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

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?

[Walter Roberson]

I seem to recall that this is an eigenvalue problem.

https://www.brown.edu/Departments/Engineering/Courses/En4/Notes/vibrations_mdof/vibrations_mdof.htm