Hello, how to solve this equation E*I*k^4-m*​v^2*k^2+2*​m*v*w*k+(m​+M)w^2=0 numerically where w is variable,

can we solve this for k numerically, sorry this is fourth order equation not two order

Thanks

Answer 1

[Star Strider]

Supply all the scalar parameters, then:

Eqn = @(w) E*I*k^2-m*​v^2*k^2+2*​m*v*w*k+(m​+M)w^2;
w0 = 42;
[w,fval] = fsolve(Eqn, w0)

Experiment with the correct value of ‘w0’ to get the correct result.

EDIT — (Dec 10 2019 at 13:18)

The Symbolic Math Toolbox produces:

w = (k*(- E*I*k^2*m - E*I*M*k^2 + 2*m^2*v^2 + M*m*v^2)^(1/2) - k*m*v)/(M + m)

or to calculate both roots:

w = [(k*(- E*I*k^2*m - E*I*M*k^2 + 2*m^2*v^2 + M*m*v^2)^(1/2) - k*m*v)/(M + m)
     -(k*(- E*I*k^2*m - E*I*M*k^2 + 2*m^2*v^2 + M*m*v^2)^(1/2) - k*m*v)/(M + m)]

[shoaib Shoaib]

Thanks , Star stider how to solve for k in term of w

[Star Strider]

The Symbolic Math Toolbox does not provide a solution for this, even using vpa and a number of other aproaches.

There are four roots for each value of ‘w’ (I cannot determine if any are complex), so this will need to be iterated over different values of ‘k0’ to find all of them. Provide a value for ‘w’, then iterate:

Eqn = @(w,k) E*I*k^2-m*v^2*k^2+2*m*v*w*k+(m+M)*w^2;
k0 = 42;
w = 3.1;
[k,fval] = fsolve(@(k)Eqn(w,k), k0)

[shoaib Shoaib]

Thank you very much, i got a coplex root how to solve the complex roots?

[Star Strider]

My pleasure.

If my Answer helped you solve your problem, please Accept it!

[Walter Roberson]

syms E I k m v w M; eqn = E*I*k^2-m*v^2*k^2+2*m*v*w*k+(m+M)*w^2
solve(eqn,k)

[shoaib Shoaib]

E*I*k^4-m*v^2*k^2+2*m*v*w*k+(m+M)*w^2;

sorry the first term is E*I*k^4, not E*I*k^2, please how to solve the fourth order equ for k in term of w

[Star Strider]

Try this:

syms E I k m v w M
Eqn =  E*I*k^4-m*v^2*k^2+2*m*v*w*k+(m+M)*w^2
[cfs,trms] = coeffs(Eqn, k)
k_vec = [ E*I, -m*v^2, 2*m*v*w, w^2*(M + m)];

producing:

cfs =
[ E*I, -m*v^2, 2*m*v*w, w^2*(M + m)]
 
trms =
[ k^4, k^2, k, 1]

for the coefficient vector and the appropriate terms, respectively. (The ‘k^3’ term is absent.)

Create an anonymous function for the ‘cfs’ vector, and supply values for all the constants and specifically for the independent variable ‘w’:

k_roots = @(w) roots([ E*I, 0, -m*v^2, 2*m*v*w, w^2*(M + m)]);
w = 42;
k_val = k_roots(w)

and that should produce the roots.

Note that there is no ‘k^3’ term, so you need to insert a 0 as the second element in the ‘k_roots’ vector to get the correct result, as I did here.

(I tried the same approach last night and could not get this result. I have no idea what the difference was between then and now.)

[shoaib Shoaib]

Thank you so much

[Star Strider]

As always, my pleasure!

[Walter Roberson]

syms E I k m v w M
Eqn =  E*I*k^4-m*v^2*k^2+2*m*v*w*k+(m+M)*w^2

sol_exact = solve(Eqn, k, 'MaxDegree', 4);   %valid for symbolic variables, gives LONG exact solutions

sol_numeric = vpasolve(Eqn, k);   %valid only if numeric values are known for everything except k, gives numeric solutions

[shoaib Shoaib]

Thanks Walter Roberson