Dynamic characteristics of infinite-length and finite-length rods with high-wave-number periodic parameters

Abstract

The dynamic characteristics of the infinite-length and finite-length rods with periodic distribution parameters are studied. The differential equation of longitudinal motion of the period-parametric rod is given. The algebraic matrix equation for the wave motion characteristics of the infinite-length periodic rod is derived based on the Bloch theorem and Fourier series. The characteristic frequencies are determined by the matrix eigenvalues which depend on the characteristic wave number and parametric wave number. Then the algebraic matrix equation for the dynamic characteristics of the finite-length periodic rod is derived based on the Galerkin method. The natural frequencies are determined by the matrix eigenvalues which depend on only the parametric wave number. An improving approach algorithm for solving the eigenvalue problem of high degree-of-freedom systems is developed based on the Rayleigh quotient. Finally, the circular cross-section rod with period-varying diameter is considered and numerical results on the dynamic characteristics are given. Large characteristic wave number and parametric wave number are considered for the infinite-length and finite-length periodic rods. The characteristic frequencies varying with the characteristic wave number and parametric wave number are shown, and the band gaps vanishing are revealed for increasing characteristic wave number. The finite-length periodic rod has the dynamic characteristics different from the infinite-length periodic rod. The effect of the term number of the displacement expansion on the natural frequencies and the natural frequencies varying with the parametric wave number and wave amplitude are shown for the finite-length periodic rod. The local resonance and periodical short descent of the natural frequencies with the increase of the parametric wave number and the different changes of the natural frequencies with the parametric wave amplitude are revealed. The above new dynamic characteristics of the infinite-length and finite-length rods with periodic distribution parameters have a potential application to period-structural design and optimization.