THE MUTUAL VARIATIONAL PRINCIPLE OF FREE WAVE PROPAGATION IN PERIODIC STRUCTURES

By taking infinite periodic beams as examples,the mutual variational principle for analyzing thefree wave propagation in periodic structures is established and demonstrated through the use of the propaga-tion constant in the present paper,and the corresponding hierarchical finite element formulation is then de-rived.Thus,it provides the numerical analysis of that problem with a firm theoretical basis of variational prin-ciples,with which one may conveniently illustrate the mathematical and physical mechanisms of the wave prop-agation in periodic structures and the relationship with the natural vibration.The solution is discussed and ex-amples are given.