4. The finite element model (Casing) The finite element model (FEM) of the casing (104 340 degrees of freedom) is shown in figure 6. The casing is supported by rubber pads, which are simulated using spring elements at the corners of the casing. The earlier model used only shell elements [5, 6] and was updated to the new one which has both solid and shell elements. The model has been compared with experimental modal testing and validated for the lower frequency modes [6]. In the current update the nodes on the hub of each bearing are connected to a centre node using rigid body elements. Thus one centre node is formed at the centre of each bearing, which will eventually capture the flexibility of the casing. This also enables the connecting of this model (a reduced version of it) with the LPM model of the internals. Figure 6: The Finite element model of the UNSW gearbox casing 5. Finite element model reduction techniques Static analyses require a detailed set of grid points or nodes to map internal stresses and strains and hence the FE models can be very large, perhaps several thousand DOFs which require large computing resources. However, dynamic analyses based upon knowledge of fundamental frequencies and their associated mode shapes need far fewer nodes or DOFs [10]. Various model reduction techniques have been developed in the past decades such as Guyan, Dynamic, CMS, IRS, SEREP etc. It is impossible to simulate the dynamics of the full system with the reduced model and every reduction transformation is a trade off between the accuracy and computational speed. The comparison of various model reduction techniques shown below is mainly taken from [11, 12]. The static reduction method also known as Irons-Guyan method [13, 14] produces smaller size system matrices by eliminating the coordinates at which no external force is applied. The reduction method is exact for static problems; however, for dynamic problems large errors may be introduced due to the fact that the DOFs eliminated may experience inertia forces, which cause their dynamic displacements to differ from the static, the deviation increasing with frequency [15]. High frequency motion is better approximated using dynamic reduction. However, the transformation matrix Tdyn depends on the choice of an appropriate initial frequency ω, which is not a trivial task. Guyan reduction is a special case of dynamic reduction, when ω = 0. Static reduction methods were further improved by O’Callahan [16], with a technique known as Improved Reduction System (IRS) method. IRS perturbs the static transformation by taking into account the inertia terms as pseudo-static forces. Although the results match the low frequency response 403
RkJQdWJsaXNoZXIy MTMzNzEzMQ==