The solution of Euler-Bernoulli beam vibration equations by Taylor matrix method

In this study, the solution of the high order variable coefficient linear differential equation of the transverse vibration of the Euler-Bernoulli beam subjected to axial loading is presented. Some general information about vibrations is given at the beginning. After reviewing the types of vibration and elements of vibrating mechanical systems, governing equation of the Euler-Bernoulli beam is derived. This equation of motion is a fourth order partial linear differential equation depending on position and time. This equation is solved for the given initial conditions under simple-simple, clamped-clamped and clamped-free boundary conditions. This study presents a new and simple Taylor polynomial matrix method for solving transverse vibration of beams. This method transforms the partial differential equation into a matrix form containing unknown Taylor coefficients. The initial and boundary conditions are imposed into this matrix equation, and then the equation is solved practically. In the study free and forced vibrations of axially loaded Euler-Bernaulli beam are considered, and general and special solutions are obtained. The method is explained with a numerical example and the results are compared with those of the exact solution. The results are shown to be compatible with each other.