Modelling and analysis of the natural oscillations of a prismatic elastic beam based on a projection approach

A regular approach to the construction of mathematical models describing the natural motions of beam-type elastic bodies within the limits of the linear theory of elasticity is developed using the method of integrodifferential relations. By employing the integral form of the equations of state, relating the stresses and strains and also the velocities and momenta, the system of partial differential equations is reduced to a denumerable system of ordinary differential-algebraic equations. A polynomial representation of the unknown functions of the displacements, stresses and momentum density along two spatial coordinates is used for this purpose. The effect of the geometric and mechanical parameters of the system on the frequencies and modes of the natural oscillations of a rectilinear elastic beam is investigated.