Modification of Rayleigh dissipation function for numerical simulation of internal damping in rod structures

Abstract

Introduction. The paper proposes a method of accounting for energy dissipation for the Timoshenko beam by constructing a damping matrix based on a modified Rayleigh function in the numerical solution of the problem. In this modification, the velocity of displacements is replaced by the velocities of linear and angular deformations. This approach allows us to take into account energy dissipation due to internal friction in the material when both its volume and shape change. The presented technique is promising in practical calculations of structures when shear stiffness has a significant impact on their stress-strain state.Materials and methods. Several proven methods of energy dissipation accounting are considered, including those that make it possible to take into account the energy loss of a moving structure during friction with the external environment (external damping) and dissipation due to friction in the material of the structure deformed in motion (internal damping). Methods for determining the damping coefficients for each of them are presented. The finite element method is used to calculate rod systems. Damping matrices are derived from the condition of stationarity of the total energy of deformation of a mechanical system in motion, including linear and angular deformation rates.Results. Damping matrices proportional to strain rates obtained on the basis of the modified dissipative Rayleigh function are given. A method for determining the damping coefficient taking into account the rates of angular deformation is proposed.Conclusions. The damping matrices presented in the paper describe the energy dissipation during vibrations of mechanical systems due to internal friction in the material. The internal damping matrix was obtained taking into account the influence of linear and angular deformation rates to simulate the dynamic behaviour of short bending structural elements, the deformation of which is described using the Timoshenko model. The performed dimensional check additionally confirms the correctness of the damping matrix construction. Moreover, the dimension of the proposed shear damping coefficient is the same as that of the widely used viscosity coefficient.

Keywords

• dissipation
• damping
• damping coefficients
• timoshenko beam
• material internal friction
• finite element method
• rayleigh dissipative function