5 Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases 43 where the matrices ([M], mass, [G], gyroscopic, [C], damping, and [K], stiffness) and vector ({q}, displacement) from Eqs. (5.25) and (5.26) are defined using Eqs. (5.13), (5.14), (5.15), and (5.16) and the linearized form of Eqs. (5.21), (5.22), (5.23), and (5.24): {q}T =[u v θ ψ] (5.27) [M] = ⎡ ⎢⎢ ⎣ md 0 0 md 0 0 0 0 0 0 0 0 Id 0 0 Id ⎤ ⎥⎥ ⎦ (5.28) [G] = ⎡ ⎢⎢ ⎣ 0 0 0 0 0 0 0 0 0 0 0 0 0 Ip −Ip 0 ⎤ ⎥⎥ ⎦ (5.29) [C] = ⎡ ⎢⎢ ⎣ cxT 0 0 cyT 0 cxC −cyC 0 0 −cyC cxC 0 cyR 0 0 cxR ⎤ ⎥⎥ ⎦ (5.30) [K] = ⎡ ⎢⎢ ⎣ kxT 0 0 kyT 0 kxC −kyC 0 0 −kyC −mdg kxC +mdg 0 kyR 0 0 kxR ⎤ ⎥⎥ ⎦ (5.31) The effect of weight has also been included in the stiffness matrix, [K]. Finally, the unbalance vector {U} [6] is {U}= ⎧ ⎪⎨ ⎪⎩ mεΩ2 cos (Ωt +δ) mεΩ2 sin(Ωt +δ) − Id −Ip βΩ2 sin(Ωt +γ) Id −Ip βΩ2 cos (Ωt +γ) ⎫ ⎪⎬ ⎪⎭ . (5.32) The eccentricity (ε) of the rotor creates an unbalance (mε). If the axis of rotation has an angle β relative to its geometric axis, it creates a rotational unbalance of magnitude (Id −Ip) β. No angular acceleration component exists at steady-state conditions, but this effect will be considered in the solution of the nonlinear cases. The phase angles δ and γ are used to provide correlation between the unbalance and rotational unbalance. 5.5 Results from the Linear Example Table 5.2 contains the rotor data that was used for the examples and came from Friswell et al. [7]. The state-space technique was used for the solution of the eigenvalues [8]. Figure 5.2 shows the Campbell diagram for the operational range from 0 to 3800 rpm. The intersection points correspond to the vertical rigid-body eigenvalues. The X value denotes the speed ( ) at which the eigenvalue was calculated, and the Y value is the actual eigenvalue result.
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