Are oscillations ubiquitous or are they merely a

Are oscillations ubiquitous or are they merely a paradigm? Superposition of 1 brain neuron activity

Energy approach to equation of motion: i. e. find trajectory x(t) if U(x) is known • • Special case of U(x)=(½)kx 2 found period T (indep of A), found x(t) = A cos(wt+f) • Found 3 other forms of A cos(wt+f) • Learned to apply initial conditions to determine A, f, and also the arbitrary parameters in other 3 forms

Energy approach to equation of motion: • Harder case of U(q)= Mg. Lcm(1 -cos q) found -> HO for small theta found T numerically measured T for pendulum • Learned to argue qualitatively about time to move certain distances, comparing T for diff U • Did NOT learn equation of motion q(t)

Complex numbers: • rectangular and polar form and Argand diag. • complex conjugate • Euler relation • solving one complex equation is actually solving 2 simultaneous equations

Free, undamped oscillators k m L No friction k m I C q x q Common notation for all T m mg

Force approach to equation of motion of FREE, UNDAMPED HARMONIC OSCILLATOR: i. e. find trajectory q(t) if F(q) is known • Special case of F(q)=-sin(q) -> small angle approx: F(q)=-q =>2 nd order DE, Found sinusoidal motion Applied initial conditions as before.

Free, damped oscillators k friction m x q Common notation for all T m mg

Force approach to equation of motion of FREE, DAMPED OSCILLATOR • Add damping force to eqn of motion • Found decaying sinusoid

FREE, DAMPED OSCILLATOR • Damping time t=1/b • measures number of oscillations in decay time • apply initial conditions, energy decay

DRIVEN, DAMPED OSCILLATOR L I Vocoswt C R 11

"Resonance" CHARGE Charge Amplitude |q 0| w 0 Charge Phase fq 0 -π/2 -π w 0 Driving Frequency w------>

CURRENT “Resonance” Current Amplitude |I 0| w 0 Current Phase f. I π/2 0 -π/2 w Driving Frequency w------>

ADMITTANCE “Resonance” Admittance Amplitude |Y 0| w 0 Addmittanceπ/2 Phase f. I 0 -π/2 Driving Frequency w------>

DRIVEN, DAMPED OSCILLATOR • can also rewrite diff eq in terms of I and solve directly (same result of course) L I Vocoswt C R

FOURIER SERIES – periodic functions are sums of sines and cosines of integer multiples of a fundamental frequency. These “basis functions are orthonormal

ODD functions f(t) = -f(-t). Their Fourier representation must also be in terms of odd functions, namely sines. Suppose we have an odd periodic function f(t) like our sawtooth wave and you have to find its Fourier series Then the unknown coefficients can be evaluated this way Integrate over the period of the fundamental Here’s the coefficient of the sin(wnt) term! Plot it on your spectrum! normalize properly the function the harmonic

Time domain Frequency domain B frequency Fundamantal freq = 2π/T Integrate over the period of the fundamental coefficient of the sin(wnt) term! normalize properly the function the harmonic

DRIVING AN OCILLATOR WITH A PERIODIC FORCING FUNCTION THAT IS NOT A PURE SINE Oscillator response Important to know where the fundamental freq of the forcing function lies in relation to the oscillator max response freq! B Forcing function frequency

DRIVING AN OCILLATOR WITH A PERIODIC FORCING FUNCTION THAT IS NOT A PURE SINE

FT - you know this time Black box Z(w) frequency Observe what (LRC) black box does to an impulse function Black box Z(w) ? Are these connected by FT? ? They’d better be you find out! This was harmonic response expt - you know what black box (LRC) does to a single freq

Series LRC Circuit Which admittance plot |Y(w)| corresponds to which free decay plot I(t)? (All plots scaled to 1 at maximum value) (A) i <-> 1, ii <-> 2 (B) i <-> 2, ii <-> 1 (i) (1) (ii) (2)

Series LRC Circuit If the admittance |Y(w)| and the phase f. I response of a series LCR circuit are as given on the left below, then which oscilloscope trace on the right corresponds to a circuit driven below resonance? Red is drive voltage, blue is current, represented by VR. (A) (B) (C) (D)

Suppose that, if you apply a (red) sinusoidal voltage across a series LRC circuit, you measure the (blue) voltage response across the resistor. Now, if you now apply a (red) square-wave voltage with the same period to the same circuit, and you measure the (blue) voltage response across the resistor, will you get this …. ? Or this? Or something else?

Suppose that, if you apply a (red) sinusoidal voltage across a series LRC circuit, you measure the (blue) voltage response across the resistor. Now, if you now apply a (red) square-wave voltage at the same frequency the same circuit, and you measure the (blue) voltage response across the resistor, will you get this …. Or this? Or something else?

The last note of the piece Mozart chose for his piano recital was a sustained middle C (256 Hz). While he listened for the last note to fade away to a ghost of its initial amplitude, he had time calculate the quality factor Q of that oscillation. If the amplitude died away to 1% of its initial value after 15 seconds, approximately what was Q? • • • 133 834 2600 5200 None of the above

EXAM FORMULAS