Objective of curve fitting is to represent a set of discrete data by a function (curve). Consider a set of discrete data as given in table.

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1 CURVE FITTING Obectve curve ttg s t represet set dscrete dt b uct curve. Csder set dscrete dt s gve tble. 3 3 = T use the dt eectvel, curve epress s tted t the gve dt set, s = + = + + = e b ler uct plml epetl Y = B Ukw cecets,,, B d b re t be determed There re tw geerl pprches r curve ttg: Lest squre regress Iterplt MM98

2 Lest squre regress The strteg s t derve sgle curve tht best represets the geerl tred the dt set. We mke ert t tersect ever dt pt. Rther, the curve s desged t llw the ptter the dt pt tke s grup. Lest squre regress s ppled t dt sets tht m ct sme errr r se. A dvdul dt pt m be crrect. Ler regress Plml regress Iterplt techque s used t t curve tht drectl psses thrugh ech the dt pts. Such dt usull rgtes rm tbles. Ler terplt Curvler terplt MM98

3 MM98 3 LEAST SQUARE METHOD A curve s tted t gve dt set such tht sum the squres the dereces betwee the gve dt set d the vlues bted rm the tted curve s mmum.. Ler Regress Dt s tted t ler rst rder uct. = + =?, =? Sum the squre the dereces c be bted s d re bted such w tht R s mmum. Ths s de s llws: 0 0 R 0 0 R Arrgg these equts, we bt, Frm ths set equts, ukw cecets d re slved. R

4 MM98. Secd Order Plml Regress Gve dt s tted t secd rder plml s, = + + =?, =?, =? Sum the squre the dereces c be bted s Derettg R wth respect t, d d equtg t zer, we bt the llwg set lgebrc equts r ukw cecets. 3 3 Slut these equts gves the ukw cecets, d. R

5 MM98 5 MULTIPLE LINEAR REGRESSION Ler regress methd c be used t bt ler uct tw r mre vrbles. Fr emple, mght be ler uct d, s, = + + Fr ths tw dmesl cse, the regress le becmes ple. Cecets re g determed b settg up the sum the squres the resduls, R Derettg wth respect t ech the ukw cecets d settg t zer, we get set ler equts r ukw cecets,, d. Arrgg these equts s mtr equt, we bt, Slut ths set ler equts gves the ukw cecets,, d the bve ler uct.

6 TRANSFORMATION FOR DATA LINEARIZATION Ler regress prvdes pwerul techque r ttg best le t dt set. Ths techque s used the relt betwee depedet d depedet vrbles s ler. Hwever, ths s t lws the cse. Fr sme cses, plml regress s pprprte. Fr thers, trsrmt c be used t epress the dt the rm tht s cmptble wth ler regress. l Lerzt e b T lerze ths epress, we tke the lgrthm bth sdes s llws: e b l l b Ths c be wrtte s Y c c t Where Y=l, c =l, c =b d t=. Ths s ler equt. Hece, bve ler regress methd c be used t d the ukw cecet c d c. MM98 6

7 EXAMPLE: Temperture T smll cpper sphere clg the r s mesured s uct tme t t eld the llwg dt set: t s T C Ft ths dt t curve T Ae t MM98 7

8 Emple: Tmt pste ws tested vscmeter d the llwg dt ere bted. Determe the lud s Newt d d ts descrptve equt. t N/m dv/d rd/s Slut: A plt these dt ppers the gure. Ths gure mples tht the lud s -Newt pseudplstc s we ssume equt shuld be the rm t K dv d Usg dt lerzt, ukws K d c be determed utlzg lest squre methd. MM98 8

9 Tkg the turl lgrthm bth sdes, we get lt = lk+ldv/d We c wrte smpler tt s, T = b +b V Where T = lt b = b = lk V = ldv/d Ukw cstts b d b c be determed usg the methd lest squre rm the trsrmed dt gve the tble belw: V =ldv/d S. T=lt S.6 b 5.b b b 58.6 Slvg r b d b, we get b =3.90 b = 0.57 Frm these, b = lk K = epb = ep 3.90 = 9.3 = b = 0.57 Fl equt r the tmt pste dv t 9.3 d 0.57 Bsed the sher stress-str curve, we cclude tht tmt pste s pseudplstc -Newt lud. MM98 9

10 EXAMPLE: Flw rte ppe s mesured r deret pressure drps d ppe dmeters. Results the mesuremets re gve the tble belw. Ft curve t the gve dt s Q BD P b D m P tm Ths tble shws the lw rte s uct dmeter D d Pressure drp P Slut: Lerzg the dt, multple ler lest squre regress methd c be used t determe ukws B, d b. We re lkg r epress the rm Q= B D P b B =?, =?, b =? Tkg the lgrhtm bth sdes the equt, we bt lq = lb + ld + blp Ths equt c be wrtte s Y = C + C + C 3 X where Y = lq, C = lb, C =, C 3 = b, = ld d = lp Nw we eed t slve C, C d C 3, d the determe B, d b C, C d C 3 c be determed pplg the lest squre regress metht MM98 0

11 LD LDp -,0397-0, ,3367-0,6935 -,00-0,8397 0,7937,557-0,0536 -,3869-0,07,3869,673 0,83 -,0788 0,339,7078,788 0, ,669 0,553885,50599, , ,08 c 9,5588 -,56065,037,05335 c 9, ,08, ,87088 c3 3,7638 MM98

12 MM98 C C C 3 Frm ths equt we bt the llwg set equts r ukws C, C d C 3. 6C 6.3C 0.C 3 = C C + 0.0C 3 = C + 0.0C + 3.8C 3 = 3.76 Slut ths set equts elds, C =.5039, C =.3039, C 3 =.005 Ukw cecets the epress we re lkg r c be bted usg the bve relts s llws: C = lb B = ep C = ep.5039 =.99 C = =.3039 C 3 = b b =.005 Therere lw rte Q, ppe s uct dmeter D, d pressure drp P c be ud s Q=.99 D.3039 P.005

13 MM98 3 LAGRANGE INERPOLATION The Lgrge terplt c be rmulted s L where L 0 Iterplt curve psses thrugh ll the dt pts csdered. Ler terplt Curvler terplt Subscrpt dctes the rder the terplt plml. Frst ler d secd rder terplt plmls c be wrtte s

14 MM98 EXAMPLE: Use Lgrge terpltg plml secd rder t t curve t pts At = =0 = =.3863 =6 =.798 Slut: Geerl terplt plml c be wrtte s L where L 0 Secd rder terplt plml c be wrtte s Substtutg the vlues, Perrmg the clcults, we bt,