tr-1441.p65
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- ああす むらかわ
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1 NAL TR-1441 NAL TR-1441 ISSN UDC TECHNICAL REPORT OF NATIONAL AEROSPACE ACE LABORATOR TORY TR-1441 NATIONAL AEROSPACE ACE LABORATOR TORY OF JAPAN AN
2 1 1 1 Self-illumination Correction for Pressure Sensitive Paints on Airplane Models in Wind Tunnel Testing Hiroki SUGIURA 1, Yoshimi IIJIMA 1, Keisuke ASAI 1 ABSTRACT One of the potential problems in applying pressure sensitive paints to a wing-body model is selfillumination of the model. A theoretical model of self-illumination correction by W. M. Ruyten was applied to two, square, painted samples and a wind tunnel test of the MU-300 model. The spatially averaged reflection coefficient of 0.75 was determined for a pressure sensitive paint PtOEP-GP197 by measurement on the two, square, painted samples at a range of included angles. As a result, the applicability of the self-illumination correction method was validated. The self-illumination effect on the MU-300 model was so small that the effect of the self-illumination correction was little. Keywords :Pressure Sensitive Paint, Self-Illumination Correction, Reflection Coefficient Ruyten 2 PtOEP-GP MU-300 I i J j A D Bi i,j A ij j M S j x i N C B i p n P received 27 November, 2001 (Fluid Science Research Center)
3 R S S T Bi x S i i C B i x ij α γ θ x ij x j x i B C n j x ji x j S i B 2 B C 2 B C Ruyten 6) corr i 2 j ref MU ),2) % 5) Ruyten 6) 2 4 MU nm 3),4) self-illumination 1 Lambert Dornier Alpha jet R
4 3 A ( ni xij)( ni x ji) Ruyten Aij = S 4 π xij BSDF (Bidirectional Scattering Distribution Function) i=j A ij 0 n i x ij 0 n j x ji 0 A ij 0 6)Le Sant x i x j A ij 0 BSDF BSDF 3 7) 3.1 a) 2 b) I p Stern-Volmer 8) I ref p 50mm 2mm 2 = A+ B (2.1) I p ref A,B ref 2 B 545mm 25mm B convergent distortion B 2 90 B B 605mm 16 (diffusive filter) j (2.3) S S i i x i i S corr i corr S i = S i Rred AS ij j (2.2) 6) A A ij j S j x i A ij 2
5 (3.5) (2.1) =100, 120, 140, nm B C B 0 S B S C PtOEP- B C GP197 (2.2) SB( γ) RredA( γ) SC( γ) = S 14 (180 ) B (3.1) CCD 60,000 =180 2 electrons nm B C ij DBi() γ = SBi() γ S (180 ) Bi (3.2) f22 T () γ = () γ () γ Bi Aij SCi (3.3) j (3.1) CCD CCD 1 5 A D Bi T Bi CCD CCD I p 3
6 5 () () γ D γ = R T Bi red Bi T Bi C B i D Bi C B i C =100, 120, 140, : Q = DBi( γ) RredTBi( γ) (3.4) γ i =180 DBi( γ) TBi( γ) =100, 120, 140, 160 γ i Rred = =100, 120, 140, 160 T 2 Bi() γ (3.5) γ i 5 =100, 120, 140 D Bi T Bi = =100, 120, 140,
7 (deg) R Q 100 1/ / / / / / / / ,120,140 1/ ,120,140 1/ ,120,140,160 1/ =100, 120, 140, ,120,140, ,120,140,160 1/ =100, 120, 140 B C BC PtOEP-GP ,120,140,160 C (3.5) R R 75% FIB7 R B B /16 B 6 (a) ,120,140, (b) 0 1/41/16 8 3% Q 30% Q C B 7 (3.4) B =180 B 6 (c) 100,120, /4 Q= m2m
8 7 6
9 B =100, 120, 140, 160 B =180 8MU-300 8% on 2 8(a) off 8(b) 2 2 PtOEP-GP197 MU-3008%
10 m 80 8, 8 3mm CCD 9 14 CCD nm on off f22 M= =0 4 80kPa 308K CCD in situ 1.2m CCD 1 10 CCD In situ
11 M=0.75 =2.5 on 2 C P 11M=0.75 =2.5 on 2 C P
12 11 12M=0.75 =2.5 off 2 C P 13
13 M=0.75 =2.5 on 2 C P C P 11 C P RGB off 2 C P MU-300 C P 12 13% 19% C P M=0.75 =2.5 19% 9 5 2% Ruyten C P 0.03 off 2 PtOEP- GP MU-300 ) 18 4 pp m=0.75 =2.5 19% )
14 13 ( ni xij) Si 31 Ω= 2 xij xij pp ) Y. Shimbo, K. Asai, H. Kanda, Y. Iijima & N. Komatsu: Evaluation of Several Calibration Techniques for Pressure Sensitive Paint in Transonic ( nj x ji) ( ni xij) Si Testing, AIAA Ω IS jcosθ = S j I 2 xij xij xij ) Y. Shimbo, K. Asai, Y. Iijima, H. Sugiura, H. Kanda, (A3) N. Komatsu, S. Kita & M. Ishiguro:Pressure Sensitive Paint Application to a Business Jet Model in B4 I J B Transonic Testing, 7th Pressure Sensitive Paint Workshop, I = J /π (A4) ) W. M. Ruyten:Correcting Luminescent Paint (A3) (A4) S j S i Measurements for Self-illumination, ICIASF '97 ( ni xij)( nj x ji) Record, International Congress on Instrumentation S 4 j SiJ π xij in Aerospace Simulation Facilities, pp.3-9, (A5) ) W. M. Ruyten: Self-illumination calibration technique for luminescent paint measurements, Review (A5) J S j S i S i of Scientific Instruments, vol.68, no.9, pp ( ni xij)( nj x ji) 3457, Aij = S 4 j π xij ) Y. Le Sant:Overview of the Self-Illumination Effect (A6) Applied to Pressure Sensitive Paint Applications, ICIASF 2001 Record, pp , ) R.C. Crites: Pressure Sensitive Paint Technique, Von Karmann Institue for Fluid Dynamics, Lecture Series , Measurement Techniques, pp.1-69, B I J j S j 15 x j S j r P i S j P i I P i ) P = θ θ φ = 2 i rd rsin d r sinθdθdφ 63 (B1) ) 32 pp A A x j x i x ji S j x ji n j x ji ( nj xji) Sjcosθ = S x ji j (A1) i S i S j x ji x j S i (A2) S j J S i I S j S i (A1) 15 S j r P i
15 S j x ji S j cos S j x j P i P i /r 2 S j P i (B1) Pi Sjcosθ I = S sinθcosθ θ φ 2 ji d d (B2) J Sj = Iπ S r I = J/ π S j S j J j (B4) 2 π π/2 Sj Isinθcosθdθdφ = Iπ S j (B3) 0 0 C MATLAB program:calculation OF REFLEX COEFFICIENT OF PSP % by Hiroki Sugiura(NAL),99/12/28. % based on W.M.Ruyten:Rev.Sci.Instrum.68(9),1997/9. clear;tic;xmax=512;ymax=512;nrm=1/4000; sby1=211;sby2=329;;sbx1=215;sbx2=335; B xbnrm=sbx2-sbx1+1;ynrm=sby2-sby1+1; Rup=0;Rdown=0;Dbsqr=0;Zdbi=0;Ztbi=0;scf=1; xbn=xbnrm;yn=ynrm; %Read Dark Image fid=fopen(' matlab reflex T dark1.drk','r'); drk1=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T dark2.drk','r'); drk2=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T dark3.drk','r'); drk3=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T dark4.drk','r'); drk4=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T dark5.drk','r'); drk5=fread(fid,[ymax,xmax],'short');status=fclose(fid); drk=(drk1+drk2+drk3+drk4+drk5)*0.2; clear drk1 drk2 drk3 drk4 drk5; % Read Image SB(180) =180 fid=fopen(' matlab reflex T t180f1.img','r'); temp1=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t180f2.img','r'); temp2=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t180f3.img','r'); temp3=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t180f4.img','r'); temp4=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t180f5.img','r'); temp5=fread(fid,[ymax,xmax],'short');status=fclose(fid); temp=(temp1+temp2+temp3+temp4+temp5)*0.2;
16 15 clear temp1 temp2 temp3 temp4 temp5; temp=temp-drk; at=temp(1:ymax/2+20,2:xmax);bt=temp(ymax/2+21:ymax,1:xmax-1); temp=[bt; at];clear at;clear bt; sb18=temp(sby1:sby2,sbx1:sbx2);clear temp; % Read Image SB(100) and SC(100) =100 v=100;disp(v) fid=fopen(' matlab reflex T t100f1.img','r'); temp1=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t100f2.img','r'); temp2=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t100f3.img','r'); temp3=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t100f4.img','r'); temp4=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t100f5.img','r'); temp5=fread(fid,[ymax,xmax],'short');status=fclose(fid); temp=(temp1+temp2+temp3+temp4+temp5)*0.2; clear temp1 temp2 temp3 temp4 temp5;temp=temp-drk; at=temp(1:ymax/2+20,2:xmax);bt=temp(ymax/2+21:ymax,1:xmax-1); temp=[bt; at];clear at;clear bt; sb10=temp(sby1:sby2,sbx1:sbx2);db10=sb10-sb18;db10=medfilt2(db10); % Resize SC(100) in Y-axis C Y dymax1=5;dymax2=12;sc10y1=sby1-dymax1;sc10y2=sby2+dymax2; djx=-11; sc10x1=round(sbx1-cos((180-v)*pi/180)*(sbx2-sbx1))+djx;sc10x2=sbx1-3; ynrm=sby2-sby1+1;xc10nrm=sc10x2-sc10x1+1; i=0;sc10y=zeros(ynrm,xc10nrm); %imshow(nrm*temp(sc10y1:sc10y2,sc10x1:sc10x2),64); for ix=sc10x1:sc10x2; dy1=round((sc10x2-ix)*dymax1/(xc10nrm-1)); dy2=round((sc10x2-ix)*dymax2/(xc10nrm-1)); i=i+1;sc10y(1:ynrm,i)=imresize(temp(sby1-dy1:sby2+dy2-1,ix),[ynrm,1]); sc10=sc10y;tic;clear temp; % Compress Image for Faster Calculation { } xbnrm=round(xbnrm/scf);ynrm=round(ynrm/scf);xc10nrm=round(xc10nrm/scf); db10=imresize(db10,[ynrm xbnrm]);sc10=imresize(sc10,[ynrm xc10nrm]); % Calculate Matrix A(100),DBi(100),and TBi(100) { A D Bi T Bi } dbi10=reshape(db10,[1 xbnrm*ynrm]);scj10=reshape(sc10,[xc10nrm*ynrm 1]); tan2v=(tan(v*pi/180))^2;tanv=tan(v*pi/180);tan2vp=tan2v/pi; a=zeros(1,ynrm*xc10nrm); tbi10=zeros(1,ynrm*xbnrm); sbrate=(xc10nrm*scf+djx)/(xc10nrm*scf)
17 for ix=1:xbnrm; ixix5t2p=(ix-0.5)*tan2vp; ixyn=(ix-1)*ynrm; for iy=1:ynrm; for jx=1:xc10nrm; jxm=jx*sbrate; ijx2jix2=(ix+jxm-1)^2+(jxm-0.5)^2*tan2v; bunsi=ixix5t2p*(jxm-0.5); jxyn=(jx-1)*ynrm; for jy=1:ynrm; a(jxyn+jy)=bunsi/(ijx2jix2+(iy-jy)^2)^2; tbi10(ixyn+iy)=a*scj10; % Calculate Denominator and Numerator of Reflex Coefficient (100) Rup=Rup+dbi10*tbi10';Rdown=Rdown+tbi10*tbi10'; Dbsqr=Dbsqr+dbi10*dbi10'; clear dbi10 tbi10 xc10nrm sb10 db10 sc10 sc10y scj10 tanv tan2vtan2vp a ix iy jx jy i j temp ; clear dy1 dy2 sc10y1 sc10y2 dymax1 dymax2; % Reset to Initial Value (*scf) xbnrm=xbn;ynrm=yn; % Read Image SB(120) and SC(120) =120 v=120;disp(v) fid=fopen(' matlab reflex T t120f1.img','r'); temp1=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t120f2.img','r'); temp2=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t120f3.img','r'); temp3=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t120f4.img','r'); temp4=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t120f5.img','r'); temp5=fread(fid,[ymax,xmax],'short');status=fclose(fid); temp=(temp1+temp2+temp3+temp4+temp5)*0.2; clear temp1 temp2 temp3 temp4 temp5;temp=temp-drk; at=temp(1:ymax/2+20,2:xmax);bt=temp(ymax/2+21:ymax,1:xmax-1); temp=[bt; at];clear at;clear bt; sb10=temp(sby1:sby2,sbx1:sbx2);db10=sb10-sb18;db10=medfilt2(db10); % Resize SC(120) in Y-axis C Y dymax1=5;dymax2=11;sc10y1=sby1-dymax1;sc10y2=sby2+dymax2; djx=-16; sc10x1=round(sbx1-cos((180-v)*pi/180)*(sbx2-sbx1))+djx;sc10x2=sbx1-3; ynrm=sby2-sby1+1;xc10nrm=sc10x2-sc10x1+1;
18 17 i=0;sc10y=zeros(ynrm,xc10nrm); for ix=sc10x1:sc10x2; dy1=round((sc10x2-ix)*dymax1/(xc10nrm-1)); dy2=round((sc10x2-ix)*dymax2/(xc10nrm-1)); i=i+1;sc10y(1:ynrm,i)=imresize(temp(sby1-dy1:sby2+dy2-1,ix),[ynrm,1]); sc10=sc10y;tic;clear temp; % Compress Image for Faster Calculation{ } xbnrm=round(xbnrm/scf);ynrm=round(ynrm/scf);xc10nrm=round(xc10nrm/scf); db10=imresize(db10,[ynrm xbnrm]);sc10=imresize(sc10,[ynrm xc10nrm]); % Calculate Matrix A(120),DBi(120),and TBi(120) { A D Bi T Bi } dbi10=reshape(db10,[1 xbnrm*ynrm]);scj10=reshape(sc10,[xc10nrm*ynrm 1]); tan2v=(tan(v*pi/180))^2;tanv=tan(v*pi/180);tan2vp=tan2v/pi; a=zeros(1,ynrm*xc10nrm); tbi10=zeros(1,ynrm*xbnrm); sbrate=(xc10nrm*scf+djx)/(xc10nrm*scf) for ix=1:xbnrm; ixix5t2p=(ix-0.5)*tan2vp; ixyn=(ix-1)*ynrm; for iy=1:ynrm; for jx=1:xc10nrm; jxm=jx*sbrate; ijx2jix2=(ix+jxm-1)^2+(jxm-0.5)^2*tan2v; bunsi=ixix5t2p*(jxm-0.5); jxyn=(jx-1)*ynrm; for jy=1:ynrm; a(jxyn+jy)=bunsi/(ijx2jix2+(iy-jy)^2)^2; tbi10(ixyn+iy)=a*scj10; % Calculate Denominator and Numerator of Reflex Coefficient (100,120) Rup=Rup+dbi10*tbi10';Rdown=Rdown+tbi10*tbi10'; Dbsqr=Dbsqr+dbi10*dbi10'; clear dbi10 tbi10 xc10nrm sb10 db10 sc10 sc10y scj10 tanv tan2vtan2vp a ix iy jx jy i j temp ; clear dy1 dy2 sc10y1 sc10y2 dymax1 dymax2; % Reset to Initial Value (*scf) xbnrm=xbn;ynrm=yn; % Read Image SB(140) and SC(140) =140 v=140;disp(v) fid=fopen(' matlab reflex T t140f1.img','r'); temp1=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t140f2.img','r'); temp2=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t140f3.img','r');
19 temp3=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t140f4.img','r'); temp4=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t140f5.img','r'); temp5=fread(fid,[ymax,xmax],'short');status=fclose(fid); temp=(temp1+temp2+temp3+temp4+temp5)*0.2; clear temp1 temp2 temp3 temp4 temp5;temp=temp-drk; at=temp(1:ymax/2+20,2:xmax);bt=temp(ymax/2+21:ymax,1:xmax-1); temp=[bt; at];clear at;clear bt; sb10=temp(sby1:sby2,sbx1:sbx2);db10=sb10-sb18;db10=medfilt2(db10); % Resize SC(140) in Y-axis C Y dymax1=4;dymax2=8;sc10y1=sby1-dymax1;sc10y2=sby2+dymax2; djx=-14; sc10x1=round(sbx1-cos((180-v)*pi/180)*(sbx2-sbx1))+djx;sc10x2=sbx1-3; ynrm=sby2-sby1+1;xc10nrm=sc10x2-sc10x1+1; i=0;sc10y=zeros(ynrm,xc10nrm); for ix=sc10x1:sc10x2; dy1=round((sc10x2-ix)*dymax1/(xc10nrm-1)); dy2=round((sc10x2-ix)*dymax2/(xc10nrm-1)); i=i+1;sc10y(1:ynrm,i)=imresize(temp(sby1-dy1:sby2+dy2-1,ix),[ynrm,1]); sc10=sc10y;tic;clear temp; % Compress Image for Faster Calculation{ } xbnrm=round(xbnrm/scf);ynrm=round(ynrm/scf);xc10nrm=round(xc10nrm/scf); db10=imresize(db10,[ynrm xbnrm]);sc10=imresize(sc10,[ynrm xc10nrm]); % Calculate Matrix A(140),DBi(140),and TBi(140) { A D Bi T Bi } dbi10=reshape(db10,[1 xbnrm*ynrm]);scj10=reshape(sc10,[xc10nrm*ynrm 1]); tan2v=(tan(v*pi/180))^2;tanv=tan(v*pi/180);tan2vp=tan2v/pi; a=zeros(1,ynrm*xc10nrm); tbi10=zeros(1,ynrm*xbnrm); sbrate=(xc10nrm*scf+djx)/(xc10nrm*scf) for ix=1:xbnrm; ixix5t2p=(ix-0.5)*tan2vp; ixyn=(ix-1)*ynrm; for iy=1:ynrm; for jx=1:xc10nrm; jxm=jx*sbrate; ijx2jix2=(ix+jxm-1)^2+(jxm-0.5)^2*tan2v; bunsi=ixix5t2p*(jxm-0.5); jxyn=(jx-1)*ynrm; for jy=1:ynrm; a(jxyn+jy)=bunsi/(ijx2jix2+(iy-jy)^2)^2; tbi10(ixyn+iy)=a*scj10;
20 19 % Calculate Denominator and Numerator of Reflex Coefficient (100,120,140) Rup=Rup+dbi10*tbi10';Rdown=Rdown+tbi10*tbi10'; Dbsqr=Dbsqr+dbi10*dbi10'; clear dbi10 tbi10 xc10nrm sb10 db10 sc10 sc10y scj10 tanv tan2vtan2vp a ix iy jx jy i j temp ; clear dy1 dy2 sc10y1 sc10y2 dymax1 dymax2; % Reset to Initial Value (*scf) xbnrm=xbn;ynrm=yn; % Read Image SB(160) and SC(160) =160 v=160;disp(v) fid=fopen(' matlab reflex T t160f1.img','r'); temp1=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t160f2.img','r'); temp2=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t160f3.img','r'); temp3=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t160f4.img','r'); temp4=fread(fid,[ymax,xmax],'short');status=fclose(fid); fid=fopen(' matlab reflex T t160f5.img','r'); temp5=fread(fid,[ymax,xmax],'short');status=fclose(fid); temp=(temp1+temp2+temp3+temp4+temp5)*0.2; clear temp1 temp2 temp3 temp4 temp5; temp=temp-drk; at=temp(1:ymax/2+20,2:xmax);bt=temp(ymax/2+21:ymax,1:xmax-1); temp=[bt; at];clear at;clear bt; sb10=temp(sby1:sby2,sbx1:sbx2); db10=sb10-sb18;db10=medfilt2(db10); % Resize SC(160) in Y-axis C Y dymax1=2;dymax2=4;sc10y1=sby1-dymax1;sc10y2=sby2+dymax2; djx=-9; sc10x1=round(sbx1-cos((180-v)*pi/180)*(sbx2-sbx1))+djx;sc10x2=sbx1-3; ynrm=sby2-sby1+1;xc10nrm=sc10x2-sc10x1+1; i=0;sc10y=zeros(ynrm,xc10nrm); for ix=sc10x1:sc10x2; dy1=round((sc10x2-ix)*dymax1/(xc10nrm-1)); dy2=round((sc10x2-ix)*dymax2/(xc10nrm-1)); i=i+1;sc10y(1:ynrm,i)=imresize(temp(sby1-dy1:sby2+dy2-1,ix),[ynrm,1]); sc10=sc10y;tic;clear temp; % Compress Image for Faster Calculation { } xbnrm=round(xbnrm/scf);ynrm=round(ynrm/scf);xc10nrm=round(xc10nrm/scf); db10=imresize(db10,[ynrm xbnrm]);sc10=imresize(sc10,[ynrm xc10nrm]);
21 % Calculate Matrix A(160),DBi(160),and TBi(160) { A D Bi T Bi } dbi10=reshape(db10,[1 xbnrm*ynrm]);scj10=reshape(sc10,[xc10nrm*ynrm 1]); tan2v=(tan(v*pi/180))^2;tanv=tan(v*pi/180);tan2vp=tan2v/pi; a=zeros(1,ynrm*xc10nrm); tbi10=zeros(1,ynrm*xbnrm); sbrate=(xc10nrm*scf+djx)/(xc10nrm*scf) for ix=1:xbnrm; ixix5t2p=(ix-0.5)*tan2vp; ixyn=(ix-1)*ynrm; for iy=1:ynrm; for jx=1:xc10nrm; jxm=jx*sbrate; ijx2jix2=(ix+jxm-1)^2+(jxm-0.5)^2*tan2v; bunsi=ixix5t2p*(jxm-0.5); jxyn=(jx-1)*ynrm; for jy=1:ynrm; a(jxyn+jy)=bunsi/(ijx2jix2+(iy-jy)^2)^2; tbi10(ixyn+iy)=a*scj10; % Calculate Denominator and Numerator of Reflex Coefficient (100,120,140,160) Rup=Rup+dbi10*tbi10';Rdown=Rdown+tbi10*tbi10'; Dbsqr=Dbsqr+dbi10*dbi10'; clear dbi10 tbi10 xc10nrm sb10 db10 sc10 sc10y scj10 tanv tan2vtan2vp a ix iy jx jy i j temp ; clear dy1 dy2 sc10y1 sc10y2 dymax1 dymax2;clear drk; % Calculate Reflex Coefficient { } R=Rup/Rdown;display(R) toc % Calculate Uncertainty of R { } qq=dbsqr-2*r*rup+r^2*rdown; urred=(qq/rdown)^0.5;display(urred)
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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