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Double polarisation in nonlinear vibrating piano strings

Tan, Jin Jack; Touzé, Cyril and Cotte, Benjamin 

The present work studies the double polarisation phenomenon observed in vibrating piano strings. From the experimental viewpoint, it is known that when a string is given an initial displacement in one transverse direction (e.g. hammer excitation in the vertical plane), the second transverse displacement (e.g. in the horizontal plane) is also excited after a few milliseconds and the amplitude can be of similar order to the first transverse displacement. This phenomenon contributes to a characteristic piano sound feature called the "double decay". The purpose of this study is to investigate the role of nonlinearities in inducing double polarisations. The nonlinear vibrations of the strings are studied with a two-degrees-of-freedom (dofs) system extracted from the Kirchhoff-Carrier string equations. The method of multiple scales is used to study the free vibrations of two polarisations having nearly equal eigenfrequencies and thus presenting a 1:1 internal resonance. For an imperfect string with slightly different eigenfrequencies between the two polarisations, it is found out that depending on the energy of the excitation, an uncoupled transverse mode can develop into a coupled mode where there is energy exchange between the two transverse polarisations. The coupled mode is stable and the string oscillates in an elliptic path. Numerical experiments are also carried out, confirming the findings of the analytical approach.