D.: Theoretical Mechanics of Particles and Continua, vol. 172. Dover Publications (2003) 11. More details of the toy gyroscope can be found at: www.gyroscopes.com 12. Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products, 7th edn. Elsevier (2007)
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