Chapter 4 Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed Hector Laos Abstract 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. Keywords Gyroscopes · Closed-form solutions · Equations of motion · Routhian · Cubic polynomial f (u) 4.1 Background In the mid-1700s, Euler made a great contribution to the dynamics of the rigid body with the first solution for the heavy symmetrical top with one point fixed [1]. In the following years, many authors continued using the Euler equations [2, 3] alongside Newtonian mechanics and created the basis for gyroscopes and their applications. Currently, the modern books of classical mechanics use the Lagrangian [4] method because it greatly simplifies the derivation of the equations of motion (EOM) [5, 6]. Even further simplification is obtained using the Routhian method [7], as shown in Udwadia and Han’s [8] application of the Routhian. For the topic of closed-form solutions for the heavy symmetrical top with one point fixed, the author has only two references available, MacMillan [9] and Fetter [10], and these sources show that this topic requires further research. The formulas derived in this paper will hopefully serve the purpose of checking the results from numerical calculations and can also be used as a component of the controls for a gyroscope system. This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http:// energy.gov/downloads/doe-public-access-plan). H. Laos ( ) Oak Ridge National Laboratory, Oak Ridge, TN, USA e-mail: laoshe@ornl.gov © The Society for Experimental Mechanics, Inc. 2022 D. S. Epp (ed.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75914-8_4 29
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