4836.sagews

#求解非线性刚度阵1,文献1的公式10h
# [1] Finite-element method for large-amplitude two-dimensional panel flutter at hypersonic speeds.pdf

#w1, s1, w2, s2, ae, x = var('w1, s1, w2, s2, ae, x ')
#本来打算将ae带入到积分中的，但是发现获得多项式太复杂。
#另一方面，ae是和θ相乘的。所以干脆令s1=ae*θ1；s2=ae*θ2

w1, s1, w2, s2, x = var('w1, s1, w2, s2, x ')
ae=1
#形函数
N1=1-3*x*x+2*x*x*x
N2=(x*(1-2*x+x*x))*ae
N3=3*x*x-2*x*x*x
N4=(x*x*x-x*x)*ae
#形函数1阶导数
N1d1=6*x*x-6*x
N2d1=(3*x*x-4*x+1)*ae
N3d1=6*x-6*x*x
N4d1=(3*x*x-2*x)*ae
#形函数2阶导数
N1d2=12*x-6
N2d2=(6*x-4)*ae
N3d2=6-12*x
N4d2=(6*x-2)*ae

f1=N1d1
f2=N2d1
f3=N3d1
f4=N4d1
g=(w1*f1+s1*f2+w2*f3+s2*f4)^2

kk11=g*f1*f1
kk12=g*f1*f2
kk13=g*f1*f3
kk14=g*f1*f4

kk21=g*f2*f1
kk22=g*f2*f2
kk23=g*f2*f3
kk24=g*f2*f4

kk31=g*f3*f1
kk32=g*f3*f2
kk33=g*f3*f3
kk34=g*f3*f4

kk41=g*f4*f1
kk42=g*f4*f2
kk43=g*f4*f3
kk44=g*f4*f4

k11=integral(kk11, x, 0, 1)
k12=integral(kk12, x, 0, 1)
k13=integral(kk13, x, 0, 1)
k14=integral(kk14, x, 0, 1)

k21=integral(kk21, x, 0, 1)
k22=integral(kk22, x, 0, 1)
k23=integral(kk23, x, 0, 1)
k24=integral(kk24, x, 0, 1)

k31=integral(kk31, x, 0, 1)
k32=integral(kk32, x, 0, 1)
k33=integral(kk33, x, 0, 1)
k34=integral(kk34, x, 0, 1)

k41=integral(kk41, x, 0, 1)
k42=integral(kk42, x, 0, 1)
k43=integral(kk43, x, 0, 1)
k44=integral(kk44, x, 0, 1)

print k11;print k12;print k13;print k14
print k21;print k22;print k23;print k24
print k31;print k32;print k33;print k34
print k41;print k42;print k43;print k44

show(k11);show(k12);show(k13);show(k14)
show(k21);show(k22);show(k23);show(k24)
show(k31);show(k32);show(k33);show(k34)
show(k41);show(k42);show(k43);show(k44)

#显示多项式分子部分
temp=420
k11=temp*k11;k12=temp*k12;k13=temp*k13;k14=temp*k14
k21=temp*k21;k22=temp*k22;k23=temp*k23;k24=temp*k24
k31=temp*k31;k32=temp*k32;k33=temp*k33;k34=temp*k34
k41=temp*k41;k42=temp*k42;k43=temp*k43;k44=temp*k44

print k11;print k12;print k13;print k14
print k21;print k22;print k23;print k24
print k31;print k32;print k33;print k34
print k41;print k42;print k43;print k44

show(k11);show(k12);show(k13);show(k14)
show(k21);show(k22);show(k23);show(k24)
show(k31);show(k32);show(k33);show(k34)
show(k41);show(k42);show(k43);show(k44)

18/35*(s1 + s2)*w1 - 18/35*(s1 + s2 + 8*w1)*w2 + 3/35*s1^2 + 3/35*s2^2 + 72/35*w1^2 + 72/35*w2^2 -6/35*(s1 + 3*w1)*w2 - 1/140*s1^2 + 1/70*s1*s2 + 6/35*s1*w1 + 1/140*s2^2 + 9/35*w1^2 + 9/35*w2^2 -18/35*(s1 + s2)*w1 + 18/35*(s1 + s2 + 8*w1)*w2 - 3/35*s1^2 - 3/35*s2^2 - 72/35*w1^2 - 72/35*w2^2 -6/35*(s2 + 3*w1)*w2 + 1/140*s1^2 + 1/70*s1*s2 - 1/140*s2^2 + 6/35*s2*w1 + 9/35*w1^2 + 9/35*w2^2 -6/35*(s1 + 3*w1)*w2 - 1/140*s1^2 + 1/70*s1*s2 + 6/35*s1*w1 + 1/140*s2^2 + 9/35*w1^2 + 9/35*w2^2 -1/70*(s1 - s2)*w1 + 1/70*(s1 - s2 - 12*w1)*w2 + 2/35*s1^2 - 1/70*s1*s2 + 1/210*s2^2 + 3/35*w1^2 + 3/35*w2^2 6/35*(s1 + 3*w1)*w2 + 1/140*s1^2 - 1/70*s1*s2 - 6/35*s1*w1 - 1/140*s2^2 - 9/35*w1^2 - 9/35*w2^2 1/70*(s1 + s2)*w1 - 1/70*(s1 + s2)*w2 - 1/140*s1^2 + 1/105*s1*s2 - 1/140*s2^2 -18/35*(s1 + s2)*w1 + 18/35*(s1 + s2 + 8*w1)*w2 - 3/35*s1^2 - 3/35*s2^2 - 72/35*w1^2 - 72/35*w2^2 6/35*(s1 + 3*w1)*w2 + 1/140*s1^2 - 1/70*s1*s2 - 6/35*s1*w1 - 1/140*s2^2 - 9/35*w1^2 - 9/35*w2^2 18/35*(s1 + s2)*w1 - 18/35*(s1 + s2 + 8*w1)*w2 + 3/35*s1^2 + 3/35*s2^2 + 72/35*w1^2 + 72/35*w2^2 6/35*(s2 + 3*w1)*w2 - 1/140*s1^2 - 1/70*s1*s2 + 1/140*s2^2 - 6/35*s2*w1 - 9/35*w1^2 - 9/35*w2^2 -6/35*(s2 + 3*w1)*w2 + 1/140*s1^2 + 1/70*s1*s2 - 1/140*s2^2 + 6/35*s2*w1 + 9/35*w1^2 + 9/35*w2^2 1/70*(s1 + s2)*w1 - 1/70*(s1 + s2)*w2 - 1/140*s1^2 + 1/105*s1*s2 - 1/140*s2^2 6/35*(s2 + 3*w1)*w2 - 1/140*s1^2 - 1/70*s1*s2 + 1/140*s2^2 - 6/35*s2*w1 - 9/35*w1^2 - 9/35*w2^2 1/70*(s1 - s2)*w1 - 1/70*(s1 - s2 + 12*w1)*w2 + 1/210*s1^2 - 1/70*s1*s2 + 2/35*s2^2 + 3/35*w1^2 + 3/35*w2^2

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{18}{35} \, {\left(s_{1} + s_{2}\right)} w_{1} - \frac{18}{35} \, {\left(s_{1} + s_{2} + 8 \, w_{1}\right)} w_{2} + \frac{3}{35} \, s_{1}^{2} + \frac{3}{35} \, s_{2}^{2} + \frac{72}{35} \, w_{1}^{2} + \frac{72}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{6}{35} \, {\left(s_{1} + 3 \, w_{1}\right)} w_{2} - \frac{1}{140} \, s_{1}^{2} + \frac{1}{70} \, s_{1} s_{2} + \frac{6}{35} \, s_{1} w_{1} + \frac{1}{140} \, s_{2}^{2} + \frac{9}{35} \, w_{1}^{2} + \frac{9}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{18}{35} \, {\left(s_{1} + s_{2}\right)} w_{1} + \frac{18}{35} \, {\left(s_{1} + s_{2} + 8 \, w_{1}\right)} w_{2} - \frac{3}{35} \, s_{1}^{2} - \frac{3}{35} \, s_{2}^{2} - \frac{72}{35} \, w_{1}^{2} - \frac{72}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{6}{35} \, {\left(s_{2} + 3 \, w_{1}\right)} w_{2} + \frac{1}{140} \, s_{1}^{2} + \frac{1}{70} \, s_{1} s_{2} - \frac{1}{140} \, s_{2}^{2} + \frac{6}{35} \, s_{2} w_{1} + \frac{9}{35} \, w_{1}^{2} + \frac{9}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{6}{35} \, {\left(s_{1} + 3 \, w_{1}\right)} w_{2} - \frac{1}{140} \, s_{1}^{2} + \frac{1}{70} \, s_{1} s_{2} + \frac{6}{35} \, s_{1} w_{1} + \frac{1}{140} \, s_{2}^{2} + \frac{9}{35} \, w_{1}^{2} + \frac{9}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{70} \, {\left(s_{1} - s_{2}\right)} w_{1} + \frac{1}{70} \, {\left(s_{1} - s_{2} - 12 \, w_{1}\right)} w_{2} + \frac{2}{35} \, s_{1}^{2} - \frac{1}{70} \, s_{1} s_{2} + \frac{1}{210} \, s_{2}^{2} + \frac{3}{35} \, w_{1}^{2} + \frac{3}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{6}{35} \, {\left(s_{1} + 3 \, w_{1}\right)} w_{2} + \frac{1}{140} \, s_{1}^{2} - \frac{1}{70} \, s_{1} s_{2} - \frac{6}{35} \, s_{1} w_{1} - \frac{1}{140} \, s_{2}^{2} - \frac{9}{35} \, w_{1}^{2} - \frac{9}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{70} \, {\left(s_{1} + s_{2}\right)} w_{1} - \frac{1}{70} \, {\left(s_{1} + s_{2}\right)} w_{2} - \frac{1}{140} \, s_{1}^{2} + \frac{1}{105} \, s_{1} s_{2} - \frac{1}{140} \, s_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{18}{35} \, {\left(s_{1} + s_{2}\right)} w_{1} + \frac{18}{35} \, {\left(s_{1} + s_{2} + 8 \, w_{1}\right)} w_{2} - \frac{3}{35} \, s_{1}^{2} - \frac{3}{35} \, s_{2}^{2} - \frac{72}{35} \, w_{1}^{2} - \frac{72}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{6}{35} \, {\left(s_{1} + 3 \, w_{1}\right)} w_{2} + \frac{1}{140} \, s_{1}^{2} - \frac{1}{70} \, s_{1} s_{2} - \frac{6}{35} \, s_{1} w_{1} - \frac{1}{140} \, s_{2}^{2} - \frac{9}{35} \, w_{1}^{2} - \frac{9}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{18}{35} \, {\left(s_{1} + s_{2}\right)} w_{1} - \frac{18}{35} \, {\left(s_{1} + s_{2} + 8 \, w_{1}\right)} w_{2} + \frac{3}{35} \, s_{1}^{2} + \frac{3}{35} \, s_{2}^{2} + \frac{72}{35} \, w_{1}^{2} + \frac{72}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{6}{35} \, {\left(s_{2} + 3 \, w_{1}\right)} w_{2} - \frac{1}{140} \, s_{1}^{2} - \frac{1}{70} \, s_{1} s_{2} + \frac{1}{140} \, s_{2}^{2} - \frac{6}{35} \, s_{2} w_{1} - \frac{9}{35} \, w_{1}^{2} - \frac{9}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{6}{35} \, {\left(s_{2} + 3 \, w_{1}\right)} w_{2} + \frac{1}{140} \, s_{1}^{2} + \frac{1}{70} \, s_{1} s_{2} - \frac{1}{140} \, s_{2}^{2} + \frac{6}{35} \, s_{2} w_{1} + \frac{9}{35} \, w_{1}^{2} + \frac{9}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{70} \, {\left(s_{1} + s_{2}\right)} w_{1} - \frac{1}{70} \, {\left(s_{1} + s_{2}\right)} w_{2} - \frac{1}{140} \, s_{1}^{2} + \frac{1}{105} \, s_{1} s_{2} - \frac{1}{140} \, s_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{6}{35} \, {\left(s_{2} + 3 \, w_{1}\right)} w_{2} - \frac{1}{140} \, s_{1}^{2} - \frac{1}{70} \, s_{1} s_{2} + \frac{1}{140} \, s_{2}^{2} - \frac{6}{35} \, s_{2} w_{1} - \frac{9}{35} \, w_{1}^{2} - \frac{9}{35} \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{70} \, {\left(s_{1} - s_{2}\right)} w_{1} - \frac{1}{70} \, {\left(s_{1} - s_{2} + 12 \, w_{1}\right)} w_{2} + \frac{1}{210} \, s_{1}^{2} - \frac{1}{70} \, s_{1} s_{2} + \frac{2}{35} \, s_{2}^{2} + \frac{3}{35} \, w_{1}^{2} + \frac{3}{35} \, w_{2}^{2}$$

216*(s1 + s2)*w1 - 216*(s1 + s2 + 8*w1)*w2 + 36*s1^2 + 36*s2^2 + 864*w1^2 + 864*w2^2 -72*(s1 + 3*w1)*w2 - 3*s1^2 + 6*s1*s2 + 72*s1*w1 + 3*s2^2 + 108*w1^2 + 108*w2^2 -216*(s1 + s2)*w1 + 216*(s1 + s2 + 8*w1)*w2 - 36*s1^2 - 36*s2^2 - 864*w1^2 - 864*w2^2 -72*(s2 + 3*w1)*w2 + 3*s1^2 + 6*s1*s2 - 3*s2^2 + 72*s2*w1 + 108*w1^2 + 108*w2^2 -72*(s1 + 3*w1)*w2 - 3*s1^2 + 6*s1*s2 + 72*s1*w1 + 3*s2^2 + 108*w1^2 + 108*w2^2 -6*(s1 - s2)*w1 + 6*(s1 - s2 - 12*w1)*w2 + 24*s1^2 - 6*s1*s2 + 2*s2^2 + 36*w1^2 + 36*w2^2 72*(s1 + 3*w1)*w2 + 3*s1^2 - 6*s1*s2 - 72*s1*w1 - 3*s2^2 - 108*w1^2 - 108*w2^2 6*(s1 + s2)*w1 - 6*(s1 + s2)*w2 - 3*s1^2 + 4*s1*s2 - 3*s2^2 -216*(s1 + s2)*w1 + 216*(s1 + s2 + 8*w1)*w2 - 36*s1^2 - 36*s2^2 - 864*w1^2 - 864*w2^2 72*(s1 + 3*w1)*w2 + 3*s1^2 - 6*s1*s2 - 72*s1*w1 - 3*s2^2 - 108*w1^2 - 108*w2^2 216*(s1 + s2)*w1 - 216*(s1 + s2 + 8*w1)*w2 + 36*s1^2 + 36*s2^2 + 864*w1^2 + 864*w2^2 72*(s2 + 3*w1)*w2 - 3*s1^2 - 6*s1*s2 + 3*s2^2 - 72*s2*w1 - 108*w1^2 - 108*w2^2 -72*(s2 + 3*w1)*w2 + 3*s1^2 + 6*s1*s2 - 3*s2^2 + 72*s2*w1 + 108*w1^2 + 108*w2^2 6*(s1 + s2)*w1 - 6*(s1 + s2)*w2 - 3*s1^2 + 4*s1*s2 - 3*s2^2 72*(s2 + 3*w1)*w2 - 3*s1^2 - 6*s1*s2 + 3*s2^2 - 72*s2*w1 - 108*w1^2 - 108*w2^2 6*(s1 - s2)*w1 - 6*(s1 - s2 + 12*w1)*w2 + 2*s1^2 - 6*s1*s2 + 24*s2^2 + 36*w1^2 + 36*w2^2

$$\renewcommand{\Bold}[1]{\mathbf{#1}}216 \, {\left(s_{1} + s_{2}\right)} w_{1} - 216 \, {\left(s_{1} + s_{2} + 8 \, w_{1}\right)} w_{2} + 36 \, s_{1}^{2} + 36 \, s_{2}^{2} + 864 \, w_{1}^{2} + 864 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-72 \, {\left(s_{1} + 3 \, w_{1}\right)} w_{2} - 3 \, s_{1}^{2} + 6 \, s_{1} s_{2} + 72 \, s_{1} w_{1} + 3 \, s_{2}^{2} + 108 \, w_{1}^{2} + 108 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-216 \, {\left(s_{1} + s_{2}\right)} w_{1} + 216 \, {\left(s_{1} + s_{2} + 8 \, w_{1}\right)} w_{2} - 36 \, s_{1}^{2} - 36 \, s_{2}^{2} - 864 \, w_{1}^{2} - 864 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-72 \, {\left(s_{2} + 3 \, w_{1}\right)} w_{2} + 3 \, s_{1}^{2} + 6 \, s_{1} s_{2} - 3 \, s_{2}^{2} + 72 \, s_{2} w_{1} + 108 \, w_{1}^{2} + 108 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-72 \, {\left(s_{1} + 3 \, w_{1}\right)} w_{2} - 3 \, s_{1}^{2} + 6 \, s_{1} s_{2} + 72 \, s_{1} w_{1} + 3 \, s_{2}^{2} + 108 \, w_{1}^{2} + 108 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-6 \, {\left(s_{1} - s_{2}\right)} w_{1} + 6 \, {\left(s_{1} - s_{2} - 12 \, w_{1}\right)} w_{2} + 24 \, s_{1}^{2} - 6 \, s_{1} s_{2} + 2 \, s_{2}^{2} + 36 \, w_{1}^{2} + 36 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}72 \, {\left(s_{1} + 3 \, w_{1}\right)} w_{2} + 3 \, s_{1}^{2} - 6 \, s_{1} s_{2} - 72 \, s_{1} w_{1} - 3 \, s_{2}^{2} - 108 \, w_{1}^{2} - 108 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}6 \, {\left(s_{1} + s_{2}\right)} w_{1} - 6 \, {\left(s_{1} + s_{2}\right)} w_{2} - 3 \, s_{1}^{2} + 4 \, s_{1} s_{2} - 3 \, s_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-216 \, {\left(s_{1} + s_{2}\right)} w_{1} + 216 \, {\left(s_{1} + s_{2} + 8 \, w_{1}\right)} w_{2} - 36 \, s_{1}^{2} - 36 \, s_{2}^{2} - 864 \, w_{1}^{2} - 864 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}72 \, {\left(s_{1} + 3 \, w_{1}\right)} w_{2} + 3 \, s_{1}^{2} - 6 \, s_{1} s_{2} - 72 \, s_{1} w_{1} - 3 \, s_{2}^{2} - 108 \, w_{1}^{2} - 108 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}216 \, {\left(s_{1} + s_{2}\right)} w_{1} - 216 \, {\left(s_{1} + s_{2} + 8 \, w_{1}\right)} w_{2} + 36 \, s_{1}^{2} + 36 \, s_{2}^{2} + 864 \, w_{1}^{2} + 864 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}72 \, {\left(s_{2} + 3 \, w_{1}\right)} w_{2} - 3 \, s_{1}^{2} - 6 \, s_{1} s_{2} + 3 \, s_{2}^{2} - 72 \, s_{2} w_{1} - 108 \, w_{1}^{2} - 108 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-72 \, {\left(s_{2} + 3 \, w_{1}\right)} w_{2} + 3 \, s_{1}^{2} + 6 \, s_{1} s_{2} - 3 \, s_{2}^{2} + 72 \, s_{2} w_{1} + 108 \, w_{1}^{2} + 108 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}6 \, {\left(s_{1} + s_{2}\right)} w_{1} - 6 \, {\left(s_{1} + s_{2}\right)} w_{2} - 3 \, s_{1}^{2} + 4 \, s_{1} s_{2} - 3 \, s_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}72 \, {\left(s_{2} + 3 \, w_{1}\right)} w_{2} - 3 \, s_{1}^{2} - 6 \, s_{1} s_{2} + 3 \, s_{2}^{2} - 72 \, s_{2} w_{1} - 108 \, w_{1}^{2} - 108 \, w_{2}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}6 \, {\left(s_{1} - s_{2}\right)} w_{1} - 6 \, {\left(s_{1} - s_{2} + 12 \, w_{1}\right)} w_{2} + 2 \, s_{1}^{2} - 6 \, s_{1} s_{2} + 24 \, s_{2}^{2} + 36 \, w_{1}^{2} + 36 \, w_{2}^{2}$$