Vibration of Nonuniform Simply-Supported Beams

The natural frequencies and corresponding mode shapes for nonuniform simply-supported beams are investigated. The deflections are approximated by a Fourier series and in addition to this normal procedure, the mass and moment of inertia functions are also approximated by Fourier expansions. These are then used in writing out the Lagrangian which can be reduced from a triple series product to a double series product. The end result is an infinite number of homogeneous linear simultaneous algebraic equations in the Fourier deflection coefficients. A finite number of equations are used in approximating the frequencies and mode shapes. Examples show good agreement between these approximations and those obtained by more exact methods.