Abstract: In this paper, the effective application of numericalinversion of Laplace transforms, based on Fourier series expansionsdeveloped by Dubner, Abate and Durbin, is studied for problems ofviscoelastic mechanics. A crucial free parameter is involved in this sort ofmethod and required to be reasonably valued for particular application inadvance since its improper choice leads to obvious errors. An optimal modelto determine the free parameter is constructed in the paper and itsapplicability is validated by the numerical inversion of two types of simplefunctions. As examples to illustrate the practical implementation ofproposed method, the quasi-static and dynamic analysis, within the scope ofaxisymmetric problem, are performed for viscoelastic laminated circularcylindrical shells under uniformly axial pressure and viscoelastic cylindersubjected to inner pressure with abrupt loading respectively. Numericalexperiments show that the optimal model yields valid free parameter, and theproduction of free parameter and calculation time t lies in a definite range.Also it is concluded that the free parameter can be set to be inverselyproportional to t with the proportional coefficient chosen among an effectiverange or determined directly by the result from optimal model for specificcalculation time, and the range of this proportional coefficient isconsidered to be irrelevant to parameter T.