Biến đổi Laplace và ứng dụng trong kỹ thuật

Nguyen Thj Phuang vd Dig Tgp chf KHOA HOC & CONG NGHE 173(13): 73-79

BIEN DOI LAPLACE VA UNG DUNG TRONG KY THUAT

NguySn Thj Phuffng', Phan Thi VSn HuyIn

Tneang Dgi hgc Ky ihual Cong nghiep - DH Thai Nguyen

TOM TAT

Dugc dit theo tSn cua nha toin hgc vi thien vin hoc nguoi Phap nil tilng Pierre Simon Laplace

(1749-1827), biSn dlil^^lace la cong cu toan hgc c6 nliiluii'ng dung trong cic ITnh vuc nhu toin

hgc, vgt \^, CO hgc..., cho nhiing b^i toan co dieu kiSn dau lien quan tdi ham cuOng bii'c, cac bai

toan gia trj bien. Ngudn goc cua img dung nay la a cho bien doi Laplace cho phep chuyen tir phep

tinh vi tich phan tr€n ham sang cac phep tinh dai so tren inh ciia ham qua bien doi Laplace, do do se

lam cho vi§c giai cic phuong trinh, hS phuong trinh vi phan, bai toan co dilu kifiii diu, vi tich phan

fro nSn don giin hon. Trong bai bio niy, chiing tfli nghiSn ciiu mfit so ling dung cQa phep biin dii

trong ky thuat. Ll'ng dung trong cac bii toin dao dSng, nghiSn cilu cic qui trinli xiy ra trong m6t

mach diSn lien tuc, tim nghiem ciia bii toin bien co sir dung sir hi trg ciia phin mim toan hgc.

Tu khda: Bien ddi Laplace, phuang trinh viphdn. phuong trinh dao dgng. mgch dien, Mallab

GIOI THIEU

Bien doi Laplace la phep bien doi tich phan

cho phep chuyen tir phep tinh vi tich phan

tren ham f{t) ciia bien thyc t sang cac phep

tinh dai so tren anh F(s) cua bien phiic s, do

do phep bien doi giiip dua cac bai toan va cac

dieu kien dau (phuang trinh, he phuong trinh

vi phan; phuang trinh, he phuang trinh tich

phan, ..). ve bai toan moi chiia ham can tim la

phuong trinh dai so trong mien phiic s. Tuy

cac he phuong trinh vi phan tuyen tinh deu co

the dugc giai bang phuong phap co dien,

nhung vcri cac bai toan ve tin higu va phan

tich he ky thuat la tuyen tinh giQ'a dau vao va

dau ra dugc bieu dien bai cac phuang trinh vi

phan tuyen tinh, phep bien doi Laplace co

ung dung dac biet Mgt khac khi diing phep

biin doi de giai phuang trinh, he phuang

trinh vi phan cac dieu kien ban dau du'pc dua

vao ngay trong qua trinh bien doi nen se xuat

hien Iu6n trong bieu thiic nghiem... Phep

bien doi nay la cong cy hieu qua de giai bai

toan vcri dilu kien ban dau thuang gap khi

nghien ciiu cac mach dien lien tuc, cac he dao

dpng CO hpc... nhit la khi ham kich dpng dau

vao ia cac ham gian doan, tuan hoan, hoac la

ham xung. Ngoai ra, phep bien doi Laplace

con CO the cung cap cong cu tinh toan, cong

Tel: 0979 472770. Email. phiiongnt!812@gmail ci

CU do thi dS viec phan tich, tong hpp va thiel

ke he thong dupe de dang hon.

Xet cac ham goc f{t) ciia bien thyc /, do la cac

ham lien tuc tirng khoang voi r > 0; 3M > 0, 3

5 > 0 sao cho 1/[0I ^ Me^W/> 0 vkf(l) = 0

khir<0 [4].

Phep bien ddi Laplace: Gia su f(t) la mot

him goc. Khi do F(s)= je'"f(t)dl la ham giai

tich trong mia mat phang Re5>j'o va dugc ggi

la anh cua f{t) qua biin doi Laplace, ky hieu

LWO] [4], [5].

Phep bien doi Laplace ngirgc: Phep bien doi

F(s) thanh f(_t) dugc ggi la phep bien doi

Laplace ngupc,XO = L"'[F(5)] [4], [5],

UNO DUNG CUA PHEP BIEN DOI

LAPLACE TRONG KY THUAT

iTng dung trong cac bai toan dao dpng

a. Dao dpng dieu hoa: Xet diem P co khoi

lugng m chuyen dpng thang tren true Ox, co

toa do x{l), va bj hiit boi lyc phuc hoi/(0 =

kx(t). Khi do

mx" = -fj(t)<^x"+(a^x = 0,vai k/m = (ii^[]]

b. Dao dpng tat dan: Xet siic can cua moi

tru'ong vdi dao dgng ti le bac nhat voi van t6c

73