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restart: with(LinearAlgebra):

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# Motion equation ( Vibration of a cracked composite beam using general solution) Edited by Adjal Yassine #

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Motion equation of flexural vibration in normalized form
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EI*W^(iv)m*omega^2*W=0;


(1) 
The general solution form of bending vibration equation
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W1:=A[1]*cosh(mu*x)+A[2]*sinh(mu*x)+A[3]*cos(mu*x)+A[4]*sin(mu*x);


(2) 
where
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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) 
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mu:=((m*omega^2*L^4/EI)^(1/4)):

Expression of crosssectional rotation , the bending moment shear force and torsional moment are given as follows respectively
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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) 
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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) 
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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) 
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W2:=A[5]*cosh(mu*x)+A[6]*sinh(mu*x)+A[7]*cos(mu*x)+A[8]*sin(mu*x);


(7) 
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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) 
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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) 
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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
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X1:=eval(subs(x=0,W1));


(11) 
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X2:=eval(subs(x=0,theta1));


(12) 
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X2:=collect(X2,mu)*(L/mu);


(13) 
The boundary condtions at free end M2(1)=S2(1)=0
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X3:=eval(subs(x=1,M2));


(14) 
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X3:=collect(X3,mu)*(L^2/mu^2/EI);


(15) 
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X4:=eval(subs(x=1,S2));


(16) 

(17) 
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X4:=collect(X4,EI)*(L^3/mu^3/EI);


(18) 
The additional boundary conditions at crack location

(19) 

(20) 
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X5:=collect(X5,EI)*(L^2/mu^2/EI);


(21) 

(22) 

(23) 
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X6:=collect(X6,EI)*(L^3/mu^3/EI);


(24) 
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X7:=combine(W2(W1+c8*S1));


(25) 
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X8:=combine (theta2(theta1+c44*M1));


(26) 
The characteristic matrix function of frequency
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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) 
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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) 
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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) 
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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) 
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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) 
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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) 
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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) 
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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) 
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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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