Estimation of the state vector and identification of the complex modulus of a beam

A non-uniform viscoelastic beam traversed by flexural waves was considered. Methods based on Timoshenkoʹs model were established for (i) estimation of its state (shear force, transverse velocity, bending moment and angular velocity) at an arbitrary section on the basis of at least four independent measurements, and (ii) identification of its complex modulus, parametric as well as non-parametric, on the basis of at least five independent measurements. From the estimated state, related useful quantities such as strain, stress and power transmission can be obtained. Experimental tests were carried out with beams made of polymethyl methacrylate and polypropylene and instrumented with pairs of strain gauges at eight non-uniformly distributed sections. Estimation of strain at one instrumented section was based on measured strains at five to seven surrounding sections, while identification of the complex modulus was based on measured strains at five to eight sections. Generally, the identified complex moduli showed fair agreement with previous results from tests involving extensional waves, while the estimated strains were in good accord with measured strains. No significant improvement in the quality of results was achieved when the number of measured strains was increased to more than five for the identification of the complex modulus and six for the estimation of state.