4834.sagews

#求解线性刚度阵1,文献1的公式10c
# [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=N1d2
f2=N2d2
f3=N3d2
f4=N4d2

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

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

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

kk41=f4*f1
kk42=f4*f2
kk43=f4*f3
kk44=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);

12 6*ae -12 6*ae 6*ae 4*ae^2 -6*ae 2*ae^2 -12 -6*ae 12 -6*ae 6*ae 2*ae^2 -6*ae 4*ae^2

$$\renewcommand{\Bold}[1]{\mathbf{#1}}12$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}6 \, \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-12$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}6 \, \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}6 \, \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}4 \, \mbox{ae}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-6 \, \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, \mbox{ae}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-12$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-6 \, \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}12$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-6 \, \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}6 \, \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, \mbox{ae}^{2}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-6 \, \mbox{ae}$$

$$\renewcommand{\Bold}[1]{\mathbf{#1}}4 \, \mbox{ae}^{2}$$