0=@mt+Yo+Y'(t),Y'(t)<<Yo -MI,Is sinYo -Bo=0 sinYo=-MI,I, Bo 元 Yo -B@/MIrIs Unstable Equilibrium Stable Equilibrium Pullout when sinyo=1Bo=MI,I, Hunting transients:sin(Yo +Y)sinyo cosy'+cos Yo siny'sinYo+y'cosYo =-MI,I,COS YoY'-By'=-(MI,I,COS7o +B)y' dt+=0i0=[MI,I.COSYo+B] d2y' Y'=A sin@ot+Az Cos@ot Stable if 02>0 (0 real) Unstable if2<0 imaginary) 6.641,Electromagnetic Fields,Forces,and Motion Lecture 13 Prof.Markus Zahn Page 14 of 146.641, Electromagnetic Fields, Forces, and Motion Lecture 13 Prof. Markus Zahn Page 14 of 14 θ ω + γ + γ γ << γ = t 't , 't m0 0 ( ) ( ) − γ − βω M I I sin = 0 rs 0 βω γ − 0 r s sin = MI I Pullout when γ ⇒ βω 0 rs sin = 1 = M I I Hunting transients: ( ) γ +γ ≈ γ γ + γ γ ≈ γ +γ γ 0 0 0 00 sin ' sin cos ' cos sin ' sin ' cos ( ) γ − γ γ − βγ − γ + β γ 2 2 rs 0 rs 0 d ' J = M I I cos ' ' = M I I cos ' dt γ +ω γ ω γ +β ⎡ ⎤ ⎣ ⎦ 2 2 2 2 0 0 rs 0 d ' ' = 0 ; = M I I cos J dt γ ω+ ω 1 02 0 ' = A sin t A cos t Stable if ω > 2 0 0 (ω0 real) Unstable if ω < 2 0 0 (ω0 imaginary)