JRW, I need more constraints.
Code:
eq0: (a0-xc)^2+(b0-yc)^2=rc^2$
eq1: (a1-xc)^2+(b1-yc)^2=rc^2$
eq2: (a2-xc)^2+(b2-yc)^2=rc^2$
eq3: (a0-x0)^2+(b0-y0)^2=r0^2$
eq4: (a1-x1)^2+(b1-y1)^2=r1^2$
eq5: (a2-x2)^2+(b2-y2)^2=r2^2$
xc,yc is the center of the coil
rc is the radius of the coil
x0,y0 are the center of roll 0. x0 is fixed/ y0 is what we want to find the equation for.
x1,y1 are the center of roll 1. x1 and y1 are fixed
x2,y2 are the center of roll 2. x2 and y2 are fixed.
a0,b0 is the contact point on the edge of roll 0.
a1,b1 is the contact point on the edge of roll 1.
a2,b2 is the contact point on the edge of roll 2.
I have 6 equations. I can only eliminate 6 variables. I have eliminated a0,b0,a1,b1,a2 and b2. We know x0, x1, y1, x2, y2 because they are fixed and we want to calculate the height of roll or so we need to calculate y0 as a function of the coil radius rc. However there are two variables xc and yc that we must eliminate too. We need to know more about the process.
BTW, I have switched to wxMaxima. It is much more powerful. Even so the formula is taxing. Here is the result I have so far.
Code:
y2 = (sqrt(((32*xc-32*x2)*yc^3+(72*xc^3-216*x2*xc^2
+(216*x2^2-36*rc^2+36*r2^2)*xc-72*x2^3
+(36*rc^2-36*r2^2)*x2)
*yc)
*sqrt(-yc^4+4*y2*yc^3
+(-6*y2^2-2*xc^2+4*x2*xc-2*x2^2+2*rc^2+2*r2^2)*yc^2
+(4*y2^3+(4*xc^2-8*x2*xc+4*x2^2-4*rc^2-4*r2^2)*y2)*yc
-y2^4+(-2*xc^2+4*x2*xc-2*x2^2+2*rc^2+2*r2^2)*y2^2-xc^4
+4*x2*xc^3+(-6*x2^2+2*rc^2+2*r2^2)*xc^2
+(4*x2^3+(-4*rc^2-4*r2^2)*x2)*xc-x2^4
+(2*rc^2+2*r2^2)*x2^2-rc^4+2*r2^2*rc^2-r2^4)
+(64*xc^2-128*x2*xc+64*x2^2)*yc^4
+(27*xc^2-54*x2*xc+27*x2^2)*(-yc^4+4*y2*yc^3
+(-6*y2^2-2*xc^2+4*x2*xc-2*x2^2
+2*rc^2+2*r2^2)
*yc^2
+(4*y2^3
+(4*xc^2-8*x2*xc+4*x2^2-4*rc^2
-4*r2^2)
*y2)
*yc-y2^4
+(-2*xc^2+4*x2*xc-2*x2^2+2*rc^2
+2*r2^2)
*y2^2-xc^4+4*x2*xc^3
+(-6*x2^2+2*rc^2+2*r2^2)*xc^2
+(4*x2^3+(-4*rc^2-4*r2^2)*x2)*xc
-x2^4+(2*rc^2+2*r2^2)*x2^2-rc^4
+2*r2^2*rc^2-r2^4)
+(32*xc^4-128*x2*xc^3+(192*x2^2-80*rc^2+80*r2^2)*xc^2
+((160*rc^2-160*r2^2)*x2-128*x2^3)*xc+32*x2^4
+(80*r2^2-80*rc^2)*x2^2-4*rc^4+8*r2^2*rc^2-4*r2^4)
*yc^2+4*xc^6-24*x2*xc^5+(60*x2^2+12*rc^2-12*r2^2)*xc^4
+((48*r2^2-48*rc^2)*x2-80*x2^3)*xc^3
+(60*x2^4+(72*rc^2-72*r2^2)*x2^2+12*rc^4-24*r2^2*rc^2+12*r2^4)
*xc^2
+(-24*x2^5+(48*r2^2-48*rc^2)*x2^3
+(-24*rc^4+48*r2^2*rc^2-24*r2^4)*x2)
*xc+4*x2^6+(12*rc^2-12*r2^2)*x2^4
+(12*rc^4-24*r2^2*rc^2+12*r2^4)*x2^2+4*rc^6-12*r2^2*rc^4
+12*r2^4*rc^2-4*r2^6)
/(2*3^(3/2))
+((27*x2-27*xc)*sqrt(-yc^4+4*y2*yc^3
+(-6*y2^2-2*xc^2+4*x2*xc-2*x2^2+2*rc^2+2*r2^2)
*yc^2
+(4*y2^3+(4*xc^2-8*x2*xc+4*x2^2-4*rc^2-4*r2^2)
*y2)
*yc-y2^4
+(-2*xc^2+4*x2*xc-2*x2^2+2*rc^2+2*r2^2)*y2^2-xc^4
+4*x2*xc^3+(-6*x2^2+2*rc^2+2*r2^2)*xc^2
+(4*x2^3+(-4*rc^2-4*r2^2)*x2)*xc-x2^4
+(2*rc^2+2*r2^2)*x2^2-rc^4+2*r2^2*rc^2-r2^4)
-16*yc^3-36*xc^2*yc+72*x2*xc*yc-36*x2^2*yc+18*rc^2*yc-18*r2^2*yc)
/54)
^(1/3)
+(4*yc^2-3*xc^2+6*x2*xc-3*x2^2-3*rc^2+3*r2^2)
/(9*(sqrt(((32*xc-32*x2)*yc^3+(72*xc^3-216*x2*xc^2
+(216*x2^2-36*rc^2+36*r2^2)*xc
-72*x2^3+(36*rc^2-36*r2^2)*x2)
*yc)
*sqrt(-yc^4+4*y2*yc^3
+(-6*y2^2-2*xc^2+4*x2*xc-2*x2^2+2*rc^2+2*r2^2)*yc^2
+(4*y2^3+(4*xc^2-8*x2*xc+4*x2^2-4*rc^2-4*r2^2)*y2)
*yc-y2^4
+(-2*xc^2+4*x2*xc-2*x2^2+2*rc^2+2*r2^2)*y2^2-xc^4
+4*x2*xc^3+(-6*x2^2+2*rc^2+2*r2^2)*xc^2
+(4*x2^3+(-4*rc^2-4*r2^2)*x2)*xc-x2^4
+(2*rc^2+2*r2^2)*x2^2-rc^4+2*r2^2*rc^2-r2^4)
+(64*xc^2-128*x2*xc+64*x2^2)*yc^4
+(27*xc^2-54*x2*xc+27*x2^2)
*(-yc^4+4*y2*yc^3
+(-6*y2^2-2*xc^2+4*x2*xc-2*x2^2+2*rc^2+2*r2^2)*yc^2
+(4*y2^3+(4*xc^2-8*x2*xc+4*x2^2-4*rc^2-4*r2^2)*y2)*yc
-y2^4+(-2*xc^2+4*x2*xc-2*x2^2+2*rc^2+2*r2^2)*y2^2-xc^4
+4*x2*xc^3+(-6*x2^2+2*rc^2+2*r2^2)*xc^2
+(4*x2^3+(-4*rc^2-4*r2^2)*x2)*xc-x2^4
+(2*rc^2+2*r2^2)*x2^2-rc^4+2*r2^2*rc^2-r2^4)
+(32*xc^4-128*x2*xc^3+(192*x2^2-80*rc^2+80*r2^2)*xc^2
+((160*rc^2-160*r2^2)*x2-128*x2^3)*xc+32*x2^4
+(80*r2^2-80*rc^2)*x2^2-4*rc^4+8*r2^2*rc^2-4*r2^4)
*yc^2+4*xc^6-24*x2*xc^5+(60*x2^2+12*rc^2-12*r2^2)*xc^4
+((48*r2^2-48*rc^2)*x2-80*x2^3)*xc^3
+(60*x2^4+(72*rc^2-72*r2^2)*x2^2+12*rc^4-24*r2^2*rc^2+12*r2^4)
*xc^2
+(-24*x2^5+(48*r2^2-48*rc^2)*x2^3
+(-24*rc^4+48*r2^2*rc^2-24*r2^4)*x2)
*xc+4*x2^6+(12*rc^2-12*r2^2)*x2^4
+(12*rc^4-24*r2^2*rc^2+12*r2^4)*x2^2+4*rc^6-12*r2^2*rc^4
+12*r2^4*rc^2-4*r2^6)
/(2*3^(3/2))
+((27*x2-27*xc)*sqrt(-yc^4+4*y2*yc^3
+(-6*y2^2-2*xc^2+4*x2*xc-2*x2^2+2*rc^2
+2*r2^2)
*yc^2
+(4*y2^3
+(4*xc^2-8*x2*xc+4*x2^2-4*rc^2-4*r2^2)*y2)
*yc-y2^4
+(-2*xc^2+4*x2*xc-2*x2^2+2*rc^2+2*r2^2)*y2^2
-xc^4+4*x2*xc^3+(-6*x2^2+2*rc^2+2*r2^2)*xc^2
+(4*x2^3+(-4*rc^2-4*r2^2)*x2)*xc-x2^4
+(2*rc^2+2*r2^2)*x2^2-rc^4+2*r2^2*rc^2-r2^4)
-16*yc^3-36*xc^2*yc+72*x2*xc*yc-36*x2^2*yc+18*rc^2*yc-18*r2^2*yc)
/54)
^(1/3))+yc/3$
I know I said we had to solve for y0 but wxMaxima wants to solve for x2. The equation would be the same just swap x0 for x2 and y0 for y2 and r0 for r2.
It would help to reduce the clutter if I had numeric values for x0,r0,x1,y1,r1,x2,y2 and r2. If all the rolls have the same diameter that would simplify things a lot.
JRW, help me help you. You can see this is not an easy problem. Those that said a cam table will be needed are right. It is easy to see the calculations are complex. I haven't even taken into account the thickness of the metal that is being rolled yet.