Week 11: 09 Nov,

2250 Lecture Record Week 11 F2009

Last Modified: November 23, 2009, 20:38 MST. Today: February 10, 2012, 09:49 MST.

Week 11, Nov 9 to 13: Sections 10.5,5.5,5.6,EPbvp3.7,7.1

09 Nov: Systems. Intro Laplace Resolvent. Undetermined Coefficients. Sections 5.5,7.1,10.5

 Undetermined Coefficients
   Which equations can be solved
   Intro to the basic trial solution method
      Laplace solution of y'' + y = 1+x [use x''(t)+x(t) = 1+t, x(0)=x'(0)=0]
      How to find the atoms in y_p(x).
      How to find the atoms in y_h(x)
   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
   EXAMPLE. How to find a shortest expression for y_p(x) using
            Laplace's method.
            Details for x''(t)+x(t) = 1+t, to
            obtain the trial solution x(t)=A+Bt
            and the answer x(t)=1+t.

 Intro to the Laplace resolvent method for 2x2 systems
   Converting a dynamical system to vector-matrix form u'=Au.
   Position-velocity substitution and the harmonic oscillator.
   Solve the systems by ch1 methods
     x' = 2x, x(0)=100,
     y' = 3y, y(0)=50.
       Answer: x=100exp(2t), y=50 exp(3t)
     x' = 2x+y, x(0)=1,
     y' = 3y, y(0)=2.
       Answer: y=2 exp(3t) and x(t) is the solution of the linear
               integrating factor problem x'=2x+2 exp(3t).

Slides: Basic undetermined coefficients, draft 4 (104.9 K, pdf, 07 Nov 2009)
Slides: Variation of parameters (109.8 K, pdf, 07 Nov 2009)

Engineering models
   The job-site cable hoist example [delayed]
   Sliding plates example  [delayed]
   Home heating example  [delayed]

10 Nov: Problem Sessions on Laplace Theory. Sections 10.1 to 10.5, EPbvp7.6

 Forward and Backward Table Applications
   Problem 10.1-18. Trig identity 2 sin (theta) cos(theta) = sin(2 theta)
     used for L(sin(3t)cos(3t)).
   Problem 10.1-28. Splitting a fraction into backward table entries.
  Partial Fractions and Backward Table Applications
    Problem 10.2-24. L(f)=1/(s(s+1)(s+2)) solved by the three methods for
    partial fractions: sampling, atom method, Heaviside cover-up.
    Problem 10.2-9. Solve x''+3x'+2x=t, x(0)=0, x'(0)=2. Get resolvent
    equation
     (s^2+3s+2)L(x)=2+L(t)
      L(x)=(1+2s^2)/(s^2(s+2)(s+1))
      L(x)=A/s + B/s^2 + C/(s+2) + D(s+1)
      L(x)=L(A+Bt+C e^{-2t} +D e^{-t})
     Solve for A,B,C,D by the sampling method.
   Shifting Theorem and u-substitution Applications
     Problem 10.3-18. L(f)=s^3/(s-4)^4.
       L(f) = (u+4)^3/u^4  where u=s-4
       L(f) = (u^3+12u^2+48u+64)/u^4
       L(f) = (1/s + 12/s^2 + 48/s^3 + 64/s^4) where s --> (s-4)
       L(f)=L(e^{4t}(1+12t+48t^2/2+64t^3/6)) by shifting thm
     Problem 10.3-8. L(f)=(s+2)/(s^2+4s+5)
       L(f) = (s+2)/((s+2)^2 + 1)
       L(f) = u/(u^2 + 1)  where u=s+1
       L(f) = s/(s^2 + 1) where s --> s+1
       L(f) = L(e^{-t} cos(t))  by shifting thm
   Second Shifting Theorem Applications
     Problem 10.5-3. L(f)=e^{-s}/(s+2)
     Problem 10.5-4. L(f) = (e^-s} - e^{2-2s})/(s-1)
     Problem 10.5-22. f(t)=t^3 pulse(t,1,2)
   Piecewise Applications
     Staircase or floor function
     Sawtooth wave
     Square wave
   Dirac Applications
     x''+x=5 Delta(t-1), x(0)=0,x'(0)=1

11 Nov: Variation of parameters. Undetermined Coefficients. Sections 5.5, 10.4

 More Laplace Examples
   Continuing 10.3, 10.5 examples from last lecture.

 Transform Terminology
   Input
   Output
   Transfer Function

 Variation of parameters
   The second order formula.
   Application to y''=1+x
   Application to y''+y=sec(x) [slides]

 Undetermined Coefficients
   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, ...
   METHOD to FIND the SOLUTION.
     Laplace solution of y'' + y = f(x) when f(x)=linear combination of atoms

     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 trial solution for y_p(x)
         Laplace theory method.
         A rule for finding y_p(x) without using Laplace theory.
             Finding the redundant trial solution from g(x) = x^n f(x).
             Finding a trial solution with fewest symbols [non-redundant trial sol].
             Finding the non-redundant trial solution from g(x) = f(x) and the
             cross-out correction rule.
             Relation between the non-redundant trial solution and the book's
             table that uses the mystery factor x^s.
      EXAMPLES.

11-12-13 Nov: Fusi and Richins

Review starts for Exam 3, using the 7:30 exam key from S2009. Solved laplace theory problems from chapter 10 dailies.

13 Nov: Laplace Resolvent. Undetermined Coefficients. Resonance. Sections 7.1, 10.3, 5.5, 5.6

Laplace resolvent method for 2x2 systems
   Model u'=Au, u(0)=u_0
         x' = 2x+y, x(0)=1,
         y' = 3y, y(0)=2.
   How to solve it by Laplace's method.
     The resolvent equations, before the answers
     are found for L(x), L(y)
       (s-2)L(x) +  (-1)L(y) = 1,
         (0)L(x) + (s-3)L(y) = 2
   A shortcut to the resolvent equations.
     Laplace resolvent formula (sI-A)L(u)=u(0)
   Problem 10.2-16. x'=x+z, y'=x+y, z'= -2x -z, x(0)=1, y(0)=z(0)=0.
     How to find the resolvent equations.
     Backward table solution.
   Problem 10.2-6. x''+4x=cos(t), x(0)=x'(0)=0.
     Resolvent equation of the dynamical system
     Solving the equation by standard Laplace methods.
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
Theory of Practical Resonance
   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.

 Projection: glass-breaking video [delayed]
   Next week: More on resonance, details of practical resonance theory.
   Next week: Calculations from 5.6.

Video: Wine glass breakage (QuickTime MOV) (96.8 K, mov, 31 Mar 2008)
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)
Extra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.

Laplace theory references

Slides: Laplace and Newton calculus. Photos. (145.3 K, pdf, 01 Nov 2009)

Slides: Intro to Laplace theory. Calculus assumed. (109.5 K, pdf, 01 Nov 2009)

Slides: Laplace rules (112.2 K, pdf, 01 Nov 2009)

Slides: Laplace table proofs (130.3 K, pdf, 01 Nov 2009)

Slides: Laplace examples (101.2 K, pdf, 07 Nov 2009)

Slides: Piecewise functions and Laplace theory (64.7 K, pdf, 01 Nov 2009)

MAPLE: Maple Lab 7. Laplace applications (84.3 K, pdf, 19 Jul 2009)

Manuscript: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008)

Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)

Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)