On The Settling And Response Times Of Underdamped Second-Order Systems

  • Abdul-Fattah Mohamed Ali
Keywords: KEY WORDS, Process control, Instrumentation, Mathematical modelling, Transient response, 2nd-order system

Abstract

The normalized settling time (ts /τ) values of oscillatory  2nd-order systems, when subjected to a step-change forcing function  (SCFF), depend on the sensitivity of the measuring instrument employed  to indicate the response (± x%). An attempt is made to mathematically  relate ts /τ to ± x% utilizing the exact, and a simplified, expression  for the lower boundary of the decay envelope (LBDE). The two obtained relationships were tested against the actual ts /τ values for a settling band range of   ±1% ≤ ±x% ≤ ±6%, covering a damping  coefficient range of 0.1 ≤ ζ ≤ 0.65. Although the relationships are not  exact, their general trend is a marginal overestimation of ts /τ. The  relationship based on the simplified LBDE was chosen for being simpler 
and slightly more accurate of the two. This led to a suggested distinction between ts /τ and the normalized response time (tR /τ) with the latter assigned the value 5/ ζ . The ratio ts /tR can thus be readily established for any ± x% value.

ABSTRAK: Masa enapan ternormal (ts/τ)  nilai ayunan system terbit kedua, apabila fungsi memaksa ubah berperingkat (step-change forcing function (SCFF)) dijalankan ke atasnya, bergantung kepada kepekaan alat pengukur yang digunakan untuk mengukur respons (± x%). Satu percubaan dijalankan secara matematik untuk mengaitkan ts/τ to ± x% dengan mempergunakan ekspresi yang tepat dan mudah, pada sempadan bawah sampul reputan (lower boundary of the decay envelope (LBDE)). Dua hubungan yang diperolehi dikaji terhadap nilai ts/t sebenar untuk julat jalur enapan ±1% ≤ ±x% ≤ ±6%, melingkungi julat pekali redaman 0.1 ≤ ζ ≤ 0.65. Walaupun hubungannya tidak tepat, trend umum merupakan penganggaran marginal ts/τ. Hubungan berdasarkan LBDE adalah berdasarkan  LBDE yang telah dipermudahkan, ia dipilih kerana ianya senang dan agak tepat antara keduanya. Ini mendorong kepada perbezaan yang disarankan antara ts/τ dan waktu respons ternormal (tR/τ), dengan nilai 5/ τ yang ditetapkan kemudiannya.

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Author Biography

Abdul-Fattah Mohamed Ali

Assistant Prof. Environmental Enginnering Department

Baghdad University / Iraq

References

REFERENCES

D.R. Coughanowr, and L.B. Koppel, "Process Systems Analysis and Control", McGraw-Hill Kogakusha, Tokyo, pp. 85-88, 1965.

A. Pollard, " Process Control", 2nd Reprint, Heinemann Educational Books, London, pp. 75-78 , 83, 1981.

G. Stephanopoulos, "Chemical Process Control", Prentice-Hall, Englewood Cliffs, New Jersey, pp. 190-191, 1984.

D.E. Seborg, T.F. Edgar, and D.A. Mellichamp, "Process Dynamics and Control", John Wiley (Asia), Singapore, pp. 117-118, 2004.

K. Ogata, “ Modern Control Engineering “ , 5th Edition, Pearson, Upper Saddle River, New Jersey, p. 180-183, 2010.

Published
2012-01-04
How to Cite
Ali, A.-F. M. (2012). On The Settling And Response Times Of Underdamped Second-Order Systems. IIUM Engineering Journal, 12(5). https://doi.org/10.31436/iiumej.v12i5.174