Dear Azharuddin Ahmed Ansari

Eigenvalues and eigenvectors allow us to "reduce" a linear operation to separate, simpler, problems. For example, if stress is applied to a "plastic" solid, the deformation can be dissected into "principle directions"- those directions in which the deformation is greatest.

Kindly look at these links:

https://orbi.uliege.be/bitstream/2268/178961/1/CAN15.pdf

https://www.slideshare.net/pgeorge3/feb10-5293671#:~:text=Section%205.1%20Eigenvalues%20and%20Eigenvectors,eigen%20meaning%20proper%20or%20characteristic.&text=the%20bridge%202.-,Natural%20frequency%20of%20the%20bridge%20is%20the%20eigenvalue%20of%20the,engineers%20when%20they%20design%20structures.

Eigen values are used for solution of equations involving many degrees of freedom and one can use the values for finding the reactions, displacements at the nodes under consideration.It is a form of matrix solution which in turn is used to solve complex equations.

You can describe the natural frequency of the bridge by the eingevalue of smaller magnitude and it is very helpful in order that you can ensure the stability of the structure. So you need to solve the equation for natural frequencies which is:

[Mass matrix] {2nd derivate of u} + [Stiffness matrix] {u}=0

Azharuddin Ahmed Ansari

If you look at the construction of suspension bridges... Each such bridge has a "natural frequency" and when the physical system that models the bridge is linearized, this frequency corresponds to the eigenvalue of the smallest magnitude.

Engineers want that this frequency to be reasonably far away from any frequencies that occur naturally, such as local wind conditions. When this fails to be the case, i.e. when the natural frequency of the bridge corresponds to wave frequencies induced by the environment, the frequencies become (nearly) additive, as opposed to the desired state of the frequency (nearly) cancelling out. The result is, for example, what occurred in the 1940 collapse of the Tacoma Narrows Bridge. Hence, Eigenvalue analysis or vibration analysis is very useful for the design of bridges.

I hope, this explanation helps!

With regards,

Pravin Kumar

Hello Azharuddin Ahmed Ansari

The natural frequencies play an important role in the design of structures subjected to moving loads. The response of a structure subjected to a moving load consists of two parts, the forced response when the load is moving on the structure and the free vibration after the load has passed. Under a number of simultaneous moving loads, for some conditions, the free vibration response due to each passing load may build up and cause high vibration amplitude i.e. it may cause resonance. This is more likely to happen in cases when the same fixed distance exists between each moving load such as a train passing over a bridge. The bridge must be designed such that these resonances do not occur. You may check my work on the dynamic response of curved beams subjected to moving loads (link below). The conditions for resonance are stated in it. You may also go through the references cited in the article.

Article In Plane Radial Vibration of Uncracked and Cracked Circular ...

Hope it helps.

Regards,

Jatin