Three-dimensional mechanical admittance: Theory and new measurement method applied to the violin bridge

The mechanical admittance (or mobility) measures the generalized velocities of a system under generalized forces, whereas the impedance measures the forces when velocities are imposed. It is shown that generally, and in cases of practical interest, the experimental determination of the impedance and that of the admittance must comply with different requirements. Therefore, one description cannot be derived from the other unless the degrees of freedom of the system which are not measured are properly dealt with. Some of the experimental methods presented in the literature are discussed along these lines. A new method is proposed for measuring locally the mobility matrix: it is based on comparison with a known mechanical impedance and requires no force measurements. A realization is presented in the case of the bridge of a violin and a quarter-size cello. Theoretical requirements are found to be met between ~450-1500 Hz for the violin and ~250-2000 Hz for the cello. Limitations of the method are found to be the influence of rotational degrees of freedom and the scarcity of modes at some frequencies. Some implications of the findings for the acoustics of the violin are also discussed.