Frequency Analysis of a rotating shaft

AIM: To find out the critical frequencies of a rotating shaft in Solidworks.

OBJECTIVES :

1. To create a 3D model of a rotating shaft.
2. To perform frequency analysis on the rotating shaft and find out the critical frequencies.

INTRODUCTION: For a rotating shaft there is a speed at which, for any small initial deflection, the centripetal force is equal to the elastic restoring force. At this point, the deflection increases greatly and the shaft is said to "whirl". Below and above this speed, this effect is very much reduced. This critical (whirling speed) is dependent on the shaft dimensions, the shaft material, and the shaft loads. The critical speed is the same as the frequency of traverse vibrations.

THEORY: Consider a rotating horizontal shaft with a central mass (m) which has a centre of gravity (G) slightly away from the geometric centroid (O),

The more general formula for the restoring traverse force of the beam is `y ((K EI) / L^ 3)` where `K` = 'a constant depending on the position of the mass and the end fixing conditions'.
Equating these forces,

This is plotted here,

This curve shows the deflection of the shaft (from the static deflection position) at any speed `ω` in terms of the critical speed.

Procedure

The material is chosen – Alloy Steel

Material Properties:

• Elastic Modulus – 1 e+11 N/m^2
• Poisson’s Ratio - 28
• Shear Modulus – 7.9 e+10 N/m^2
• Mass Density – 770 kg/m^3
• Tensile strength – 723825600 N/m^2
• Yield Strength – 620422000 N/m^2

Dimensions

• The shaft of given dimensions are designed and extruded in Solidworks.
• A new FEA study is initiated in Solidworks simulation and the material is defined as alloy steel.
• Shaft with the given conditions are extruded using Solidworks and the required boundary conditions are placed i.e bearing fixture on both ends of the shaft.
• Frequency study is initiated and the following post-processing plots were obtained along with the various resonant frequencies for the given rotating shaft.

Post-Processing Data

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Conclusion

From the above frequency analysis, it is understood that the shaft should not be operated in the given frequency range 0.159416Hz, 1043.6Hz, 1044Hz, 3905.1JHz and 3907.5Hz and would result in resonance phenomenon and failure of the shaft, however, mode 1 resonant frequency is a bit peculiar as it is of an extremely low Hertz range.