r5.05.07 Gyroscopic matrices of straight beams and disks

Summary:

This document presents the formulation of gyroscopic damping and stiffness matrices for girder, solid and discrete elements.

Beams are straight beams only (Elements POU_D_T and POU_D_E). The section is constant over length and circular in shape. The material is homogeneous and isotropic.

Disks are cylinders of revolution whose axis coincides with the axis of the beam. The disk is assumed to be undeformable.

The hypotheses adopted are as follows:

• Timoshenko hypothesis: transverse shear and all inertia terms are taken into account. This hypothesis is to be used for low levels of movement (Elements POU_D_T).

• Euler hypothesis: transverse shear is neglected. This hypothesis is verified for strong impulses (Elements POU_D_E).

The natural speed of rotation (along the axis of the beam) may be constant or variable.

In Code_Aster, the adopted convention defines the positive direction along the axis of rotation as being the usual trigonometric direction of rotation.

• 1. The beam element with a constant circular cross section
  • 1.1. Defining landmarks
  • 1.2. specifications
  • 1.3. Calculation of the kinetic energy of the Timoshenko beam
  • 1.4. Interpolation functions
  • 1.5. Calculation of equilibrium equations
• 2. The circular disk
  • 2.1. Calculation of the kinetic energy of the disk
  • 2.2. Calculation of equilibrium equations
• 3. The solid 3D element
  • 3.1. Position in motion in the rotating coordinate system
  • 3.2. Expression of kinetic and potential energies
  • 3.3. Calculation of the equilibrium equations of the rotating system
• 4. Description of the versions