Computational Solution Techniques

even further from the axis of rotation, thereby increasing the imbalance. This geometrical nonlinearity is not accounted for in the approach described above, as the imbalance force is considered to be a function of time

U=U(t) , but not of displacement U=U(t, q) . (2.8) Depending on the rotor system to be analyzed, the assumption of a small imbalance may or may not be be appropriate. In many cases, the rotor is well balanced or the lateral displacement is limited by the housing or adjacent parts. In other cases, like in the rotor system presented in Chapter 3, it is essential to incorporate the geometrical nonlinearity caused by lateral displacement due to the imbalance force.

2.2 Computational Solution Techniques

A non-problem-specific summary of solution techniques for Equations (2.1) and (2.2) can be found in [144]. Generally speaking, it is important to distinguish between solving the analysis in the time domain by direct integration of the equilibrium equations or by solving the problem in the frequency domain, for example, by using modal computations [37]. In what follows, the two different approaches will be outlined briefly with regard to peculiarities that arise from rotation.

2.2.1 Solution in the Frequency Domain

A frequency solution is only possible when the system matrices are time-independent. Sum-marizing the previous section, this is only true if the following assumptions are met:

• the rotor has a well defined axis of rotation,

• the rotor is rotating at a constant spin speed Ω₀ (steady-state),

• all rotations and displacements are small except for the rotation about the axis of rotation,

• the imbalance of the rotor is small,

• either the rotor or the supports show isotropic stiffness behavior,

• in an inertial reference frame, the rotor needs to be axial symmetric.

In other words, a solution in the frequency domain is only possible with linear systems (or at least linearized systems) operating in a steady state.

2 Selected Theoretical Foundations of Rotordynamics

Most frequency domain approaches use the eigenvalues and mode shapes to calculate the free and the forced vibrations of a rotor system. The eigenvalues (λ) and mode shapes (ϕ^R) are obtained by solving the characteristic polynomial of Equations (2.1) and (2.2), which is in the inertial frame:

λ²M+λ(D+G(Ω)) + (K+N(Ω))

ϕ^R=0 (2.9)

Equation 2.9 describes a quadratic eigenvalue problem that is difficult to solve directly. One solution is to transform the equation of motion from the configuration space into a state-space representation, thereby converting the system of n coupled second-order differential equations into a set of 2n first-order differential equations [91]:

A ˙z+Bz=R (2.10)

The characteristic polynomial to calculate the eigenvalues (λs) and mode shapes (ψ^R) be-comes accordingly:

[λ_sA+B]ψ^R =0 (2.13)

Due to the non-symmetrical system matrices, which are a special characteristic of rotordy-namic analyses, the eigenvalues and mode shapes appear in conjugate complex pairs. Solv-ing Equation (2.13) therefore requires a complex eigensolver, which is nowadays available in many but not necessarily in all programs.

A common alternative to converting the problem description into a state-space representation is to use the subspace-projection method in order to extract the complex eigenvalues and complex mode shapes (see i.e. [1, 23, 105]). Here, in a first step, the symmetric eigenvalue problem is solved by ignoring matricesDandGin Equations (2.1), respectively, matrixC in Equation (2.2), as well as any unsymmetric contributions to the stiffness matrix K. In doing so, the eigenvalues λ_r of the reduced system become purely imaginary numbers λ_r =iω, in

2.2 Computational Solution Techniques

which ω stands for the undamped natural frequencies, and the eigenvalue problem is now:

−ω²M+K

ϕ_r =0 (2.14)

In the next step, the following transformation prescription is used to project the original matrices on the subspace spanned by the real eigenvectors ϕ_r of the reduced system:

M_r = [ϕr,1,ϕ_r,2, ...,ϕ_r,n]^T M[ϕr,1,ϕ_r,2, ...,ϕ_r,n] D_r = [ϕr,1,ϕ_r,2, ...,ϕ_r,n]^T D[ϕr,1,ϕ_r,2, ...,ϕ_r,n] K_r = [ϕr,1,ϕ_r,2, ...,ϕ_r,n]^T K[ϕr,1,ϕ_r,2, ...,ϕ_r,n] G_r(Ω) = [ϕr,1,ϕ_r,2, ...,ϕ_r,n]^T G(Ω) [ϕr,1,ϕ_r,2, ...,ϕ_r,n] N_r(Ω) = [ϕr,1,ϕ_r,2, ...,ϕ_r,n]^T N(Ω) [ϕr,1,ϕ_r,2, ...,ϕ_r,n]

(2.15)

Now, the projected eigenvalue problem can be expressed in the following form:

λ²_rM_r+λ_r(Dr+G_r(Ω)) + (Kr+N_r(Ω))

φ^R_r =0 (2.16)

Since rotordynamics is typically interested in the lower vibration modes, high-frequency modes can be ignored, thereby reducing the problem size. “Typically, the number of eigen-vectors is relatively small; a few hundred is common” [23]. This reduction method is called Rayleigh-Ritz condensation by some authors [19]. Equation (2.16) can be solved by using the QZ algorithm, which is a solution method for a generalized unsymmetrical eigenvalue problem [93]. The such obtained complex eigenvalues λ_r of the projected system are an ap-proximation of the eigenvalues λ of the original system in Equation (2.9). The eigenvectors of the original system are approximated likewise by:

ϕ^R_k = [ϕr,1,ϕ_r,2, ...,ϕ_r,n]φ^R_r,k (2.17) where ϕ^R_k is the approximation of the k-th eigenvector of the original system.

2.2.2 Solution through Direct Time Integration

The previous chapter summed up the requirements that need to be fulfilled, or rather the simplifications that need to be acceptable, when performing the rotordynamic analysis in the frequency domain. In the case of a nonlinear system or in a transient analysis, the solution must be obtained through direct time integration. Since the equation of motion of

2 Selected Theoretical Foundations of Rotordynamics

a rotating system contains matrices that depend on the spin-speed, and, in general, contains additional matrices compared to a structure at rest, this is computationally very costly, but is sometimes unavoidable.

The most commonly used numerical integration algorithms for rotordynamics are the New-mark family methods and the Wilson-θ method [14]. The former is widely used for implicit integration in terms of the average acceleration method, as well as for explicit integration in terms of the central difference method.