Lars Hillström

Non-parametric identification of viscoelastic materials from wave propagation experiments

We investigate how the frequency-dependent wave propagation coefficient and complex modulus for a linearly viscoelastic material can be estimated from wave propagation experiments. The strains at different sections of a bar specimen are measured as functions of time. The time series are transformed into the frequency domain, where a non-parametric identification is made. A thorough analysis of the quality of the non-parametric estimate is made, in which approximate expressions for the covariance matrices of the wave propagation coefficient and the complex modulus are derived. The validity of these expressions are confirmed by numerical studies and real experiments.

On the use of flexural wave propagation experiments for identification of complex modulus

In this paper, we investigate the nonparametric estimation of the frequency dependent complex modulus of a viscoelastic material. The strains due to flexural wave propagation in a bar specimen are registered at different cross sections. The time domain data is transformed into frequency domain using discrete Fourier transform and a nonlinear least squares algorithm is then employed to estimate the complex modulus at each frequency. Inherent numerical problems due to associated ill-conditioned matrices are treated with special care. An analysis of the quality of the nonlinear least squares estimate is also carried out. The validity of the theoretical results are confirmed by numerical studies and experimental tests.

Parametric identification of viscoelastic materials from time and frequency domain data

In this paper a parametric model of the elasticity modulus of a viscoelastic material (polypropylene) is determined by measured strains from wave propagation experiments. The identification of the model parameters is carried out both in the time and in the frequency domain. The advantages and limitations of either approach is discussed. Our main result is that a good representation of the elasticity modulus can be obtained by a general standard linear solid with three elements or seven free parameters. The model is valid over the same frequency range as that of the data, which in the present case extends from 100 Hz to 10 kHz. Such a model makes it possible to perform direct time simulations by time-stepping algorithms.

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 Timoshenkos 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.

Identification of viscoelastic materials

Abstract The frequency-dependent complex modulus for a linearly viscoelastic material is estimated from wave propagation experiments. In these experiments, strains at different sections of a bar specimen are measured as functions of time. The time series are then transformed into the frequency domain, where the identification is made. Two identification techniques, a non-parametric and a parametric, are considered in the paper. Analyses of the quality of the estimates given by the two different techniques are carried out, and approximate expressions for the covariance matrices of the complex modulus are provided. The validity of these expressions are confirmed by numerical studies as well as by real experiments. A comparison between the results of the non-parametric and parametric approaches is made. The results of the comparison are that the parametric model describes the complex modulus quite well and that the variances for the parametric estimates are considerably smaller.

Computationally efficient estimation of wave propagation functions from 1-D wave experiments on viscoelastic materials

Least-squares based non-parametric estimation of the wave propagation functions of a viscoelastic material is considered in this paper. Widely used nonlinear least-squares-based algorithms are often computationally expensive and suffer from numerical problems. In this paper, we propose a class of subspace estimators which assume equidistant sensor configuration. The proposed estimator is computationally economical and numerically robust. Analytical expressions for the estimation accuracy have been derived. It is also shown that the subspace estimator achieves the optimal accuracy under the optimal weighting. The algorithm is employed on simulated data as well as on real experimental data. The results therefrom are shown to confirm the analytical results.

Computationally efficient estimation of wave propagation functions of viscoelastic materials

Computationally efficient estimation of wave propagation functions of viscoelastic materials