1-4 Theforced Harmonic OscillatorDriven Harmonic motionm fD=Fcosotmx=f,-yx-kxFx+2βx+002x =cosotm x = Ae-βt cos(oot +α)+ Ao cos(ot + p)* A, α initial condition.SASA0
1-4 The forced Harmonic Oscillator m x f x kx = D − − •• • t m F x 2 x x cos 2 + + 0 = •• • cos( ) cos( ) 0 0 = + + + − x Ae t A t t A , initial condition. Driven Harmonic motion: f D = F cost S t 0 S 0 t
After long timesteady statex = Ao cos(ot + @)F-2βoAom=arctanm/(w002-02)2+4p2020200VelocityOFV=0Ao =m/(002 -w2)2 +4β202OA-2o元arctanPv08-02200
2 2 2 2 2 0 0 ( ) 4 − + = = m w F V A 2 2 0 2 arctan 2 − − v = − Velocity: After long time steady state. cos( ) x = A0 t + 2 2 2 2 2 0 0 ( − ) + 4 = m w F A 2 2 0 2 arctan − − = A 0