Frequency Analysis of a rotating shaft

Aim :- To find out the critical frequencies of a rotating shaft in solidworks

Abstract :- In this project we will model the shaft, as shown in the fig.1 , using solidworks part modelling and then carry out fine element analysis (FEA) to find the critical frequencies for different modes.

INTRODUCTION :-

Terms –

.1.) Natural frequency - All physical structures have natural frequencies. These are the frequencies at which the structure will tend to vibrate when subjected to certain external forces. These frequencies are dependent on the way mass and stiffness are distributed within the structure.

2.) Resonance - Resonance is a phenomenon in which a dynamic force drives a structure to vibrate at its natural frequency. When a structure is in resonance, a small force can produce a large vibration response.

All physical objects have multiple natural frequencies and can resonate under the right conditions. Sometimes the natural frequencies are excited by external forces acting on the object, which creates vibration. These vibrations may be so small that they cannot be seen by the human eye. Sometimes, they are quite large and easily observable.

3.) Mode shape - Each natural frequency is associated with a certain shape, called mode shape, that the model tends to assume when vibrating at that frequency.

4.) Mass Participation - The Mass Participation associated with each mode represents the amount of system mass participating in that mode. Therefore, a mode with a large effective mass is usually a significant contributor to the system’s response.

4.1) Effective Mass Participation Factor (EMPF) - The effective mass participation factor represents the percentage of the system mass that participates in a particular mode. It provides a measure of the energy contained within each resonant mode. A mode with a large EMPF is usually a significant contributor to the dynamic response of a system. In a frequency analysis, when we plot the mode shapes, the magnitude of the deformation is meaningless; it’s only the shape of the mode that tells us how the system will vibrate.

4.2) Cumulative Effective Mass Participation Factor (CEMPF) - The CEMPF is a measure of how the EMPF builds up as additional modes are included in the analysis. In general, the number of modes considered must contribute to a CEMPF of at least 80% of the system mass in the direction of excitation.

Fig. 1 :-Referance figure

Table 1 :- Alloy Steel Properties

Alloy Steel Properties
Property	Value	Unit
Elastic Modulus	2.10E+11	N/m^2
Poisson's Ratio	0.28	N/A
Shear Modulus	7.90E+10	N/m^2
Mass Density	7700	kg/m^3
Tensile Strength	7.24E+08	N/m^2
Compressive Strength		N/m^2
Yield Strength	6.2E+08	N/m^2
Thermal Expansion Coefficient	1.30E-05	/K
Thermal Conductivity	50	W/(m·K)
Specific Heat	460	J/(kg·K)
Material Damping Ratio		N/A

Simulation :-

1.) Model setup –

• Make the shaft model as in fig.1 with madeup dimensions.

fig. - Rotating-shaft 3D model

• create a new frequency study under solidworks simulation.
• Go to the properties of the study and from optoions tab check the option of ‘Automatic Solver Selection’.

2.) Boundary conditions -

• Define material to the part as Alloy steel with properties as shown in table-1.
• Add bearing support as a fixture to the stepped ends of the shaft.
• Now, apply torque of 1000N-m. as external load to the shaft.
• Create mesh of size 5mm.

fig. - Shaft with boundary conditions

3.) Run the simulation.

RESULTS & DISCUSSION :-

1.) Mode 1 -

The first mode shape is only rotation w.r.t. its longitudinal axis.

2.) Mode 2 -

The second mode shape is a half- wave with movement in the global YZ - plane.

3.) Mode 3 -

The third mode shape have movement in the global YZ - plane.

4.) Mode 4 -

The fourth mode shape is a full wave with movement in the global Y- direction.

5.) Mode 5 -

The fifth mode shape is a full wave with movement in the global Z- direction.

6.) Resonant Frequency list -

The resonant frequencies of the rotating shaft as obtained by the simulation

7.) Frequency vs. Mode no. -

This plot shows a graphical representation of the natural frequencies calculated by the analysis.

8.) Mass Participation -

It shows the amount of system mass participating in a particular direction for all the five modes.

9.) Frequency vs. EMPF -

It is also called as the second frequency response graph, Effective Mass Participation Factor (EMPF), helps us visualize how the mass is vibrating in X, Y and Z directions, for each natural frequency.

The reason for the EMPF you get for each mode and each direction is a reflection of how the center of gravity of the structure gets excited by that mode.

• For mode 1 there is almost no deformation/ movement, so we expect zero EMPF in all the 3 directions.
• For mode 2, there is a significant movement in YZ - plane, therefore, EMPF is zero in X - direction and 36.4% of system's mass participating in z- direction and relatively high 54.4% in Y- direction.
• For mode 3, there is again significant movement in YZ - plane, therefore, EMPF is zero in X - direction and 36.4% of system's mass participating in Y- direction and relatively high 54.4% in Z- direction.
• Mode 4 & 5 are symmetric modes, therefore EMPF is zero for all the 3 directions.

10.) Frequency vs. CEMPF -

It is also called as the third frequency response graph, Cumulative Effective Mass Participation Factor (CEMPF), helps us visualize the accumulation of mass participating in vibration as we calculate an increasing number of natural frequencies of our designs.