NS_BEg_vl_tr -- Sage

%auto
implicit_multiplication (True)

var ('P q0 q1 l E J h')

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(P, q_{0}, q_{1}, l, E, J, h\right)$

var ('a0 a1 a2 a3')

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(a_{0}, a_{1}, a_{2}, a_{3}\right)$

var ('c0 c1 c2 c3')

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(c_{0}, c_{1}, c_{2}, c_{3}\right)$

wpart(x) = c0 + c1 x + c2 x^2 + c3 x^3
wpart

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ c_{3} x^{3} + c_{2} x^{2} + c_{1} x + c_{0}$

wpart2(x) = diff (wpart(x), x, 2)
wpart2

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 6 \, c_{3} x + 2 \, c_{2}$

wpart4(x) = diff (wpart(x), x, 4)
wpart4

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 0$

s2 = solve (-2 h^2/l^2 c2 == q0/(E J), c2)
s2

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[c_{2} = -\frac{l^{2} q_{0}}{2 \, E J h^{2}}\right]$

cc2 = s2[0].rhs()
cc2

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{l^{2} q_{0}}{2 \, E J h^{2}}$

s3 = solve (-6 h^2/l^2 c3 == q1/(l E J), c3)
s3

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[c_{3} = -\frac{l q_{1}}{6 \, E J h^{2}}\right]$

cc3 = s3[0].rhs()
cc3

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{l q_{1}}{6 \, E J h^{2}}$

w(x) = a0 + a1 x + a2 sinh (h/l x) + a3 cosh (h/l x) + cc2 x^2 + cc3 x^3
w

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ a_{1} x + a_{3} \cosh\left(\frac{h x}{l}\right) + a_{2} \sinh\left(\frac{h x}{l}\right) + a_{0} - \frac{l^{2} q_{0} x^{2}}{2 \, E J h^{2}} - \frac{l q_{1} x^{3}}{6 \, E J h^{2}}$

w1(x) = diff (w(x), x)
w1

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{a_{2} h \cosh\left(\frac{h x}{l}\right)}{l} + \frac{a_{3} h \sinh\left(\frac{h x}{l}\right)}{l} + a_{1} - \frac{l^{2} q_{0} x}{E J h^{2}} - \frac{l q_{1} x^{2}}{2 \, E J h^{2}}$

w2(x) = diff (w(x), x, 2)
w2

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{a_{3} h^{2} \cosh\left(\frac{h x}{l}\right)}{l^{2}} + \frac{a_{2} h^{2} \sinh\left(\frac{h x}{l}\right)}{l^{2}} - \frac{l^{2} q_{0}}{E J h^{2}} - \frac{l q_{1} x}{E J h^{2}}$

w(0) == 0

$\newcommand{\Bold}[1]{\mathbf{#1}}a_{0} + a_{3} = 0$

w1(0) == 0

$\newcommand{\Bold}[1]{\mathbf{#1}}a_{1} + \frac{a_{2} h}{l} = 0$

w(l) == 0

$\newcommand{\Bold}[1]{\mathbf{#1}}a_{1} l + a_{3} \cosh\left(h\right) + a_{2} \sinh\left(h\right) + a_{0} - \frac{l^{4} q_{0}}{2 \, E J h^{2}} - \frac{l^{4} q_{1}}{6 \, E J h^{2}} = 0$

w1(l) == 0

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{a_{2} h \cosh\left(h\right)}{l} + \frac{a_{3} h \sinh\left(h\right)}{l} + a_{1} - \frac{l^{3} q_{0}}{E J h^{2}} - \frac{l^{3} q_{1}}{2 \, E J h^{2}} = 0$

aa = solve ([w(0) == 0, 
             w1(0) == 0, 
             w(l) == 0, 
             w1(l) == 0], 
            [a0, a1, a2, a3])

aa[0][0]

$\newcommand{\Bold}[1]{\mathbf{#1}}a_{0} = -\frac{{\left({\left(q_{1} {\left(\cosh\left(h\right) + 2\right)} + 3 \, q_{0} {\left(\cosh\left(h\right) + 1\right)}\right)} h - 6 \, q_{0} \sinh\left(h\right) - 3 \, q_{1} \sinh\left(h\right)\right)} l^{4}}{6 \, {\left(E J h^{4} \sinh\left(h\right) + {\left(E J \cosh\left(h\right)^{2} - E J \sinh\left(h\right)^{2} - 2 \, E J \cosh\left(h\right) + E J\right)} h^{3}\right)}}$

aa0 = (aa[0][0]).rhs().simplify_full()
aa0

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{3 \, {\left(h \cosh\left(h\right) + h - 2 \, \sinh\left(h\right)\right)} l^{4} q_{0} + {\left(h \cosh\left(h\right) + 2 \, h - 3 \, \sinh\left(h\right)\right)} l^{4} q_{1}}{6 \, {\left(E J h^{4} \sinh\left(h\right) - 2 \, E J h^{3} \cosh\left(h\right) + 2 \, E J h^{3}\right)}}$

aa[0][1]

$\newcommand{\Bold}[1]{\mathbf{#1}}a_{1} = \frac{{\left({\left(3 \, q_{0} \sinh\left(h\right) + q_{1} \sinh\left(h\right)\right)} h - 6 \, q_{0} {\left(\cosh\left(h\right) - 1\right)} - 3 \, q_{1} {\left(\cosh\left(h\right) - 1\right)}\right)} l^{3}}{6 \, {\left(E J h^{3} \sinh\left(h\right) + {\left(E J \cosh\left(h\right)^{2} - E J \sinh\left(h\right)^{2} - 2 \, E J \cosh\left(h\right) + E J\right)} h^{2}\right)}}$

aa1 = (aa[0][1]).rhs().simplify_full()
aa1

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, {\left(h \sinh\left(h\right) - 2 \, \cosh\left(h\right) + 2\right)} l^{3} q_{0} + {\left(h \sinh\left(h\right) - 3 \, \cosh\left(h\right) + 3\right)} l^{3} q_{1}}{6 \, {\left(E J h^{3} \sinh\left(h\right) - 2 \, E J h^{2} \cosh\left(h\right) + 2 \, E J h^{2}\right)}}$

aa[0][2]

$\newcommand{\Bold}[1]{\mathbf{#1}}a_{2} = -\frac{{\left({\left(3 \, q_{0} \sinh\left(h\right) + q_{1} \sinh\left(h\right)\right)} h - 6 \, q_{0} {\left(\cosh\left(h\right) - 1\right)} - 3 \, q_{1} {\left(\cosh\left(h\right) - 1\right)}\right)} l^{4}}{6 \, {\left(E J h^{4} \sinh\left(h\right) + {\left(E J \cosh\left(h\right)^{2} - E J \sinh\left(h\right)^{2} - 2 \, E J \cosh\left(h\right) + E J\right)} h^{3}\right)}}$

aa2 = (aa[0][2]).rhs().simplify_full()
aa2

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{3 \, {\left(h \sinh\left(h\right) - 2 \, \cosh\left(h\right) + 2\right)} l^{4} q_{0} + {\left(h \sinh\left(h\right) - 3 \, \cosh\left(h\right) + 3\right)} l^{4} q_{1}}{6 \, {\left(E J h^{4} \sinh\left(h\right) - 2 \, E J h^{3} \cosh\left(h\right) + 2 \, E J h^{3}\right)}}$

aa[0][3]

$\newcommand{\Bold}[1]{\mathbf{#1}}a_{3} = \frac{{\left({\left(q_{1} {\left(\cosh\left(h\right) + 2\right)} + 3 \, q_{0} {\left(\cosh\left(h\right) + 1\right)}\right)} h - 6 \, q_{0} \sinh\left(h\right) - 3 \, q_{1} \sinh\left(h\right)\right)} l^{4}}{6 \, {\left(E J h^{4} \sinh\left(h\right) + {\left(E J \cosh\left(h\right)^{2} - E J \sinh\left(h\right)^{2} - 2 \, E J \cosh\left(h\right) + E J\right)} h^{3}\right)}}$

aa3 = (aa[0][3]).rhs().simplify_full()
aa3

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, {\left(h \cosh\left(h\right) + h - 2 \, \sinh\left(h\right)\right)} l^{4} q_{0} + {\left(h \cosh\left(h\right) + 2 \, h - 3 \, \sinh\left(h\right)\right)} l^{4} q_{1}}{6 \, {\left(E J h^{4} \sinh\left(h\right) - 2 \, E J h^{3} \cosh\left(h\right) + 2 \, E J h^{3}\right)}}$

wa = w.subs (a0 = aa0, a1 = aa1, a2 = aa2, a3 = aa3)
wa

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{{\left(3 \, {\left(h \sinh\left(h\right) - 2 \, \cosh\left(h\right) + 2\right)} l^{3} q_{0} + {\left(h \sinh\left(h\right) - 3 \, \cosh\left(h\right) + 3\right)} l^{3} q_{1}\right)} x}{6 \, {\left(E J h^{3} \sinh\left(h\right) - 2 \, E J h^{2} \cosh\left(h\right) + 2 \, E J h^{2}\right)}} - \frac{l^{2} q_{0} x^{2}}{2 \, E J h^{2}} - \frac{l q_{1} x^{3}}{6 \, E J h^{2}} + \frac{{\left(3 \, {\left(h \cosh\left(h\right) + h - 2 \, \sinh\left(h\right)\right)} l^{4} q_{0} + {\left(h \cosh\left(h\right) + 2 \, h - 3 \, \sinh\left(h\right)\right)} l^{4} q_{1}\right)} \cosh\left(\frac{h x}{l}\right)}{6 \, {\left(E J h^{4} \sinh\left(h\right) - 2 \, E J h^{3} \cosh\left(h\right) + 2 \, E J h^{3}\right)}} - \frac{{\left(3 \, {\left(h \sinh\left(h\right) - 2 \, \cosh\left(h\right) + 2\right)} l^{4} q_{0} + {\left(h \sinh\left(h\right) - 3 \, \cosh\left(h\right) + 3\right)} l^{4} q_{1}\right)} \sinh\left(\frac{h x}{l}\right)}{6 \, {\left(E J h^{4} \sinh\left(h\right) - 2 \, E J h^{3} \cosh\left(h\right) + 2 \, E J h^{3}\right)}} - \frac{3 \, {\left(h \cosh\left(h\right) + h - 2 \, \sinh\left(h\right)\right)} l^{4} q_{0} + {\left(h \cosh\left(h\right) + 2 \, h - 3 \, \sinh\left(h\right)\right)} l^{4} q_{1}}{6 \, {\left(E J h^{4} \sinh\left(h\right) - 2 \, E J h^{3} \cosh\left(h\right) + 2 \, E J h^{3}\right)}}$

wp = wa.subs (h = l sqrt (P/(E J)))
wp

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -\frac{q_{0} x^{2}}{2 \, P} - \frac{q_{1} x^{3}}{6 \, P l} + \frac{{\left(3 \, {\left(l \sqrt{\frac{P}{E J}} \sinh\left(l \sqrt{\frac{P}{E J}}\right) - 2 \, \cosh\left(l \sqrt{\frac{P}{E J}}\right) + 2\right)} l^{3} q_{0} + {\left(l \sqrt{\frac{P}{E J}} \sinh\left(l \sqrt{\frac{P}{E J}}\right) - 3 \, \cosh\left(l \sqrt{\frac{P}{E J}}\right) + 3\right)} l^{3} q_{1}\right)} x}{6 \, {\left(E J l^{3} \left(\frac{P}{E J}\right)^{\frac{3}{2}} \sinh\left(l \sqrt{\frac{P}{E J}}\right) - 2 \, P l^{2} \cosh\left(l \sqrt{\frac{P}{E J}}\right) + 2 \, P l^{2}\right)}} - \frac{{\left(3 \, {\left(l \sqrt{\frac{P}{E J}} \cosh\left(l \sqrt{\frac{P}{E J}}\right) + l \sqrt{\frac{P}{E J}} - 2 \, \sinh\left(l \sqrt{\frac{P}{E J}}\right)\right)} l^{4} q_{0} + {\left(l \sqrt{\frac{P}{E J}} \cosh\left(l \sqrt{\frac{P}{E J}}\right) + 2 \, l \sqrt{\frac{P}{E J}} - 3 \, \sinh\left(l \sqrt{\frac{P}{E J}}\right)\right)} l^{4} q_{1}\right)} \cosh\left(x \sqrt{\frac{P}{E J}}\right)}{6 \, {\left(2 \, E J l^{3} \left(\frac{P}{E J}\right)^{\frac{3}{2}} \cosh\left(l \sqrt{\frac{P}{E J}}\right) - 2 \, E J l^{3} \left(\frac{P}{E J}\right)^{\frac{3}{2}} - \frac{P^{2} l^{4} \sinh\left(l \sqrt{\frac{P}{E J}}\right)}{E J}\right)}} + \frac{{\left(3 \, {\left(l \sqrt{\frac{P}{E J}} \sinh\left(l \sqrt{\frac{P}{E J}}\right) - 2 \, \cosh\left(l \sqrt{\frac{P}{E J}}\right) + 2\right)} l^{4} q_{0} + {\left(l \sqrt{\frac{P}{E J}} \sinh\left(l \sqrt{\frac{P}{E J}}\right) - 3 \, \cosh\left(l \sqrt{\frac{P}{E J}}\right) + 3\right)} l^{4} q_{1}\right)} \sinh\left(x \sqrt{\frac{P}{E J}}\right)}{6 \, {\left(2 \, E J l^{3} \left(\frac{P}{E J}\right)^{\frac{3}{2}} \cosh\left(l \sqrt{\frac{P}{E J}}\right) - 2 \, E J l^{3} \left(\frac{P}{E J}\right)^{\frac{3}{2}} - \frac{P^{2} l^{4} \sinh\left(l \sqrt{\frac{P}{E J}}\right)}{E J}\right)}} + \frac{3 \, {\left(l \sqrt{\frac{P}{E J}} \cosh\left(l \sqrt{\frac{P}{E J}}\right) + l \sqrt{\frac{P}{E J}} - 2 \, \sinh\left(l \sqrt{\frac{P}{E J}}\right)\right)} l^{4} q_{0} + {\left(l \sqrt{\frac{P}{E J}} \cosh\left(l \sqrt{\frac{P}{E J}}\right) + 2 \, l \sqrt{\frac{P}{E J}} - 3 \, \sinh\left(l \sqrt{\frac{P}{E J}}\right)\right)} l^{4} q_{1}}{6 \, {\left(2 \, E J l^{3} \left(\frac{P}{E J}\right)^{\frac{3}{2}} \cosh\left(l \sqrt{\frac{P}{E J}}\right) - 2 \, E J l^{3} \left(\frac{P}{E J}\right)^{\frac{3}{2}} - \frac{P^{2} l^{4} \sinh\left(l \sqrt{\frac{P}{E J}}\right)}{E J}\right)}}$

wp0 = wp.subs (l = 5., q0 = 25, q1 = 50., E = 3e7, J = 0.3^4/12, P = 2500.)
wp0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -0.000666666666666667 \, x^{3} - 0.00500000000000000 \, x^{2} + 0.00763552805639238 \, x + 0.116591704806706 \, \cosh\left(0.351364184463153 \, x\right) - 0.0217310938166866 \, \sinh\left(0.351364184463153 \, x\right) - 0.116591704806706$

wp0(2.5)

$\newcommand{\Bold}[1]{\mathbf{#1}}0.00373114122928955$

pw0 = plot (-wp0, 0., 5., aspect_ratio = 500)
pw0

M(x) = -E*J*w2(x)

Ma = M.subs (a2 = aa2, a3 = aa3)
Ma

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -\frac{1}{6} \, E J {\left(\frac{{\left(3 \, {\left(h \cosh\left(h\right) + h - 2 \, \sinh\left(h\right)\right)} l^{4} q_{0} + {\left(h \cosh\left(h\right) + 2 \, h - 3 \, \sinh\left(h\right)\right)} l^{4} q_{1}\right)} h^{2} \cosh\left(\frac{h x}{l}\right)}{{\left(E J h^{4} \sinh\left(h\right) - 2 \, E J h^{3} \cosh\left(h\right) + 2 \, E J h^{3}\right)} l^{2}} - \frac{{\left(3 \, {\left(h \sinh\left(h\right) - 2 \, \cosh\left(h\right) + 2\right)} l^{4} q_{0} + {\left(h \sinh\left(h\right) - 3 \, \cosh\left(h\right) + 3\right)} l^{4} q_{1}\right)} h^{2} \sinh\left(\frac{h x}{l}\right)}{{\left(E J h^{4} \sinh\left(h\right) - 2 \, E J h^{3} \cosh\left(h\right) + 2 \, E J h^{3}\right)} l^{2}} - \frac{6 \, l^{2} q_{0}}{E J h^{2}} - \frac{6 \, l q_{1} x}{E J h^{2}}\right)}$

w2p0(x) = diff (wp0(x), x, 2)
w2p0(x)

$\newcommand{\Bold}[1]{\mathbf{#1}}-0.00400000000000000 \, x + 0.0143940376304575 \, \cosh\left(0.351364184463153 \, x\right) - 0.00268285108847983 \, \sinh\left(0.351364184463153 \, x\right) - 0.0100000000000000$

Mp0(x) = -E*J*w2p0(x)
Mp0 = Mp0.subs (E = 3e7, J = 0.3^4/12)
Mp0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 81.0000000000000 \, x - 291.479262016764 \, \cosh\left(0.351364184463153 \, x\right) + 54.3277345417166 \, \sinh\left(0.351364184463153 \, x\right) + 202.500000000000$

Mp0(0.)

$\newcommand{\Bold}[1]{\mathbf{#1}}-88.9792620167641$

Mp0(5.)

$\newcommand{\Bold}[1]{\mathbf{#1}}-109.368494645193$

Mp0(2.5)

$\newcommand{\Bold}[1]{\mathbf{#1}}47.7482685957978$

pm0 = plot (-Mp0, 0., 5., aspect_ratio = 1/62.5)
pm0

wl4(x) = (q0 + q1/l x)/(E J)
wl4

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{q_{0} + \frac{q_{1} x}{l}}{E J}$

var ('C0 C1 C2 C3')

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(C_{0}, C_{1}, C_{2}, C_{3}\right)$

wl3(x) = integrate (wl4, x) + C0
wl3

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ C_{0} + \frac{2 \, q_{0} x + \frac{q_{1} x^{2}}{l}}{2 \, E J}$

wl2(x) = integrate (wl3, x) + C1
wl2

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ C_{0} x + C_{1} + \frac{3 \, q_{0} x^{2} + \frac{q_{1} x^{3}}{l}}{6 \, E J}$

wl1(x) = integrate (wl2, x) + C2
wl1

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{1}{2} \, C_{0} x^{2} + C_{1} x + C_{2} + \frac{4 \, q_{0} x^{3} + \frac{q_{1} x^{4}}{l}}{24 \, E J}$

wl0(x) = integrate (wl1, x) + C3
wl0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{1}{6} \, C_{0} x^{3} + \frac{1}{2} \, C_{1} x^{2} + C_{2} x + C_{3} + \frac{5 \, q_{0} x^{4} + \frac{q_{1} x^{5}}{l}}{120 \, E J}$

bb = solve ([wl0(0) == 0, 
             wl1(0) == 0, 
             wl0(l) == 0, 
             wl1(l) == 0], 
            [C0, C1, C2, C3])

wl(x) = wl0(x).subs (bb[0]).simplify_full()
wl

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{5 \, l q_{0} x^{4} + q_{1} x^{5} - {\left(10 \, l^{2} q_{0} + 3 \, l^{2} q_{1}\right)} x^{3} + {\left(5 \, l^{3} q_{0} + 2 \, l^{3} q_{1}\right)} x^{2}}{120 \, E J l}$

wl0(x) = wl(x).subs (l = 5., q0 = 25, q1 = 50., E = 3e7, J = 0.3^4/12)
wl0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \left(4.11522633744856 \times 10^{-6}\right) \, x^{5} + 0.0000514403292181070 \, x^{4} - 0.000823045267489712 \, x^{3} + 0.00231481481481482 \, x^{2}$

wl0(2.5)

$\newcommand{\Bold}[1]{\mathbf{#1}}0.00401877572016461$

pwl0 = plot (-wl0, 0., 5., aspect_ratio = 500, color = 'green')
pwl0

pw0 + pwl0

wl2(x) = diff (wl0, x, 2)
wl2

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 0.0000823045267489712 \, x^{3} + 0.000617283950617284 \, x^{2} - 0.00493827160493827 \, x + 0.00462962962962963$

Ml0(x) = -E*J*wl2 (x)
Ml0 = Ml0.subs (E = 30000000, J = (3/10)^4/12)
Ml0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -1.66666666666667 \, x^{3} - 12.5000000000000 \, x^{2} + 100.000000000000 \, x - 93.7500000000000$

N (Ml0 (0))

$\newcommand{\Bold}[1]{\mathbf{#1}}-93.7500000000000$

N (Ml0 (5))

$\newcommand{\Bold}[1]{\mathbf{#1}}-114.583333333333$

pml0 = plot (-Ml0, 0., 5., aspect_ratio = 1/62.5, color = 'green')
pml0

pm0 + pml0

NS_BEg_vl_tr

1076 days ago by fresl