on continuous elastic foundations (i.e. the continuous reactions due to external loading is distributed along the length of the beam) Dynamic bending of beams The dynamic bending of beams, also known as flexural vibrations of beams, was first investigated by Daniel Bernoulli in the late 18th century. Bernoulli's equation of motion of a vibrating beam tended to overestimate the natural frequencies of beams and was improved marginally by Rayleigh in 1877 by the addition of a mid-plane rotation. In 1921 Stephen Timoshenko improved the theory further by incorporating the effect of shear on the dynamic response of bending beams. This allowed the theory to be used for problems involving high frequencies of vibration where the dynamic Euler–Bernoulli theory is inadequate. The Euler-Bernoulli and Timoshenko theories for the dynamic bending of beams continue to be used widely by engineers. Euler–Bernoulli theory The Euler–Bernoulli equation for the dynamic bending of slender, isotropic, homogeneous beams of constant cross-section under an applied transverse load is where is the Young's modulus, is the area moment of inertia of the cross-section, is the deflection of the neutral axis of the beam, and is mass per unit length of the beam. Free vibrations For the situation where there is no transverse load on the beam, the bending equation takes the form Free, harmonic vibrations of the beam can then be expressed as and the bending equation can be written as The general solution of the above equation is where are constants and Timoshenko–Rayleigh theory In 1877, Rayleigh proposed an improvement to the dynamic Euler–Bernoulli beam theory by including the effect of rotational inertia of the cross-section of the beam. Timoshenko improved upon that theory in 1922 by adding the effect of shear into the beam equation. Shear deformations of the normal to the mid-surface of the beam are allowed in the Timoshenko–Rayleigh theory. The equation for the bending of a linear elastic, isotropic, homogeneous beam of constant cross-section under these assumptions is where is the polar moment of inertia of the cross-section, is the mass per unit length of the beam, is the density of the beam, is the cross-sectional area, is the shear modulus, and is a shear correction factor. For materials with Poisson's ratios () close to 0.3, the shear correction factor are approximately Free vibrations For free, harmonic vibrations the Timoshenko–Rayleigh equations take the form This equation can be solved by noting that all