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Oscillations and ResonancesInstructor Charles MylesLee EunMo Outline of the talk The Harmonic Oscillator. Nonlinear Oscillations Nonlinear Resonance Parametric Resonance The Harmonic Oscillator 1 Basic equations of motion and. 0 0 t Ce iw0t C e iw0t t A sin w0 t B cos w0 t 2 DampingThe equation of motions has an additional term.which comes from the damping force d 2 d 2 0 0 2 2 0.The underdamped case 02 t max sin t The critically damped case 2 t At B exp The overdamped case 2 0. t A exp t A exp t Equation of motion of a damped and driven harmonic oscillatord 2 d 2 0 a cos 2 ft. t A sin 2 ft B sin 2 ft Where A 2 02 2 f 2 2 f 2 0 2 f a 2 f .0 2 f 2 The amplitude of oscillations depend on the driving frequency It has its maximum when the driving frequency matches theeigenfrequency This phenomenon is called resonance. In the underdamped case max 2 t 2 2 f 2 tan 2 0 2 t .The width of resonance line is proportional toIn the critically damped and overdampedcase the resonance line disappears 2 Nonlinear Oscillations.the total energysin 01 d 2 2E 0 cos max 2 E 0 cos .2 E 0 cos The canonical form of the complete ellipticintegral of the first kind K 3 Nonlinear ResonanceNonlinear resonance seems not to be so much.different from the linear resonance of a harmonicoscillator But both the dependency of theeigenfrequency of a nonlinear oscillator on theamplitude and the nonharmoniticity of theoscillation lead to a behavior that is impossible in.harmonic oscillators namely the foldover effect andsuperharmonic resonance respectively Both effectsare especially important in the case of weakThe foldover effect got its name from the bending of theresonance peak in a amplitude versus frequency plot .This bending is due to the frequency amplitude relationwhich is typical for nonlinear oscillators Nonlinear oscillators do not oscillate sinusoidal Theiroscillation is a sum of harmonic i e sinusoidal oscillationswith frequencies which are integer multiples of the.fundamental frequency i e the inverse of the period of thenonlinear oscillation This is the well known theorem of JeanBaptiste Joseph Fourier 1768 1830 which says that periodicfunctions can be written as infinite sums so called Fourierseries of sine and cosine functions . 1 The foldover effectg 9 81m sec2 l 1m 0 4 sec 1 2 Superharmonic Resonanceg 9 81m sec2 l 1m 0 1sec 1 4 Parametric Resonance.Parametric resonance is a resonance phenomenon different fromnormal resonance and superharmonic resonance because it is aninstability phenomenon 1 The instabilityThe onset of first order parametric resonance can.be approximated analytically very well by theMathieu equation parametric resonance conditionThis instability threshold has a minimum just at the parametricresonance condition f 0.The minimum ac 2 f 2 Parametrically excited oscillationsg 9 81m sec 2 l 1m 0 1sec 1 A 0 07mInstructor : Charles Myles Lee, EunMo Outline of the talk The Harmonic Oscillator Nonlinear Oscillations Nonlinear Resonance Parametric Resonance The Harmonic Oscillator The amplitude of oscillations depend on the driving frequency. It has its maximum when the driving frequency matches the eigenfrequency. This phenomenon is called resonance 2.