Flexural vibration systems with gyroscopic spinners

Abstract

In this paper, we study the spectral properties of a finite system of flexural elements connected by gyroscopic spinners. We determine how the eigenfrequencies and eigenmodes of the system depend on the gyricity of the spinners. In addition, we present a transient numerical simulation that shows how a gyroscopic spinner attached to the end of a hinged beam can be used as a 'stabilizer', reducing the displacements of the beam. We also discuss the dispersive properties of an infinite periodic system of beams with gyroscopic spinners at the junctions. In particular, we investigate how the band-gaps of the structure can be tuned by varying the gyricity of the spinners. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.

Keywords: chirality; dispersion; flexural waves; gyrobeam; gyroscopic spinners.

Figure 1.

Finite system of beams attached to gyroscopic spinners and with a hinge at one end.

Figure 2.

(a) Eigenfrequencies of the system in figure 1 versus the gyricity of the spinners; (b) a magnification of the results presented in (a) for 0≤ω≤30 rad s⁻¹; (c) mode shape of the structure in the limit when Ω → ∞ for ω = 3.45 rad s⁻¹.

Figure 3.

(a) Single beam hinged at z = 0 and with a gyroscopic spinner at z = L; (b) trajectory O−P of the tip of the beam in (a), subjected to an initial velocity in the x-direction.

Figure 4.

Periodic system made of elastic beams connected by gyroscopic spinners. The positive directions of the displacements and rotations of the structure in the time-harmonic regime are shown in both the yz- and xz-planes.

Figure 5.

Dispersion curves of the periodic structure in figure 4 when (a) Ω = 100 rad s⁻¹, (b) Ω = 500 rad s⁻¹, (c) Ω = 1060.14 rad s⁻¹ = Ω* and (d) Ω = 1500 rad s⁻¹. The length of the periodic cell is L = 6 m. The dashed lines in (a)–(d) represent the case Ω = 0. We note that the scales of the vertical axes in (a) and (b) are different from those in (c) and (d).

Figure 6.

Vibrational modes of the periodic system in figure 4, calculated for Ω = 500 rad s⁻¹, k = π/12 1/m and (a) ω = 79.565 rad s⁻¹ = ω^(a), (b) ω = 231.757 rad s⁻¹ = ω^(b), (c) ω = 507.826 rad s⁻¹ = ω^(c), (d) ω = 605.634 rad s⁻¹ = ω^(d).