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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| Format: | Article |
| Language: | English |
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IIUM Press, International Islamic University Malaysia
2012-01-01
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| Series: | International Islamic University Malaysia Engineering Journal |
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| Online Access: | http://journals.iium.edu.my/ejournal/index.php/iiumej/article/view/174 |
| Summary: | 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 |
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| ISSN: | 1511-788X 2289-7860 |