## 2280 12:55pm Lectures Week 7 S2018

Last Modified: February 23, 2018, 12:38 MST.    Today: March 19, 2019, 15:32 MDT.
```Topics
Sections 7.1 to 7.6
The textbook topics, definitions, examples and theoremsEdwards-Penney Ch 3, 7.1 to 7.5 (21.0 K, txt, 22 Feb 2015)```
```Tuesday:   Resonance, Section 3.6. Circuits, Section 3.7.
Applications
Pure Resonance x''+x=cos(t), frequency matching
Solution explosion, unbounded solution x=(1/2) t sin t.
Practical Resonance: x'' + x = cos(omega t) with omega near 1
Large amplitude harmonic oscillationsPDF: Pure resonance y = x sin(x) (74.7 K, pdf, 18 Mar 2013)
Resonance examples: Soldiers marching in cadence, Tacoma narrows bridge,
Wine Glass Experiment. Theodore Von Karman and vortex shedding.
Cable model of the Tacoma bridge, year 2000. Resonance explanations.
Millenium Foot-Bridge London
Beats x''+x=cos(2t)
Graphics for beats [x=sin(10 t)sin(t/2)], slowly-oscillating envelope,
rapidly oscillating harmonic with time-varying amplitude.PDF: Beats y=sin(10x)sin(x/2) (68.9 K, pdf, 18 Mar 2013)
Theory of Practical ResonanceSlides: Forced vibrations and resonance (253.0 K, pdf, 08 Mar 2014)
The equation is
mx''+cx'+kx=F_0 cos(omega t)
THEOREM. The limit of x_h(t) is zero at t=infinity
THEOREM. x_p(t) = C(omega) cos(omega t - phi)
C(omega) = F_0/Z, Z^2 = A^2+B^2, A and B are the
undetermined coefficient answers for trial solution
x(t) = A cos(omega t) + B sin(omega t).
THEOREM. The output x(t) = x_h(t) + x_p(t) is graphically
just x_p(t) = C(omega) cos(omega t - phi) for large t.
Therefore, x_p(t) is the OBSERVABLE output.
THEOREM. The amplitude C(omega) is maximized over all possible
input frequencies omega>0 by the single choice
omega = sqrt(k/m - c^2/(2m^2)).
DEFINITION. The practical resonance frequency is the number omega
defined by the above square root expression.
Circuits EPbvp3.7 and Electrical resonance
Derivation from mechanical problems 5.6.
THEOREM: omega = 1/sqrt(LC).
REVIEW
Impedance, reactance.
amplitude
Transfer function.
Input and output equation..

Wine Glass Experiment
The lab table setup
Speaker.
Frequency generator with adjustment knob.
Amplifier with volume knob.
Wine glass.
x(t)=deflection from equilibrium of the radial component of the
glass rim, represented in polar coordinates, orthogonal to
the speaker front.
mx'' + cx' + kx = F_0 cos(omega t)  The model of the wine glass
m,c,k are properties of the glass sample itself
F_0 = volume knob adjustment
omega = frequency generator knob adjustment

Slides: Basic undetermined coefficients (147.6 K, pdf, 14 Feb 2018)Slides: Variation of parameters (164.5 K, pdf, 03 Mar 2012)Slides: Forced vibrations and resonance, Millenium Bridge, Wine Glass, Tacoma Narrows (253.0 K, pdf, 08 Mar 2014)Slides: Resonance and undetermined coefficients, cafe door, pet door, phase-amplitude (178.0 K, pdf, 08 Mar 2014)Slides: Electrical circuits (112.8 K, pdf, 19 Feb 2016)

References from week 6

Chapter 3 references. Sections 3.4, 3.5, 3.6. Forced/Unforced oscillations.Slides: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014)Slides: Forced damped vibrations (263.9 K, pdf, 10 Feb 2016)Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012) Slides: phase-amplitude, cafe door, pet door, damping classification (136.0 K, pdf, 08 Mar 2014)  Slide: Drawing for Exercise 3.4-34 (26.0 K, pdf, 23 Feb 2018)

Resonance Videos
Projection: glass-breaking video. Wine glass experiment. Tacoma narrows.
Video: Wine glass breakage (avi) (260.5 K, avi, 18 Feb 2015)
2015 Video: Glass breakage in slow motion, MIT
2009 Video: Glass breakage in slow motion, MIT (same video)
Video: Same 2009 Glass Breakage, local copy (12992.3 K, mp4, 16 Feb 2016)
Video: Wine glass experiment (12mb mpg, 2min) (12493.8 K, mpg, 01 Apr 2008)       Video: Tacoma Narrows Bridge Nov 7, 1940 (18mb mpg, 4min) (18185.8 K, mpg, 01 Apr 2008)

```