FREQUENCY ANALYSIS OF A ROTATING SHAFT

INTRODUCTION:

Natural Frequencies ae those frequencies with which a body oscillates when there is no repeated or continuous load acting on it.

So, before we use any part in any assembly to tend to study the frequency at which it naturally vibrates and make sure that the operating range of the model does not fall in that range.

This is to make sure that the amplitude of vibration of the model and assembly doesn’t multiply to produce displacement large enough to disturb the assembly.

Since the body when acted upon by loads vibrate at frequencies. When these bodies vibrate at the natural frequencies, it can undergo large deformations which are often destructive.

This phenomenon is termed as Resonance. Often, we try to avoid resonance while designing a component.

So, it is very crucial to perform frequency analysis on the part before using it in any mechanical system.

Below it shows some of the engineering accidents due to Resonance

MODES OF VIBRATION:

The material subjected to natural frequencies have a tendency to vibrate in any direction this tendency can be determined by looking at the mode shapes and the vibrating frequencies.

Displacements produced at the natural frequencies of the vibrating body is referred to as mode shapes

Modes are the properties of a particular structure that depends on:

1) Material properties (such as the stiffness, damping, etc)

2) Boundary conditions applied to the structure.

The modes are defined by the resonant frequency, mode shapes, and modal damping.

If the material of the structure changes, then the mode of vibration also changes.

What is Effective mass participation factor:

“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.”

When investigating EMPF, occasionally the user will come across a mode shape with zero or null EMPF in all three directions (X, Y and Z). This can lead to some head-scratching, especially if the mode shape shows significant excitation in one of the directions that is reporting a null EMPF.

We will be performing frequency analysis of Rotating shafts to make sure that the amplitude of vibration of the model doesn’t multiply to produce displacement large enough to disturb the assembly and hence to avoid resonance and we will look at the effect of the effective mass participation factor with different modes of frequencies.

So, we can increase or decrease the mass in order to reduce the effects of vibrations.

OBJECTIVES:

1) To perform Frequency analysis of rotating shaft by finding out the critical frequencies (resonance frequencies) by determining different mode shapes.

2) To List the Resonant Frequencies and Effective mass participation factors.

3) To plot the Resonant Frequencies V/s modes and Resonance Frequencies V/s Effective mass participation factor.

PROCEDURE:

The Entire Process of FEA analysis is divided into 3 parts

1) Pre-Processing

Geometry Creation

Generation of Mesh

Assignment of Material properties

Boundary Conditions

2) Processing

Governing Equations are applied to the discretized elements and selected variables at selected points are calculated

3) Post-Processing

After the calculation of the displacements across the nodes the Stress and Strains can be calculated at any point.

Create the required Contours and Plots

Interpret the Results

Modify the geometry to get the optimized results

PRE-PROCESSING:

GEOMETRY AND MATERIAL PROPERTIES:

Rotating shaft is assigned Alloy steel

Properties: Alloy-Steel

BOUNDARY CONDITIONS:

Apply a bearing fixture on the stepped face of the rod at the two ends

MESH:

PROCESSING:

Governing Equations are applied to the discretized elements and selected variables at selected points are calculated.

POST-PROCESSING:

RESULTS AND CONTOURS

1^ST MODE SHAPE:

The first mode shape is shown below and the critical frequency is very low because the plate at the center is placed at one of the antinodes and hence which is not allowing it to have movement in any direction.

ANIMATION:

2^ND MODE SHAPE:

The 2^nd mode shape is half-wave and gives the 1^st critical mode shape with movement in y-direction which has 2 nodes and 1 anti-node.

ANIMATION:

3^RD MODE SHAPE:

The 3^rd mode shape is half-wave and gives the 1^st critical mode shape with movement in x-direction which has 2 nodes and 1 anti-node.

ANIMATION:

4^TH MODE SHAPE:

The 4^th mode shape is full-wave and gives 2^nd critical mode shape with movement in the y-direction which has 3 nodes and 2 antinodes.

ANIMATION:

5^TH MODE SHAPE:

The 5^th mode shape is full-wave and gives 2^nd critical mode shape with movement in the x-direction which has 3 nodes and 2 antinodes.

ANIMATION:

RESONANT FREQUENCY:

EQUIVALENT MASS PARTICIPATION FACTOR (EMPF):

RESONANT FREQUENCY V/S MODES:

Resonant Frequency V/s Effective mass participation factor (EMPF):

OBSERVATIONS AND CONLCUSION:

1) The first mode shape has very low critical frequency because the plate at the center is placed at one of the antinodes and hence which is not allowing it to have movement in any direction. For a symmetric mode, the CG experiences no movement in the direction of symmetry. As from the above table of EMPF, the percentage of EFFECTIVE mass participating in the mode, the solution will show zero in all directions.

2) The 2^nd mode shape is half-wave and gives the 1^st critical mode shape with movement in Y direction which has 2 nodes and 1 anti-node. The Effective mass participation factor (EMPF) output for mode shape two makes intuitive sense. The motion is all in the global Y direction, so we expect zero EMPF in the X and Z directions and a relatively large number for EMPF in the Y direction. In this case, 82.049% of the system’s mass is participating in this mode shape in the Y direction.

3) The 3^rd mode shape is half-wave and gives the 1^st critical mode shape with movement in X direction which has 2 nodes and 1 anti-node. The Effective mass participation factor (EMPF) output for mode shape two makes intuitive sense. The motion is all in the global X direction, so we expect zero EMPF in the Y and Z directions and a relatively large number for EMPF in the X-direction. In this case, 82.05% of the system’s mass is participating in this mode shape in the X-direction.

4) The 4^th mode shape is full-wave and gives 2^nd critical mode shape with movement in the Y direction which has 3 nodes and 2 antinodes. Here with mode shape 4 (the full-wave), the motion is all in the global Y direction. We might expect zero EMPF in the X and Z directions, and a relatively large number for Effective mass participation factor (EMPF) in the Y direction. But the EMPF in Y is also zero. The reason is that the Effective mass participation factor (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. Here the mode 4 is symmetric and For a symmetric mode, the CG experiences no movement in the direction of symmetry. So as a percentage of EFFECTIVE mass participating in the mode, the solution will show zero. In mode shape 4 the movement in the negative Y direction to the left of the inflection point in the center of the beam is canceled out by the movement in the positive Y direction to the right of the inflection point, resulting in zero movement of the CG. Therefore, the EMPF for that mode is null and the results are correct.

5) The 5^th mode shape is full-wave and gives 2^nd critical mode shape with movement in the X direction which has 3 nodes and 2 antinodes. Here with mode shape 4 (the full-wave), the motion is all in the global X direction. We might expect zero EMPF in the Y and Z directions, and a relatively large number for Effective mass participation factor (EMPF) in the X-direction. But the EMPF in X is also zero. The reason is that the Effective mass participation factor (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. Here the mode 5 is symmetric and For a symmetric mode, the CG experiences no movement in the direction of symmetry. So as a percentage of EFFECTIVE mass participating in the mode, the solution will show zero. In mode shape 4 the movement in the negative X direction to the left of the inflection point in the center of the beam is canceled out by the movement in the positive X direction to the right of the inflection point, resulting in zero movements of the CG. Therefore, the EMPF for that mode is null and the results are correct.