Fnd the natural frequencies - find the amplitude ratio -

Question

Question - Given the equation of motion of a 2-DOF system:

J₁θ¨₁(t) + 2k_tθ₁(t) - k_tθ₂(t) = 0

J₂θ¨₂(t) + 2k_tθ₁(t) + 2k_tθ₂(t) = 0

(1) Rewrite them into matrix form:

(2) From above matrix form, define the coefficient matrix of as J, define the coefficient matrix of as K, define θ = , We will have a simplified matrix of form: _____.

In order to have non-zero solution, we need det (-ω²J + K) = 0, express that in its detailed matrix form.

(3) Given J₁ = J₂ = 1 kg.m², k_t =1 N.m/rad,  find the natural frequencies ω₁ and ω₂.

(4) Given J₁ = J₂ = 1 kg.m², k_t = 1N.m/rad, find the amplitude ratio r₁ and r₂.