Homework #7 (ASEN5010, Spring 2005)(Due at Start of Class on Wednesday, 16 March 2005)Stability Regions of Kelvin’s GyrostatProblem 7.1 Derivation of Linearized EquationsStarting with the governing equations of motion in the form:J ·˙ω + ω × (J · ω + hs) = 0 (1)with the following conditionsω =(00Ä3)H = J · ω + hs= λω, hs= hsb3(2)and utilizing the general linearized equation set handed out previously in carrying out Homework#5,derive the following linearized equation:J3¨θ3= 0·J100 J2¸½¨θ1¨θ2¾+ Ä3·0 −(J2+ J1− λ)+(J2+ J1− λ) 0¸½˙θ1˙θ2¾+ Ä23·(λ − J2) 00 (λ − J1)¸½θ1θ2¾= 0(3)(Hint: Show that, for the present problem, λ corresponds to J3in the standard linearized equation.)1Problem 7.2: Stability analysis7.2.1 Obtain the characteristic equation of (3) in the form:s4+ a2s2+ a4= 0 (4)7.2.2 Utilizing the Routh-Hurwitz condition, determine the stability condition asa2> 0, a4> 0, a22− 4a4> 0 (5)7.2.3 From the stability condition obtained from Problem 7.2.2, express the stability conditions interms of (Ignore equation (6.117), elegant but complex, and use your simpler expressions!)x = (J3− J2)/J1, y = (J3− J1)/J2,ˆÄ =hs/(J1J2)1/2Ä3J2J1=(1 − x)(1 − y)(provided J1+ J2− J36= 0)(6)where λ = J3+ hs/Ä3is to be utilized.Hint: UsehsJ2Ä=hs√J1J2ÄsJ1J2=ˆÄs(1 − y)(1 − x)hsJ1Ä=hs√J1J2ÄsJ2J1=ˆÄs(1 − x)(1 − y)(7)7.2.4 ForˆÄ = [ −0.5, −0.2, 0.2, 0.5 ], and the ranges of (−1 < x < 1) and (−1 < y < 1),plot the stable regions.(Stable regions must satisfy all of the above three conditions.)Hint: Plot the three conditions for a givenˆÄ in x vs. y on the same figure. Then find the