Calculate the difference between minimum values of a parabola and straight line (from a plot)

Question

Alina Abdikadyr 2023-2-10

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1910540-calculate-the-difference-between-minimum-values-of-a-parabola-and-straight-line-from-a-plot

评论： Les Beckham 2023-2-10

Hello everyone, please could you help me with a code.

How I can calculate the values from a plot? I need the difference between straight line (P) and between the minimum value of a parabola (P) for each curve.

So, for example in this curve, the difference between the straight blue curve and the first parabola, then the next straight line and green parabola and so on.

My code is:

clear all; close all
W = 60000;
S = 28.2;
AR=7;
cd0 = 0.02;
k = 0.04;
RC=0.51;
clalpha = 2*pi;
Psl=741000; 
hv=0:1:10;
cdminp=4*cd0;
clminp=sqrt(3*cd0/k);
Vmin=sqrt(2*W/(1.225*28.2*clminp));
D=0.5*1.225*Vmin^2*S*cdminp;
Pminreq=D*Vmin;
deltaPgiven=RC*W;
figure(1);hold on; xlabel('V');ylabel('P')
hv=0:1:10;
for k1 = 1:numel(hv)
    h = hv(k1);
    i=0;
    for alpha = 1:0.25:15
        i=i+1;
        rho(i)=1.225*exp(-h/10.4);
        cl(i) = clalpha * alpha * pi/180;
        V(i) = sqrt(2*W/rho(i)/S/cl(i));
        L(i) = 0.5 * rho(i) * V(i) * V(i) * S * cl(i);
        cd(i) = cd0 + k * cl(i) * cl(i);
        D(i) = 0.5 * rho(i) * V(i) * V(i) * S * cd(i);
        clcd(i) = cl(i)/cd(i);
        p(i) = D(i)*V(i);
        Ph(i)=Psl*(rho(i)/1.225).^0.75;
    end
    figure(1); plot(V,p)
    hold on 
    plot(V,Ph);
end

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Answer 1

Les Beckham 2023-2-10

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链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1910540-calculate-the-difference-between-minimum-values-of-a-parabola-and-straight-line-from-a-plot#answer_1168795

编辑：Les Beckham 2023-2-10

在 MATLAB Online 中打开

Maybe this?

W = 60000;

S = 28.2;

AR=7;

cd0 = 0.02;

k = 0.04;

RC=0.51;

clalpha = 2*pi;

Psl=741000;

hv=0:1:10;

cdminp=4*cd0;

clminp=sqrt(3*cd0/k);

Vmin=sqrt(2*W/(1.225*28.2*clminp));

D=0.5*1.225*Vmin^2*S*cdminp;

Pminreq=D*Vmin;

deltaPgiven=RC*W;

figure(1);hold on; xlabel('V');ylabel('P')

hv=0:1:10;

for k1 = 1:numel(hv)

h = hv(k1);

i=0;

for alpha = 1:0.25:15

i=i+1;

rho(i)=1.225*exp(-h/10.4);

cl(i) = clalpha * alpha * pi/180;

V(i) = sqrt(2*W/rho(i)/S/cl(i));

L(i) = 0.5 * rho(i) * V(i) * V(i) * S * cl(i);

cd(i) = cd0 + k * cl(i) * cl(i);

D(i) = 0.5 * rho(i) * V(i) * V(i) * S * cd(i);

clcd(i) = cl(i)/cd(i);

p(i) = D(i)*V(i);

Ph(i)=Psl*(rho(i)/1.225).^0.75;

end

figure(1); plot(V,p)

hold on

plot(V,Ph);

[pmin, imin] = min(p); % find the min p

deltas(k1) = Ph(1) - pmin; % calculate the difference

tolerance = 5000; % or whatever you want

if (abs(deltas(k1) - 300000) < tolerance)

fprintf('delta = %8.1f at h = %4.1f, rho = %.5f, V = %.2f, Ph = %.1f, p = %.1f\n', ...

deltas(k1), h, rho(imin), V(imin), Ph(imin), p(imin))

end

delta = 302330.3 at h = 4.0, rho = 0.83387, V = 64.31, Ph = 555317.0, p = 252986.7

legend(compose('h = %.1f', hv), 'location', 'northwest')

grid on

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Alina Abdikadyr 2023-2-10

在 MATLAB Online 中打开

Yes, you inderstand it correctly, where I should put this piece of code? I implmented it here:

clear all; close all
W = 60000;
S = 28.2;
AR=7;
cd0 = 0.02;
k = 0.04;
RC=0.51;
clalpha = 2*pi;
Psl=741000; 
hv=0:1:10;
cdminp=4*cd0;
clminp=sqrt(3*cd0/k);
Vmin=sqrt(2*W/(1.225*28.2*clminp));
D=0.5*1.225*Vmin^2*S*cdminp;
Pminreq=D*Vmin;
deltaPgiven=RC*W;
figure(1);hold on; xlabel('V');ylabel('P')
hv=0:1:10;
for k1 = 1:numel(hv)
    h = hv(k1);
    i=0;
    for alpha = 1:0.25:15
        i=i+1;
        rho(i)=1.225*exp(-h/10.4);
        cl(i) = clalpha * alpha * pi/180;
        V(i) = sqrt(2*W/rho(i)/S/cl(i));
        L(i) = 0.5 * rho(i) * V(i) * V(i) * S * cl(i);
        cd(i) = cd0 + k * cl(i) * cl(i);
        D(i) = 0.5 * rho(i) * V(i) * V(i) * S * cd(i);
        clcd(i) = cl(i)/cd(i);
        p(i) = D(i)*V(i);
        Ph(i)=Psl*(rho(i)/1.225).^0.75;
    end
    figure(1); plot(V,p)
    hold on 
    plot(V,Ph);
    [pmin, imin] = min(p); % find the min p
    deltas(k1) = Ph(1) - pmin; % calculate the difference
tolerance = 1000; % or whatever you want
if (abs(deltas(k1) - 30000) < tolerance)
    fprintf('delta = %8.1f at h = %4.1f, rho = %.5f, V = %.2f, Ph = %.1f, p = %.1f\n', ...
        deltas(k1), h, rho(imin), V(imin), Ph(imin), p(imin))
end
legend(compose('h = %.1f', hv), 'location', 'northwest')
grid on
end

Alina Abdikadyr 2023-2-10

Thank you! What does the tolerance value mean?

Les Beckham 2023-2-10

It just represents how close to 300000 the delta has to be to make the code print the results. Adjust as desired.

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Answer 2

Torsten 2023-2-10

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1910540-calculate-the-difference-between-minimum-values-of-a-parabola-and-straight-line-from-a-plot#answer_1168820

编辑：Torsten 2023-2-10

在 MATLAB Online 中打开

syms h V W S rho cd0 k cl Psl
eqn = V == sqrt(2*W/rho/S/cl);
cl = solve(eqn,cl);
Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.
cd = cd0 + k * cl^2;
D = 0.5 * rho * V * V * S * cd;
p = D*V;
Vmin = solve(diff(p,V)==0,V);
pmin = subs(p,V,Vmin);
Ph = Psl*(rho/1.225)^0.75;
hnum = 0:0.01:10;
Wnum = 60000;
Snum = 28.2;
rhonum = 1.225*exp(-hnum/10.4);
cd0num = 0.02;
knum = 0.04;
Pslnum = 741000;
for i = 1:numel(hnum)
  Phnum(i) = double(subs(Ph,[rho Psl],[rhonum(i),Pslnum]));
  pm = double(subs(pmin,[W S rho cd0 k],[Wnum Snum rhonum(i) cd0num,knum]));
  pminnum(i) = pm(end);
  deltaP(i) = Phnum(i)-pminnum(i);
end
format long
deltaP.'
ans = 1001×1
1.0e+05 *

   5.322767552311991
   5.316422010284389
   5.310079836471793
   5.303741027866013
   5.297405581460750
   5.291073494251587
   5.284744763235998
   5.278419385413340
   5.272097357784856
   5.265778677353669
idx = deltaP > 2.8e5 & deltaP < 3.2e5;  % select those deltaP with 2.8e5 <= deltaP < = 3.2e5
[hnum(idx).' deltaP(idx).']                 % Show these values together with the corresponding h values
ans = 77×2
1.0e+05 *

   0.000036700000000   3.196913122178178
   0.000036800000000   3.191616062324753
   0.000036900000000   3.186321382077705
   0.000037000000000   3.181029079029593
   0.000037100000000   3.175739150774377
   0.000037200000000   3.170451594907418
   0.000037300000000   3.165166409025479
   0.000037400000000   3.159883590726722
   0.000037500000000   3.154603137610705
   0.000037600000000   3.149325047278384

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Alina Abdikadyr 2023-2-10

Thank you very much! But If I will have more h values, so may I display the deltaP values that are almost equal to 3?

Torsten 2023-2-10

编辑：Torsten 2023-2-10

在 MATLAB Online 中打开

Add the lines

idx = deltaP > 2.8e5 & deltaP < 3.2e5;  % select those deltaP with 2.8e5 <= deltaP < = 3.2e5
[hnum(idx) deltaP(idx)]                 % Show these values together with the corresponding h values

at the end of the code.

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Calculate the difference between minimum values of a parabola and straight line (from a plot)

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