2101.sagews

# Getting the volume, mass, inertia, centrifugals of a triagle in space with constant depth d.
#                          Y ^ 
#       pmax                 |
#         O--                |     X
#        /|  \        Global O----->
#       / |   --            /
#      /  |     \          * Z
#  p3 O-  |      --O p1
#       \ |    ---/
#        -O---/
#       pmin (= Local)

x = var('x')
y = var('y')
z = var('z')

xcgpmin = var('xcgpmin')
ycgpmin = var('ycgpmin')
zcgpmin = var('zcgpmin')

rho = var('rho')
d = var('d')

xp1 = var('xp1')
yp1 = var('yp1')
ypmax = var('ypmax')

"Volume = "; integrate( integrate( integrate( 1 , z, 0, d ) , y, yp1 / xp1 * x, ypmax + (yp1 - ypmax) / xp1 * x ) , x, 0, xp1 )

'Volume = ' 1/2*d*xp1*ypmax

"Mass = "; rho * integrate( integrate( integrate( 1 , z, 0, d ) , y, yp1 / xp1 * x, ypmax + (yp1 - ypmax) / xp1 * x ) , x, 0, xp1 )

'Mass = ' 1/2*d*rho*xp1*ypmax

"Ixx = "; rho * integrate( integrate( integrate( (ycgpmin + y)^2 + (zcgpmin + z)^2 , z, 0, d ) , y, yp1 / xp1 * x, ypmax + (yp1 - ypmax) / xp1 * x ) , x, 0, xp1 )

'Ixx = ' 1/12*(6*d^2*xp1*ypmax*zcgpmin + d*xp1*ypmax^3 + 6*d*xp1*ypmax*zcgpmin^2 + (4*d*xp1*ycgpmin + d*xp1*yp1)*ypmax^2 + (2*d^3*xp1 + 6*d*xp1*ycgpmin^2 + 4*d*xp1*ycgpmin*yp1 + d*xp1*yp1^2)*ypmax)*rho

"Iyy = "; rho * integrate( integrate( integrate( (xcgpmin + x)^2 + (zcgpmin + z)^2 , z, 0, d ) , y, yp1 / xp1 * x, ypmax + (yp1 - ypmax) / xp1 * x ) , x, 0, xp1 )

'Iyy = ' 1/12*(6*d^2*xp1*ypmax*zcgpmin + 6*d*xp1*ypmax*zcgpmin^2 + (4*d*xcgpmin*xp1^2 + d*xp1^3 + 2*(d^3 + 3*d*xcgpmin^2)*xp1)*ypmax)*rho

"Izz = "; rho * integrate( integrate( integrate( (xcgpmin + x)^2 + (ycgpmin + y)^2 , z, 0, d ) , y, yp1 / xp1 * x, ypmax + (yp1 - ypmax) / xp1 * x ) , x, 0, xp1 )

'Izz = ' 1/12*(xp1*ypmax^3 + (4*xp1*ycgpmin + xp1*yp1)*ypmax^2 + (6*xcgpmin^2*xp1 + 4*xcgpmin*xp1^2 + xp1^3 + 6*xp1*ycgpmin^2 + 4*xp1*ycgpmin*yp1 + xp1*yp1^2)*ypmax)*d*rho

"I = "; rho * integrate( integrate( integrate( (xcgpmin + x)^2 + (ycgpmin + y)^2 + (zcgpmin + z)^2 , z, 0, d ) , y, yp1 / xp1 * x, ypmax + (yp1 - ypmax) / xp1 * x ) , x, 0, xp1 )

'I = ' 1/12*(6*d^2*xp1*ypmax*zcgpmin + d*xp1*ypmax^3 + 6*d*xp1*ypmax*zcgpmin^2 + (4*d*xp1*ycgpmin + d*xp1*yp1)*ypmax^2 + (4*d*xcgpmin*xp1^2 + d*xp1^3 + 6*d*xp1*ycgpmin^2 + 4*d*xp1*ycgpmin*yp1 + d*xp1*yp1^2 + 2*(d^3 + 3*d*xcgpmin^2)*xp1)*ypmax)*rho

"Cxy = "; rho * integrate( integrate( integrate( (xcgpmin + x) * (ycgpmin + y) , z, 0, d ) , y, yp1 / xp1 * x, ypmax + (yp1 - ypmax) / xp1 * x ) , x, 0, xp1 )

'Cxy = ' 1/24*((4*xcgpmin*xp1 + xp1^2)*ypmax^2 + 2*((2*xcgpmin*xp1 + xp1^2)*yp1 + 2*(3*xcgpmin*xp1 + xp1^2)*ycgpmin)*ypmax)*d*rho

"Cyz = "; rho * integrate( integrate( integrate( (ycgpmin + y) * (zcgpmin + z) , z, 0, d ) , y, yp1 / xp1 * x, ypmax + (yp1 - ypmax) / xp1 * x ) , x, 0, xp1 )

'Cyz = ' 1/12*(d^2 + 2*d*zcgpmin)*(xp1*ypmax^2 + (3*xp1*ycgpmin + xp1*yp1)*ypmax)*rho

"Cxz = "; rho * integrate( integrate( integrate( (xcgpmin + x) * (zcgpmin + z) , z, 0, d ) , y, yp1 / xp1 * x, ypmax + (yp1 - ypmax) / xp1 * x ) , x, 0, xp1 )

'Cxz = ' 1/12*(3*xcgpmin*xp1 + xp1^2)*(d^2 + 2*d*zcgpmin)*rho*ypmax