Frequency Analysis of a Rotating Shaft

Aim:

To perform Frequency analysis to find out the critical frequencies and mode shapes of a rotating shaft in Solidworks.

Introduction:

(1) Frequencies and mode shapes:

Natural frequencies or eigenfrequencies are certain discrete frequencies at which a system is prone to vibrate.

Resonance is a phenomenon in which frequency of a periodically applied force is equal or close to a natural frequency of the system on which it acts and this can be destructive to a system.

When vibrating at a certain frequency, a structure deforms into a corresponding shape, called mode shape.

These frequencies and mode shapes are critical when designing a system and hence need to be evaluated.

This type of analyses are Frequency analyses or also called Modal Analyses.

(2) Frequency Analysis:

It can only provide the shape of the mode and the true size of the deformation can only be determined if an actual excitation is known together with damping properties.

Each mode is described in terms of its modal parameters : natural frequency, the modal damping factor and characteristic displacement pattern, namely mode shape. Each corresponds to a natural frequency.

The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration.

It is common to use the Finite element method (FEM) to perform this analysis because, like other calculations using the FEM, the object being analyzed can have arbitrary shape and the results of the calculations are acceptable.

(3) Continuous systems:

Every system can be described in terms of a stiffness matrix that connects the displacements (or system response) and forces (or system inputs). These frequencies are known as natural frequencies of the system and are provided by the eigenvectors of the stiffness matrix. These frequencies are also known as resonant frequencies.

The resonant frequencies related to mechanical structures are known as mechanical resonance.

A continuous system — such as a general solid, beam, shaft etc — will exhibit eigenfrequencies that depend on geometry, material properties, and constraints.

A continuous system has an infinite number of natural frequencies. For all practical purposes, only a limited number of modes are of interest. Higher modes are unlikely to be excited to a significant extent and often have a higher damping.

(4) Frequency analysis on a shaft:

A shaft is a rotating mechanical component used to transmit power from one part to the other.

Machine elements such as gears, pulleys, flywheels, clutches and sprockets are mounted on various types of shafts and are used to transmit power from the driving device such as motor or engine.

When rotating at high speeds, a shaft is subjected to vibrations and so while designing it, the knowledge about its natural frequencies is important.

Various mode shapes at different frequencies give information about the load conditions in which a shaft can be operated. This also helps to avoid the loads that can be critical for operation of the shaft.

Case setup and Procedure:

A shaft with a circular disc at the centre and bearings on both the ends, is modelled.

The dimensions used are:

Disc diameter = 300 mm

Disc extrusion length = 20 mm

Shaft diameter = 60 mm

Shaft extrusion length = 800 mm

Bearing diameter = 40 mm

Bearing extrusion length = 30 mm

Alloy Steel is used as the material for whole model. Bearing fixtures are applied on both the ends of the model (purple color elements) and a Fine mesh is used for meshing as shown below:

Under the Frequency dialog box, select Number of frequencies as 5 and leave rest of the options with default selections and click Ok.

Now Run the study and wait for the results to be displayed.

Results:

Mode Shape 1:

Mode Shape 2:

Mode Shape 3:

Mode Shape 4:

Mode Shape 5:

Resonant Frequencies:

Study name : Shaft Frequency Analysis
Mode No.	Frequency(Rad/sec)	Frequency(Hertz)	Period(Seconds)
1	0	0	1.00E+32
2	799.69	127.27	0.0078571
3	799.71	127.28	0.0078568
4	3429	545.75	0.0018323
5	3429.5	545.82	0.0018321

Explanation:

For the first mode, resonant frequency is 0 Hz with a maximum amplitude of 4.158e-01

The second and third modes have almost the same resonant frequency of around 127 Hz and same maximum amplitude of 2.283e-01

Similarly, fourth and fifth modes have frequencies closer to each other and maximum amplitude of 4.98e-01

At these frequencies the shaft tends to fail due to greater amplitudes. Hence the shaft should be operated in such a way that it does not reach or exceed resonant frequencies.

Note that these frequencies can vary depending on the geometry and the ways in which it is constrained.

Conclusion:

A shaft was modelled similar to the one given in problem statement and frequency analysis was carried out based on the given conditions. Also, the 5 mode shapes and resonant frequencies were listed out and explained.

References:

https://www.sciencedirect.com/topics/engineering/modal-analysis

https://en.wikipedia.org/wiki/Modal_analysis_using_FEM

https://www.comsol.co.in/multiphysics/eigenfrequency-analysis

https://www.simscale.com/blog/2016/12/what-is-modal-analysis/