First five natural frequency of cracked nanoarch

Greetings,

I am working on a project ''Vibration of nanoarch''. I am solving an algebaic equation having one variable. The equation has infinite many solutions. I am interested in getting first five positive solutions. I can get the first positive solution. Now I have to calculate the next four. Any guidance in this regard will be much appreciated.

clear all;
syms logw;
w=exp(logw);
E=10^7; ro=0.1;h=0.01289;b=0.1016;
I=b*h^3/12;
Z=b*h;
U=ro*h;
q=0;R=30;a=0;B=1;p=0;
K=14.859*w^2%(U*R^4*w^2)/E*I; %U=mu
A=sqrt(220.789881*(q)^2*w^4+59.436*q*w^2+59.436*w^2);%sqrt((K*q)^2+4*K*q+4*K);%
u=sqrt((-2-K*q+A)/2);
v=sqrt((2+K*q+A)/2)*1i;
K11=sinh(u*a);
K12=sin(v*a);
K13=-sinh(u*(a-B));
K14=-sin(v*(a-B));
K21=u*cosh(u*a);
K22=v*cos(v*a);
K23=-u*cosh(u*a)-p*(u^2+1)*(sinh(u*(a-B)))-u*cosh(v*(a-B));
K24=-v*cos(v*a)+p*(1-v^2)*(sin(v*(a-B)))-v*cos(v*(a-B));
K31=u^2*sinh(u*a);
K32=-v^2*sin(v*a);
K33=-u^2*(sinh(u*(a-B)));
K34=-v^2*sin(v*(a-B));
K41=u^3*cosh(u*a);
K42=-v^3*cos(v*a);
K43=-u^3*(cosh(u*(a-B)));
K44=v^3*(cos(v*(a-B)));
Matrix=[K11 K12 K13 K14;K21 K22 K23 K24;K31 K32 K33 K34;K41 K42 K43 K44];
makta=det(Matrix);
F1=simplify(makta);
F=matlabFunction(F1);
logw1=[];
for j=-10:10
logww=fzero(F,j);           
logw1=[logw1;logww];
end
w1=exp(logw1);
wn=min(w1(w1>0))
chiang1=wn*sqrt((ro*Z*R^4)/E*I)

Answer 1

[Stephan]

Try to keep working symbolic, by using solve:

syms logw k;
w=exp(logw);
E=10^7; ro=0.1;h=0.01289;b=0.1016;
I=b*h^3/12;
Z=b*h;
U=ro*h;
q=0;R=30;a=0;B=1;p=0;
K=14.859*w^2;%(U*R^4*w^2)/E*I; %U=mu
A=sqrt(220.789881*(q)^2*w^4+59.436*q*w^2+59.436*w^2);%sqrt((K*q)^2+4*K*q+4*K);%
u=sqrt((-2-K*q+A)/2);
v=sqrt((2+K*q+A)/2)*1i;
K11=sinh(u*a);
K12=sin(v*a);
K13=-sinh(u*(a-B));
K14=-sin(v*(a-B));
K21=u*cosh(u*a);
K22=v*cos(v*a);
K23=-u*cosh(u*a)-p*(u^2+1)*(sinh(u*(a-B)))-u*cosh(v*(a-B));
K24=-v*cos(v*a)+p*(1-v^2)*(sin(v*(a-B)))-v*cos(v*(a-B));
K31=u^2*sinh(u*a);
K32=-v^2*sin(v*a);
K33=-u^2*(sinh(u*(a-B)));
K34=-v^2*sin(v*(a-B));
K41=u^3*cosh(u*a);
K42=-v^3*cos(v*a);
K43=-u^3*(cosh(u*(a-B)));
K44=v^3*(cos(v*(a-B)));
Matrix=[K11 K12 K13 K14;K21 K22 K23 K24;K31 K32 K33 K34;K41 K42 K43 K44];
makta=det(Matrix);
F1=simplify(makta)

% See the link to the documentation
[sol,parameters,conditions] = solve(F1,logw,'ReturnConditions',true)
assume(conditions)
restriction = [k > 0, k < 6];
solk = solve(restriction,parameters);
valx = subs(sol,parameters,solk)

[shahid mubasshar]

Thank you so much Stephen for editing the code and sending the link. I get the the complex result by using the code you sent. Moreover, when I change some parameters which I had put zero initially, the code doesn't give me explicit solution. Now I am putting p, a, q greater than zero.

syms logw k;
w=exp(logw);
E=10^7; ro=0.1;h=0.01289;b=0.1016;
I=b*h^3/12;
Z=b*h;
U=ro*h;
q=1;R=30;a=0.4;B=1.5;p=0.3424;
K=14.859*w^2;%(U*R^4*w^2)/E*I; %U=mu
A=sqrt(220.789881*(q)^2*w^4+59.436*q*w^2+59.436*w^2);%sqrt((K*q)^2+4*K*q+4*K);%
u=sqrt((-2-K*q+A)/2);
v=sqrt((2+K*q+A)/2)*1i;
K11=sinh(u*a);
K12=sin(v*a);
K13=-sinh(u*(a-B));
K14=-sin(v*(a-B));
K21=u*cosh(u*a);
K22=v*cos(v*a);
K23=-u*cosh(u*a)-p*(u^2+1)*(sinh(u*(a-B)))-u*cosh(v*(a-B));
K24=-v*cos(v*a)+p*(1-v^2)*(sin(v*(a-B)))-v*cos(v*(a-B));
K31=u^2*sinh(u*a);
K32=-v^2*sin(v*a);
K33=-u^2*(sinh(u*(a-B)));
K34=-v^2*sin(v*(a-B));
K41=u^3*cosh(u*a);
K42=-v^3*cos(v*a);
K43=-u^3*(cosh(u*(a-B)));
K44=v^3*(cos(v*(a-B)));
Matrix=[K11 K12 K13 K14;K21 K22 K23 K24;K31 K32 K33 K34;K41 K42 K43 K44];
makta=det(Matrix);
F1=simplify(makta)

% See the link to the documentation
[sol,parameters,conditions] = solve(F1,logw,'ReturnConditions',true)
assume(conditions)
restriction = [k > 0, k < 6];
solk = solve(restriction,parameters);
valx = subs(sol,parameters,solk)