2920.sagews

var('m g d O a b l Odot adot bdot ldot t')

#angular velocity in inertial frame
omega = vector([0,0,O])

#angular velocity in tip mass location frame
COS1 = matrix([[sin(a),0,-cos(a)],[cos(a),0,sin(a)],[0,-1,0]])
COS2 = matrix([[cos(b),0,sin(b)],[0,1,0],[-sin(b),0,cos(b)]])
omega = COS1*omega+vector([0,0,adot])
omega = omega*COS2+vector([0,bdot,0])
show(omega)

vel = omega.cross_product(vector([l,0,0]))+vector([0,0,-d*O])*COS2+vector([ldot,0,0])
show(vel)

T = 1/2*m*vel*vel
show(T)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(-O \cos\left(a\right) \cos\left(b\right) - \mbox{adot} \sin\left(b\right),\,O \sin\left(a\right) + \mbox{bdot},\,-O \sin\left(b\right) \cos\left(a\right) + \mbox{adot} \cos\left(b\right)\right)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(O d \sin\left(b\right) + \mbox{ldot},\,-{\left(O \sin\left(b\right) \cos\left(a\right) - \mbox{adot} \cos\left(b\right)\right)} l,\,-O d \cos\left(b\right) - {\left(O \sin\left(a\right) + \mbox{bdot}\right)} l\right)

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, {\left(O \sin\left(b\right) \cos\left(a\right) - \mbox{adot} \cos\left(b\right)\right)}^{2} l^{2} m + \frac{1}{2} \, {\left(O d \cos\left(b\right) + {\left(O \sin\left(a\right) + \mbox{bdot}\right)} l\right)}^{2} m + \frac{1}{2} \, {\left(O d \sin\left(b\right) + \mbox{ldot}\right)}^{2} m

V = -m*g*l*cos(a)*cos(b)

L = T-V
show(L)

#solving Lagrange for a
L_adot = diff(L,adot)
L_a = diff(L,a)
O = function('O',t)
a = function('a',t)
l = function('l',t)
b = function('b',t)
L_adot = L_adot.subs(Odot = diff(O,t),adot = diff(a,t),bdot = diff(b,t),ldot = diff(l,t),O = O,l = l,a=a,b=b)
L_adot = diff(L_adot,t)
L_a = L_a.subs(Odot = diff(O,t),adot = diff(a,t),bdot = diff(b,t),ldot = diff(l,t),O = O,l = l,a=a,b=b)
f = L_adot-L_a
A = solve(f==0,diff(diff(a,t),t))
show(A)

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, {\left(O \sin\left(b\right) \cos\left(a\right) - \mbox{adot} \cos\left(b\right)\right)}^{2} l^{2} m + g l m \cos\left(a\right) \cos\left(b\right) + \frac{1}{2} \, {\left(O d \cos\left(b\right) + {\left(O \sin\left(a\right) + \mbox{bdot}\right)} l\right)}^{2} m + \frac{1}{2} \, {\left(O d \sin\left(b\right) + \mbox{ldot}\right)}^{2} m

\newcommand{\Bold}[1]{\mathbf{#1}}\left[D[0, 0]\left(a\right)\left(t\right) = -\frac{\sin\left(a\left(t\right)\right) \sin\left(b\left(t\right)\right)^{2} \cos\left(a\left(t\right)\right) O\left(t\right)^{2} l\left(t\right) - \sin\left(a\left(t\right)\right) \cos\left(a\left(t\right)\right) O\left(t\right)^{2} l\left(t\right) - \sin\left(b\left(t\right)\right) \cos\left(a\left(t\right)\right) \cos\left(b\left(t\right)\right) l\left(t\right) D[0]\left(O\right)\left(t\right) - 2 \, {\left(\sin\left(b\left(t\right)\right) \cos\left(a\left(t\right)\right) \cos\left(b\left(t\right)\right) O\left(t\right) - \cos\left(b\left(t\right)\right)^{2} D[0]\left(a\right)\left(t\right)\right)} D[0]\left(l\right)\left(t\right) - {\left(d \cos\left(a\left(t\right)\right) O\left(t\right)^{2} - g \sin\left(a\left(t\right)\right)\right)} \cos\left(b\left(t\right)\right) + {\left(\sin\left(b\left(t\right)\right)^{2} \cos\left(a\left(t\right)\right) O\left(t\right) l\left(t\right) - \cos\left(a\left(t\right)\right) \cos\left(b\left(t\right)\right)^{2} O\left(t\right) l\left(t\right) - 2 \, \sin\left(b\left(t\right)\right) \cos\left(b\left(t\right)\right) l\left(t\right) D[0]\left(a\right)\left(t\right) - \cos\left(a\left(t\right)\right) O\left(t\right) l\left(t\right)\right)} D[0]\left(b\right)\left(t\right)}{\cos\left(b\left(t\right)\right)^{2} l\left(t\right)}\right]

#solving Lagrange for b
var('m g d O a b l Odot adot bdot ldot t')
L_bdot = diff(L,bdot)
L_b = diff(L,b)
O = function('O',t)
a = function('a',t)
l = function('l',t)
b = function('b',t)
L_bdot = L_bdot.subs(Odot = diff(O,t),adot = diff(a,t),bdot = diff(b,t),ldot = diff(l,t),O = O,l = l,a=a,b=b)
L_bdot = diff(L_bdot,t)
L_b = L_b.subs(Odot = diff(O,t),adot = diff(a,t),bdot = diff(b,t),ldot = diff(l,t),O = O,l = l,a=a,b=b)
f = L_bdot-L_b
B = solve(f==0,diff(diff(b,t),t))
show(B)

\newcommand{\Bold}[1]{\mathbf{#1}}\left[D[0, 0]\left(b\right)\left(t\right) = \frac{\sin\left(b\left(t\right)\right)^{2} \cos\left(a\left(t\right)\right) O\left(t\right) l\left(t\right) D[0]\left(a\right)\left(t\right) - \cos\left(a\left(t\right)\right) \cos\left(b\left(t\right)\right)^{2} O\left(t\right) l\left(t\right) D[0]\left(a\right)\left(t\right) - \cos\left(a\left(t\right)\right) O\left(t\right) l\left(t\right) D[0]\left(a\right)\left(t\right) - 2 \, {\left(\sin\left(a\left(t\right)\right) O\left(t\right) + D[0]\left(b\right)\left(t\right)\right)} D[0]\left(l\right)\left(t\right) - {\left(d \cos\left(b\left(t\right)\right) + \sin\left(a\left(t\right)\right) l\left(t\right)\right)} D[0]\left(O\right)\left(t\right) - {\left(d \sin\left(a\left(t\right)\right) O\left(t\right)^{2} - {\left(\cos\left(a\left(t\right)\right)^{2} O\left(t\right)^{2} l\left(t\right) - l\left(t\right) D[0]\left(a\right)\left(t\right)^{2}\right)} \cos\left(b\left(t\right)\right) + g \cos\left(a\left(t\right)\right)\right)} \sin\left(b\left(t\right)\right)}{l\left(t\right)}\right]

#solving Lagrange for l
var('m g d O a b l Odot adot bdot ldot t')
L_ldot = diff(L,ldot)
L_l = diff(L,l)
O = function('O',t)
a = function('a',t)
l = function('l',t)
b = function('b',t)
L_ldot = L_ldot.subs(Odot = diff(O,t),adot = diff(a,t),bdot = diff(b,t),ldot = diff(l,t),O = O,l = l,a=a,b=b)
L_ldot = diff(L_ldot,t)
L_l = L_l.subs(Odot = diff(O,t),adot = diff(a,t),bdot = diff(b,t),ldot = diff(l,t),O = O,l = l,a=a,b=b)
f = L_ldot-L_b
C = solve(f==0,diff(diff(l,t),t))
show(C)

\newcommand{\Bold}[1]{\mathbf{#1}}\left[D[0, 0]\left(l\right)\left(t\right) = \sin\left(b\left(t\right)\right)^{2} \cos\left(a\left(t\right)\right) O\left(t\right) l\left(t\right)^{2} D[0]\left(a\right)\left(t\right) - \cos\left(a\left(t\right)\right) \cos\left(b\left(t\right)\right)^{2} O\left(t\right) l\left(t\right)^{2} D[0]\left(a\right)\left(t\right) + d \cos\left(b\left(t\right)\right) O\left(t\right) D[0]\left(l\right)\left(t\right) - d \sin\left(b\left(t\right)\right) D[0]\left(O\right)\left(t\right) - {\left(d \sin\left(b\left(t\right)\right) O\left(t\right) l\left(t\right) + d \cos\left(b\left(t\right)\right) O\left(t\right)\right)} D[0]\left(b\right)\left(t\right) - {\left(d \sin\left(a\left(t\right)\right) O\left(t\right)^{2} l\left(t\right) + g \cos\left(a\left(t\right)\right) l\left(t\right) - {\left(\cos\left(a\left(t\right)\right)^{2} O\left(t\right)^{2} l\left(t\right)^{2} - l\left(t\right)^{2} D[0]\left(a\right)\left(t\right)^{2}\right)} \cos\left(b\left(t\right)\right)\right)} \sin\left(b\left(t\right)\right)\right]