Accounting for Transducer Dynamics in the Measurement of Friction

In many studies of sliding interfaces, the measurement of friction is complicated by quasi-harmonic vibrations of the transducer system. An analytical technique is introduced which accounts for the dynamic characteristics of a force transducer under periodic excitation, and is used to compute the forcing function in the sliding interface. The force transducer is modeled as an elastic cantilever-beam with an attached rigid mass. The forcing function is obtained by solving the time-dependent, fourth-order partial differential equation of Euler-Bernoulli beam theory. The solution is facilitated by the application of Fourier series expansions in time and eigenfunction expansions in space. Results of the method are compared to previous analyses of friction-induced vibration in which the elasticity of the transducer is modeled as a simple spring and the rigid body as a lumped mass, leading to a single degree-of-freedom (DOF) governing equation. It is found that a single DOF calculation based on instantaneous measurement of displacement agrees surprisingly well with the results of the Euler-Bernoulli analysis. A single DOF model based on instantaneous measurement of strain and a static displacement-strain calibration factor agrees well with the Euler-Bernoulli analysis for a low frequency range but deviates at higher frequencies.