Periodic behavior in oscillatory systems of one and two dimensions

Abstract

In this work, we study the temporal periodic behavior of well-known one and two-dimensional mechanical systems as the vibrating homogenous string, vibrating square membrane and vibrating circular membrane. When an initial configuration of position and velocity of the system is imposed, the temporal evolution of the superposition of the natural modes of vibration is analyzed according to the non-zero Fourier or Bessel-Fourier coefficients. The relations between the temporal frequencies of the normal modes are used to verify the periodicity of the mechanical movement. Numerical results show the temporal evolution of the systems and the periodicity or non-periodicity of the composed movement is verified.