Lab 1

Euler’s relation

Differential equation of damped harmonic oscillator and derivation

Overdamping

Critical damping

Matlab Ch1-3:

Numerical accuracy

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Lab 2

Air friction in differential equation:  linear vs quadratic

Converting F = ma (second order) to two first order differential equations

Since friction depends on speed, should it affect most the x or y motion of a homerun baseball hit?

Matlab Ch4-6:

x=a:b:c  

cell edge grid

cell center grid

Greek letters, subscripts, superscripts in strings

hold on

Lab 3

How differential equations make a curve

Curvature and second derivative

Transient vs. steady state of driven oscillation

Effect of damping on resonance curve

Matlab Ch7-8:

Row, column selection (slicing) with “:”, e.g. A(:,2)

Range e.g. A(2,1:10)

Lab 4

Loop and logic programming structures

Phase space of pendulum, with and without damping.

Lab 5

Roundoff  and subtraction errors

Forward and centered first derivatives for data spaced on finite intervals

Derive second derivative approximation

Midpoint rule, Simpson's rule

Lab 6

Aliasing, critical frequency, and relation to data sampling rate.

Uncertainty relation between pulse time width and frequency width

Relation between total sampling time T and frequency spacing Dn

Windowing

Lab 7

Parametric oscillator: know simplest differential equation (7.1), physical example, and what physical parameter is varying?

Why parametric pumping can act like an amplifier: Why initial velocity or displacement is necessary.

How multiplied oscillatory terms (as in parametric amplifier or nonlinear terms) create new frequencies.  (Complex form is easiest).

Perturbation theory used in nonlinear equations to get first order effects

Lab 8

Nonlinear differential equation

Write down both exact (nonlinear) and small angle approximation (linear) diffeq for pendulum

Dependence of pendulum period on amplitude

Why pumping a pendulum at single frequency can’t give large amplitudes

Matlab Ch 14-15:

Concept of how to use Fminsearch and a square error function to solve systems of nonlinear equations

Lab 9

Attractors

Limit cycles

Strange attractors

Entrainment

Intermittency

1/f noise

Butterfly effect

Matlab Ch 16:

Euler’s method: why it's almost never used to solve a differential equation

predictor-corrector method: what are we predicting in the prediction step?

Lab 10

“Normal modes” of two coupled pendulums

Energy transfer between coupled oscillators not in a normal mode.

Entrainment in clocks on the same wall

Lab 11

Ponderomotive force

Einstein’s principle of equivalence and how we used it on shaken pendulum.

Lab 12

Writing distance between two masses in terms of Cartesian coordinates of the two.

Difference between paths of two masses orbiting and drifting, and in center of mass frame

Planar motion comes from conservation of L

Elliptical motion consequence of 1/r^2 law

Lab 13

Hysteresis

Bistability

Why hysteresis occurs when frequency depends on amplitude (see i, ii on pg 66)