The application of Chebyshev polynomials to the solution of the nonprismatic Timoshenko beam vibration problem

Chebyshev series approximation is applied to solve the problem of vibration of the nonprismatic Timoshenko beam resting on a two-parameter elastic foundation. As a result, a system of equations (whose coefficients have a closed form) for calculating the coefficients of the sought solution is obtained. The method is used to solve the free vibration problem for simple supported and clamped–free nonprismatic beams. The results are compared with the results obtained by other authors. Also the nonprismatic beam stability problem is solved and the results are compared with those obtained for Euler beams. To demonstrate the method's applicability to more complex systems the problem of stability of a frame system is solved.