PSI - Issue 24

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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. Models for rotordynamic analysis In order to study the dynamic characteristics of the pump different models, with an increasing level of complexity were developed. They will be discussed in the following subsections. 4.1. Rigid model The firstly developed model is the rigid one (Fig. 3); the only elements with finite stiffness are the supports that are assumed as isotropic and each of them is modeled by two perpendicular linear springs in the transverse plane. To correctly compute the equivalent stiffness of the supports a series of elastic elements should be considered: the bearings, the bushing, the casing and finally the stiffness of the connection to the frame; however, in this particular case, the only bearing stiffness was considered as a reasonable hypothesis. The system degrees of freedom (DOF) u related to the lateral dynamics are the components of translation and rotation around the global center of mass G: = [x G , y G , θ X G , θ Y G ] T (1)

Fig. 3. The rigid model: all the components are assumed as rigid and each support is modeled by two springs.

Assuming a constant rotational speed Ω and small displacements the system is governed by 4 ordinary differential equations: m ̈ G + 2k G + k(a 2 − a 1 ) Y G = X ( ) m ̈ G + 2k G + k(a 1 − a 2 ) X G = Y ( ) I d ̈ X G + I p Ω ̇ y + k(a 1 − a 2 ) G + k(a 2 1 + a 2 2 ) X G = X ( ) I d ̈ Y G − I p Ω ̇ y + k(a 2 − a 1 ) G + k(a 2 1 + a 2 2 ) Y G = Y ( ) (2) where m indicates the total mass, I d and I p are respectively the diametric and the polar moment of inertia and k the radial stiffness of the bearings. X ( ) and Y ( ) are the components of the resultant load in X and Y direction while X ( ) and Y ( ) are the components of the resultant moment (due to unbalance, fluid dynamics forces and bearing