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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4.3. Multibody with flexible components The more complex model developed for the simulation of the dynamic behavior of the pump is a fully flexible multibody model that is capable to perform transient analysis. For this aim the software ANSYS Workbench was employed, setting all the components as flexible. In order to take into account the flexible behavior in multibody simulation the Craig-Brampton approach is widely applied because it allows to reduce the DOF of the entire system combining the static (Guyan reduction) and dynamic (CMS, Component Mode Synthesis) condensation techniques, as explained by Géradin et al. (2001). Firstly the master or interface nodes (subscript m) and the slave or internal (subscript s) must be defined for each component so the overall DOF are in the vector = [ ] whose dimension is n . By applying the static and dynamic reduction the overall DOF is reduced to ̌ = [ ] where contains the participation coefficients corresponding to the first s modes computed keeping the master nodes fixed. Thus the original n DOF are reduced to ( + ) with p static modes and s components modes, with ( + ) ≪ and the dynamic equation of motion, similar to the previous ones, is ̌ ̌̈ + ( ̌ − Ω ̌ ) ̌̇ + ̌ ̌ = ̌ (10) Joint element types are the most common tools used for defining the connections among the components and the supports in multibody analysis. In this case, a fixed joint was used for the connections shaft-rotor, shaft-insert, insert-impeller, a bushing for the shaft-ground connection. The bushing-joint is very suitable for modeling an elastic support because it allows to set some degrees of freedom free (such as the rotation along the axis) but at the same time it completely describes the stiffness and the damping properties with suitable coefficients (also a non-linear relationship between stiffness and displacement can be included if needed). One of the greatest advantages related to this approach is the capability to include any load time-history by importing it in the program and, in addition, to easily and rapidly include unbalances thanks to the option “ Follower Load . ” The proper choice of the time integration step strictly depends on the characteristic time of the applied loads. Figure 9 shows some features of the multibody model.

Fig. 9. Reduced model with condensed parts and intefaces on the left, applied forces: rotating unbalances and sinusoidal forces at the bearings (frequency=BPFO, in antiphase) on the right. After solving the reduced dynamic system (10), the “ Expansion Pass ” procedu re is needed in order to obtain the results for all the nodes. Once this process is ended, the program can provide stress and strain field over time in all the components, nodal displacements and deflection of the axis, time-history of the forces on bearings and the mutual actions of the bodies upon each other. Even if a huge amount of information can be evaluated, the computational time required is very high compared to the previous methods (around 7 to 8 hours with a common PC) and so this approach is suggested only for a detailed analysis in the last design phases. In addition, ANSYS post-processing phase has been integrated with Matlab functions that provide the radial fluctuations of the selected nodes (in Fig. 10 an example related to a node on the rotor), to directly evaluate the