SEM712 Structural Response Of Wired Automotive Frame For Abaqus Simulation

Calculate Young’s modulus of the material. You know the moments of inertia from geometry, you can calculate the displacement under a known load, and then solve for ‘E’ (this is much easier to calculate with a straight piece of wire!).

 

Answer:

Introduction to Modal Analysis 

Abaqus Simulation

Objective

This is report is documented for Task 3 in “Project 1 Structural responses of an arbitrary wire” in SEM712: CAE and Finite Element Analysis subject. Please see Appendix for the details.

Geometry

As per the problem statement, we need to select arbitrary wire model. For the same I have taken reference image from Wire manufacturer website^[1]. Figure1 shows the image of the same. Taking the image as reference for the model, I have assumed the dimensions of the geometry in scalable proportionately. Figure 2 gives the details of the same. The diameter of the Wire is 3mm.

Material and Section Properties

Standard Steel Properties are applied to the wire. The mechanical properties are defined in consistent unit system as below: 

Mass Density	Young’s Modulus	Poisson’s Ratio
7.85E-009	21000	0.3

The Wire Section Properties are defined as below: 

Section Type	Material	Profile Type	Profile radius
Beam	Steel	Circular	r = 1.5 mm

Element Name	Element Type	Number of Elements	Number of Nodes
B31	Linear Line Element	238	239

Vibration Frequency of the First Mode of Wire with concentrated mass 

Details of concentrated Mass	Mass in Abaqus Consistent Unit	Frequency of First Mode (Hz)	Reference	Abaqus Input File Name	Abaqus Results File Name
2C Coin   (5.20 g)	5.2E-06	8.3179	Figure 6	LC01.inp	LC01.odb
5C Coin   (2.83g)	2.83E-06	8.4554	Figure 7	LC02.inp	LC02.odb
10C Coin (5.65 g)	5.65E-06	8.2923	Figure 8	LC03.inp	LC03.odb
25 g Weight	2.5E-05	7.3227	Figure 9	LC04.inp	LC04.odb
50 g Weight	5.0E-05	6.3607	Figure 10	LC05.inp	LC05.odb

 Displacement of the structure when loaded at the opposite end with a known load

Details of known load	Mass in Abaqus Consistent Unit	Displacement	Reference	Abaqus Input File Name	Abaqus Results File Name
15 g	1.5E-05	1.7262E+03	Figure 11	LC06.inp	LC06.odb

Conclusion

From the series of Experiments we conducted we conclude following: 

• As the weight of the concentrated mass increases the frequency of the vibration decreases. This can be also understood by the equation of the frequency   .  So as “m” increases “ f “ decreases. 

Using ABAQUS solve the following problems numerically (for the bent wire):    

1. Vibrationfrequency of the first mode of vibration of:
2. Just the wire, and
3. Wire with a concentrated mass attached 
4. Displacement of the structure when loaded at the opposite end with a known load.  

Using your brain and a pen, solve the following problems analytically (for the bent wire):   

1. Vibrationfrequency of the first mode (ball?park figure required only) of: 
2. Just the wire, and 
3. Wire with a concentrated mass attached  

1. Displacement of the structure when loaded at the opposite end with a known load  (exact solution or ball?park answer can be provided. It will be marked accordingly).

About Modal Analysis

Modal analysis technique is used to determine the vibration characteristics (i.e., natural frequencies and mode shapes) of linear elastic structures. It is the most fundamental of all dynamic analysis types. Modal Analysis allows the design to avoid resonant vibrations or to vibrate at a specified frequency. It not only gives engineers an idea of how the design will respond to different types of dynamic loads but also helps in calculating solution controls for other dynamic analyses. For Modal Analysis in FEA software it is assumed that the (i) structure is linear (i.e.[M] and [K] matrices are constant) and (ii) no loads (force, displacements, pressures or temperatures) are allowed, i.e., free vibration. Because a structure’s vibration characteristics determine how it responds to any type of dynamic load, it is generally recommended to perform a modal analysis first before trying any other dynamic analysis. 

Mathematical Form 

The linear equation of motion for free, undamped vibration is 

Assume harmonic motion 

Substituting   {u} and {} in the governing equation gives an eigenvalue equation: 

This equality is satisfied if

It implies the solution of the equation is trivial and has no vibration which is not possible. 

Or if det ([K] - [M]) = {0} 

This is an eigenvalue problem which may be solved up to n roots (, …,).

These roots are the eigenvalues of the equation. 

For each root (eigenvalue), there is a corresponding eigenvector ({ 

Eigen frequencies and Mode Shapes

In Modal Analysis the eigenvalues are the square of the natural circular frequency of the structure  whereas the eigenvectors represent the corresponding mode shapes . Mode shapes can be normalized either to the mass matrix or to unity, where the largest component of the vector  is set to 1. Abaqus displays results normalized to the mass matrix.   

1

Because of this normalization, the shape of Degree of Freedom (DOF) solution has real meaning.

The square roots of the eigenvalues are, the structure’s natural circular frequencies (rad/s). Natural frequencies f_i can then calculated as  (cycles /s).

The eigenvectors  represent the mode shapes, i.e. the shape assumed by the structure when vibrating at frequency.

References

• Wire Application reference from Wire Manufacturer https://www.wiroforms.com/products.php
• Abaqus/CAE Vibrations Tutorial https://academy.3ds.com/sites/default/files/pdf/Abaqus_Vibrations_Tutorial.pdf
• Theory of Vibrations