Frequency Analysis of Rotating Shaft Using Solid Works
Aim
- Conduct a frequency analysis and apply a bearing fixture on the stepped face of the rod at the two ends and perform the analysis using SOLID WORKS
Objective
- Find out 5 mode shapes and list the resonant frequencies
- create the model shown below
Introduction
Natural frequency is the frequency at which a system tends to oscillate in the absence of any driving or damping force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all parts of the system move sinusoidally with that same frequency).
If the oscillating system is driven by an external force at the frequency close to a natural frequency of the system, this frequency is called resonant frequency. Resonance describes the phenomena of amplification that occurs when the frequency of a periodically applied force is in harmonic proportion to a natural frequency of the system on which it acts. When an oscillating force is applied at a resonant frequency of a dynamical system, the system will oscillate at a higher amplitude than when the same force is applied at other, non-resonant frequencies which can fail the components.
Thus to design a system operating at a frequency other than the natural frequencies, the natural frequency of the system is necessary. This project involves the study of the mode shapes and determining the critical frequencies of the shaft.
Model Created
Theory
Every system has characteristic mass and inherent stiffness. Thus any given system can be reduced to a spring and a mass system. When this system is initially given a displacement, its vibratory motion can be represented by
`m (d^2 x) /dt^2 +kx=0`
Solving the equation for natural frequency we get
`ω_n=√(k/m)`
where,
k= Stifffnes of the system
m=mass of the system
Whirling of Shafts
Consider a typical shaft, carrying a rotor (disk) mounted between two bearings as shown in the figure. Let us assume that the overall mass of the shaft is negligible compared to that of the rotor (disk) and hence we can consider it as a simple torsional spring. The rotor (disk) section has a geometric centre i.e., the centre of the circular cross-section and the mass centre due to the material distribution. These two may or may not coincide in general, leading to eccentricity as indicated in the figure. The eccentricity could be due to internal material defects, manufacturing errors etc. As the shaft rotates, the eccentricity implies that the mass of the rotor rotating with some eccentricity will cause in-plane centrifugal force. Due to the flexibility of the shaft, the shaft will be pulled away from its centre line as indicated in the figure. Let us assume that the air-friction damping force is negligible. The centrifugal force for a given speed is thus balanced by the internal resistance force in the shaft spring and the system comes to an equilibrium position with the shaft in a bent configuration as indicated in the figure. Thus the shaft is rotating about its axis and the plane containing the bent shaft and the line of bearings rotates about an axis coinciding with the line of bearings. We consider here only the case, wherein these two rotational speeds are identical, called the synchronous whirl.
Procedure
Modelling: A shaft with the disk at the centre is modelled in SOLIDWORKS.
Material: Alloy Steel
Mesh Type: Fine Mesh
Results
| Mode No. | Freq (Hertz) | X direction | Y direction | Z direction |
| 1 | 0 | 1.16E-12 | 2.89E-14 | 2.03E-32 |
| 2 | 198.76 | 0.90465 | 0.011834 | 3.03E-12 |
| 3 | 198.76 | 0.011835 | 0.90465 | 2.47E-13 |
| 4 | 1,082.30 | 1.96E-10 | 4.88E-11 | 6.71E-10 |
| 5 | 1,082.30 | 1.59E-10 | 1.50E-13 | 3.12E-10 |
| Sum X = 0.91649 | Sum Y = 0.91649 | Sum Z = 9.8595e-10 |
Plots Obtained by Frequency Analysis
Conclusion
We obtain natural frequencies at different modes. Thus the shaft is operated at speeds other than the natural frequencies