Oscillations and Resonances PHYS 5306 Instructor Charles Myles

Oscillations and Resonances PHYS 5306 Instructor : Charles Myles Lee, Eun. Mo

Outline of the talk The Harmonic Oscillator ¡ Nonlinear Oscillations ¡ Nonlinear Resonance ¡ Parametric Resonance ¡ 2

The Harmonic Oscillator (1). Basic equations of motion and solutions Solution 3

(2). Damping The equation of motions has an additional term which comes from the damping force: Soultion The underdamped case: 4

The critically damped case: The overdamped case: 5

(3). Resonance Equation of motion of a damped and driven harmonic oscillator Solution Where 6

¡ ¡ ¡ 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 7

In the underdamped case . The width of resonance line is proportional to In the critically damped and overdamped case the resonance line disappears 8

2. Nonlinear Oscillations the total energy 9

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The canonical form of the complete elliptic integral of the first kind K 11

3. Nonlinear Resonance Nonlinear resonance seems not to be so much different from the (linear) resonance of a harmonic oscillator. But both, the dependency of the eigenfrequency of a nonlinear oscillator on the amplitude and the nonharmoniticity of the oscillation lead to a behavior that is impossible in harmonic oscillators, namely, the foldover effect and superharmonic resonance, respectively. Both effects are especially important in the case of weak damping. The foldover effect got its name from the bending of the resonance peak in a amplitude versus frequency plot. This bending is due to the frequency-amplitude relation which is typical for nonlinear oscillators. 12

Nonlinear oscillators do not oscillate sinusoidal. Their oscillation is a sum of harmonic (i. e. , sinusoidal) oscillations with frequencies which are integer multiples of the fundamental frequency (i. e. , the inverse of the period of the nonlinear oscillation). This is the well-known theorem of Jean Baptiste Joseph Fourier (1768 -1830) which says that periodic functions can be written as (infinite) sums (so-called Fourier series) of sine and cosine functions. 13

(1) The foldover effect 14

(2). Superharmonic Resonance 15

4. Parametric Resonance Parametric resonance is a resonance phenomenon different from normal resonance and superharmonic resonance because it is an instability phenomenon. 16

• 1. The instability The onset of first-order parametric resonance can be approximated analytically very well by the ansatz: Mathieu equation 17

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parametric resonance condition This instability threshold has a minimum just at the parametric resonance condition The minimum reads 19

2. Parametrically excited oscillations 20