Aim :

To find crictical frequency of shaft .

Objective :

1. To prepare 3d model of shaft .

2. perform a frequency analysis of shaft and finding the critical frequancy of shaft .

Theory :

Consider a typical shaft, carrying a rotor (disk) mounted between two bearings as shown in Fig. Let us assume that the overall mass of the shaft is negligible compared to that of the rotor and hence we can consider it as a simple torsional spring. The rotor section has a geometric centre. 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 Fig. 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 central line as indicated in the figure.

Fig. shows the shaft and rotor system undergoing synchronous whirl in four different positions in a single revolution. Let us write down the force equilibrium equation as follows, following the notation shown in Fig. :

centrifugal force = m*omega^2*(r+e);

shaft resistance = k*r

k = 48 *E*I/L^3

thus from above equalibrium equation we can get the natural frequency value omega = (k/m)^0.5

Model creation :

below is the sketch of the model in front plane as the dimensions are given below

3D model

now we can revolve the sketch and we can get the our desired model .

Frequency Analysis study :

Material applied :

Model Reference Properties Components Name: Alloy Steel Model type: Linear Elastic Isotropic Default failure criterion: Max von Mises Stress Yield strength: 6.20422e+08 N/m^2 Tensile strength: 7.23826e+08 N/m^2 Mass density: 7700 kg/m^3 Elastic modulus: 2.1e+11 N/m^2 Poisson's ratio: 0.28 Thermal expansion coefficient: 1.3e-05 /Kelvin SolidBody 1 Curve Data:N/A

Boundary condition :

we have applied the bearing fixture at the both step side cyllinder so that there will be the rotating freedom is available so that we can find the critical frequency of the rotating shaft . as shown in figure we have applied bearing fixture at the face of the cyllinder highlited in blue colour.

Mesh information :

Mesh type	Solid Mesh
Mesher Used:	Standard mesh
Automatic Transition:	Off
Include Mesh Auto Loops:	Off
Jacobian points	4 Points
Element Size	2.5 mm
Tolerance	0.125 mm
Mesh Quality Plot	High

Total Nodes	49089
Total Elements	32306
Maximum Aspect Ratio	3.6439
% of elements with Aspect Ratio < 3	99.9
% of elements with Aspect Ratio > 10	0
% of distorted elements(Jacobian)	0

as we can see from the mesh information that 99.9% of our elements having aspect ratio less than 3 so that we have excellent meshing .

Aspect ratio is the parameter to check the quality of the mesh genarally it is basically the ratio of the longest edge of the tetrahedral to the normal from vertex to opposite face .

Results :

Name	Type	Min	Max
Amplitude1	AMPRES: Resultant Amplitude Plot for Mode Shape: 1(Value = 0.128635 Hz)	1.210e-03 Node: 2943	2.764e+00 Node: 44122
Part2-Frequency analysis of shaft-Amplitude-Amplitude1

Name	Type	Min	Max
Amplitude2	AMPRES: Resultant Amplitude Plot for Mode Shape: 2(Value = 887.201 Hz)	1.811e-03 Node: 2929	1.545e+00 Node: 29257
Part2-Frequency analysis of shaft-Amplitude-Amplitude2

Name	Type	Min	Max
Amplitude3	AMPRES: Resultant Amplitude Plot for Mode Shape: 3(Value = 887.286 Hz)	1.737e-03 Node: 1516	1.545e+00 Node: 29252
Part2-Frequency analysis of shaft-Amplitude-Amplitude3

Name	Type	Min	Max
Amplitude4	AMPRES: Resultant Amplitude Plot for Mode Shape: 4(Value = 3989.16 Hz)	6.239e-03 Node: 2951	2.210e+00 Node: 2206
Part2-Frequency analysis of shaft-Amplitude-Amplitude4

Name	Type	Min	Max
Amplitude5	AMPRES: Resultant Amplitude Plot for Mode Shape: 5(Value = 3990.12 Hz)	6.766e-03 Node: 6182	2.211e+00 Node: 44464
Part2-Frequency analysis of shaft-Amplitude-Amplitude5

Mode List

Frequency Number	Rad/sec	Hertz	Seconds
1	0.80824	0.12863	7.774
2	5574.4	887.2	0.0011271
3	5575	887.29	0.001127
4	25065	3989.2	0.00025068
5	25071	3990.1	0.00025062

Mass Participation (Normalized)

Mode Number	Frequency(Hertz)	X direction	Y direction	Z direction
1	0.12863	4.2039e-44	5.2649e-13	1.3386e-12
2	887.2	3.6271e-11	0.20761	0.66306
3	887.29	8.9618e-11	0.66304	0.20762
4	3989.2	2.0915e-09	2.5843e-10	2.083e-09
5	3990.1	1.6531e-07	1.3862e-09	1.1627e-10
		Sum X = 1.6753e-07	Sum Y = 0.87065	Sum Z = 0.87069

List of resonant frequency :

Conclusion :

we have run our study on the alloy steel and we can find that the resonant frequancy of our 0.12Hz , 887Hz and 3990 Hz so it is better to avoid this freqency in our opearation .

at this frequency resonance occur which will result in higher deformation and our component will fail , so we can provide damper and springs to control our frequency range .

Refrence:

https://nptel.ac.in/content/storage2/courses/112101096/download/lecture-25.pdf