# 2250-1 7:30am Lectures Week 11 S2014

Last Modified: March 24, 2014, 04:44 MDT.    Today: July 18, 2018, 12:34 MDT.
``` Edwards-Penney, sections 5.6, 10.1, 10.2, 10.3, 10.4
The textbook topics, definitions and theoremsEdwards-Penney 5.5, 5.6 (15.5 K, txt, 19 Dec 2013)Edwards-Penney 10.1, 10.2, 10.3, 10.4, 10.5 (20.5 K, txt, 19 Dec 2013)```

### Week 11: Sections 5.6, 10.1, 10.2, 10.3, 10.4

#### Monday (after the Break): Mechanical oscillators. Resonance. Beats. Sections 10.1, 10.2

```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.

Chapter 5 referencesSlides: Electrical circuits (112.9 K, pdf, 08 Mar 2014)Slides: Unforced vibrations 2008 (647.6 K, pdf, 28 Feb 2014)Slides: Forced damped vibrations (264.0 K, pdf, 08 Mar 2014)Slides: Forced vibrations and resonance (253.0 K, pdf, 08 Mar 2014)Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012)Slides: Resonance and undetermined coefficients (178.0 K, pdf, 08 Mar 2014)MOVIE: Water glass shattering due to resonant sound waves. (96.8 K, mov, 21 Mar 2013)
Lecture: Basic Laplace theory.
Read ch6, ch7, ch8, ch9 later.
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, 04 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, 19 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, 04 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012)
Problems 10.1: 18, 22, 28
```
```More exam 2 review, intro to problems 1,2,34,5

Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 04 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 04 Mar 2012) Slides: Laplace examples (149.1 K, pdf, 04 Mar 2012) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: Optional Maple Lab 7. Laplace applications (151.6 K, pdf, 18 Mar 2014) Slides: Laplace resolvent method (88.1 K, pdf, 04 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, 28 Nov 2010) Text: Laplace theory problem notes S2013 (17.7 K, txt, 18 Mar 2014) Text: Final exam study guide (8.0 K, txt, 20 Apr 2014) Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012)```

#### Tuesday: Laplace Theory. Tables, Rules of Laplace's Calculus. Sections 10.1,10.2,10.3.

```   Problems 10.1: 18, 22, 28
Exam review, Problem 5.
Text: Laplace theory problem notes S2013 (17.7 K, txt, 18 Mar 2014)

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, 04 Mar 2012)  Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014)
Solving differential equations by Laplace's method.
Basic Theorems of Laplace Theory
Functions of exponential order
Existence theorem for Laplace integrals
Euler solution atoms have a Laplace integral
Lerch's theorem
Linearity.
The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
Shift theorem L(exp(at)f(t)) = L(f(t))|s->(s-a)
Parts theorem L(y')=sL(y)-y(0)

Slide: Solving y' = -1, y(0)=2 with Laplace's method
Examples:

```

#### Wednesday: Piecewise functions. More Laplace theory. Sections 10.4, 10.5

```Exam Review, Problem 4.

Forward table

Parts formula derivation.
Laplace's method and quadrature for higher order equations and systems
Solving x'' + 4x = t exp(-t), x(0)=1, x'(0)=0 by the Laplace method
DEF: Piecewise Continuous Function
Functions of exponential order.
Existence of the Laplace integral.
One-sided and two-sided Laplace integral
Freeway example, suspension collides with a ramp.

Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 04 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012) Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 04 Mar 2012) Slides: Laplace examples (149.1 K, pdf, 04 Mar 2012) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969) Manuscript: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014) Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 10 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: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010) Text: Laplace theory problem notes F2008 (17.7 K, txt, 18 Mar 2014) Text: Final exam study guide (8.0 K, txt, 20 Apr 2014)```

#### Thursday: Rebecca

```Lab 11.
The Week 11 Lab will have one Laplace problem. Details Wed-Thu of Week 11.
The rest of the Thursday Lab time will be dedicated to exam review.
There is no Lab 10, because that was the week of Spring Break.
Lab 9 problems (2 problems)  are due Thursday. The 3rd problem was changed to Lab Extra Credit,
due April 9. See week 9 in the CALENDAR for the PDF link.
```

#### Thursday: Special Exam 2 review 4pm in WEB L102 by Doctor G

```  Exam 2 review for problems 1,2,3,4,5. Largely questions, some problem presentations.
Study Sample Exam 2 and Sample Exam 2 Solution key for Friday, Mar 21.
```

#### Friday: Midterm 2

The exam starts as early as 7:00am and ends at 8:25am.