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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
differential-equations vibration transfer

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