How to do dynamic bending of beams; A detailed Explanation.

The dynamic bending of beams;

The dynamic bending of beams, also known as flexural vibrations of beams, was earliest investigated by Daniel Bernoulli in the late 18th century. Bernoulli's equation of motion of a vibrating beam tended to overrate the natural frequencies of beams and was enhanced marginally by Rayleigh in 1877 by the adding up of a mid-plane rotation. In 1921 Stephen Timoshenko improved the theory promote by incorporating the effect of shear on the dynamic reaction of bending beams. This allowed the theory to be used for problems linking high frequencies of vibration where the dynamic Euler-Bernoulli theory is insufficient. The Euler-Bernoulli and Timoshenko theories for the dynamic bending of beams carry on to be used generally by engineers.