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Question: Doing some Substitution

Question:Doing some Substitution

9009134 110 Maple 2018
substitution differential-equations
April 03 2019
0

How I can do following procedure?

Thank you.

 

Substitution of Eqs. (5,6,7) into Eqs. (1)–(4), gives the new equation as functions of the generalized coordinates,
u_m,n(t);  v_m,n ( t), and w_m,n ( t). These expressions are then inserted in the Lagrange equations (see Eq. (8)) resulting into a set of N second-order coupled ordinary differential equations with both quadratic   and cubic nonlinearities.

In Eq (8) q are generalized coordinate such as u,v,w  and q = {`u__m,n`(t), `v__m,n`(t), `w__m,n`(t)}^T.

\where the elements of the vector,q_i are the time-dependent generalized coordinates.

L_Maple.mw
 

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U = (1/2)*(int(int(int(E*(`∂`(u(x, y, t))/`∂`(x)+(1/2)*(`∂`(w(x, y, t))/`∂`(x))^2+`∂`(w(x, y, t))/`∂`(x)*(`∂`(w__0(x, y, t))/`∂`(x))-z*(diff(w(x, y, t), x, x))+v(x, y, t)*(`∂`(v(x, y, t))/`∂`(y)+(1/2)*(`∂`(w(x, y, t))/`∂`(y))^2+`∂`(w(x, y, t))/`∂`(y)*(`∂`(w__0(x, y, t))/`∂`(y))-z*(diff(w(x, y, t), y, y))))*(`∂`(u(x, y, t))/`∂`(x)+(1/2)*(`∂`(w(x, y, t))/`∂`(x))^2+`∂`(w(x, y, t))/`∂`(x)*(`∂`(w__0(x, y, t))/`∂`(x))-z*(diff(w(x, y, t), x, x)))/(-nu^2+1)+E*(`∂`(nu(x, y, t))/`∂`(y)+(1/2)*(`∂`(w(x, y, t))/`∂`(y))^2+`∂`(w(x, y, t))/`∂`(y)*(`∂`(w__0(x, y, t))/`∂`(y))-z*(diff(w(x, y, t), y, y))+v(x, y, t)*(`∂`(u(x, y, t))/`∂`(x)+(1/2)*(`∂`(w(x, y, t))/`∂`(x))^2+`∂`(w(x, y, t))/`∂`(x)*(`∂`(w__0(x, y, t))/`∂`(x))-z*(diff(w(x, y, t), x, x))))*(`∂`(v(x, y, t))/`∂`(y)+(1/2)*(`∂`(w(x, y, t))/`∂`(y))^2+`∂`(w(x, y, t))/`∂`(y)*(`∂`(w__0(x, y, t))/`∂`(y))-z*(diff(w(x, y, t), y, y)))/(-nu^2+1)+E*(`∂`(u(x, y, t))/`∂`(y)+`∂`(v(x, y, t))/`∂`(x)+`∂`(w(x, y, t))/`∂`(x)*(`∂`(w(x, y, t))/`∂`(y))+`∂`(w__0(x, y, t))*`∂`(w(x, y, t))/(`∂`(x)*`∂`(y))+`∂`(w__0(x, y, t))*`∂`(w(x, y, t))/(`∂`(x)*`∂`(y))-2*z*(diff(w(x, y, t), x, y)))^2/(2*(1+nu))+E*l^2*(diff(w(x, y, t), x, y))^2/(1+nu)+E*l^2*(diff(w(x, y, t), x, y))^2/(1+nu)+E*l^2*(diff(w(x, y, t), y, y)-(diff(w(x, y, t), x, x)))^2/(2*(1+nu))+E*l^2*(diff(v(x, y, t), y, y)-(diff(u(x, y, t), x, x)))^2/(8*(1+nu))+E*l^2*(diff(v(x, y, t), x, y)-(diff(u(x, y, t), y, y)))^2/(8*(1+nu)), z = -(1/2)*h .. (1/2)*h), y = 0 .. b), x = 0 .. a))

U = (1/2)*(int(int((1/12)*(-E*(-v(x, y, t)*(diff(diff(w(x, y, t), y), y))-(diff(diff(w(x, y, t), x), x)))*(diff(diff(w(x, y, t), x), x))/(-nu^2+1)-E*(-v(x, y, t)*(diff(diff(w(x, y, t), x), x))-(diff(diff(w(x, y, t), y), y)))*(diff(diff(w(x, y, t), y), y))/(-nu^2+1)+4*E*(diff(diff(w(x, y, t), x), y))^2/(2+2*nu))*h^3+E*(`∂`(u(x, y, t))/`∂`(x)+(1/2)*`∂`(w(x, y, t))^2/`∂`(x)^2+`∂`(w(x, y, t))*`∂`(w__0(x, y, t))/`∂`(x)^2+v(x, y, t)*(`∂`(v(x, y, t))/`∂`(y)+(1/2)*`∂`(w(x, y, t))^2/`∂`(y)^2+`∂`(w(x, y, t))*`∂`(w__0(x, y, t))/`∂`(y)^2))*(`∂`(u(x, y, t))/`∂`(x)+(1/2)*`∂`(w(x, y, t))^2/`∂`(x)^2+`∂`(w(x, y, t))*`∂`(w__0(x, y, t))/`∂`(x)^2)*h/(-nu^2+1)+E*(`∂`(nu(x, y, t))/`∂`(y)+(1/2)*`∂`(w(x, y, t))^2/`∂`(y)^2+`∂`(w(x, y, t))*`∂`(w__0(x, y, t))/`∂`(y)^2+v(x, y, t)*(`∂`(u(x, y, t))/`∂`(x)+(1/2)*`∂`(w(x, y, t))^2/`∂`(x)^2+`∂`(w(x, y, t))*`∂`(w__0(x, y, t))/`∂`(x)^2))*(`∂`(v(x, y, t))/`∂`(y)+(1/2)*`∂`(w(x, y, t))^2/`∂`(y)^2+`∂`(w(x, y, t))*`∂`(w__0(x, y, t))/`∂`(y)^2)*h/(-nu^2+1)+E*(`∂`(u(x, y, t))/`∂`(y)+`∂`(v(x, y, t))/`∂`(x)+`∂`(w(x, y, t))^2/(`∂`(x)*`∂`(y))+2*`∂`(w__0(x, y, t))*`∂`(w(x, y, t))/(`∂`(x)*`∂`(y)))^2*h/(2+2*nu)+2*E*l^2*(diff(diff(w(x, y, t), x), y))^2*h/(1+nu)+E*l^2*(diff(diff(w(x, y, t), y), y)-(diff(diff(w(x, y, t), x), x)))^2*h/(2+2*nu)+E*l^2*(diff(diff(v(x, y, t), y), y)-(diff(diff(u(x, y, t), x), x)))^2*h/(8+8*nu)+E*l^2*(diff(diff(v(x, y, t), x), y)-(diff(diff(u(x, y, t), y), y)))^2*h/(8+8*nu), y = 0 .. b), x = 0 .. a))

 (1)
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T = rho*h*(int(int((`∂`(u(x, y, t))/`∂`(t))^2+(`∂`(v(x, y, t))/`∂`(t))^2+(`∂`(w(x, y, t))/`∂`(t))^2, y = 0 .. b), x = 0 .. a))

T = rho*h*(int(int(`∂`(u(x, y, t))^2/`∂`(t)^2+`∂`(v(x, y, t))^2/`∂`(t)^2+`∂`(w(x, y, t))^2/`∂`(t)^2, y = 0 .. b), x = 0 .. a))

 (2)
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F = (1/2)*c*(int(int((`∂`(u(x, y, t))/`∂`(t))^2+(`∂`(v(x, y, t))/`∂`(t))^2+(`∂`(w(x, y, t))/`∂`(t))^2, y = 0 .. b), x = 0 .. a))

F = (1/2)*c*(int(int(`∂`(u(x, y, t))^2/`∂`(t)^2+`∂`(v(x, y, t))^2/`∂`(t)^2+`∂`(w(x, y, t))^2/`∂`(t)^2, y = 0 .. b), x = 0 .. a))

 (3)
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W = int(int(w(x, y, t)*f__1(x, y, t)*cos(omega*t), y = 0 .. b), x = 0 .. a)

W = int(int(w(x, y, z)*f__1(x, y, z)*cos(omega*t), y = 0 .. b), x = 0 .. a)

 (4)
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u(x, y, t) = sum(sum(`u__m,n`(t)*sin(m*Pi*x/a)*sin(n*Pi*y/b), n = 1 .. N), m = 1 .. M)

u(x, y, t) = -(1/4)*(cos(Pi*y*N/b)*cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*y*N/b)*cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y/b)+cos(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)-cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)-cos(Pi*y*N/b)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)+cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)+sin(Pi*x/a)*sin(Pi*y/b)*cos((M+1)*Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)+sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)+sin(Pi*y/b)*sin((M+1)*Pi*x/a))*`u__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))+(1/4)*(-cos(Pi*y*N/b)*sin(Pi*y/b)*sin(Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin(Pi*x/a)+sin(Pi*y*N/b)*sin(Pi*x/a)+sin(Pi*y/b)*sin(Pi*x/a))*`u__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))

 (5)
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v(x, y, t) = sum(sum(`v__m,n`(t)*sin(m*Pi*x/a)*sin(n*Pi*y/b), n = 1 .. N), m = 1 .. M)

v(x, y, t) = -(1/4)*(cos(Pi*y*N/b)*cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*y*N/b)*cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y/b)+cos(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)-cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)-cos(Pi*y*N/b)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)+cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)+sin(Pi*x/a)*sin(Pi*y/b)*cos((M+1)*Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)+sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)+sin(Pi*y/b)*sin((M+1)*Pi*x/a))*`v__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))+(1/4)*(-cos(Pi*y*N/b)*sin(Pi*y/b)*sin(Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin(Pi*x/a)+sin(Pi*y*N/b)*sin(Pi*x/a)+sin(Pi*y/b)*sin(Pi*x/a))*`v__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))

 (6)
> 

w(x, y, t) = sum(sum(`w__m,n`(t)*sin(m*Pi*x/a)*sin(n*Pi*y/b), n = 1 .. N), m = 1 .. M)

w(x, y, t) = -(1/4)*(cos(Pi*y*N/b)*cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*y*N/b)*cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y/b)+cos(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)-cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)*cos(Pi*y/b)-cos(Pi*y*N/b)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)-cos(Pi*x/a)*sin(Pi*y/b)*sin((M+1)*Pi*x/a)+cos((M+1)*Pi*x/a)*sin(Pi*x/a)*sin(Pi*y*N/b)+sin(Pi*x/a)*sin(Pi*y/b)*cos((M+1)*Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin((M+1)*Pi*x/a)+sin(Pi*y*N/b)*sin((M+1)*Pi*x/a)+sin(Pi*y/b)*sin((M+1)*Pi*x/a))*`w__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))+(1/4)*(-cos(Pi*y*N/b)*sin(Pi*y/b)*sin(Pi*x/a)-sin(Pi*y*N/b)*cos(Pi*y/b)*sin(Pi*x/a)+sin(Pi*y*N/b)*sin(Pi*x/a)+sin(Pi*y/b)*sin(Pi*x/a))*`w__m,n`(t)/((cos(Pi*x/a)-1)*(cos(Pi*y/b)-1))

 (7)
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diff(`∂`(T(x, y, t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`), t)-`∂`(T(x, y, t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`)+`∂`(U(x, y, t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`)+`∂`(U(x, y, t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`)+`∂`(F(x, y, t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`) = `∂`(W(x, y, t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`), j = 1, () .. (), N

(D(`∂`))(T(x, y, t))*(diff(T(x, y, t), t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`)-`∂`(T(x, y, t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`)+2*`∂`(U(x, y, t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`)+`∂`(F(x, y, t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`) = `∂`(W(x, y, t))/`∂`(`#mscripts(mi("q"),mi("j"),none(),none(),mo("."),none(),none())`), j = 1, () .. (), N

 (8)
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NULL


 

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