Exact free vibration analysis of Timoshenko stepped shaft carrying elastically supported eccentric masses with application on SWRO mechanical system

The aim of the study is to investigate the exact natural frequencies and their associated modal shapes for an axially loaded Timoshenko stepped multi-span shaft or beam system. Each span is carrying an elastically supported eccentric mass and resting on Pasternak elastic foundation. The end and intermediate conditions governing the lateral deflections and slope due to bending of the system have been formulated. Application models have been chosen to verify the reliability of the generalized system frequency equation. Some results of previous works are validated using the present analysis. Real application example of a combined shaft assembly for high pressure boosting device for sea water reverse osmosis (SWRO) desalination plant is investigated. Experimental tests for extended real shaft are carried out. The comparison between the analytical results and the experimental ones, for the first two modes of vibration, shows good agreement with minimal errors. These errors do not exceed 5%. In addition, the results of the proposed shaft model can be easily obtained for any number of spans and attachments. The present model and analysis is important for industrial applications, such as mechanical, civil, naval and for wider range of sea water desalination machinery.