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2280 8:05am Lectures Week 6 S2015

Last Modified: February 22, 2015, 13:22 MST.    Today: November 17, 2017, 16:15 MST.
Topics
  Sections 3.1 to 3.6
  The textbook topics, definitions, examples and theorems
Edwards-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.

Monday: President's Day Holiday. No classes.

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 oscillations
PDF: 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 Resonance
Slides: 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. Steady-state current 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 references
Slides: 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.
 Reading: Chapter 7.
          Read ch4, ch5, ch6 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, 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 Theory
Slides: 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)