## 25 Reputation

0 years, 203 days

## Transfer matrix method for vibration ana...

Maple

Program of Transfer matrix method for solving the vibrational behavior of a cracked beam. For more details and comprehension check the paper entitled ''A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions ''

## Stiffness matrix implementation of a fi...

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 (1)
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Stiffness matrix implementation of a finite element beam based on Euler-Bernoulli theory stiffnesmatrice.mw

## Free vibration of a cracked composite be...

Maple

 > restart: with(LinearAlgebra):
 > # Motion equation (  Vibration of a cracked composite beam using general solution)  Edited by Adjal Yassine #
 > ####################################################################

Motion equation of flexural  vibration in normalized form

 > EI*W^(iv)-m*omega^2*W=0;
 (1)

The general solution form of bending vibration equation

 > W1:=A[1]*cosh(mu*x)+A[2]*sinh(mu*x)+A[3]*cos(mu*x)+A[4]*sin(mu*x);
 (2)

where

 > E:=2682e6;L:=0.18;h:=0.004;b:=0.02;rho:=2600;area=b*h;m:=rho*h*b;II:=(h*b^3)/12:
 (3)
 > mu:=((m*omega^2*L^4/EI)^(1/4)):

Expression of cross-sectional rotation , the bending moment shear  force and torsional moment  are given as follows respectively

 > theta1 := (1/L)*(A[1]*mu*sinh(mu*x)+A[2]*mu*cosh(mu*x)-A[3]*mu*sin(mu*x)+A[4]*mu*cos(mu*x));
 (4)
 > M1:= (EI/L^2)*(A[1]*mu^2*cosh(mu*x)+A[2]*mu^2*sinh(mu*x)-A[3]*mu^2*cos(mu*x)-A[4]*mu^2*sin(mu*x));
 (5)
 > S1:= (-EI/L^3)*(A[1]*mu^3*sinh(mu*x)+A[2]*mu^3*cosh(mu*x)+A[3]*mu^3*sin(mu*x)-A[4]*mu^3*cos(mu*x));
 (6)
 >
 > W2:=A[5]*cosh(mu*x)+A[6]*sinh(mu*x)+A[7]*cos(mu*x)+A[8]*sin(mu*x);
 (7)
 >
 > theta2:= (1/L)*(A[5]*mu*sinh(mu*x)+A[6]*mu*cosh(mu*x)-A[7]*mu*sin(mu*x)+A[8]*mu*cos(mu*x));
 (8)
 > M2:= (EI/L^2)*(A[5]*mu^2*cosh(mu*x)+A[6]*mu^2*sinh(mu*x)-A[7]*mu^2*cos(mu*x)-A[8]*mu^2*sin(mu*x));
 (9)
 > S2:= -(EI/L^3)*(A[5]*mu^3*sinh(mu*x)+A[6]*mu^3*cosh(mu*x)+A[7]*mu^3*sin(mu*x)-A[8]*mu^3*cos(mu*x));
 (10)
 >

The boundary conditions at fixed end W1(0)=Theta(0)=0

 > X1:=eval(subs(x=0,W1));
 (11)
 > X2:=eval(subs(x=0,theta1));
 (12)
 > X2:=collect(X2,mu)*(L/mu);
 (13)
 >

The boundary condtions at free end M2(1)=S2(1)=0

 > X3:=eval(subs(x=1,M2));
 (14)
 > X3:=collect(X3,mu)*(L^2/mu^2/EI);
 (15)
 > X4:=eval(subs(x=1,S2));
 (16)
 > X4:=collect(X4,mu);
 (17)
 > X4:=collect(X4,EI)*(L^3/mu^3/EI);
 (18)
 >

The additional boundary conditions at crack location

 > X5:=combine(M1-M2);
 (19)
 > X5:=collect(X5,mu);
 (20)
 > X5:=collect(X5,EI)*(L^2/mu^2/EI);
 (21)
 > X6:=combine(S1-S2);
 (22)
 > X6:=collect(X6,mu);
 (23)
 > X6:=collect(X6,EI)*(L^3/mu^3/EI);
 (24)
 >
 > X7:=combine(W2-(W1+c8*S1));
 (25)
 > X8:=combine (theta2-(theta1+c44*M1));
 (26)
 >

The characteristic matrix function of frequency

 > FD8:=subs(A[1]=1,A[3]=0,X1);FD12:=subs(A[1]=0,A[3]=0,X1);FD13:=subs(A[1]=0,A[3]=1,X1);FD14:=subs(A[1]=0,A[3]=0,X1);FD15:=subs(A[1]=0,A[3]=0,X1);FD16:=subs(A[1]=0,A[3]=0,X1);FD17:=subs(A[1]=0,A[3]=0,X1);FD18:=subs(A[1]=0,A[3]=0,X1);
 (27)
 > FD21:=subs(A[2]=0,A[4]=0,X2);FD22:=subs(A[2]=1,A[4]=0,X2);FD23:=subs(A[2]=0,A[4]=0,X2);FD24:=subs(A[2]=0,A[4]=1,X2);FD25:=subs(A[2]=0,A[4]=0,X2);FD26:=subs(A[2]=0,A[4]=0,X2);FD27:=subs(A[2]=0,A[4]=0,X2);FD28:=subs(A[2]=0,A[4]=0,X2);
 (28)
 >
 > FD31:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X3);FD32:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X3);FD33:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X3);FD34:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X3);FD35:=subs(A[5]=1,A[6]=0,A[7]=0,A[8]=0,X3);;FD36:=subs(A[5]=0,A[6]=1,A[7]=0,A[8]=0,X3);FD37:=subs(A[5]=0,A[6]=0,A[7]=1,A[8]=0,X3);FD38:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=1,X3);
 (29)
 > FD41:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X4);FD42:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X4);FD43:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X4);FD44:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X4);FD45:=subs(A[5]=1,A[6]=0,A[7]=0,A[8]=0,X4);FD46:=subs(A[5]=0,A[6]=1,A[7]=0,A[8]=0,X4);FD47:=subs(A[5]=0,A[6]=0,A[7]=1,A[8]=0,X4);FD48:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=1,X4);
 (30)
 >
 > FD51:=subs(A[1]=1,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X5);FD52:=subs(A[1]=0,A[2]=1,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X5);FD53:=subs(A[1]=0,A[2]=0,A[3]=1,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X5);FD54:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=1,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X5);FD55:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=1,A[6]=0,A[7]=0,A[8]=0,X5);FD56:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=1,A[7]=0,A[8]=0,X5);FD57:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=1,A[8]=0,X5);FD58:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=1,X5);
 (31)
 > FD61:=subs(A[1]=1,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X6);FD62:=subs(A[1]=0,A[2]=1,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X6);FD63:=subs(A[1]=0,A[2]=0,A[3]=1,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X6);FD64:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=1,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X6);FD65:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=1,A[6]=0,A[7]=0,A[8]=0,X6);FD66:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=1,A[7]=0,A[8]=0,X6);FD67:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=1,A[8]=0,X6);FD68:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=1,X6);
 (32)
 >
 > FD71:=subs(A[1]=1,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X7);FD72:=subs(A[1]=0,A[2]=1,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X7);FD73:=subs(A[1]=0,A[2]=0,A[3]=1,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X7);FD74:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=1,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X7);FD75:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=1,A[6]=0,A[7]=0,A[8]=0,X7);FD76:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=1,A[7]=0,A[8]=0,X7);FD77:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=1,A[8]=0,X7);FD78:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=1,X7);
 (33)
 > FD81:=subs(A[1]=1,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X8);FD82:=subs(A[1]=0,A[2]=1,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X8);FD83:=subs(A[1]=0,A[2]=0,A[3]=1,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X8);FD84:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=1,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X8);FD85:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=1,A[6]=0,A[7]=0,A[8]=0,X8);FD86:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=1,A[7]=0,A[8]=0,X8);FD87:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=1,A[8]=0,X8);FD88:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=1,X8);
 (34)
 >
 > MM:=matrix(8,8,[[FD11,FD12,FD13,FD14,FD15,FD16,FD17,FD18],[FD21,FD22,FD23,FD24,FD25,FD26,FD27,FD28],[FD31,FD32,FD33,FD34,FD35,FD36,FD37,FD38],[FD41,FD42,FD43,FD44,FD45,FD46,FD47,FD48],[FD51,FD52,FD53,FD54,FD55,FD56,FD57,FD58],[FD61,FD62,FD63,FD64,FD65,FD66,FD67,FD68],[FD71,FD72,FD73,FD74,FD75,FD76,FD77,FD78],[FD81,FD82,FD83,FD84,FD85,FD86,FD87,FD88]]);
 (35)

Program end

 >
 >
 >

 > restart: with(LinearAlgebra):
 > # Motion equation (  Vibration of a cracked composite beam using general solution)  Edited by Adjal Yassine #
 > ####################################################################

Motion equation of flexural  vibration in normalized form

 > EI*W^(iv)-m*omega^2*W=0;
 (1)

The general solution form of bending vibration equation

 > W1:=A[1]*cosh(mu*x)+A[2]*sinh(mu*x)+A[3]*cos(mu*x)+A[4]*sin(mu*x);
 (2)

where

 > E:=2682e6;L:=0.18;h:=0.004;b:=0.02;rho:=2600;area=b*h;m:=rho*h*b;II:=(h*b^3)/12:
 (3)
 > mu:=((m*omega^2*L^4/EI)^(1/4)):

Expression of cross-sectional rotation , the bending moment shear  force and torsional moment  are given as follows respectively

 > theta1 := (1/L)*(A[1]*mu*sinh(mu*x)+A[2]*mu*cosh(mu*x)-A[3]*mu*sin(mu*x)+A[4]*mu*cos(mu*x));
 (4)
 > M1:= (EI/L^2)*(A[1]*mu^2*cosh(mu*x)+A[2]*mu^2*sinh(mu*x)-A[3]*mu^2*cos(mu*x)-A[4]*mu^2*sin(mu*x));
 (5)
 > S1:= (-EI/L^3)*(A[1]*mu^3*sinh(mu*x)+A[2]*mu^3*cosh(mu*x)+A[3]*mu^3*sin(mu*x)-A[4]*mu^3*cos(mu*x));
 (6)
 >
 > W2:=A[5]*cosh(mu*x)+A[6]*sinh(mu*x)+A[7]*cos(mu*x)+A[8]*sin(mu*x);
 (7)
 >
 > theta2:= (1/L)*(A[5]*mu*sinh(mu*x)+A[6]*mu*cosh(mu*x)-A[7]*mu*sin(mu*x)+A[8]*mu*cos(mu*x));
 (8)
 > M2:= (EI/L^2)*(A[5]*mu^2*cosh(mu*x)+A[6]*mu^2*sinh(mu*x)-A[7]*mu^2*cos(mu*x)-A[8]*mu^2*sin(mu*x));
 (9)
 > S2:= -(EI/L^3)*(A[5]*mu^3*sinh(mu*x)+A[6]*mu^3*cosh(mu*x)+A[7]*mu^3*sin(mu*x)-A[8]*mu^3*cos(mu*x));
 (10)
 >

The boundary conditions at fixed end W1(0)=Theta(0)=0

 > X1:=eval(subs(x=0,W1));
 (11)
 > X2:=eval(subs(x=0,theta1));
 (12)
 > X2:=collect(X2,mu)*(L/mu);
 (13)
 >

The boundary condtions at free end M2(1)=S2(1)=0

 > X3:=eval(subs(x=1,M2));
 (14)
 > X3:=collect(X3,mu)*(L^2/mu^2/EI);
 (15)
 > X4:=eval(subs(x=1,S2));
 (16)
 > X4:=collect(X4,mu);
 (17)
 > X4:=collect(X4,EI)*(L^3/mu^3/EI);
 (18)
 >

The additional boundary conditions at crack location

 > X5:=combine(M1-M2);
 (19)
 > X5:=collect(X5,mu);
 (20)
 > X5:=collect(X5,EI)*(L^2/mu^2/EI);
 (21)
 > X6:=combine(S1-S2);
 (22)
 > X6:=collect(X6,mu);
 (23)
 > X6:=collect(X6,EI)*(L^3/mu^3/EI);
 (24)
 >
 > X7:=combine(W2-(W1+c8*S1));
 (25)
 > X8:=combine (theta2-(theta1+c44*M1));
 (26)
 >

The characteristic matrix function of frequency

 > FD8:=subs(A[1]=1,A[3]=0,X1);FD12:=subs(A[1]=0,A[3]=0,X1);FD13:=subs(A[1]=0,A[3]=1,X1);FD14:=subs(A[1]=0,A[3]=0,X1);FD15:=subs(A[1]=0,A[3]=0,X1);FD16:=subs(A[1]=0,A[3]=0,X1);FD17:=subs(A[1]=0,A[3]=0,X1);FD18:=subs(A[1]=0,A[3]=0,X1);
 (27)
 > FD21:=subs(A[2]=0,A[4]=0,X2);FD22:=subs(A[2]=1,A[4]=0,X2);FD23:=subs(A[2]=0,A[4]=0,X2);FD24:=subs(A[2]=0,A[4]=1,X2);FD25:=subs(A[2]=0,A[4]=0,X2);FD26:=subs(A[2]=0,A[4]=0,X2);FD27:=subs(A[2]=0,A[4]=0,X2);FD28:=subs(A[2]=0,A[4]=0,X2);
 (28)
 >
 > FD31:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X3);FD32:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X3);FD33:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X3);FD34:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X3);FD35:=subs(A[5]=1,A[6]=0,A[7]=0,A[8]=0,X3);;FD36:=subs(A[5]=0,A[6]=1,A[7]=0,A[8]=0,X3);FD37:=subs(A[5]=0,A[6]=0,A[7]=1,A[8]=0,X3);FD38:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=1,X3);
 (29)
 > FD41:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X4);FD42:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X4);FD43:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X4);FD44:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=0,X4);FD45:=subs(A[5]=1,A[6]=0,A[7]=0,A[8]=0,X4);FD46:=subs(A[5]=0,A[6]=1,A[7]=0,A[8]=0,X4);FD47:=subs(A[5]=0,A[6]=0,A[7]=1,A[8]=0,X4);FD48:=subs(A[5]=0,A[6]=0,A[7]=0,A[8]=1,X4);
 (30)
 >
 > FD51:=subs(A[1]=1,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X5);FD52:=subs(A[1]=0,A[2]=1,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X5);FD53:=subs(A[1]=0,A[2]=0,A[3]=1,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X5);FD54:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=1,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X5);FD55:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=1,A[6]=0,A[7]=0,A[8]=0,X5);FD56:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=1,A[7]=0,A[8]=0,X5);FD57:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=1,A[8]=0,X5);FD58:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=1,X5);
 (31)
 > FD61:=subs(A[1]=1,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X6);FD62:=subs(A[1]=0,A[2]=1,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X6);FD63:=subs(A[1]=0,A[2]=0,A[3]=1,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X6);FD64:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=1,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X6);FD65:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=1,A[6]=0,A[7]=0,A[8]=0,X6);FD66:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=1,A[7]=0,A[8]=0,X6);FD67:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=1,A[8]=0,X6);FD68:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=1,X6);
 (32)
 >
 > FD71:=subs(A[1]=1,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X7);FD72:=subs(A[1]=0,A[2]=1,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X7);FD73:=subs(A[1]=0,A[2]=0,A[3]=1,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X7);FD74:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=1,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X7);FD75:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=1,A[6]=0,A[7]=0,A[8]=0,X7);FD76:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=1,A[7]=0,A[8]=0,X7);FD77:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=1,A[8]=0,X7);FD78:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=1,X7);
 (33)
 > FD81:=subs(A[1]=1,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X8);FD82:=subs(A[1]=0,A[2]=1,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X8);FD83:=subs(A[1]=0,A[2]=0,A[3]=1,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X8);FD84:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=1,A[5]=0,A[6]=0,A[7]=0,A[8]=0,X8);FD85:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=1,A[6]=0,A[7]=0,A[8]=0,X8);FD86:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=1,A[7]=0,A[8]=0,X8);FD87:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=1,A[8]=0,X8);FD88:=subs(A[1]=0,A[2]=0,A[3]=0,A[4]=0,A[5]=0,A[6]=0,A[7]=0,A[8]=1,X8);
 (34)
 >
 > MM:=matrix(8,8,[[FD11,FD12,FD13,FD14,FD15,FD16,FD17,FD18],[FD21,FD22,FD23,FD24,FD25,FD26,FD27,FD28],[FD31,FD32,FD33,FD34,FD35,FD36,FD37,FD38],[FD41,FD42,FD43,FD44,FD45,FD46,FD47,FD48],[FD51,FD52,FD53,FD54,FD55,FD56,FD57,FD58],[FD61,FD62,FD63,FD64,FD65,FD66,FD67,FD68],[FD71,FD72,FD73,FD74,FD75,FD76,FD77,FD78],[FD81,FD82,FD83,FD84,FD85,FD86,FD87,FD88]]);
 (35)

Program end

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