help solving polynomial equation

clc

clear

syms G J E I l b q Cla ru Ru U w

beta=((E*I)/(G*J));

landa=l/b;

K=((G*J)/l)*[ ((256*beta)/(landa^2)), 0 ((-80*beta)/(landa^2)) , 0;

0, 16/3 , 0 , -8/3;

((-80*beta)/(landa^2)), 0, ((28*beta)/(landa^2)), 0;

0, -8/3, 0, 7/3];

M=2*b^4*l*Ru*[64/105 4/15 -16/105 1/15

4/15 2/15 -1/15 1/60

-16/105 -1/15 4/105 -1/60

1/15 1/60 -1/60 1/30];

A=pi*b*(l/6)*[0 4*b 0 4*b

0 8 0 8

0 b 0 b

0 2 0 2];

B=pi*b*(l/6)*[-4*b 0 -4*b 0

-8 8*b -8 8*b

-b 0 -b 0

-2 2*b -2 2*b];

C=pi*b^2*(l/6)*[0 -b^2/2 0 -b^2/2

-4 0 -4 0

0 -b^2/8 0 -b^2/8

-1 0 -1 0];

Meq=M-ru*C;

Keq=K-ru*U^2*A;

Deq=-ru*U*B;

H=(-w^2)*Meq+(1i*w)*Deq+Keq;

v=det(H)

d=solve(v==0,w)

the above code didnt give the roots of w as function of U, pleas help why ?

Answer 1

[Torsten]

det(H) is a polynomial of degree 10 in w. Since the roots of polynomials of degree > 4 can't explicitly be given, you will have to use the numerical "roots" or the symbolic "root".

https://de.mathworks.com/help/symbolic/sym.root.html

https://de.mathworks.com/help/matlab/ref/roots.html

Before, you will have to give numerical values to all other symbolic variables involved.

[mammad dehghan]

i can give numerical values to other symbolic variables exept U i want roots in term of U.

[Torsten]

Then do after the line v = det(H)

vsubs = subs(v,G,...,J,...,E,...etc);
R = solve(vsubs,U);
Unum = vpa(R )