I am finding natural frequencies(w) of a beam. The matrix of 8*8 order, I need to find the w values for which the determinant of the matrix becomes zero?I gave the solution

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Mani S
Mani S 2021-9-21
评论： Walter Roberson 2021-9-21
syms w
a = 2e-3;
X = 98.4-3;
h = 20e-3;
b = 20e-3;
A = b*h;
l = 500e-3;
I = (b*(h^3))/12;
p = 2700;
E= 69e9;
v12 = 0.33;
L = ((w^2*p*A*l^4)/(E*I))^(1/4);
j = 0.016;
Kt = 5.375588477512108e+05;
%j = 1.8624*(a /h)^2 - 3.95*(a/h)^3 + 16.375*(a/h)^4 - 37.226*(a/h)^5 + 76.81*(a/h)^6 - 126.9*(a/h)^7+ 172*(a/h)^8 - 143.97*(a/h)^9 + 66.56*(a/h)^10
%Kt = ((E*I)/(6*(1-v12^2)*h))*(1/j)
Q = [1 0 1 0 0 0 0 0;...
0 1 0 1 0 0 0 0;...
cosh((L*X)/l) sinh((L*X)/l) cos((L*X)/l) sin((L*X)/l) -cosh((L*X)/l) -sinh( (L*X)/l) -cos((L*X)/l) -sin((L*X)/l) ;...
sinh((L*X)/l) cosh((L*X)/l) sin((L*X)/l) -cos((L*X)/l) -sinh((L*X)/l) -cosh((L*X)/l) -sin((L*X)/l) cos((L*X)/l) ;...
cosh((L*X)/l) sinh((L*X)/l) -cos((L*X)/l) -sin((L*X)/l) -cosh((L*X)/l) -sinh((L*X)/l) cos((L*X)/l) sin((L*X)/l);...
0 0 0 0 cosh(L) sinh(L) -cos(L) -sin(L);...
0 0 0 0 sinh(L) cosh(L) sin(L) -cos(L);...
cosh((L*X)/l)+((Kt*l)/(E*I*L))*sin((L*X)/l) sinh((L*X)/l)+((Kt*l)/(E*I*L))*cosh((L*X)/l) -cos((L*X)/l)-((Kt*l)/(E*I*L))*sin((L*X)/l) -sin((L*X)/l)+((Kt*l)/(E*I*L))*cos((L*X)/l) ((Kt*l)/(E*I*L))*sinh((L*X)/l) ((Kt*l)/(E*I*L))*cosh((L*X)/l) ((Kt*l)/(E*I*L))*sin((L*X)/l) ((Kt*l)/(E*I*L))*cos((L*X)/l)];
det(Q)
A = det(Q) ==0;
solve (A,w)
the solutions to this problem are
w1= 12.164Hz,
w2= 76.5471Hz,
w3= 214.6394Hz,
w4=420.4588Hz.
Could you please expain, how to get these values?

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