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

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.

Answer 1

[Walter Roberson]

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.

[Mani S]

Sir, that was same thing I found. I coudn't get correct solutions. what could be the reason? how to solve this problem?

[Walter Roberson]

syms w real
Z = @(v) sym(v);   %convert to symbolic number
a = Z(2e-3);
X = Z(150-3);
h = Z(20e-3);
b = Z(20e-3);
A = b*h;
l = Z(500e-3);
I = (b*(h^3))/12;
p = Z(2700);
E= Z(69e9);
v12 = Z(33)/100;
j = Z(16)/1000;
Kt = Z('5375588477512108')/1e10;
L = ((w^2*p*A*l^4)/(E*I))^(Z(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 = simplify(det(Q))

detQ = 

sol3 = vpasolve(detQ, -1)

sol3 = 

subs(detQ, w, sol3)

ans = 

fplot(detQ, [-1e-2 1e-2])

%if you look carefully at sigma_6, the negative and positive branches are
%the same
W = linspace(-10000, realmin, 200);
detQW = subs(detQ, w, W);
fake_for_plot = double(log10(abs(detQW)) .* sign(detQW));  %map log of negative to negative of log
plot(W, fake_for_plot)

This last plot does hint that there might not be any other zeros.

Note that the -2000 here does not mean 10^-2000: it means that the value was on the order of -10^2000 . It is not exactly a log plot, it is a sign-and-log plot .

[Mani S]

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]

format long g
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)

ans = 

A = det(Q);
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 -1.40928207189411e+56 0.00309918730337934 9.22212011300043e-46 76.5471 -9.56483609835847e+135 76.5471 0 214.6394 0 214.167971428571 -4.77344389972784e+281 420.4588 -Inf NaN NaN

fplot([A,0], expects(3)+[-1/100 1/100]); %ylim([-100 100])

vpa(subs(A, w, expects(3)-[1 1/10 1/100 1/1000]))

ans = 

[Walter Roberson]

Summary of the above: None of the positions where zeros are expected, are even close to reality, at least not for the equations you programmed.

I know the 214.6394 looks like it has produced a 0, but look at the plot and see that it cannot plot near there, and see the values when A is examined nearby. Either there is a quite high frequency cycle, or else the results are dominated by numeric noise.

[Mani S]

Respected walter Roberson, there were some mistakes,I have modified matrix. please try this time

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)
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.

[Walter Roberson]

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