# Our Latest Topics Relating to Simplified Modeling of Rotating Systems

• Osami Matsushita
• Masato Tanaka
• Masao Kobayashi
• Patrick Keogh
• Hiroshi Kanki
Chapter
Part of the Mathematics for Industry book series (MFI)

## Abstract

This chapter is divided into three parts, which include our latest areas of interest. (1) Guyan reduction or mode synthesis: With respect to vibration analysis for actual machines, many types of codes, 3D-FEM and 1D-FEM (e.g., MyROT introduced in R1_Chap. 12), are available for engineering design based upon multi-DOF (degree of freedom) modeling. However, case studies of vibration problems show the benefits for understanding the vibration mechanisms and their practical and potential solutions using simplified models. In this book, Guyan reduction and the mode synthesis technique are strongly recommended for this purpose. (2) System instability in oil film bearing rotors: This model order reduction (MOR) is applied to a flexible rotor supported by oil film bearings to analyze system instability, often referred to as oil whip, oil whirl, and casing whirl. Stability analysis is generally performed by using complex eigenvalue codes in the 1D-FEM modeling. However, this chapter yields an alternative way without complex eigenvalue calculation, i.e., how to predict the limiting speed for stability and the resultant unstable frequency. (3) Blade modeling coupled with shaft vibration: The MOR by the mode synthesis technique is applied to a flexible blading system to analyze the vibration coupled with the rotating shafting system. Uncoupled eigen frequencies and modes may be analyzed initially for the rotating blading system only by a 3D-FEM code, based upon the rotating coordinates. As the result, the corresponding 2-DOF models for each eigen mode of blading are proposed by two methods depending upon the number of the nodal diameter, κ = 0 and κ = 1. The former is coupled with torsional and axial shaft vibration models and the latter with bending (lateral) shaft vibration models, respectively. These blade reduced models are combined into shaft vibration data by a 1D-FEM code for the evaluation of the coupling effect. The effectiveness of these procedures is confirmed by numerical examples.

## Keywords

Model order reduction (MOR) Guyan reduction Mode synthesis Eigen frequency Oil whirl/whip λ model Blade–shaft coupled vibration 1D-FEM 3D-FEM

© Springer Japan KK, part of Springer Nature 2019

## Authors and Affiliations

• Osami Matsushita
• 1
Email author
• Masato Tanaka
• 2
• Masao Kobayashi
• 3
• Patrick Keogh
• 4
• Hiroshi Kanki
• 5