Linearized and nonlinear dynamics of bowed bars

Abstract

Friction-excited instruments have been for many decades an inexhaustible source of physical delight. In the last decades, spectacular improvements in computational power and in numerical methods enabled simulations of the self-excited nonlinear regimes with considerable realism and detail. The authors of the present paper have achieved many such nonlinear simulations, using a powerful modal approach, and decided to investigate here in detail the characteristics of the linearized and nonlinear regimes of bowed bars. After stating the theoretical modeling approach for the nonlinear problem, we derive a corresponding linearized model, from which the complex eigenvalues and eigenvectors are computed as a function of the bowing parameters (friction parameters, normal force, and tangential velocity, accounting for the bowing location). We thus obtain plots of the modal frequencies, damping values, and complex mode shapes, as a function of the bowing parameters, as well as stability charts for each one of the system modes. When compared with the nonlinear motion regimes, these results offer interesting information concerning the stability behavior of the system, and further insight when addressing the postinstability nonlinear limit-cycle responses. Several differently tuned bars are addressed in order to assert the configurations leading to optimal playability.