Chapter 5 Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases Hector Laos Abstract Centuries ago, the prolific mathematician Leonhard Euler (1707–1783) wrote down the equations of motion (EOM) for the heavy symmetrical top with one point fixed. The resulting set of equations turned out to be nonlinear and had a limited number of closed-form solutions. Today, tools such as transfer matrix and finite elements enable the calculation of the rotordynamic properties for rotorbearing systems. Some of these tools rely on the “linearized” version of the EOM to calculate the eigenvalues, unbalance response, or transients in these systems. In fact, industry standards mandate that rotors be precisely balanced to have safe operational characteristics. However, in some cases, the nonlinear aspect of the EOM should be considered. The purpose of this chapter is to show examples of how the linear vs. nonlinear formulations differ. This chapter also shows how excessive unbalance is capable of dramatically altering the behavior of the system and can produce chaotic motions associated with the “jump” phenomenon. Keywords Rotordynamics · Rigid body · Nonlinear · Equations of motion · Generalized forces 5.1 Definition of the Equations of Motion The EOMs are defined using the Lagrange equations [1]: d dt ∂L ∂ ˙qi − ∂L ∂qi =Fqi , (5.1) where the Lagrangian L (L=TD−UD)is defined as the difference between the kinetic energy (TD) and the potential energy (UD). Also, i (1 ≤i ≤N) is the number of degrees of freedom, the generalized independent coordinates are qi, and the generalized forces are Fqi . The generalized coordinates in the vertical rigid rotor are u, v, θ, ψ, as shown in Fig. 5.1. The Newtonian approach (not the Lagrangian) will be used for the external forces including springs and dampers Fqi =−Fu, −Fv, −Fθ, −Fψ and for the effect of the rotor weight. This will cancel out the potential energy (UD) in the Lagrangian, and the expression in Eq. (5.1) is reduced to the following [2]: 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_5 39
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