| Using Hamilton's principle, the finite element equations for both non rotating and rotating shafts are given, so that not only bending but also torsional, axial and other possible modes in a 3-D domain can be analysed simultaneously through the corresponding developed codes. The computed frequencies of bending-like 3-D modes are in very good agreement with those measured. Use of a rotating frame of reference gives a clear view of the shaft whirl as a result of bending vibration. Forward and backward whirls in this frame are found to be able to exist simultaneously along the shaft, which is reported for the first time. Resonant whirl frequencies or natural frequencies of rotating shafts can be determined with respect to the rotating or fixed frame. Critical speeds and unbalance response can also be consistently evaluated using the 3-D modelling. For this chain-like structure, reduction procedures such as the developed transfer matrix method based on finite element models and dynamic reduction method, can be implemented without any loss of accuracy. A distributed fluid/shaft interaction model has been given for rotor-bearing analysis in accordance with the 3-D shaft model. Distributed damping effects on the peak forced response frequency have also been discussed. Good agreement with some results from the ABAQUS software, testing and analytical solutions of uniform beams shows the validity of the written codes which are used throughout the research. |
