Dissipation and Damping

The Bar-Resonator Interaction in Mallet Percussion Instruments: A Multi-Modal Model and Experimental Validation

“…When we account for the radiation damping (R R > 0) the eigenfrequencies and mode shapes become complex, which leads to a non-diagonal damping matrix C in the modal space and (formally) prevents a decoupling of the equations of motion. However, provided that modal damping is weak and the eigenfrequencies are well separated from each other, it can be shown [21] that the inter-modal damping coefficients ζ rm are negligible, as well as their (first-order) effect on the mode shapes. Therefore, the equations of motion for the dissipative system can be assumed decoupled while maintaining the (real) mode shapes of the equivalent conservative system (assuming R R = 0), as demonstrated in detail in [21].…”

“…However, provided that modal damping is weak and the eigenfrequencies are well separated from each other, it can be shown [21] that the inter-modal damping coefficients ζ rm are negligible, as well as their (first-order) effect on the mode shapes. Therefore, the equations of motion for the dissipative system can be assumed decoupled while maintaining the (real) mode shapes of the equivalent conservative system (assuming R R = 0), as demonstrated in detail in [21]. Under these simplifying, but reasonable, assumptions we can develop the particle displacement w(y, t) in terms of the (real) acoustic modes ϕ r (y) as…”