Whirling of Shafts

INTRODUCTION

All bodies which have mass and elasticity exhibit the phenomenon of vibration. Mass is the inherent property of the body and the elasticity causes relative movement between parts. When external forces displace the body, internal forces due to elasticity will try to bring body to its initial position. All the structures designed to support high speed engines, machineries and power generation equipments are subjected to vibration. Similarly, Shafts are the single most important power transmission tool used. Shafts are expected to turn at the certain speed, similarly it will also have the natural frequency of vibration, when the natural frequency of the shaft matches that of the angular speed with which the shaft rotates, it undergoes resonance. If the shaft operate in resonance, shaft can snap due to stresses developed. Hence it is vital that the shaft is designed to operate nowhere closer to natural frequency of shaft to prevent resonance.

Natural frequency is dependent on mass, material, end fixtures.

In this study, shaft with the disc is held between two bearing fixtures, the 5 modes of frequency is analysed for the three cases,

case 1: Shaft and disc with negligible weight and no torque

case 2: Shaft and disc with negligible weight with applied torque

case 3: Shaft and disc with weight and applied torque

FEA SIMULATION

GEOMETERY :

This is the sketch of the geometry that is taken to extrude.

The shaft of diameter 37.81 mm is considered with length of 1m. The disc of 30mm thickness, 145mm diameter. Bearing seat of 23.6mm diameter and length of 25mm is considered on each side.

MESHING : Automatic meshing feature is utilised.

SOLVER SETUP AND INTIALISATION:

Cases\Parameters	Torque (Nm)	Mass of shaft(kg)	Mass of disc(Kg)	Gravity	Solver
Case 1	0	0	0	0	FFEPLUS
Case 2	100	0	4	0	Direct sparse
Case 3	100	2	2	9.81ms-2	Direct Sparse

RESULTS AND PLOTS

Mode shapes are as shown below

However, the mode shapes give limited info, hence the frequncies could be compared across the cases

Modes\Cases	Case 1 Frequency(Hz)	Case 2 Frequency(Hz)	Case 3 Frequency(Hz)
Mode 1	248.05	248.06	181.59
Mode 2	248.15	248.15	181.66
Mode 3	438.37	438.37	250.79
Mode 4	1026.6	1026.6	714.61
Mode 5	1026.7	1026.7	714.77

DISCUSSION ON RESULTS

1) Frequency analysis of the rotating shafts prevent destructive rupture of shaft and damage to equipement and personnel

2) Speed of operation of shaft should be carefully considered while deciding design of the shaft

3) If the distance between geometric center of disc and center of gravity is e, x is the defletion, then ratio of x/e can give important relations (x/e also known as Amplitude ratio)

- x/e is positive when frequency of operation is less than critical frequency, Geomteric center will lie between axis joining the bearing fixture and center of gravity of disc

- x/e is negative when frequency of operation is more than critical frequency, Geometric center will lie outside center of gravity of disc and hence results in heavier side closer to Bearing fixture axis and hence less vibrations

- x/e is infinity when frequency of operation is equal to critical frequency.