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2280 8:05am Lectures Week 6 S2016

Last Modified: February 17, 2016, 20:57 MST.    Today: November 17, 2017, 19:40 MST.
Topics
  Sections 3.1 to 3.6
  The textbook topics, definitions, examples and theorems
Edwards-Penney Ch 3, 3.1 to 3.4 (16.5 K, txt, 04 Jan 2015)
Edwards-Penney Ch 3, 3.5 to 3.7 (17.7 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. Beats.

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 (112.8 K, pdf, 19 Feb 2016)
Slides: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014)
Slides: Forced damped vibrations (263.9 K, pdf, 10 Feb 2016)
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)
2015 Video: Glass breakage in slow motion, MIT (0.0 K, 31 Dec 1969)
2009 Video: Glass breakage in slow motion, MIT (same video) (0.0 K, 31 Dec 1969)
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)
Thursday Exam Review: Intro to problems 1,2,3,4,5

Tue-Wed: Undetermined Coefficients. Sections 4.1,3.5

    REVIEW: Undetermined Coefficients
   Which equations can be solved
   THEOREM. Solution y_h(x) is a linear combination of atoms.
   THEOREM. Solution y_p(x) is a linear combination of atoms.
   THEOREM. (superposition)  y = y_h + y_p
Slides: Basic undetermined coefficients, draft 5 (156.3 K, pdf, 29 Mar 2013)
Slides: Variation of parameters (164.5 K, pdf, 03 Mar 2012) EXAMPLE. How to find a shortest expression for y_p(x) using Details for x''(t)+x(t) = 1+t the trial solution x(t)=A+Bt the answer x_p(t)=1+t. BASIC METHOD. Given a trial solution with undetermined coefficients, find a system of equations for d1, d2, ... and solve it. Report y_p as the trial solution with substituted answers d1, d2, d3, ... THEORY. y = y_h + y_p, and each is a linear combination of atoms. How to find the homogeneous solution y_h(x) from the characteristic equation. How to determine the form of the shortest trial solution for y_p(x) METHOD. A rule for finding y_p(x) from f(x) and the DE. Finding a trial solution with fewest symbols. Rule I. Assume the right side f(x) of the differential equation is a linear combination of atoms. Make a list of all distinct atoms that appear in the derivatives f(x), f'(x), f''(x), ... . Multiply these k atoms by undetermined coefficients d_1, ... , d_k, then add to define a trial solution y. This rule FAILS if one or more of the k atoms is a solution of the homogeneous differential equation. Rule II. If Rule I FAILS, then break the k atoms into groups with the same base atom. Cycle through the groups, replacing atoms as follows. If the first atom in the group is a solution of the homogeneous differential equation, then multiply all atoms in the group by factor x. Repeat until the first atom is not a solution of the homogeneous differential equation. Multiply the constructed k atoms by symbols d_1, ... , d_k and add to define trial solution y. Explanation: The relation between the Rule I + II trial solution and the book's table that uses the mystery factor x^s. EXAMPLES. y'' = x y'' + y = x exp(x) y'' - y = x exp(x) y'' + y = cos(x) y''' + y'' = 3x + 4 exp(-x) THEOREM. Suppose a list of k atoms is generated from the atoms in f(x), using Rule I. Then the shortest trial solution has exactly k atoms. EXAMPLE. How to find a shortest trial solution using Rules I and II. Details for x''(t)+x(t) = t^2 + cos(t), obtaining the shortest trial solution x(t)=d1+d2 t+d3 t^2+d4 t cos(t) + d5 t sin(t). How to use dsolve() in maple to check the answer. EXAMPLE. Suppose the DE has order n=4 and the homogeneous equation has solution atoms cos(t), t cos(t), sin(t), t sin(t). Assume f(t) = t^2 + cos(t). What is the shortest trial solution? EXAMPLE. Suppose the DE has order n=2 and the homogeneous equation has solution atoms cos(t), sin(t). Assume f(t) = t^2 + t cos(t). What is the shortest trial solution? EXAMPLE. Suppose the DE has order n=4 and the homogeneous equation has solution atoms 1, t, cos(t), sin(t). Assume f(t) = t^2 + t cos(t). What is the shortest trial solution?

Wednesday-Friday: 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 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.. 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, draft 5 (156.3 K, pdf, 29 Mar 2013)
Slides: Variation of parameters (164.5 K, pdf, 03 Mar 2012)
Slides: Resonance and undetermined coefficients (178.0 K, pdf, 08 Mar 2014) Variation of Parameters and Undetermined Coefficients references
Slides: Basic undetermined coefficients, draft 5 (156.3 K, pdf, 29 Mar 2013)
Slides: Variation of parameters (164.5 K, pdf, 03 Mar 2012) Systems of Differential Equations references
Manuscript: Systems of DE examples and theory (730.9 K, pdf, 10 Apr 2014)
Slides: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016)
Slides: Laplace second order systems (273.7 K, pdf, 14 Mar 2016)
Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016)
Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)
Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008) Oscillations. Mechanical and Electrical.
Slides: Electrical circuits (112.8 K, pdf, 19 Feb 2016)
Slides: Forced damped vibrations (263.9 K, pdf, 10 Feb 2016)
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)
Slides: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014)

Monday: 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. (188.3 K, pdf, 14 Mar 2016) 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. (144.9 K, pdf, 13 Mar 2016) 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. (188.3 K, pdf, 14 Mar 2016)
Slides: Intro to Laplace theory. Calculus assumed. (144.9 K, pdf, 13 Mar 2016)
Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014)
Slides: Laplace rules (144.9 K, pdf, 14 Mar 2016) Problems 7.1: 18, 22, 28
Laplace theory references
 
Slides: Laplace and Newton calculus. Photos. (188.3 K, pdf, 14 Mar 2016)
Slides: Intro to Laplace theory. Calculus assumed. (144.9 K, pdf, 13 Mar 2016)
Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014)
Slides: Laplace rules (144.9 K, pdf, 14 Mar 2016)
Slides: Laplace table proofs (151.9 K, pdf, 14 Mar 2016)
Slides: Laplace examples (133.7 K, pdf, 14 Mar 2016)
Slides: Piecewise functions and Laplace theory (98.5 K, pdf, 14 Mar 2016)
MAPLE: Optional Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969)
Slides: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016)
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 (273.7 K, pdf, 14 Mar 2016)

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. (188.3 K, pdf, 14 Mar 2016)
Slides: Laplace rules (144.9 K, pdf, 14 Mar 2016)
Slides: Intro to Laplace theory. Calculus assumed. (144.9 K, pdf, 13 Mar 2016)
Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014)
  Laplace theory references
 
Slides: Laplace and Newton calculus. Photos. (188.3 K, pdf, 14 Mar 2016)
Slides: Intro to Laplace theory. Calculus assumed. (144.9 K, pdf, 13 Mar 2016)
Slides: Laplace rules (144.9 K, pdf, 14 Mar 2016)
Slides: Laplace table proofs (151.9 K, pdf, 14 Mar 2016)
Slides: Laplace examples (133.7 K, pdf, 14 Mar 2016)
Slides: Piecewise functions and Laplace theory (98.5 K, pdf, 14 Mar 2016)
Manuscript: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014)
Slides: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016)
Slides: Laplace second order systems (273.7 K, pdf, 14 Mar 2016)
Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016)
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 (0.0 K, pdf, 31 Dec 1969)
Transparencies: Ch7 Laplace solutions 7.1 to 7.4 (from EP 2250 book) (1068.7 K, pdf, 22 Feb 2015)