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On the monotonicity of the relaxation spectrum of fractional Maxwell model of viscoelastic materials

Stankiewicz A.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

2018

Vol. 7, No 1

117--125

This article focuses on the relaxation spectrum of fractional Maxwell model, which is a generalization of classic viscoelastic Maxwell model to non-integer order derivatives. The analytical formula for the spectrum of relaxation frequencies is derived. Theoretical analysis of the relaxation spectrum monotonicity is conducted by using simple analytical methods and illustrated by means of numerical examples. The necessary and sufficient conditions for the existence and uniqueness of the maximum of relaxation spectrum are stated and proved. The analytical formulas for minimum and maximum of the relaxation spectrum are derived. Also, a few useful properties concerning the relaxation spectrum monotonicity and concavity are given in the mathematical form of simple inequalities expressed directly in terms of the fractional Maxwell model parameters, which can be used to simplify the calculations and analysis.

Identification of parameters of the fractional Maxwell model

Lewandowski R., Chorążyczewski B.

Vibrations in Physical Systems

2008

Vol. 23

223--228

The passive dampers are often modeled using the either classical or fractional rheological models. An important problem, bounded with the fractional models, is an estimation of the model parameters from the experimental data. The process of parameter identification is an inverse problem which is underdetermined and can be ill conditioned. The new method of parameters identification of the fractional Maxwell model is proposed. The parameters are estimated using results obtained from dynamical tests. Results of example calculation based on artificial and experimental data are presented.