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

The normalized settling time (ts/t) values of oscillatory second 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/t to ± x% utilizing...

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Main Author: Abdul-Fattah Mohamed Ali
Format: Article
Language:English
Published: IIUM Press, International Islamic University Malaysia 2012-01-01
Series:International Islamic University Malaysia Engineering Journal
Subjects:
Online Access:http://journals.iium.edu.my/ejournal/index.php/iiumej/article/view/174
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spelling doaj-f3064954b481481d8f144be4acb930ba2020-11-25T03:17:54ZengIIUM Press, International Islamic University MalaysiaInternational Islamic University Malaysia Engineering Journal1511-788X2289-78602012-01-0112510.31436/iiumej.v12i5.174On The Settling And Response Times Of Underdamped Second-Order SystemsAbdul-Fattah Mohamed Ali The normalized settling time (ts/t) values of oscillatory second 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/t 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/t values for a settling bands range of ±1% = ±x% = ±6%, covering a damping coefficient range of 0.1 = ? = 0.65. Although the relationships are not exact, their general trend is a marginal overestimation of ts/t. 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/t and the normalized response time (tR/t), with the latter assigned the value 5/ ? . The ratio ts/tR can thus be readily established for any ± x% value. ABSTRAK: Masa enapan ternormal (ts/t)  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/t 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 = ? = 0.65. Walaupun hubungannya tidak tepat, trend umum merupakan penganggaran marginal ts/t. 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/t  dan waktu respons ternormal (tR/t), dengan nilai 5/ ? yang ditetapkan kemudiannya. KEYWORDS: process control; instrumentation; mathematical modelling; transient response; 2nd-order system http://journals.iium.edu.my/ejournal/index.php/iiumej/article/view/174KEY WORDSProcess controlInstrumentationMathematical modellingTransient response2nd-order system
collection DOAJ
language English
format Article
sources DOAJ
author Abdul-Fattah Mohamed Ali
spellingShingle Abdul-Fattah Mohamed Ali
On The Settling And Response Times Of Underdamped Second-Order Systems
International Islamic University Malaysia Engineering Journal
KEY WORDS
Process control
Instrumentation
Mathematical modelling
Transient response
2nd-order system
author_facet Abdul-Fattah Mohamed Ali
author_sort Abdul-Fattah Mohamed Ali
title On The Settling And Response Times Of Underdamped Second-Order Systems
title_short On The Settling And Response Times Of Underdamped Second-Order Systems
title_full On The Settling And Response Times Of Underdamped Second-Order Systems
title_fullStr On The Settling And Response Times Of Underdamped Second-Order Systems
title_full_unstemmed On The Settling And Response Times Of Underdamped Second-Order Systems
title_sort on the settling and response times of underdamped second-order systems
publisher IIUM Press, International Islamic University Malaysia
series International Islamic University Malaysia Engineering Journal
issn 1511-788X
2289-7860
publishDate 2012-01-01
description The normalized settling time (ts/t) values of oscillatory second 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/t 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/t values for a settling bands range of ±1% = ±x% = ±6%, covering a damping coefficient range of 0.1 = ? = 0.65. Although the relationships are not exact, their general trend is a marginal overestimation of ts/t. 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/t and the normalized response time (tR/t), with the latter assigned the value 5/ ? . The ratio ts/tR can thus be readily established for any ± x% value. ABSTRAK: Masa enapan ternormal (ts/t)  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/t 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 = ? = 0.65. Walaupun hubungannya tidak tepat, trend umum merupakan penganggaran marginal ts/t. 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/t  dan waktu respons ternormal (tR/t), dengan nilai 5/ ? yang ditetapkan kemudiannya. KEYWORDS: process control; instrumentation; mathematical modelling; transient response; 2nd-order system
topic KEY WORDS
Process control
Instrumentation
Mathematical modelling
Transient response
2nd-order system
url http://journals.iium.edu.my/ejournal/index.php/iiumej/article/view/174
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