resonances of the full system better than static reduction, the IRS reduced stiffness matrix is stiffer than the Guyan reduced matrix and the mass matrix is less suited for orthogonality checks. The IRS transformation matrix TIRS depends on the reduced mass and stiffness matrices obtained by static reduction. To minimise the error produced by this scheme, IRS could be extended to the Iterated IRS method [17]. Even though the algorithm converges to yield the eigenvalues of the full system, the method results in a too stiff structure [4]. 5.1 Component Mode Synthesis (CMS): Craig-Bampton Method Dynamic condensation, as an efficient method for model reduction, was proposed in 1965 [3]. One such method known as the Component Mode Synthesis (CMS) technique [18-20] consists of dividing the complex structure into smaller substructures (or superelements) and recovering afterwards the dynamic behavior of the original structure by assembling the superelements and considering the equilibrium of nodes at the interfaces between the various components. The dynamic analyses of large structures are often carried out using superelements based on the substructuring principle. In such algorithms the reduction takes place by creating master degrees of freedom. The Craig-Bampton (CB) method [21] is a dynamic reduction method used to reduce the size of the finite element models. In this method, the motion of the whole structure is represented as a combination of boundary points (so called master degrees of freedom) and modes of the structure assuming that the master degrees of freedom are held fixed. Unlike Guyan reduction [13], which only accounts for the stiffness matrix, Craig-Bampton accounts for both the mass and stiffness. Furthermore, it enables defining the frequency range of interest by identifying the modes of interest and including these as a part of the transformation matrix. The decomposition of the model into both physical DOFs (master DOFs) and modal coordinates allows the flexibility of connecting the finite elements to other substructures, while maintaining a reasonably good result within a required frequency range. The algorithmic scheme of CMS is divided into three categories namely, fixed-interface, free-interface, and the residual-flexible free interface. In this paper only the fixed-interface CMS method is considered. The CMS algorithm does have certain disadvantages and the static term (Guyan) constitutes the source of the largest amount of information loss. The error can be compensated to some extent by increasing the number of CB modes, where coupling of the reduced order model (ROM) into the Multi-Body System (MBS) is not intended [11]. 6. Combined (LPM + reduced FEM) model Fig 7 shows the flowchart to obtain the total response at the measurement point by combining the LPM and reduced FE model of the casing. The reduced mass and stiffness matrices ([Mred] [Kred]) of the casing (obtained using Craig-Bampton based CMS method) were combined with the LPM model in the Simulink® environment. The combination was achieved by re-writing the equations of motion at the connecting nodes. The LPM model of the internals had 22 DOFs. The reduced casing model had 124 DOFS which were made up of 24 master degrees of freedom corresponding to 4 bearing centre nodes (each with 6 DOFs) plus 100 modes with master DOFs fixed. This resulted in a new combined model with a total of 146 DOFs (46 physical DOFs and 100 modal or generalized coordinates). The modal damping ]= 4% was introduced. 404
RkJQdWJsaXNoZXIy MTMzNzEzMQ==