2280 8:05am Lectures Week 6 S2015

Last Modified: February 22, 2015, 13:22 MST.    Today: March 19, 2019, 15:04 MDT.
```Topics
Sections 3.1 to 3.6
The textbook topics, definitions, examples and theoremsEdwards-Penney Ch 3, 3.1 to 3.4 (16.2 K, txt, 04 Jan 2015)Edwards-Penney Ch 3, 3.5 to 3.7 (17.4 K, txt, 02 Jan 2015)```
This is a 4-day week with Monday holiday.

Tuesday (after the holiday): Mechanical oscillators. Resonance. Beats.

```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.
Slides:
Shock-less auto.
Rolling wheel on a spring.
Swinging rod.
Mechanical watch.
Bike trailer.
Physical pendulum.

Chapter 3 referencesSlides: Resonance and undetermined coefficients, cafe door, pet door, phase-amplitude (178.0 K, pdf, 08 Mar 2014)Slides: Electrical circuits (112.9 K, pdf, 08 Mar 2014)Slides: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014)Slides: Forced damped vibrations (264.0 K, pdf, 08 Mar 2014)Slides: Forced vibrations and resonance, Millenium Bridge, Wine Glass, Tacoma Narrows (253.0 K, pdf, 08 Mar 2014)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)
Projection: glass-breaking video. Wine glass experiment. Tacoma narrows.
Video: Wine glass breakage (avi) (260.5 K, avi, 18 Feb 2015)       Video: Glass breakage in slow motion, MIT
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)
```
```Thursday Exam Review: Intro to problems 1,2,3,4,5
```

Friday: Intro to Laplace Theory

```Lecture: Basic Laplace theory.
Direct Laplace transform == Laplace integral.
Def: Direct Laplace
transform == Laplace integral
== int(f(t)exp(-st),t=0..infinity)
== L(f(t)).
Introduction and History of Laplace's method
Photos of Newton and Laplace: portraits of the Two Greats.Slides: Laplace and Newton calculus. Photos of Newton and Laplace. (200.2 K, pdf, 03 Mar 2012)
The method of quadrature for higher order equations and systems.
Calculus for chapter one quadrature versus the Laplace calculus.
The Laplace integrator dx=exp(-st)dt.
The abbreviation L(f(t)) for the Laplace integral of f(t).
Lerch's cancelation law and the fundamental theorem of calculus.
Intro to Laplace TheorySlides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 18 Mar 2012)
A Brief Laplace Table
1, t, t^2, t^n, exp(at), cos(bt), sin(bt)
Some Laplace rules: Linearity, Lerch
Laplace's L-notation and the forward table
Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 18 Mar 2012) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012)
Problems 7.1: 18, 22, 28
```
```Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 18 Mar 2012) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 03 Mar 2012) Slides: Laplace examples (149.1 K, pdf, 03 Mar 2012) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: Optional Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969) Slides: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012) Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008) Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008) Manuscript: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014) Manuscript: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 22 Feb 2015) Text: Laplace theory problem notes S2013 (0.0 K, txt, 31 Dec 1969) Text: Final exam study guide (0.0 K, txt, 31 Dec 1969) Slides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012)```

Next Week

Mon-Tue-Wed-Fri: Laplace Theory. Tables, Rules of Laplace's Calculus. Sections 7.1,7.2,7.3.

```   Problems 7.1: 18, 22, 28
Exam review, Problem 5.
Text: Laplace theory problem notes S2013 (0.0 K, txt, 31 Dec 1969)

History of the Laplace Transform
REF: Deakin (1981), Development of the Laplace transform 1737 to 1937
EULER
LAPLACE 1784
End of WWII 1945
Fourier Transform
Mellin Transform and Gamma function
Laplace transform: one-sided and 2-sided transform
Applications: DE, PDE, difference equations, functional equations
Diffusion equation for spatial diffusion problems

A brief Laplace table.
Forward table.
Backward table.
Extensions of the Table.
Laplace rules.
Linearity.
The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
Shift theorem.
Parts theorem.
Finding Laplace integrals using Laplace calculus.
Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012)  Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 18 Mar 2012) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014)```
```  Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 18 Mar 2012) Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 03 Mar 2012) Slides: Laplace examples (149.1 K, pdf, 03 Mar 2012) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) Manuscript: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014) Slides: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 09 Apr 2014) Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008) Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008) Manuscript: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014) Manuscript: Laplace theory 2008 (500.9 K, pdf, 16 Mar 2014) Transparencies: Ch7 Laplace solutions 7.1 to 7.4 (from EP 2250 book) (1068.7 K, pdf, 22 Feb 2015)```