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

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.

1. Timoshenko S.P., Gere J. Mechanics of Materials. Moscow, Mir Publ., 1976; 669. (rus.).

2. Sidorov V.N., Badina E.S., Detina E.P. Numerical modeling for oscillations of composite frames accounting for time-nonlocal damping. Mechanics of Composite Materials and Structures. 2022; 28(4):543-552. DOI: 10.33113/mkmk.ras.2022.28.04.543_552.08. EDN ATPNWH. (rus.).

3. Bazoune A. Combined influence of rotary inertia and shear coefficient on flexural frequencies of Timoshenko beam: numerical experiments. Acta Mechanica. 2023; 234(10):4997-5013. DOI: 10.1007/s00707-023-03648-6

4. Onyia M.E., Rowland-Lato E.O. Finite element analysis of timoshenko beam using energy separation principle. International Journal of Engineering Research and Technology. 2020; 13(1):28. DOI: 10.37624/ijert/13.1.2020.28-35

5. Dudaev M. Timoshenko beam stiffness matrix in finite element analysis of turbomachine dynamic behavior. Proceedings of Irkutsk State Technical University. 2014; 6(89):59-65. EDN SGIVXX. (rus.).

6. Bathe K.J., Wilson E.L. Numerical methods in finite element analysis. Prentice-Hall Inc, 1976.

7. Sumali H., Carne T.G. Air-Drag Damping on Micro-Cantilever Beams. Sandia National Laboratories M/S 1070Albuquerque, NM 87185-1070, 2007.

8. Reiner М. Rheology. Moscow, Nauka, 1965; 223. (rus.).

9. Shitikova M.V., Krusser A.I. Models of viscoelastic materials : a review on historical development and formulation. Advanced Structured Materials. 2022; 285-326. DOI: 10.1007/978-3-031-04548-6_14

10. Berendeyev N.N., Zimin N.V., Leontyev N.V., Lyubimov A.K., Smirnov I.A., Storozhev E.V. Determining damping characteristics of a compound structure. Problems of Strength and Plasticity. 2013; 75(4):323-331. EDN RWPMBH (rus.).

11. Arora V., Adhikari S., Vijayan K. FRF-based finite element model updating for non-viscous and non-proportionally damped systems. Journal of Sound and Vibration. 2023; 552:117639. DOI: 10.1016/j.jsv.2023.117639

12. Sorokin E.S. Method of taking into account inelastic resistance of the material when calculating structures for vibrations. Studies on the Dynamics of Structures. Moscow, Gosstroyizdat Publ., 1951. (rus.).

13. Sorokin E.S. To the theory of internal friction in oscillations of elastic systems. Moscow, State publishing house of literature on construction, architecture and building materials, 1960; 130. (rus.).

14. Barabash M.S., Pikul A.V. Material damping in dynamic analysis of structures. International Journal for Computational Civil and Structural Engineering. 1970; 13(3):13-18. DOI: 10.22337/1524-5845-2017-13-3-13-18

15. Potapov V.D. On the stability of a rod under deterministic and stochastic loading with allowance for nonlocal elasticity and nonlocal material damping. Journal of Machinery Manufacture and Reliability. 2015; 1:9-16. EDN TKTLDH (rus.).

16. Sidorov V., Shitikova M., Badina E., Detina E. Review of nonlocal-in-time damping models in the dynamics of structures. Axioms. 2023; 12(7):676. DOI: 10.3390/axioms12070676

17. Ghavanloo E., Shaat M. General nonlocal Kelvin–Voigt viscoelasticity: Application to wave propagation in viscoelastic media. Acta Mechanica. 2022; 233(1):57-67. DOI: 10.1007/s00707-021-03104-3

18. Shepit’ko E.S. Non-local material damping model for the calculation of rod elements : thesis of candidate of technical sciences. Moscow, 2019; 119. (rus.).

19. Sidorov V.N., Badina E.S., Detina E.P. Nonlocal in time model of material damping in composite structural elements dynamic analysis. International Journal for Computational Civil and Structural Engineering. 2021; 17(4):14-21. DOI: 10.22337/2587-9618-2021-17-4-14-21

20. Sidorov V.N., Badina E.S. Finite element method in problems of stability and vibrations of rod structures: examples of calculations in Mathcad and MATLAB : study guide. Moscow, ASV Publishing House, 2021; 173. (rus.).