40 H. Laos Fig. 5.1 Rigid-body rotor d dt ∂TD ∂ ˙qi − ∂TD ∂qi =Fqi (5.2) For the sake of clarity, the definition of the terms in Eq. (5.2), TD and Fqi , will be explained in different sections. 5.2 Kinetic Energy of a Rigid-Body Rotor The kinetic energy of a rigid-body rotor is as follows [3]: TD = 1 2 md ˙u 2 + ˙ v 2 + ˙ w 2 + 1 2 Id ω 2 x +ω 2 y + 1 2 Ip ω 2 z. (5.3) Figure 5.1 shows the vertical rigid-body rotor with its system of coordinates, the location of the center of gravity (CG), and the parameters that will be used to define the displacements and rotations at the CG. In the present work, only the lateral displacements—u, v—and the rotations—θ, ψ, φ—were considered. All lateral displacement and rotations coincide with the CG. The axial movement wis assumed to be decoupled from the lateral displacements. Therefore, ˙wwill not be considered any further in this chapter. The rotor spin angle φ will be kept because the transient unbalance is a function of φ and its derivatives. The angular velocity vector {ω} will be defined as a function of the angular velocities of the Euler angles: ˙θ, ˙ψ, ˙φ. {ω}T = ωx ωy ωz (5.4) The process consists of a series of rotations starting from an initial axis {Y} as shown in the following sequence [3–5]: {ω}= ˙ψ {Y}+ ˙θ {x1}+ ˙φ, (5.5) and
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