Effect of resonant absorption in viscous and dry vibrating contact: Mathematical models and theory connected with slow dynamics and friction welding

• Mathematical model of heating of thin layer located between two vibrating surfaces is studied.
• It is shown that resonant behavior of energy loss must be taken into account.
• Possibility of modelling in various technologies, e.g. in friction welding, is discussed.
• Models that are nonlinearly self-adjoint are found.
• New conservation laws are constructed for self-adjoint models using symmetries.

Process of heating of thin layer located between two vibrating surfaces is studied. Energy loss goes on due to viscous or dry friction. Optimal quantities of shear viscosity and friction corresponding to maximum energy loss are determined. Resonant behavior of loss must be taken into account in the description of “slow dynamics” of rocks and materials exposed to high-intensity seismic or acoustic irradiation as well as in various technologies. Bonding of materials by linear friction welding, widely used in propulsion engineering, can exemplify such a technology.