Gawronski, W.

Sawicki, Jerzy T.

English

EngagedScholarship @ Cleveland State University

Comments on Response Errors of Non-proportionally Lightly Damped Structures

Cleveland-Marshall College of Law

Cleveland State UniversityEngagedScholarship@CSUMechanical Engineering Faculty Publications Mechanical Engineering Department2-19-1998Comments on Response Errors of Non-proportionally Lightly Damped StructuresW. GawronskiCalifornia Institute of TechnologyJerzy T. SawickiCleveland State University, j.sawicki@csuohio.eduFollow this and additional works at: https://engagedscholarship.csuohio.edu/enme_facpubPart of the Mechanical Engineering CommonsHow does access to this work benefit you? Let us know!Publisher's StatementNOTICE: this is the author’s version of a work that was accepted for publication in Journal of Soundand Vibration. Changes resulting from the publishing process, such as peer review, editing,corrections, structural formatting, and other quality control mechanisms may not be reflected in thisdocument. Changes may have been made to this work since it was submitted for publication. Adefinitive version was subsequently published in Journal of Sound and Vibration, 210, 2,(02-19-1998); 10.1006/jsvi.1997.1272This Letter to the Editor is brought to you for free and open access by the Mechanical Engineering Department at EngagedScholarship@CSU. It hasbeen accepted for inclusion in Mechanical Engineering Faculty Publications by an authorized administrator of EngagedScholarship@CSU. For moreinformation, please contact library.es@csuohio.edu.Original CitationGawronski, W.K., and Sawicki, J.T., (1998) Comments on Response Errors of Non-proportionally Lightly Damped Structures. Journalof Sound and Vibration, 210(2), 285-285, doi: 10.1006/jsvi.1997.1272.AUTHORS’ REPLY W. Gawronski Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, U.S.A. and J. T. Sawicki Department of Mechanical Engineering, Cleveland State University, Cleveland, Ohio 44115, U.S.A. The authors wish to thank Dr Shahruz for his comments [1] on the authors’ letter [2], in which he questioned: (1) that the assumptions of lbk l/lbi l = 1 and l�K l/l�i l =1 lead to the erroneous conclusions regarding the output error; (2) that the two-norm was mistakenly used for the step response; and (3) that the error can exceed the output, as shown in the example of [1]. Regarding the ﬁrst question. In our letter [2] we did not explain that the assumption lbk l/lbi l = 1 denotes the worst case of equation (7). Indeed, consider lbkl/lbi l«1. In this case the kth mode is excited lightly, and the kth error is negligible at the output. Similarly, the ratio l�k l/l�i l = 1 denotes the worst case in equation (7) of reference [2]. Certainly, for l�k l/l�i l«1 the ith mode is excited lightly, and the ith error is negligible at the output. In order to explain the second question, note that the two-norm was ﬁnite in our example. We used the two-norm of the step response in a limited time segment, up to the moment when the motion is stationary. In our case it was from 0 to 10 s in Example 1, and from 0 to 50 s in Example 2, as in Figure 4 of reference [2]. Now we turn to the last question. The example of reference [1] shows that errors under speciﬁc conditions are substantial. The magnitude of the error could be considered small or large, depending on the signal it is compared to. In the discussed example, the frequency of the harmonic excitation was equal to the ﬁrst resonance frequency (of 1 rad/s). Therefore the ﬁrst mode response is dominant (of amplitude 250), while the responses of the remaining modes are negligible (of amplitudes 0·25 and 0·1). The output as a combination of all modal responses is dominated by the ﬁrst mode response in the example. The errors of the ﬁrst mode, as well as the errors of the remaining modes, are negligible when compared to the output (less than 0·1%). Finally, the claim in reference [1] that ‘‘the error in modal co-ordinates does not provide a deﬁnitive measure of size in the physical co-ordinates’’ is not true if the output error is considered: output does not depend on a choice of co-ordinates. In conclusion, we consider an output error as a measure of the system performance. We thank Dr Shahruz for pointing out the inconsistencies in explaining the problem assumptions and in conditions of norm computation. references 1. S. M. Shahruz 1998 Journal of Sound and Vibration 210, 279–284. Comments on ‘‘Response errors of non-proportionally lightly damped structures’’. 2. W. Gawronski and J. T. Sawicki 1997 Journal of Sound and Vibration 200, 543–550. Response errors of non-proportionally lightly damped structures. 

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Comments on Response Errors of Non-proportionally Lightly Damped Structures

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