Aim: -

The study aims to find the mode shapes and resonant frequencies of a rotating shaft.

Introduction: -

Frequency: -

Frequency, in a general sense, is defined as the number of times an event occurs per unit time. However, a more scientific definition is The number of times a point on a wave passes through a fixed reference point in one second is known as frequency. The formula for the frequency is shown below.

; where f = frequency ,and T = Time

Natural Frequency: -

The Natural Frequencies of a structure are the frequencies at which the structure naturally tends to vibrate if it is subjected to a disturbance. This vibration will cause the structure harm and sometimes the structure might even fail.

Modes and Modes Shapes: -

The deformed shape of the structure at a specific natural frequency of vibration is termed its normal mode of vibration.

Resonance: -

In physics, resonance describes the phenomenon of increased amplitude that occurs when the frequency of a periodically applied force is equal or close to a natural frequency of the system on which it acts. At resonance, the vibration amplitudes are so high that there is a chance that the system might fail.

Objective/Tasks: -

1. To create a model of a rotating shaft.
2. Run a frequency analysis on the shaft and get the mode shapes.
3. Compare the mode shapes and get resonance frequencies for all the modes.

Analysis: Frequency Analysis of a rotating shaft

Procedure: -

The procedure used for the analysis is as follows

1. Create a sketch for the rotating shaft and revolve it 360 degrees.
2. Start a new study. Under it, select Frequency analysis and name it Frequency 1.
3. Apply the Alloy Steel Material
4. Apply the fixtures.
5. Generate a mesh with appropriate parameters.
6. Run the study.
7. Compare the mode shape and get resonance frequencies.

Analysis Setup: -

Model: -

The model was created in two steps using SOLIDWORKS 2018 modeling software which are given below

1. First of all, a sketch with appropriate dimensions was made as shown in figure 1.
2. In the second step, the model was completed by revolving the sketch by 360 degrees about the axis of the shaft using the Revolve tool as shown in figure 2. Figure 3. shows the model of the shaft.

Figure 1. Sketch of the shaft.

Figure 2. Revolve Tool.

Figure 3. Model of the shaft.

Material Selection: -

The material assigned to the model for this simulation or analysis was Alloy steel. The properties of alloy steel are shown in figure 4.

Figure 4. Properties of Alloy steel.

Fixures: -

The model was given two bearing fixtures or support at the stepped faces to encompass the bearing to be fitted in that place in practice. Figure 5. Shows the bearing support fixtures.

Figure 5. Fixtures in the model.

Mesh: -

A mesh was created using the parameters shown in figure 6.

Figure 6. Mesh Parameters.

The meshed model is shown in figure 7.

Figure 7. Meshed model.

Study Settings: -

The study settings were kept as the default settings. They are shown in figure 8.

The number of frequencies and modes shapes is kept at 5. The Bonding option is selected as automatic. The solver used by the SOLIDWORKS simulation wizard is FFEPlus.

Figure 8. Study Settings.

Results: -

There were in total 5 mode shapes.

Mode shape 1:

The mode shape plot obtained is shown in figure 9. The maximum and minimum amplitudes are 2.352e-01 and 9.090e-06.

Figure 9.Frequency mode shape 1.

Mode shape 2:

The mode shape plot obtained is shown in figure 10. The maximum and minimum amplitudes are 1.434e-01 and 1.545e-05.

Figure 10. Frequency mode shape 2.

Mode shape 3:

The mode shape plot obtained is shown in figure 11. The maximum and minimum amplitudes are 1.434e-01 and 2.209e-05.

Figure 11.Frequency mode shape 3.

Mode shape 4:

The mode shape plot obtained is shown in figure 12. The maximum and minimum amplitudes are 3.153e-01 and 1.519e-04.

Figure 12. Frequency mode shape 4.

Mode shape 5:

The mode shape plot obtained is shown in figure 13. The maximum and minimum amplitudes are 3.153e-01 and 5.701e-06.

Figure 13.Frequency mode shape 5.

Resonance Frequencies: -

The resonance frequencies for different mode shapes are shown in table 1.

Table 1. Resonance Frequency.

Mode No.	Frequency(Rad/sec)	Frequency(Hertz)	Period(Seconds)
1	0.017678	0.0028136	355.42
2	937.87	149.27	0.0066994
3	937.91	149.27	0.0066991
4	3469.5	552.18	0.001811
5	3469.8	552.23	0.0018108

Natural Frequency vs Mode no.plot: -

The plot for Natural Frequency vs Mode No. is shown in figure 14.

Figure 14. Natural Frequency plot on a frequency vs mode no. Graph.

Conclusion: -

The resonance frequencies obtained are 0.028136, 149.27, 149.27, 552.18, and 552.23. Apart from these frequencies, the shaft can be operated without any problem. But whenever the frequency is equal to any of the above-mentioned ones, the system will go in resonance and high vibrations amplitude are obtained. This fails in the system. This can easily be prevented by providing Dampers in the system. The material also plays an important role in the determination of the resonant frequency. If a more suitable material is used, the resonant frequency can be increased such that it can never be reached under working conditions.