TY - GEN
T1 - Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed
AU - Laos, Hector
N1 - Publisher Copyright:
© 2022, The Society for Experimental Mechanics, Inc.
PY - 2022
Y1 - 2022
N2 - The equations of motion (EOM) for the heavy symmetrical top with one point fixed are highly nonlinear. The literature describes the numerical methods that are used to resolve this classical system, including modern tools, such as the Runge−Kutta fourth−order method. Finding the derivate of closed-form solutions for the EOM is more difficult and, as mentioned in the literature, discovering the solution is not always possible for all the EOM. Fortunately, a few examples are available that serve as a guide to move further in this topic. The purpose of this paper is to find a methodology that will produce the solutions for a given subset of EOMs that fulfill certain requisites. This paper summarizes the literature available on this topic and then follows with the derivation of the EOM using the Euler−Lagrange method. The Routhian method will be used to reduce the size of the expression, and it continues with the formulation of the classical cubic function, f(u), through a novel process. The roots of f(u) are of the utmost importance in finding the EOM closed-form solution, and once the final roots are selected, the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process, and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.
AB - The equations of motion (EOM) for the heavy symmetrical top with one point fixed are highly nonlinear. The literature describes the numerical methods that are used to resolve this classical system, including modern tools, such as the Runge−Kutta fourth−order method. Finding the derivate of closed-form solutions for the EOM is more difficult and, as mentioned in the literature, discovering the solution is not always possible for all the EOM. Fortunately, a few examples are available that serve as a guide to move further in this topic. The purpose of this paper is to find a methodology that will produce the solutions for a given subset of EOMs that fulfill certain requisites. This paper summarizes the literature available on this topic and then follows with the derivation of the EOM using the Euler−Lagrange method. The Routhian method will be used to reduce the size of the expression, and it continues with the formulation of the classical cubic function, f(u), through a novel process. The roots of f(u) are of the utmost importance in finding the EOM closed-form solution, and once the final roots are selected, the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process, and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.
KW - Closed-form solutions
KW - Cubic polynomial f(u)
KW - Equations of motion
KW - Gyroscopes
KW - Routhian
UR - http://www.scopus.com/inward/record.url?scp=85116211115&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-75914-8_4
DO - 10.1007/978-3-030-75914-8_4
M3 - Conference contribution
AN - SCOPUS:85116211115
SN - 9783030759131
T3 - Conference Proceedings of the Society for Experimental Mechanics Series
SP - 29
EP - 38
BT - Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021
A2 - Epp, David S.
PB - Springer
T2 - 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021
Y2 - 8 February 2021 through 11 February 2021
ER -