PIt p2t C P-P 12 =-0±ji e- ( pe/ope on) Ja c ale )-jo(e/o +e o ) Une-at a sin ot+ cos ( tI Qo Ue sinat cos ot inβ c =Ue( sin ot+cos∞t) β= arcto oo uge sin(ot+β)V= Ke sin(ot+β)V
( ) 2 ( ) 2 1 0 2 1 2 1 0 1 2 t j t j t p t p t C e P e P e j U P e P e P P U u − − − − = − − = P12 = − j 0 [ ( ) ( )] 2 0 t j t j t j t j t e e e j e e j U − − − − − − + − = ] 2 ( ) 2 ( ) [ 0 j t j t j t j t t e e j e e U e − − − + + − = ( sin cos ) 0 U e t t t + = − ( sin cos ) 0 0 0 0 U e t t t + = − U e t V Ke t V t t sin( ) sin( ) 0 0 + = + = − − sin t cos t = = = arctg sin cos 0 0
另解 =Aepl tae t C (4;e+A2e-) 式中A=P2-P1 0 2 P2-P p1,p2共轭1,A2也共轭 cosa t+sina t C e ar (A Jot em t e Jat e =cos@ t-Jsina t e(a,+a,)coso t+j(a-a,sin ot 实数 虚数 e-a [A cos ot+ Bsin ot] A=Uo P2+P1 Uoe lcos ot+sino B=JP-P =Ue a sin(ot+b)r 同前 Ke sin(0t+β)
实数 虚数 e t j t e t j t j t j t cos sin cos sin = − = + − [(A A )cos (A A )sin ] 1 2 1 2 e t j t t = + + − − p1 ，p2 共轭 A1 ，A2 也共轭 A A (A A ) 1 2 1 2 1 2 p t p t t j t j t C u e e e e e − − = + = + = sin( + ) − Ke t t (A A ) 1 2 t j t j t C u e e e − − = + e [Acos t Bsin t] t = + − 0 2 1 2 A1 U P P P − = 0 2 1 1 A2 U P P P − − 式中 = 另解 0 0 2 1 2 1 0 U U P P P P B j A U = − + = = [cos sin ] 0 U e t t t = + − 同前 U e t V t sin( ) 0 0 + = −
也可直接求A、BlC=e“[ Acos ot+ Bsin at 由初始条件 uc(0)=U0→>A=U0d duc=-ae /Acos ot+Bsinat du +e-oAsinat +oBcosat (0)=0→-aA+OB=0 dt A=U 0 B 0 e Uo(cos ot+sin ot) 或 U(-cos ot +sin at) 0 == Ue u sin(t+β)
(0 ) 0 A B 0 d d = → − + = + t uC 由初始条件 0 A 0 uC (0 ) = U → = U + u e [Acos t Bsin t] t C = + − A U0 B U0 = = (cos sin ) 0 u e U t t t C = + − 或 sin( ) 0 0 + = − U e t t [ A in B os ] [Acos Bsin ] d d e s t c t e t t t u t C t + − + = − + − − ( cos sin ) 0 0 0 0 u e U t t t C + = − 0 也可直接求A、B