NS_BEg_tl -- Sage

var ('P q0 l E J h')

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

var ('a0 a1 a2 a3')

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

w(x) = a0 + a1*x + a2*sin(h*x/l) + a3*cos(h*x/l) + (q0*l^2)/(2*E*J*h^2)*x^2
w

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

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

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

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

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

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

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

w1(l) == 0

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

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

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

aa[0][0]

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

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

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

((h*cos(h) + h - 2*sin(h))/(h*sin(h) + 2*cos(h) - 2)).simplify_trig()

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{h \cos\left(h\right) + h - 2 \, \sin\left(h\right)}{h \sin\left(h\right) + 2 \, \cos\left(h\right) - 2}$

var ('h2')
aa0_2hh = ((h*cos(h) + h - 2*sin(h))/(h*sin(h) + 2*cos(h) - 2)).subs (h = 2*h2)
aa0_2hh

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{h_{2} \cos\left(2 \, h_{2}\right) + h_{2} - \sin\left(2 \, h_{2}\right)}{h_{2} \sin\left(2 \, h_{2}\right) + \cos\left(2 \, h_{2}\right) - 1}$

aa0_2hh.simplify_trig()

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\cos\left(h_{2}\right)}{\sin\left(h_{2}\right)}$

aa0 = -(q0*l^4)/(2*E*J*h^3) * cot (h/2)
aa0

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

aa[0][1]

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

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

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{l^{3} q_{0}}{2 \, E J h^{2}}$

aa[0][2]

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

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

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{l^{4} q_{0}}{2 \, E J h^{3}}$

aa[0][3]

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

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

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

aa3 = -aa0
aa3

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

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

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

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

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

wp0 = wp.subs (l = 5, q0 = 50, E = 30000000, J = (3/10)^4/12, P = 2500)
wp0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{9}{200} \, \sqrt{10} \cos\left(\frac{1}{9} \, \sqrt{10} x\right) \cot\left(\frac{5}{18} \, \sqrt{10}\right) + \frac{1}{100} \, x^{2} - \frac{9}{200} \, \sqrt{10} \cot\left(\frac{5}{18} \, \sqrt{10}\right) + \frac{9}{200} \, \sqrt{10} \sin\left(\frac{1}{9} \, \sqrt{10} x\right) - \frac{1}{20} \, x$

wp0 = wp.subs (l = 5., q0 = 50., E = 30000000., J = 0.3^4/12, P = 2500.)
wp0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 0.0100000000000000 \, x^{2} - 0.0500000000000000 \, x + 0.118019299116261 \, \cos\left(0.351364184463153 \, x\right) + 0.142302494707577 \, \sin\left(0.351364184463153 \, x\right) - 0.118019299116261$

wp0(0.)

$\newcommand{\Bold}[1]{\mathbf{#1}}0.000000000000000$

wp0(5.)

$\newcommand{\Bold}[1]{\mathbf{#1}}1.04083408558608 \times 10^{-17}$

wp0(2.5)

$\newcommand{\Bold}[1]{\mathbf{#1}}0.00435513114356016$

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

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

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

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

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

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

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -0.0145702838415137 \, \cos\left(0.351364184463153 \, x\right) - 0.0175682092231577 \, \sin\left(0.351364184463153 \, x\right) + 0.0200000000000000$

Mp0(x) = -E*J*w2p0(x)
Mp0 = Mp0.subs (E = 30000000., J = 0.3^4/12)
Mp0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 295.048247790652 \, \cos\left(0.351364184463153 \, x\right) + 355.756236768943 \, \sin\left(0.351364184463153 \, x\right) - 405.000000000000$

Mp0(0.)

$\newcommand{\Bold}[1]{\mathbf{#1}}-109.951752209347$

Mp0(5.)

$\newcommand{\Bold}[1]{\mathbf{#1}}-109.951752209348$

Mp0(2.5)

$\newcommand{\Bold}[1]{\mathbf{#1}}57.1860756495529$

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

wl4(x) = q0 / (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

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

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

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

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

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

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

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{1}{6} \, C_{0} x^{3} + \frac{1}{2} \, C_{1} x^{2} + \frac{q_{0} x^{4}}{24 \, E J} + C_{2} x + C_{3}$

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

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[C_{0} = -\frac{l q_{0}}{2 \, E J}, C_{1} = \frac{l^{2} q_{0}}{12 \, E J}, C_{2} = 0, C_{3} = 0\right]\right]$

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

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

wl0 = wl.subs (l = 5, q0 = 50, E = 30000000, J = (3/10)^4/12)
wl0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{1}{9720} \, x^{4} - \frac{1}{972} \, x^{3} + \frac{5}{1944} \, x^{2}$

wl0 = wl.subs (l = 5., q0 = 50., E = 30000000., J = 0.3^4/12)
wl0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 0.000102880658436214 \, x^{4} - 0.00102880658436214 \, x^{3} + 0.00257201646090535 \, x^{2}$

wl0(2.5)

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

wp0(2.5)

$\newcommand{\Bold}[1]{\mathbf{#1}}0.00435513114356016$

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

pw0 + pwl0

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

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{l^{2} q_{0} - 6 \, l q_{0} x + 6 \, q_{0} x^{2}}{12 \, E J}$

wl2_0(x) = diff (wl0, x, 2)
wl2_0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 0.00123456790123457 \, x^{2} - 0.00617283950617284 \, x + 0.00514403292181070$

Ml0(x) = -E*J*wl2_0 (x)
Ml0 = Ml0.subs (E = 30000000., J = 0.3^4/12)
Ml0

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -25.0000000000000 \, x^{2} + 125.000000000000 \, x - 104.166666666667$

Ml0(0.)

$\newcommand{\Bold}[1]{\mathbf{#1}}-104.166666666667$

Ml0(5.)

$\newcommand{\Bold}[1]{\mathbf{#1}}-104.166666666667$

Ml0(2.5)

$\newcommand{\Bold}[1]{\mathbf{#1}}52.0833333333333$

Mp0(0.); Mp0(2.5); Mp0(5.)

$\newcommand{\Bold}[1]{\mathbf{#1}}-109.951752209347$

$\newcommand{\Bold}[1]{\mathbf{#1}}57.1860756495529$

$\newcommand{\Bold}[1]{\mathbf{#1}}-109.951752209348$

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

pm0 + pml0

NS_BEg_tl

5 days ago by fresl