# hw09 - φ ˙ θ 2 ˙ φ 2 2(1 cos φ ˙ θ ˙ φ i V...

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AE 352: Aerospace Dynamics II, Fall 2008 Homework 9 Due Wednesday, November 19 Problem 1. For the matrix K , and the vector q K = 3 1 0 1 3 1 0 1 2 q = 1 - 2 1 (a) Evaluate Kq , q T K , q T K T , and 1 2 q T Kq . (b) Determine whether the matrix K is positive deﬁnite. (c) Determine whether the matrix K - 2 I is positive deﬁnite, where I is the identity matrix. (d) Show that for an arbitrary nonzero real vector q having components a,b, and c , the quadratic form V = 1 2 q T Kq > 0 for the K matrix given above. (Hint: V can be expressed as a sum of squared terms) Problem 2. For the compound pendulum (Greenwood Problem 6-7), T = 1 2 ml 2 h (3 + 2 cos
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Unformatted text preview: φ ) ˙ θ 2 + ˙ φ 2 + 2(1 + cos φ ) ˙ θ ˙ φ i V =-mgl [2 cos θ + cos( θ + φ )] (a) Determine all the static equilibrium values of q T = ( θ, φ ). Draw the conﬁgura-tion of each equilibria. (b) Determine the K matrix for each static equilibrium. (c) Determine the stability of each equilibrium. (d) Determine the M matrix for each stable equilibrium. (e) Write out the two scalar equations of motion at the stable equilibrium represent-edc by the matrix equation M ¨ q + Kq = 0 Page 1 of 1...
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