PSI - Issue 24

7

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

516

defects). In (2) the damping terms introduced by the supports are omitted just to have a shorter expression. Adopting a compact matrix form and including damping, the system (2) can be written as ̈ + ( − Ω ) ̇ + = (3) where the dimensions of the matrices are 4x4. Applying the Laplace Transform the differential system (3) in the time domain is transformed into an algebraic system in the complex variable s. Considering for simplicity initial null conditions one obtains: (s) = (s) (s) (4) where (s) is the transfer function matrix of the system. For s=jω the 4x4 frequency response matrix is obtained. For example, in Fig. 4 the magnitude plots of |H 14 | and |H 11 | are respectively shown with peaks corresponding to the natural frequencies; due to the gyroscopic effect, the first natural whirl frequencies related to the conical motions of the axis are split into two different values: the lower one for the backward whirl (BW) conical motion and the higher one for the forward whirl (FW). On the other hand, the natural whirl frequencies related to the cylindrical motions have almost the same value for both BW ad FW precessions since the gyroscopic moment does not affect them; in other words the respective poles coincide. The plot axes are dimensionless for confidentiality reasons and, in particular, frequencies and rotational speeds, thoughout the paper, are reported as ratios of the pump operating rotational frequency or speed.

|H11|

|H14|

̅

̅

0

2.5

5

7.5

10

12.5

0

2.5

5

7.5

10

12.5

Fig. 4. Example of frequency response |H14| and |H11| (see nomenclature).

As it will be seen better in the next section, in this case the first natural frequencies are mostly influenced by the stiffness of the supports and not by the flexibility of the components; therefore, the first BW and FW modes of the real system are very similar to pure rigid conical motions. Thus, the rigid model can be used to quickly estimate the first BW and FW frequencies of the system. Especially for the preliminary phase of the project, the rigid model represents an important tool to perform sensitivity analysis by varying the main parameters of the system; a Matlab code has been created for this purpose, where the input quantities are parameterized in vectors. Since the computational cost required for a single set of parameters is really low a huge amount of configurations can be evaluated this way in a reduced time. However, this type of approach cannot be used to evaluate natural frequencies related to higher order modes and so a more complicated mathematical model that includes at least the shaft flexibility is needed.