A new method is presented to determine the mode shapes and frequencies of uniform systems consisting of chains of masses and springs of arbitrary number with arbitrary boundary conditions. Instead of the classical eigenproblem approach, the system is analysed in terms of circulating waves and associated phase lags. The phasor conditions for the establishment of standing waves determine the vibration modes. The conditions fully specify their shapes and frequencies, and lead to simple, explicit expressions for the components of the modal vectors and the associated natural frequencies. In addition, the form of the phasor diagrams of the modes gives insight into the modal behaviour. The orthogonality of mode shapes also readily emerges. Examples are presented for different boundary conditions. Although not presented, it is possible to extend the approach to non-uniform lumped systems and to forced frequency responses.