Adjal yassine

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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 ''
 

restart; with(LinearAlgebra)NULLNULL

W11 := A[1]*cos(nu*x)+A[2]*sin(nu*x)+A[3]*cosh(nu*x)+A[4]*sinh(nu*x);

theta11 := -A[1]*nu*sin(nu*x)+A[2]*nu*cos(nu*x)+A[3]*nu*sinh(nu*x)+A[4]*nu*cosh(nu*x)  NULL

M11 := EI*(-A[1]*nu^2*cos(nu*x)-A[2]*nu^2*sin(nu*x)+A[3]*nu^2*cosh(nu*x)+A[4]*nu^2*sinh(nu*x))
S11 := EI*(A[1]*nu^3*sin(nu*x)-A[2]*nu^3*cos(nu*x)+A[3]*nu^3*sinh(nu*x)+A[4]*nu^3*cosh(nu*x))
 

MD11 := subs(A[1] = 1, A[2] = 0, A[3] = 0, A[4] = 0, W11); MD12 := subs(A[1] = 0, A[2] = 1, A[3] = 0, A[4] = 0, W11); MD13 := subs(A[1] = 0, A[2] = 0, A[3] = 1, A[4] = 0, W11); MD14 := subs(A[1] = 0, A[2] = 0, A[3] = 0, A[4] = 1, W11)
NULL

MD21 := subs(A[1] = 1, A[2] = 0, A[3] = 0, A[4] = 0, theta11); MD22 := subs(A[1] = 0, A[2] = 1, A[3] = 0, A[4] = 0, theta11); MD23 := subs(A[1] = 0, A[2] = 0, A[3] = 1, A[4] = 0, theta11); MD24 := subs(A[1] = 0, A[2] = 0, A[3] = 0, A[4] = 1, theta11)
 

MD31 := subs(A[1] = 1, A[2] = 0, A[3] = 0, A[4] = 0, M11); MD32 := subs(A[1] = 0, A[2] = 1, A[3] = 0, A[4] = 0, M11); MD33 := subs(A[1] = 0, A[2] = 0, A[3] = 1, A[4] = 0, M11); MD34 := subs(A[1] = 0, A[2] = 0, A[3] = 0, A[4] = 1, M11)
 

MD41 := subs(A[1] = 1, A[2] = 0, A[3] = 0, A[4] = 0, S11); MD42 := subs(A[1] = 0, A[2] = 1, A[3] = 0, A[4] = 0, S11); MD43 := subs(A[1] = 0, A[2] = 0, A[3] = 1, A[4] = 0, S11); MD44 := subs(A[1] = 0, A[2] = 0, A[3] = 0, A[4] = 1, S11)
 

 

TM := Matrix(4, 4, [[MD11, MD12, MD13, MD14], [MD21, MD22, MD23, MD24], [MD31, MD32, MD33, MD34], [MD41, MD42, MD43, MD44]])

C := Matrix(4, 4, [[1, 0, 0, 0], [0, 1, c44, 0], [0, 0, 1, 0], [0, 0, 0, 1]])NULL

NULL    TM2 := subs(x = 0, TM); TM3 := subs(x = L, TM)

with(MTM)

TM4 := inv(TM)NULLNULL 

TM5 := inv(TM2)NULL

Y11 := MatrixMatrixMultiply(TM3, TM4)

    Y22 := MatrixMatrixMultiply(C, TM)   

Y33 := MatrixMatrixMultiply(Y11, Y22)

Y44 := MatrixMatrixMultiply(Y33, TM5)NULL

BB11 := Y44[3, 3]

BB12 := Y44[3, 4]

BB21 := Y44[4, 3]

BB22 := Y44[4, 4] 

BB := Matrix(2, 2, [[BB11, BB12], [BB21, BB22]]) 

NULL

R11 := det(BB) 

NULL

L := .18; L1 := 1; h := 0.5e-2; b := 0.2e-1; rho := 957.5; area = b.h; m := rho*h*b; EI := 0.2682e10*b*h^3mu := ((m.(omega^2))*L^4/EI)^(1/4); x := .5; c44 := .1; c11 := 0NULLNULL

plot(R11, omega = 1 .. 100)

 

 

 

``

NULL

NULL


 

Download transfer.mw


 

with(LinearAlgebra); restart

v1 := x^3*a[3]+x^2*a[2]+x*a[1]+a[0]:

r := diff(v1, x):

v[i] := subs(x = 0, v1):

theta[i] := subs(x = 0, r):

v[j] := subs(x = L, v1):

theta[j] := subs(x = L, r):

MD11 := subs(a[0] = 1, a[1] = 0, a[2] = 0, a[3] = 0, v[i]):

MS11 := subs(a[0] = 0, a[1] = 0, a[2] = 0, a[3] = 0, theta[i]):

MT11 := subs(a[0] = 1, a[1] = 0, a[2] = 0, a[3] = 0, v[j]):

MR11 := subs(a[0] = 1, a[1] = 0, a[2] = 0, a[3] = 0, theta[j]):

TM := Matrix(4, 4, [[MD11, MD12, MD13, MD14], [MS11, MS12, MS13, MS14], [MT11, MT12, MT13, MT14], [MR11, MR12, MR13, MR14]]):

with(MTM):

TM1 := inv(TM):

b := Typesetting:-delayDotProduct(TM1, c):

c := `<,>`(v[1], theta[1], v[2], theta[2]):

a[0] := b(1):

a[1] := b(2):

a[2] := b(3):

a[3] := b(4):

vshape := x^3*a[3]+x^2*a[2]+x*a[1]+a[0]:

XX1 := collect(vshape, v[1]):

XX2 := collect(XX1, theta[1]):

XX3 := collect(XX2, v[2]):

XX4 := collect(XX3, theta[2]):

v[1] := 1:

NN1 := XX4:

v[1] := 0:

NN2 := XX4:

v[1] := 0:

NN3 := XX4:

v[1] := 0:

NN4 := XX4:

Mat := `<,>`([NN1, NN2, NN3, NN4]):

Mat1 := diff(Mat, x):

Mat2 := diff(Mat1, x):

segma := Mat2:

KK := EI*expand(Typesetting:-delayDotProduct(segma, transpose(segma))):

KG := int(KK, x = 0 .. L)

KG := Matrix(4, 4, {(1, 1) = 12*EI/L^3, (1, 2) = 6*EI/L^2, (1, 3) = -12*EI/L^3, (1, 4) = 6*EI/L^2, (2, 1) = 6*EI/L^2, (2, 2) = 4*EI/L, (2, 3) = -6*EI/L^2, (2, 4) = 2*EI/L, (3, 1) = -12*EI/L^3, (3, 2) = -6*EI/L^2, (3, 3) = 12*EI/L^3, (3, 4) = -6*EI/L^2, (4, 1) = 6*EI/L^2, (4, 2) = 2*EI/L, (4, 3) = -6*EI/L^2, (4, 4) = 4*EI/L})

(1)

``

NULL


 

Download stiffnesmatrice.mw

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


 

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;

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);

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:

0.2682e10

 

.18

 

0.4e-2

 

0.2e-1

 

2600

 

area = 0.8e-4

 

.20800

(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));

(A[1]*mu*sinh(mu*x)+A[2]*mu*cosh(mu*x)-A[3]*mu*sin(mu*x)+A[4]*mu*cos(mu*x))/L

(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));

EI*(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))/L^2

(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));

-EI*(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))/L^3

(6)

 

W2:=A[5]*cosh(mu*x)+A[6]*sinh(mu*x)+A[7]*cos(mu*x)+A[8]*sin(mu*x);

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));

(A[5]*mu*sinh(mu*x)+A[6]*mu*cosh(mu*x)-A[7]*mu*sin(mu*x)+A[8]*mu*cos(mu*x))/L

(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));

EI*(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))/L^2

(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));

-EI*(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))/L^3

(10)

 

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

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

A[1]+A[3]

(11)

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

(mu*A[2]+mu*A[4])/L

(12)

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

A[2]+A[4]

(13)

 

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

X3:=eval(subs(x=1,M2));

EI*(A[5]*mu^2*cosh(mu)+A[6]*mu^2*sinh(mu)-A[7]*mu^2*cos(mu)-A[8]*mu^2*sin(mu))/L^2

(14)

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

cosh(mu)*A[5]+sinh(mu)*A[6]-cos(mu)*A[7]-sin(mu)*A[8]

(15)

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

-EI*(A[5]*mu^3*sinh(mu)+A[6]*mu^3*cosh(mu)+A[7]*mu^3*sin(mu)-A[8]*mu^3*cos(mu))/L^3

(16)

X4:=collect(X4,mu);

-EI*(cosh(mu)*A[6]+sinh(mu)*A[5]-cos(mu)*A[8]+sin(mu)*A[7])*mu^3/L^3

(17)

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

-cosh(mu)*A[6]-sinh(mu)*A[5]+cos(mu)*A[8]-sin(mu)*A[7]

(18)

 

The additional boundary conditions at crack location

X5:=combine(M1-M2);

(EI*cosh(mu*x)*mu^2*A[1]-EI*cosh(mu*x)*mu^2*A[5]+EI*sinh(mu*x)*mu^2*A[2]-EI*sinh(mu*x)*mu^2*A[6]-EI*cos(mu*x)*mu^2*A[3]+EI*cos(mu*x)*mu^2*A[7]-EI*sin(mu*x)*mu^2*A[4]+EI*sin(mu*x)*mu^2*A[8])/L^2

(19)

X5:=collect(X5,mu);

(EI*cosh(mu*x)*A[1]-EI*cosh(mu*x)*A[5]+EI*sinh(mu*x)*A[2]-EI*sinh(mu*x)*A[6]-cos(mu*x)*EI*A[3]+A[7]*cos(mu*x)*EI-A[4]*sin(mu*x)*EI+A[8]*sin(mu*x)*EI)*mu^2/L^2

(20)

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

A[1]*cosh(mu*x)-A[5]*cosh(mu*x)+A[2]*sinh(mu*x)-A[6]*sinh(mu*x)-A[3]*cos(mu*x)+A[7]*cos(mu*x)-A[4]*sin(mu*x)+A[8]*sin(mu*x)

(21)

X6:=combine(S1-S2);

(-EI*cosh(mu*x)*mu^3*A[2]+EI*cosh(mu*x)*mu^3*A[6]-EI*sinh(mu*x)*mu^3*A[1]+EI*sinh(mu*x)*mu^3*A[5]+EI*cos(mu*x)*mu^3*A[4]-EI*cos(mu*x)*mu^3*A[8]-EI*sin(mu*x)*mu^3*A[3]+EI*sin(mu*x)*mu^3*A[7])/L^3

(22)

X6:=collect(X6,mu);

(-EI*cosh(mu*x)*A[2]+EI*cosh(mu*x)*A[6]-EI*sinh(mu*x)*A[1]+EI*A[5]*sinh(mu*x)+cos(mu*x)*A[4]*EI-cos(mu*x)*A[8]*EI-sin(mu*x)*EI*A[3]+sin(mu*x)*A[7]*EI)*mu^3/L^3

(23)

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

-cosh(mu*x)*A[2]+cosh(mu*x)*A[6]-sinh(mu*x)*A[1]+sinh(mu*x)*A[5]+cos(mu*x)*A[4]-cos(mu*x)*A[8]-sin(mu*x)*A[3]+sin(mu*x)*A[7]

(24)

 

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

(EI*cosh(mu*x)*c8*mu^3*A[2]+EI*sinh(mu*x)*c8*mu^3*A[1]-EI*cos(mu*x)*c8*mu^3*A[4]+EI*sin(mu*x)*c8*mu^3*A[3]-cosh(mu*x)*A[1]*L^3+cosh(mu*x)*A[5]*L^3-sinh(mu*x)*A[2]*L^3+sinh(mu*x)*A[6]*L^3-cos(mu*x)*A[3]*L^3+cos(mu*x)*A[7]*L^3-sin(mu*x)*A[4]*L^3+sin(mu*x)*A[8]*L^3)/L^3

(25)

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

(-EI*cosh(mu*x)*c44*mu^2*A[1]-EI*sinh(mu*x)*c44*mu^2*A[2]+EI*cos(mu*x)*c44*mu^2*A[3]+EI*sin(mu*x)*c44*mu^2*A[4]-L*cosh(mu*x)*mu*A[2]+L*cosh(mu*x)*mu*A[6]-L*sinh(mu*x)*mu*A[1]+L*sinh(mu*x)*mu*A[5]-L*cos(mu*x)*mu*A[4]+L*cos(mu*x)*mu*A[8]+L*sin(mu*x)*mu*A[3]-L*sin(mu*x)*mu*A[7])/L^2

(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);

1

 

0

 

1

 

0

 

0

 

0

 

0

 

0

(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);

0

 

1

 

0

 

1

 

0

 

0

 

0

 

0

(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);

0

 

0

 

0

 

0

 

cosh(mu)

 

sinh(mu)

 

-cos(mu)

 

-sin(mu)

(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);

0

 

0

 

0

 

0

 

-sinh(mu)

 

-cosh(mu)

 

-sin(mu)

 

cos(mu)

(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);

cosh(mu*x)

 

sinh(mu*x)

 

-cos(mu*x)

 

-sin(mu*x)

 

-cosh(mu*x)

 

-sinh(mu*x)

 

cos(mu*x)

 

sin(mu*x)

(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);

-sinh(mu*x)

 

-cosh(mu*x)

 

-sin(mu*x)

 

cos(mu*x)

 

sinh(mu*x)

 

cosh(mu*x)

 

sin(mu*x)

 

-cos(mu*x)

(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);

(EI*sinh(mu*x)*c8*mu^3-cosh(mu*x)*L^3)/L^3

 

(EI*cosh(mu*x)*c8*mu^3-sinh(mu*x)*L^3)/L^3

 

(EI*sin(mu*x)*c8*mu^3-L^3*cos(mu*x))/L^3

 

(-EI*cos(mu*x)*c8*mu^3-sin(mu*x)*L^3)/L^3

 

cosh(mu*x)

 

sinh(mu*x)

 

cos(mu*x)

 

sin(mu*x)

(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);

(-EI*cosh(mu*x)*c44*mu^2-L*sinh(mu*x)*mu)/L^2

 

(-EI*sinh(mu*x)*c44*mu^2-L*cosh(mu*x)*mu)/L^2

 

(EI*cos(mu*x)*c44*mu^2+L*sin(mu*x)*mu)/L^2

 

(EI*sin(mu*x)*c44*mu^2-L*cos(mu*x)*mu)/L^2

 

sinh(mu*x)*mu/L

 

cosh(mu*x)*mu/L

 

-sin(mu*x)*mu/L

 

cos(mu*x)*mu/L

(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]]);

MM := Matrix(8, 8, {(1, 1) = FD11, (1, 2) = 0, (1, 3) = 1, (1, 4) = 0, (1, 5) = 0, (1, 6) = 0, (1, 7) = 0, (1, 8) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (2, 4) = 1, (2, 5) = 0, (2, 6) = 0, (2, 7) = 0, (2, 8) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (3, 5) = cosh(mu), (3, 6) = sinh(mu), (3, 7) = -cos(mu), (3, 8) = -sin(mu), (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 0, (4, 5) = -sinh(mu), (4, 6) = -cosh(mu), (4, 7) = -sin(mu), (4, 8) = cos(mu), (5, 1) = cosh(mu*x), (5, 2) = sinh(mu*x), (5, 3) = -cos(mu*x), (5, 4) = -sin(mu*x), (5, 5) = -cosh(mu*x), (5, 6) = -sinh(mu*x), (5, 7) = cos(mu*x), (5, 8) = sin(mu*x), (6, 1) = -sinh(mu*x), (6, 2) = -cosh(mu*x), (6, 3) = -sin(mu*x), (6, 4) = cos(mu*x), (6, 5) = sinh(mu*x), (6, 6) = cosh(mu*x), (6, 7) = sin(mu*x), (6, 8) = -cos(mu*x), (7, 1) = (EI*sinh(mu*x)*c8*mu^3-cosh(mu*x)*L^3)/L^3, (7, 2) = (EI*cosh(mu*x)*c8*mu^3-sinh(mu*x)*L^3)/L^3, (7, 3) = (EI*sin(mu*x)*c8*mu^3-L^3*cos(mu*x))/L^3, (7, 4) = (-EI*cos(mu*x)*c8*mu^3-sin(mu*x)*L^3)/L^3, (7, 5) = cosh(mu*x), (7, 6) = sinh(mu*x), (7, 7) = cos(mu*x), (7, 8) = sin(mu*x), (8, 1) = (-EI*cosh(mu*x)*c44*mu^2-L*sinh(mu*x)*mu)/L^2, (8, 2) = (-EI*sinh(mu*x)*c44*mu^2-L*cosh(mu*x)*mu)/L^2, (8, 3) = (EI*cos(mu*x)*c44*mu^2+L*sin(mu*x)*mu)/L^2, (8, 4) = (EI*sin(mu*x)*c44*mu^2-L*cos(mu*x)*mu)/L^2, (8, 5) = sinh(mu*x)*mu/L, (8, 6) = cosh(mu*x)*mu/L, (8, 7) = -sin(mu*x)*mu/L, (8, 8) = cos(mu*x)*mu/L})

(35)

Program end

 

NULL

 

``


 

Download Vibration_of_a_cracked_composite_beam.mw
 

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;

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);

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:

0.2682e10

 

.18

 

0.4e-2

 

0.2e-1

 

2600

 

area = 0.8e-4

 

.20800

(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));

(A[1]*mu*sinh(mu*x)+A[2]*mu*cosh(mu*x)-A[3]*mu*sin(mu*x)+A[4]*mu*cos(mu*x))/L

(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));

EI*(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))/L^2

(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));

-EI*(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))/L^3

(6)

 

W2:=A[5]*cosh(mu*x)+A[6]*sinh(mu*x)+A[7]*cos(mu*x)+A[8]*sin(mu*x);

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));

(A[5]*mu*sinh(mu*x)+A[6]*mu*cosh(mu*x)-A[7]*mu*sin(mu*x)+A[8]*mu*cos(mu*x))/L

(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));

EI*(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))/L^2

(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));

-EI*(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))/L^3

(10)

 

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

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

A[1]+A[3]

(11)

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

(mu*A[2]+mu*A[4])/L

(12)

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

A[2]+A[4]

(13)

 

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

X3:=eval(subs(x=1,M2));

EI*(A[5]*mu^2*cosh(mu)+A[6]*mu^2*sinh(mu)-A[7]*mu^2*cos(mu)-A[8]*mu^2*sin(mu))/L^2

(14)

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

cosh(mu)*A[5]+sinh(mu)*A[6]-cos(mu)*A[7]-sin(mu)*A[8]

(15)

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

-EI*(A[5]*mu^3*sinh(mu)+A[6]*mu^3*cosh(mu)+A[7]*mu^3*sin(mu)-A[8]*mu^3*cos(mu))/L^3

(16)

X4:=collect(X4,mu);

-EI*(cosh(mu)*A[6]+sinh(mu)*A[5]-cos(mu)*A[8]+sin(mu)*A[7])*mu^3/L^3

(17)

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

-cosh(mu)*A[6]-sinh(mu)*A[5]+cos(mu)*A[8]-sin(mu)*A[7]

(18)

 

The additional boundary conditions at crack location

X5:=combine(M1-M2);

(EI*cosh(mu*x)*mu^2*A[1]-EI*cosh(mu*x)*mu^2*A[5]+EI*sinh(mu*x)*mu^2*A[2]-EI*sinh(mu*x)*mu^2*A[6]-EI*cos(mu*x)*mu^2*A[3]+EI*cos(mu*x)*mu^2*A[7]-EI*sin(mu*x)*mu^2*A[4]+EI*sin(mu*x)*mu^2*A[8])/L^2

(19)

X5:=collect(X5,mu);

(EI*cosh(mu*x)*A[1]-EI*cosh(mu*x)*A[5]+EI*sinh(mu*x)*A[2]-EI*sinh(mu*x)*A[6]-cos(mu*x)*EI*A[3]+A[7]*cos(mu*x)*EI-A[4]*sin(mu*x)*EI+A[8]*sin(mu*x)*EI)*mu^2/L^2

(20)

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

A[1]*cosh(mu*x)-A[5]*cosh(mu*x)+A[2]*sinh(mu*x)-A[6]*sinh(mu*x)-A[3]*cos(mu*x)+A[7]*cos(mu*x)-A[4]*sin(mu*x)+A[8]*sin(mu*x)

(21)

X6:=combine(S1-S2);

(-EI*cosh(mu*x)*mu^3*A[2]+EI*cosh(mu*x)*mu^3*A[6]-EI*sinh(mu*x)*mu^3*A[1]+EI*sinh(mu*x)*mu^3*A[5]+EI*cos(mu*x)*mu^3*A[4]-EI*cos(mu*x)*mu^3*A[8]-EI*sin(mu*x)*mu^3*A[3]+EI*sin(mu*x)*mu^3*A[7])/L^3

(22)

X6:=collect(X6,mu);

(-EI*cosh(mu*x)*A[2]+EI*cosh(mu*x)*A[6]-EI*sinh(mu*x)*A[1]+EI*A[5]*sinh(mu*x)+cos(mu*x)*A[4]*EI-cos(mu*x)*A[8]*EI-sin(mu*x)*EI*A[3]+sin(mu*x)*A[7]*EI)*mu^3/L^3

(23)

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

-cosh(mu*x)*A[2]+cosh(mu*x)*A[6]-sinh(mu*x)*A[1]+sinh(mu*x)*A[5]+cos(mu*x)*A[4]-cos(mu*x)*A[8]-sin(mu*x)*A[3]+sin(mu*x)*A[7]

(24)

 

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

(EI*cosh(mu*x)*c8*mu^3*A[2]+EI*sinh(mu*x)*c8*mu^3*A[1]-EI*cos(mu*x)*c8*mu^3*A[4]+EI*sin(mu*x)*c8*mu^3*A[3]-cosh(mu*x)*A[1]*L^3+cosh(mu*x)*A[5]*L^3-sinh(mu*x)*A[2]*L^3+sinh(mu*x)*A[6]*L^3-cos(mu*x)*A[3]*L^3+cos(mu*x)*A[7]*L^3-sin(mu*x)*A[4]*L^3+sin(mu*x)*A[8]*L^3)/L^3

(25)

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

(-EI*cosh(mu*x)*c44*mu^2*A[1]-EI*sinh(mu*x)*c44*mu^2*A[2]+EI*cos(mu*x)*c44*mu^2*A[3]+EI*sin(mu*x)*c44*mu^2*A[4]-L*cosh(mu*x)*mu*A[2]+L*cosh(mu*x)*mu*A[6]-L*sinh(mu*x)*mu*A[1]+L*sinh(mu*x)*mu*A[5]-L*cos(mu*x)*mu*A[4]+L*cos(mu*x)*mu*A[8]+L*sin(mu*x)*mu*A[3]-L*sin(mu*x)*mu*A[7])/L^2

(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);

1

 

0

 

1

 

0

 

0

 

0

 

0

 

0

(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);

0

 

1

 

0

 

1

 

0

 

0

 

0

 

0

(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);

0

 

0

 

0

 

0

 

cosh(mu)

 

sinh(mu)

 

-cos(mu)

 

-sin(mu)

(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);

0

 

0

 

0

 

0

 

-sinh(mu)

 

-cosh(mu)

 

-sin(mu)

 

cos(mu)

(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);

cosh(mu*x)

 

sinh(mu*x)

 

-cos(mu*x)

 

-sin(mu*x)

 

-cosh(mu*x)

 

-sinh(mu*x)

 

cos(mu*x)

 

sin(mu*x)

(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);

-sinh(mu*x)

 

-cosh(mu*x)

 

-sin(mu*x)

 

cos(mu*x)

 

sinh(mu*x)

 

cosh(mu*x)

 

sin(mu*x)

 

-cos(mu*x)

(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);

(EI*sinh(mu*x)*c8*mu^3-cosh(mu*x)*L^3)/L^3

 

(EI*cosh(mu*x)*c8*mu^3-sinh(mu*x)*L^3)/L^3

 

(EI*sin(mu*x)*c8*mu^3-L^3*cos(mu*x))/L^3

 

(-EI*cos(mu*x)*c8*mu^3-sin(mu*x)*L^3)/L^3

 

cosh(mu*x)

 

sinh(mu*x)

 

cos(mu*x)

 

sin(mu*x)

(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);

(-EI*cosh(mu*x)*c44*mu^2-L*sinh(mu*x)*mu)/L^2

 

(-EI*sinh(mu*x)*c44*mu^2-L*cosh(mu*x)*mu)/L^2

 

(EI*cos(mu*x)*c44*mu^2+L*sin(mu*x)*mu)/L^2

 

(EI*sin(mu*x)*c44*mu^2-L*cos(mu*x)*mu)/L^2

 

sinh(mu*x)*mu/L

 

cosh(mu*x)*mu/L

 

-sin(mu*x)*mu/L

 

cos(mu*x)*mu/L

(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]]);

MM := Matrix(8, 8, {(1, 1) = FD11, (1, 2) = 0, (1, 3) = 1, (1, 4) = 0, (1, 5) = 0, (1, 6) = 0, (1, 7) = 0, (1, 8) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (2, 4) = 1, (2, 5) = 0, (2, 6) = 0, (2, 7) = 0, (2, 8) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (3, 5) = cosh(mu), (3, 6) = sinh(mu), (3, 7) = -cos(mu), (3, 8) = -sin(mu), (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 0, (4, 5) = -sinh(mu), (4, 6) = -cosh(mu), (4, 7) = -sin(mu), (4, 8) = cos(mu), (5, 1) = cosh(mu*x), (5, 2) = sinh(mu*x), (5, 3) = -cos(mu*x), (5, 4) = -sin(mu*x), (5, 5) = -cosh(mu*x), (5, 6) = -sinh(mu*x), (5, 7) = cos(mu*x), (5, 8) = sin(mu*x), (6, 1) = -sinh(mu*x), (6, 2) = -cosh(mu*x), (6, 3) = -sin(mu*x), (6, 4) = cos(mu*x), (6, 5) = sinh(mu*x), (6, 6) = cosh(mu*x), (6, 7) = sin(mu*x), (6, 8) = -cos(mu*x), (7, 1) = (EI*sinh(mu*x)*c8*mu^3-cosh(mu*x)*L^3)/L^3, (7, 2) = (EI*cosh(mu*x)*c8*mu^3-sinh(mu*x)*L^3)/L^3, (7, 3) = (EI*sin(mu*x)*c8*mu^3-L^3*cos(mu*x))/L^3, (7, 4) = (-EI*cos(mu*x)*c8*mu^3-sin(mu*x)*L^3)/L^3, (7, 5) = cosh(mu*x), (7, 6) = sinh(mu*x), (7, 7) = cos(mu*x), (7, 8) = sin(mu*x), (8, 1) = (-EI*cosh(mu*x)*c44*mu^2-L*sinh(mu*x)*mu)/L^2, (8, 2) = (-EI*sinh(mu*x)*c44*mu^2-L*cosh(mu*x)*mu)/L^2, (8, 3) = (EI*cos(mu*x)*c44*mu^2+L*sin(mu*x)*mu)/L^2, (8, 4) = (EI*sin(mu*x)*c44*mu^2-L*cos(mu*x)*mu)/L^2, (8, 5) = sinh(mu*x)*mu/L, (8, 6) = cosh(mu*x)*mu/L, (8, 7) = -sin(mu*x)*mu/L, (8, 8) = cos(mu*x)*mu/L})

(35)

Program end

 

NULL

 

``


 

Download Vibration_of_a_cracked_composite_beam.mwVibration_of_a_cracked_composite_beam.mwVibration_of_a_cracked_composite_beam.mw

 

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