4835.sagews

#求解非线性刚度阵1,文献1的公式10e, 10f, 10g
# [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 ')
#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

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=210
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)

-6/35*ae*s1 - 6/35*ae*s2 - 54/35*w1 + 54/35*w2 -1/70*(6*ae*s1 - ae*s2 + 12*w1 - 12*w2)*ae 6/35*ae*s1 + 6/35*ae*s2 + 54/35*w1 - 54/35*w2 1/70*(ae*s1 - 6*ae*s2 - 12*w1 + 12*w2)*ae -1/70*(6*ae*s1 - ae*s2 + 12*w1 - 12*w2)*ae 1/105*(6*ae*s1 - ae*s2 - 9*w1 + 9*w2)*ae^2 1/70*(6*ae*s1 - ae*s2 + 12*w1 - 12*w2)*ae -1/210*(2*ae*s1 + 2*ae*s2 - 3*w1 + 3*w2)*ae^2 6/35*ae*s1 + 6/35*ae*s2 + 54/35*w1 - 54/35*w2 1/70*(6*ae*s1 - ae*s2 + 12*w1 - 12*w2)*ae -6/35*ae*s1 - 6/35*ae*s2 - 54/35*w1 + 54/35*w2 -1/70*(ae*s1 - 6*ae*s2 - 12*w1 + 12*w2)*ae 1/70*(ae*s1 - 6*ae*s2 - 12*w1 + 12*w2)*ae -1/210*(2*ae*s1 + 2*ae*s2 - 3*w1 + 3*w2)*ae^2 -1/70*(ae*s1 - 6*ae*s2 - 12*w1 + 12*w2)*ae -1/105*(ae*s1 - 6*ae*s2 + 9*w1 - 9*w2)*ae^2

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{6}{35} \, \mbox{ae} s_{1} - \frac{6}{35} \, \mbox{ae} s_{2} - \frac{54}{35} \, w_{1} + \frac{54}{35} \, w_{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{70} \, {\left(6 \, \mbox{ae} s_{1} - \mbox{ae} s_{2} + 12 \, w_{1} - 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{6}{35} \, \mbox{ae} s_{1} + \frac{6}{35} \, \mbox{ae} s_{2} + \frac{54}{35} \, w_{1} - \frac{54}{35} \, w_{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{70} \, {\left(\mbox{ae} s_{1} - 6 \, \mbox{ae} s_{2} - 12 \, w_{1} + 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{70} \, {\left(6 \, \mbox{ae} s_{1} - \mbox{ae} s_{2} + 12 \, w_{1} - 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{105} \, {\left(6 \, \mbox{ae} s_{1} - \mbox{ae} s_{2} - 9 \, w_{1} + 9 \, w_{2}\right)} \mbox{ae}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{70} \, {\left(6 \, \mbox{ae} s_{1} - \mbox{ae} s_{2} + 12 \, w_{1} - 12 \, w_{2}\right)} \mbox{ae}$$

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

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{6}{35} \, \mbox{ae} s_{1} + \frac{6}{35} \, \mbox{ae} s_{2} + \frac{54}{35} \, w_{1} - \frac{54}{35} \, w_{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{70} \, {\left(6 \, \mbox{ae} s_{1} - \mbox{ae} s_{2} + 12 \, w_{1} - 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{6}{35} \, \mbox{ae} s_{1} - \frac{6}{35} \, \mbox{ae} s_{2} - \frac{54}{35} \, w_{1} + \frac{54}{35} \, w_{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{70} \, {\left(\mbox{ae} s_{1} - 6 \, \mbox{ae} s_{2} - 12 \, w_{1} + 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{70} \, {\left(\mbox{ae} s_{1} - 6 \, \mbox{ae} s_{2} - 12 \, w_{1} + 12 \, w_{2}\right)} \mbox{ae}$$

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

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{70} \, {\left(\mbox{ae} s_{1} - 6 \, \mbox{ae} s_{2} - 12 \, w_{1} + 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{105} \, {\left(\mbox{ae} s_{1} - 6 \, \mbox{ae} s_{2} + 9 \, w_{1} - 9 \, w_{2}\right)} \mbox{ae}^{2}$$

-36*ae*s1 - 36*ae*s2 - 324*w1 + 324*w2 -3*(6*ae*s1 - ae*s2 + 12*w1 - 12*w2)*ae 36*ae*s1 + 36*ae*s2 + 324*w1 - 324*w2 3*(ae*s1 - 6*ae*s2 - 12*w1 + 12*w2)*ae -3*(6*ae*s1 - ae*s2 + 12*w1 - 12*w2)*ae 2*(6*ae*s1 - ae*s2 - 9*w1 + 9*w2)*ae^2 3*(6*ae*s1 - ae*s2 + 12*w1 - 12*w2)*ae -(2*ae*s1 + 2*ae*s2 - 3*w1 + 3*w2)*ae^2 36*ae*s1 + 36*ae*s2 + 324*w1 - 324*w2 3*(6*ae*s1 - ae*s2 + 12*w1 - 12*w2)*ae -36*ae*s1 - 36*ae*s2 - 324*w1 + 324*w2 -3*(ae*s1 - 6*ae*s2 - 12*w1 + 12*w2)*ae 3*(ae*s1 - 6*ae*s2 - 12*w1 + 12*w2)*ae -(2*ae*s1 + 2*ae*s2 - 3*w1 + 3*w2)*ae^2 -3*(ae*s1 - 6*ae*s2 - 12*w1 + 12*w2)*ae -2*(ae*s1 - 6*ae*s2 + 9*w1 - 9*w2)*ae^2

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-36 \, \mbox{ae} s_{1} - 36 \, \mbox{ae} s_{2} - 324 \, w_{1} + 324 \, w_{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-3 \, {\left(6 \, \mbox{ae} s_{1} - \mbox{ae} s_{2} + 12 \, w_{1} - 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}36 \, \mbox{ae} s_{1} + 36 \, \mbox{ae} s_{2} + 324 \, w_{1} - 324 \, w_{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}3 \, {\left(\mbox{ae} s_{1} - 6 \, \mbox{ae} s_{2} - 12 \, w_{1} + 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-3 \, {\left(6 \, \mbox{ae} s_{1} - \mbox{ae} s_{2} + 12 \, w_{1} - 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left(6 \, \mbox{ae} s_{1} - \mbox{ae} s_{2} - 9 \, w_{1} + 9 \, w_{2}\right)} \mbox{ae}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}3 \, {\left(6 \, \mbox{ae} s_{1} - \mbox{ae} s_{2} + 12 \, w_{1} - 12 \, w_{2}\right)} \mbox{ae}$$

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

$$\renewcommand{\Bold}[1]{\mathbf{#1}}36 \, \mbox{ae} s_{1} + 36 \, \mbox{ae} s_{2} + 324 \, w_{1} - 324 \, w_{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}3 \, {\left(6 \, \mbox{ae} s_{1} - \mbox{ae} s_{2} + 12 \, w_{1} - 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-36 \, \mbox{ae} s_{1} - 36 \, \mbox{ae} s_{2} - 324 \, w_{1} + 324 \, w_{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-3 \, {\left(\mbox{ae} s_{1} - 6 \, \mbox{ae} s_{2} - 12 \, w_{1} + 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}3 \, {\left(\mbox{ae} s_{1} - 6 \, \mbox{ae} s_{2} - 12 \, w_{1} + 12 \, w_{2}\right)} \mbox{ae}$$

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

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-3 \, {\left(\mbox{ae} s_{1} - 6 \, \mbox{ae} s_{2} - 12 \, w_{1} + 12 \, w_{2}\right)} \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, {\left(\mbox{ae} s_{1} - 6 \, \mbox{ae} s_{2} + 9 \, w_{1} - 9 \, w_{2}\right)} \mbox{ae}^{2}$$