This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: is the spring constant, b the (constant) damping coeﬃcient, v = x t the velocity and a = v t = x tt the acceleration. 1. Show that this 2nd order ODE is equivalent to the 1st order linear system of ODEs ± x v ² t = ± 1k mb m ²± x v ² 2. Assume that we are using Forward Euler to solve this system numerically, with a timestep equal to Δ t . If λ 1 , λ 2 ∈ C are the complex eigenvalues of the matrix ± 1 Δ tk Δ t m 1b Δ t m ² show that the condition for stability is k λ 1 k < 1 and k λ 2 k < 1 3. Show that if b 2 < 4 km (such spring systems are referred to as underdamped ), then the eigenvalues of the matrix above are given as λ 1 , 2 = 1b Δ t 2 m ± i Δ t 2 m p 4 kmb 2 4. Show that if b 2 < 4 km the condition for stability is Δ t < b/k . 2...
View
Full Document
 Fall '07
 Fedkiw
 Derivative, Vector Space, Velocity, Constant of integration

Click to edit the document details