Natural frequencies and mode shapes can be obtained from existing engineering structures like pile group or single pile subjected to dynamic loading using a dynamic sensor. However, finite element analysis can be used to calculate the mode shapes and natural frequencies. A semi-empirical formula can be used, Nevertheless, modelling using the Finite element method is much more realistic and efficient. You can use the Plaxis 3D software package.

You can perform model analysis (linear analysis) using the Winkler model considering pile as a beam element and soil as a spring. SAP 2000; L-Pile software can be used.

ABAQUS software offer free vibration analysis option. this type of numerical analysis will help to account soil-structure interaction effect on the natural frequency of the system. Following simple example shows how to create simple model by ABAQUS to compute the required number of the natural frequencies of the system.

https://www.youtube.com/watch?v=ujDeH6Pnaio&ab_channel=TrendingMechVideos

Hello Sp Chaurasiya

We should differentiate between vertical vibration and lateral vibration.

The determination of the natural frequency of a pile or system is not at an

easy task because the dynamic resistance of the soil has to be accounted for.

To understand the complexity of the problem you should refer to the paper

Lateral Vibration of Piles and Pile Groups in Non-Homogeneous Transversely Isotropic Media

Hamed Moghaddasi ; Babak Shahbodagh ; and Nasser Khalili

International Journal of Geomechanics, Volume 20 Issue 8 - August 2020