Keywords:
vibrations, damping, approximation theory, matrix algebra, differential equations, eigenvalues and eigenfunctions
Topics:
Approximation, Damping, Differential equations, Eigenfunctions, Eigenvalues, Matrix algebra, Vibration, Vibration analysis
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
Y¨t+eCtΛ2−C2e−CtYt=0,
Y0=X0,Y˙0=CX0+X˙0
(1)
As 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 eCtΛ2−C2eCt is time-invariant within short time duration jh<t<j+1hj=0,1,2,…,N where h is the time step, i.e.,
eCt[Λ2−C2]e−Ct≈ejhC[Λ2−C2]...
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