[S0021-8936(00)03803-4]

We thank Mr. S. Shahruz for his interest in our paper and welcome his comment. In our paper we consider a response bound as a bound with time evolution. Instead of Eq. (7) in Mr. Shahruz’s discussion, our original response bound (Eq. (49) in 1) reads
$‖yt‖⩽min{2λm−1KE0,e−ut2λm−1K′E0*}.$
(1)
The term $e−ut$ appearing in this response bound plays an important role. For the maximum amplitude of the response $maxt‖yt‖,$ we prefer to call it amplitude bound. S. Shahruz and P. Mahavamana’s results in paper (2) and some results from W. Schiehlen and the first author of this closure in papers (3,4) are for the amplitude bounds. Here we would like to point out that the procedure listed in S. Shahruz’s discussion should be to compute our response bound given in Eq. (1) above. It may not be meaningful for the amplitude bounds. In Eq. (29) of our paper, we gave an amplitude bound
$maxlimt‖yt‖⩽2λm−1KE0$
(2)
which also follows directly from Eq. (1) in this closure. We can see that for the computation of this amplitude bound, most operations in the procedure listed in S. Shahruz’s discussion are not necessary. Compared with Mr. Shahruz and Mr. Mahavamana’s amplitude bound given in Eq. (8) of Mr. Shahruz’s discussion, we have the opinion that our amplitude bound is not harder to compute since either the computation of the smallest undamped frequency $ω1$ or the determination whether the matrix $DM−1K+KM−1D$ is positive semi-definite costs extra time. Though he showed their amplitude bounds are tighter than ours for two examples, we do not think this conclusion holds in general. Let us choose a simple example to explain this point. If we change the mass matrix in Eq. (9) of Mr. Shahruz’s discussion to
$M=10001orM=0.1001$
(3)
and the numerical values of the damping matrix and the stiffness matrix remain unchanged, then Mr. Shahruz and Mr. Mahavamana’s amplitude bounds for both cases B1 and B2 become 10. However, our amplitude bounds remain unchanged. They are still 2.51 for the case B1 and 1.15 for the case B2. It is not difficult to find examples which show neither method to be superior.

Besides, we would like to state that although Mr. Shahruz and Mr. Mahavamana’s paper about amplitude bounds for some nonclassically damped systems was published in December 1998 in the Journal of Sound and Vibration, their results were not known to the authors since our paper was received by the ASME Applied Mechanics Division on Aug. 24, 1998 and the final revision of the paper was received on Jan. 19, 1999. Therefore, a comparison with their results was not possible (and maybe not even reasonable since in our paper we discussed mainly response bounds with time evolution and not amplitude bounds).

In conclusion, we agree with Mr. Shahruz that no upper bound can be expected to be tight for all systems. In fact, in our paper we also stated that K. Yae and D. Inman’s response bounds given in paper (5) are in some cases better than ours. But in contrary to Mr. Shahruz we think that improvements on the response bounds are meaningful and do not consider only the amplitude bounds to be important. Mr. Shahruz stated that our response bounds are neither easily computable nor are tight. We hope that we have been able to contribute to this interesting field of research and that in the future more easily computable and tighter response bounds will be developed.

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