[Paper Review] Effect of Elasticity of Shafts, Bearings, Casing and Couplings on the Critical Rotational Speeds of a Gearbox

Emmanuel Rigaud, J. Sabot|ArXiv.org|Jan 3, 2007

Gear and Bearing Dynamics AnalysisEngineering5 references21 citations

TL;DR

This study investigates how the elasticity of shafts, bearings, casing, and couplings affects the critical rotational speeds of a gearbox with a helical gear pair. Using a finite element model with detailed component stiffness representations—including a 12×12 mesh stiffness matrix and multi-directional bearing elements—it demonstrates that accurate prediction of critical speeds requires modeling all gearbox components collectively, as resonant excitation of modes with high mesh-related potential energy leads to peak dynamic mesh forces.

ABSTRACT

The aim of this study is to analyse the influence of the mechanical characteristics of the set of components on the critical rotational speeds of a gearbox. The case of a gearbox fitted out with a helical gear pair was considered. The shafts and the casing were discretised using the finite element method. The elastic coupling between the toothed wheels was characterised by a 12 x12 stiffness matrix. The bearings were modelled using radial, axial and rotational stiffness elements. The calculation of the vibration response induced by the static transmission error showed that the highest dynamic mesh forces correspond to a resonant excitation of modes which have a high potential energy associated with the mesh stiffness. The numerical simulations performed showed that a realistic prediction of the critical rotational speeds should take account of all the components of the gearbox.

Motivation & Objective

• To analyze the influence of mechanical flexibility in gearbox components on critical rotational speeds.
• To address the limitation of simplified models that neglect the elasticity of non-gear components.
• To improve dynamic mesh force prediction by accounting for coupled system modes with high potential energy related to mesh stiffness.
• To demonstrate that realistic critical speed prediction requires modeling all major gearbox components, not just the gear pair.
• To validate the importance of including coupling and casing flexibility in vibration analysis of geared systems.

Proposed method

• The shafts and casing were discretized using the finite element method (FEM) for structural modeling.
• The gear mesh was represented by a 12×12 stiffness matrix to capture coupled translational and rotational degrees of freedom.
• Bearings were modeled using independent radial, axial, and rotational stiffness elements to reflect their dynamic behavior.
• The dynamic response was calculated under excitation from static transmission error to simulate real meshing forces.
• Modal analysis was performed to identify critical speeds corresponding to resonant excitation of flexible modes.
• Numerical simulations evaluated the relationship between mode shape potential energy and dynamic mesh force amplitude.

Experimental results

Research questions

• RQ1 How does the elasticity of shafts, bearings, casing, and couplings collectively influence the critical rotational speeds of a gearbox?
• RQ2 Which modes of vibration contribute most significantly to dynamic mesh force amplification?
• RQ3 To what extent do traditional simplified models fail to predict critical speeds accurately compared to a full-component model?
• RQ4 How does the potential energy associated with mesh stiffness correlate with resonant dynamic loading?
• RQ5 What is the impact of including coupling and casing flexibility on the overall dynamic behavior of the gearbox system?

Key findings

• The highest dynamic mesh forces occurred at rotational speeds that resonantly excite modes with high potential energy associated with the gear mesh stiffness.
• Critical rotational speeds were significantly altered when all components—including shafts, bearings, casing, and couplings—were modeled with their actual elastic properties.
• Neglecting the elasticity of non-gear components led to inaccurate predictions of critical speeds and dynamic load amplification.
• The 12×12 stiffness matrix effectively captured the coupled dynamic behavior of the helical gear mesh, enabling precise force prediction.
• Modal shapes with high mesh-related potential energy were identified as primary contributors to resonance and high dynamic loading.
• Full-system modeling is essential for realistic prediction of critical speeds, as individual component flexibility interacts nonlinearly in the dynamic response.