On The Settling And Response Times Of Underdamped Second-Order Systems
DOI:
https://doi.org/10.31436/iiumej.v12i5.174Keywords:
KEY WORDS, Process control, Instrumentation, Mathematical modelling, Transient response, 2nd-order systemAbstract
The normalized settling time (ts /?) 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 /? 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 /? values for a settling band 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 /?. 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 /? and the normalized response time (tR /?) 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/?) 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/? 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/?. 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/? dan waktu respons ternormal (tR/?), dengan nilai 5/ ? yang ditetapkan kemudiannya.
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