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find the values of a variable for which the determinant of the matrix become zero?

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Mani S
Mani S on 16 Sep 2021
Commented: Walter Roberson on 22 Sep 2021 at 9:28
a = 2e-3;
X = 150-3;
h = 20e-3;
b = 20e-3;
A = b*h;
l = 500e-3;
I = (b*(h^3))/12;
p = 2700;
E= 69e9;
v12 = 0.33;
j = 0.016;
Kt = 5.375588477512108e+05;
L = ((w^2*p*A*l^4)/(E*I))^(1/4);
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,L)
here I am finding the natural frequencies of beam.
I need to find w values for which the determinant of Q become zero, there will be n number of w values.
please help me to find the solution
Thank you.

Accepted Answer

Walter Roberson
Walter Roberson on 16 Sep 2021
syms w real
a = 2e-3;
X = 150-3;
h = 20e-3;
b = 20e-3;
A = b*h;
l = 500e-3;
I = (b*(h^3))/12;
p = 2700;
E= 69e9;
v12 = 0.33;
j = 0.016;
Kt = 5.375588477512108e+05;
L = ((w^2*p*A*l^4)/(E*I))^(1/4);
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)];
detQ = det(Q)
detQ = 
sol3 = vpasolve(detQ, -1)
sol3 = 
subs(detQ, w, sol3)
ans = 
fplot(detQ, [-1e-2 1e-2])
sol1 = vpasolve(detQ, -1000)
sol1 = 
subs(detQ, w, sol1)
ans = 
0.0
vpa(subs(detQ, w, -1000))
ans = 
3.8828224224878183459368954124443e+1083
vpa(subs(diff(detQ,w),w,sol1))
ans = 
1.7469061892324894642718882345277e+1084
There might be additional solutions, but you can see that the slope is impossibly steep... I am not convinced that the solutin near -1000 is real: if you bump the digits up to 50, -1000.0 shows as the solution but substituting that back in shows a distinct non-zero value.
  7 Comments
Walter Roberson
Walter Roberson on 22 Sep 2021 at 9:28
No, none of the
format long g
syms w
a = 2e-3;
X = 98.4e-3;
h = 10e-3;
b = 20e-3;
A = b*h;
l = 820e-3;
I = (b*(h^3))/12;
p = 2700;
E= 72e9;
v12 = 0.33;
L = ((w^2*p*A*l^4)/(E*I))^(1/4);
j = 0.016;
%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)
Kt =
140276.06329256
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)
ans = 
A = det(Q);
%solve (A,w)
expects = [12.164, 76.5471, 214.6394, 420.4588] .';
digits(50)
for K = 1 : length(expects)
W = expects(K);
got(K,1) = subs(A, w, W);
sol = vpasolve(A, w, W);
if isempty(sol)
nearby(K,1) = nan;
val_nearby(K,1) = nan;
else
nearby(K,1) = sol;
val_nearby(K,1) = subs(A, w, sol);
end
end
double([expects, got, nearby, val_nearby])
ans = 4×4
12.164 -20185.0170574006 77.9685880444843 -1.44749574417154e-53 76.5471 -289.096868327054 77.9685880444843 -1.44749574417154e-53 214.6394 25085.7641362872 77.9685880444843 -1.44749574417154e-53 420.4588 22421.5359853579 491.06556340064 3.07614015113951e-53
fplot([A], expects(3)+[-1/100 1/100]); %ylim([-100 100])
testlocs = linspace(expects(3)-1/10, expects(3)+1/10, 50);
vals_at_test = double(subs(A, w, testlocs));
plot(testlocs, vals_at_test, '-*');
fplot(A, [1 2500])
Actual zeros: near 78, 491, 1300, 2600, possibly others

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