Wave propagation and free vibration of a Timoshenko beam mounted on the viscoelastic Pasternak foundation modeled by fraction-order derivatives

There is a second frequency spectrum existing for the classic Timoshenko beam with hinged supports at both ends. However, it is usually assumed that the second frequency spectrum is unphysical. In this paper, the modified Timoshenko beam is studied. The modified Timoshenko beam is mounted upon the viscoelastic foundation. The flexural wave propagation and the free-vibration problem are investigated. The viscoelastic foundation is modeled by the standard solid model or Zenner model with the fraction-order derivative. It is found that there are two kinds of flexural waves that are not only dispersive but also attenuated. In contrast to the classical Timoshenko beam, there is only one frequency spectrum in the modified Timoshenko beam. Furthermore, complex-valued natural frequencies are induced by the viscoelastic foundation. The imaginary part of the complex natural frequency reflects the attenuation properties associated with time. The dispersion and attenuated curves of flexural waves and the complex-valued natural frequency of the first three orders are provided in the numerical results. The influences of the viscoelastic foundation are discussed based on the numerical results.