Closure to “Discussion on ‘Vibration Analysis of Non-Classically Damped Linear Systems’ ” (2004, ASME J. Vib. Acoust., 126)

Abstract

We appreciate S. M. Shahruz for his interest in our paper and for the useful comments on the numerical method for solving the transformed system Ÿ(t)+eCt(Λ2−C2)e−CtY(t)=0,Display FormulaY(0)=X0, Ẏ(0)=CX0+Ẋ0As pointed out by Shahruz, the numerical method proposed in our paper is an approximate approach. The approximation comes from the assumption that the time-varying matrix eCt(Λ2−C2)eCt is time-invariant within short time duration jh<t<(j+1)h(j=0,1,2,[[ellipsis]],N) where h is the time step, i.e., Display FormulaeCt[Λ2−C2]e−Ct≈ejhC[Λ2−C2]e−jhC,jh<t<(j+1)hIn this way Eq. (1) is approximated by Display FormulaŸ(t)+ejhC[Λ2−C2]e−jhCY(t)=0Initial Conditions, jh<t<(j+1)hThe accuracy of this numerical approach depends on the time step length h. The shorter the time step length h is, the better the accuracy is.