Localisation of local nonlinearities in structural dynamics using spatially incomplete measured data

This paper presents a procedure to localise nonlinear elements using spatially incomplete measured frequency response data from the structural vibration test. The method does not require measurements of all the responses associated with nonlinear elements and the information about the types of nonlinear elements. In this procedure, the Craig-Bampton reduction method is employed to reduce the dynamic equation onto the measured region and to project the nonlinear forces onto the measured degrees of freedom (DOFs), which are then called reduced nonlinear forces (RNFs). It is shown that the reduced nonlinear forces are the sum of the measured nonlinear forces and the projections of the unmeasured nonlinear forces through the transpose of linear constraint modes. Therefore, by analysing and comparing the magnitude of the reduced nonlinear forces obtained from experiments with the linear constraint modes, we can localise the nonlinear elements without directly measuring their responses. Numerical simulations of a discrete system with two nonlinearities and experimental data from a clamped beam with a nonlinear connection are used to validate the localisation procedure.