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How do I write the equation V in the pdf as a MATLAB equation. I am messing up somewhere and I appreciate any help I can get.

Here is my code ,

syms

p = 1000;

L = 10;

a = 5;

B = 10;

W = 100000;

V_w = a*L*B+(((L^2)*B)/2);

V = B*(L*cos(x)*a+((L^2).*cos(x).*sin(x))/2+((L*cos(x))^2)/2);

theta_max = solve(V/V_w < 1 ,x);

Star Strider
on 19 Apr 2019

You need to add ‘x’ to your syms call:

syms x

then with this solve call:

theta_max = solve(V/V_w ,x)

you get:

theta_max =

pi/2

pi

Walter Roberson
on 19 Apr 2019

It would not be solve(v/V_w, x) as that would be solve(v/V_w == 0,x) rather than solve(v/V_w < 1,x)

Star Strider
on 19 Apr 2019

When I ran it with the inequality, that produces an empty result, no solution.

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Walter Roberson
on 19 Apr 2019

There is a trick to solving inequalities: convert them to equalities.

P/R < 1 implies P < R implies P + dP == R for some positive dP. That is, if you had to add something positive to P to get R, then it implies that P is strictly less than R.

(P+dP)==R implies P/R + dP/R == 1, which implies P/R == 1 - dP/R . So particular dP answers the question of how much less than one P/R is, proportional to R.

Q = @(v) sym(v,'r');

syms x %do not use syms x real, it will not find solutions

p = Q(1000);

L = Q(10);

a = Q(5);

B = Q(10);

W = Q(100000);

V_w = a*L*B+(((L^2)*B)/2);

V = B*(L*cos(x)*a+((L^2).*cos(x).*sin(x))/2+((L*cos(x))^2)/2);

syms dV

theta_max = solve(V + dV == V_w, x, 'MaxDegree', 4);

theta_maxt = simplify( rewrite(theta_max, 'atan') );

The result you get out in theta_maxt will be a series of four long expressions involving atan. Each one of them answers the question of "What would the x have to be for V + dP to exactly equal V_w ?". Or in terms of the constants, "Given an exact amount less than 1000, what would x be for V to be exactly 1000 minus that amount?"

Now, depending on the difference from 1000 that you choose, some of the four theta_maxt might be imaginary. They involved roots of a quartic (degree 4 polynomial), so either 0, 2, or 4 of the solutions will be real valued. There are breakpoints at approximately dV = 839 point something, 1000 exactly, and 1361 point something (yes, there are solutions where x is negative.) The range between 839 and 1000 has four real-valued solutions.

When I put the original inequality to Maple as an inequality, Maple responds with four solutions, all involving arctan() of a particular constant, which I will call C for the moment. They are (arctan(C)-4*pi, 2*pi) exclusive, arctan(C)-2*pi,0) exclusive, (arctan(C),2*pi) exclusive, and (arctan(C)+2*pi,4*pi). The arctan(C) value is approximately 0.57 .

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