# Find Eigenvalues of ODE45 Solution MATLAB

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Jonathan Frutschy il 9 Apr 2024
I have the following non-linear ODE:
I have the following ODE45 solution:
fun = @(t,X)odefun(X,K,C,M,F(t),resSize);
The input matrices are stiffness K(X), damping C, mass M, and force F. resSize is the total number of masses in the system.
I would like to find the system's eigenvalues using either the Jacobian matrix, transfer function, or any other viable method.
I have tried using:
This is tricky since my K matrix is a function handle of X. In other words, K=@(X). X represents a displacement vector of each mass in the system (x_1(t),x_2(t),...x_resSize(t)), where resSize is the total number of masses. My X_answer matrix is a double with dimensions of t_ode by resSize, where each row is the displacement vector of each mass in double form. Is there some way to substitute X_answer into my function handle for K so I can use polyeig()? If not, how would I go about finding my system's transfer function or Jacobian matrix so that I can find it's eigenvalues?
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Sam Chak il 2 Mag 2024
The input force consists of the sum of a slower wave and a faster wave.
I'm curious about the scientific basis or any journal paper that describes the method of determining the system's resonance by computing the state-dependent eigenvalues at each time step. Is this related to the design of a Tuned Mass Damper for a high-rise building?
Jonathan Frutschy il 3 Mag 2024
@Sam Chak This is intended for a classified research project. While I can't divulge the details, a tuned mass damper on a high rise building is a great application!

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### Risposta accettata

Torsten il 9 Apr 2024
Modificato: Torsten il 9 Apr 2024
z1'= z2
z2' = ( F-(c1+c2)*z2-( ((k1*gamma1+k2*gamma2)*z1^2+k1+k2)*z1+(-k2*gamma2*z1^2-k2)*z3 ) )/m1
z3' = z4
z4' = (-(c2+c3)*z4-( (-k2*gamma2*z3^2-k2)*z1+((k2*gamma2+k3*gamma3)*z3^2+k2+k3)*z3))/m2
Linearize the right hand side by computing the 4x4 Jacobian (with respect to z1 - z4) and compute its eigenvalues in the course of the simulation.
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