PSI - Issue 24

8

Vincenzo D’Addio et al. / Procedia Structural Integrity 24 (2019) 510–525 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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4.2. Model with flexible shaft The best way to include the flexibility of the shaft is by building a FEM model where the shaft is meshed with beam elements and all the other rotating parts are fixed on it (such as disks, rotors and bladed impellers) as concentrated point masses having the corresponding inertial properties, ref. Krämer (1993). In particular, in the present work the FEM model has been set up using the commercial software ANSYS in the environment Mechanical APDL, referring to ANSYS Inc. (2012) rotordynamics guide; the model is shown in Fig. 5

where the following elements are employed:  beam elements (Beam188) for the shaft;

 mass elements (Mass21) for the rotor and the impeller;  spring-damper elements (Combin14) for the supports.

Fig. 5. Flexible shaft: FEM model scheme. The point masses contribute to form the mass and gyroscopic matrices, and , while the spring-damper elements for the supports involve the damping and the stiffness ones, and . Indicating with = [ 1 … … ] the global vector of the nodal displacements, the equation of the entire rotating system in a matrix form can be written as (3), where the matrices dimensions depend on the number of nodes. To evaluate modes and natural whirl frequencies of the system in free motion, the forcing terms are set equal to zero, i.e. p = 0 in (3); looking for solutions such as = ̅e jω n and simplifying by dividing by e jω n the system (3) becomes the eigenvalue problem: [−[ ω n 2 + ( − Ω )jω n + ] ̅ = (5) Figure 6 presents some of the rotor first modal shapes at the operating speed  . It is noteworthy that the first two modes are substantially rigid modes mainly influenced by the compliance of the supports while the higher modes are greatly influenced by the shaft flexibility. The solution of (5) for different values of the rotational speed Ω gives the corresponding natural whirl frequencies in the Campbell diagram. Figure 7 shows an example of such a diagram: the colored and dashed curves represent the trend of the BW frequencies while the continuous ones the FW frequencies (the splitting being related to the gyroscopic effect); instead the bundle of black lines passing by the origin represents the fundamental frequencies of the principal excitations of the system. As a result, the intersections between the curves of the natural frequencies and the lines of the excitations can be potential critical working points: the primary (intersections with 1X) and the secondary critical speeds (intersections with nX).