A general analytical method is developed for the vibrations of two beams coupled together at an arbitrary angle.The stiffness of a joint can take any value from zero to infinity to better model many real-world coupling conditions.Both flexural and longitudinal waves are included to account for the cross-coupling effects at the junctions.Each displacement component is here invariantly expressed,regardless of the coupling or boundary conditions,as a Fourier series supplemented by several closed-form functions to ensure the uniform convergence of the series expansions.Examples are presented to compare the current solution with finite element and experimental results.