Consider the double spring-mass system presented in class (equations below). m21 m2 k k2 4 21...

 

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Consider the double spring-mass system presented in class (equations below). m21 m2²₂ k₁ k2 4 Ţ21 Ţ12₂ m₁₁+k₁z₁+k₂ (1₁7₂) = 0 m₂₂+k₂ (t₂-₁) = 0 Write these coupled second order linear differential equations as (a) A single fourth order ODE in z (b) A single fourth order ODE in 2 (c) A system of four coupled linear ODEs in terms of the positions and velocities of each mass. Please write this as a matrix ODE. For the remaining parts you may assume that k₁k₂ = m₁ = m₂ = 1. What are the eigenvalues of the matrix system of ODEs in part (c)? You can use Matlab to compute these. Use Matlab to simulate the system from t0 tot 15 using ode45 with a sufficiently small time-step, and plot the results; start with an initial condition r₁(0) = ₂(0) = 1 and (0) = ₂(0) ⠀⠀⠀ 0. Do the simulations agree with your intuition based on the eigenvalues? - Consider the double spring-mass system presented in class (equations below). m21 m2²₂ k₁ k2 4 Ţ21 Ţ12₂ m₁₁+k₁z₁+k₂ (1₁7₂) = 0 m₂₂+k₂ (t₂-₁) = 0 Write these coupled second order linear differential equations as (a) A single fourth order ODE in z (b) A single fourth order ODE in 2 (c) A system of four coupled linear ODEs in terms of the positions and velocities of each mass. Please write this as a matrix ODE. For the remaining parts you may assume that k₁k₂ = m₁ = m₂ = 1. What are the eigenvalues of the matrix system of ODEs in part (c)? You can use Matlab to compute these. Use Matlab to simulate the system from t0 tot 15 using ode45 with a sufficiently small time-step, and plot the results; start with an initial condition r₁(0) = ₂(0) = 1 and (0) = ₂(0) ⠀⠀⠀ 0. Do the simulations agree with your intuition based on the eigenvalues? -

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