Google Search in:
Week 12: 16 Nov,  17 Nov,  18 Nov,  19 Nov,  20 Nov,

2250 Lecture Record Week 12 F2009

Last Modified: November 28, 2009, 06:06 MST.    Today: October 24, 2017, 01:36 MDT.

Week 12, Nov 16 to 20: Sections 5.5,5.6,EPbvp3.7,6.1,7.1

16 Nov: Undetermined Coefficients. Practical Resonance. Glass Video. Sections 5.5,5.6

 Undetermined Coefficients
     THEOREM. Suppose a list of k atoms is generated from the
              atoms in f(x), by adding related lower-power atoms.
              Then the shortest trial solution has exactly k atoms.
     EXAMPLE. How to find a shortest trial solution using a
              shortcut derived from the cross-out method.

              Details for x''(t)+x(t) = t^2 + cos(t),
              to obtain 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?
 Problem 5.4-20
    Solving 2x''+16x'+40x=0, x(0)=5, x'(0)=4.
      Solution Atoms=e^{-4t} cos 2t, e^{-4t} sin 2t. Under-damped.
    Solving 2x''+0x'+40x=0, x(0)=5, x'(0)=4.
      Solution Atoms=cos( sqrt(20)t), sin( sqrt(20)t). Under-damped.
    Solving for constants c_1, c_2 in the two models.
      Ans: model 1, c1=5,c2=12; model 2, c1=5,c2=4/sqrt(20)
    Phase-amplitude form of the solution in each of the two cases.
    Issues with graphics: maple, mathematica, matlab, by hand.

 Problem 5.4-34
   Details about the graph
   How to use problems 32, 33
   How p and omega1 are defined from the quadratic formula.
   The fps system of units and why the mass is m=3.125 slugs

Theory of Practical Resonance
   The equation for practical resonance is
     mx''+cx'+kx=F_0 cos(omega t)
   THEOREM. The limit of x_h(t) is zero at t=infinity
   THEOREM. The trial solution for x_p(t) is
                 x(t) = A cos(omega t) + B sin(omega t).
            The atoms cos(omega t), sin(omega t) are not solutions
            of the homogeneous equation
                 mx''+cx'+kx=0.
pre>
 Projection: glass-breaking video. Wine glass experiment. Tacoma narrows.

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)

Slides: Basic undetermined coefficients, draft 4 (104.9 K, pdf, 07 Nov 2009)
Slides: Variation of parameters (109.8 K, pdf, 07 Nov 2009)
Slides: Resonance and undetermined coefficients (143.3 K, pdf, 07 Nov 2009)

17 Nov: Sections EPbvp3.7,6.1,6.2

Variation of Parameters 
  How to calculate y_p(x) from the five parameters
    y1(x)
    y2(x)
    W(x) = y1(x)y2'(x)-y1'(x)y2(x)
    a(x) = coefficient in the DE of y''
    f(x) = input or forcing term, the RHS of the DE

Undetermined Coefficient Examples
  Textbook problems from 5.5

Circuits EPbvp3.7:
  Electrical resonance.
  Impedance, reactance.
  Steady-state current
  amplitude.
  Transfer function.
  Input and output equation.
Reference:
     Edwards-Penney, Differential Equations and Boundary Value
     Problems, 4th edition, section 3.7 [math 2280 textbook].
     Extra pages supplied by Pearson with bookstore copies of
     the 2250 textbook. Also available as a xerox copy in case
     your book came from elsewhere. Check-out the 2280 book in
     the math center.


 

18 Nov: Sections 5.6, EPbvp7.6

 Undetermined Coefficients
   Further examples
 Delta function problems
   First shifting theorem
   Second shifting theorem
   Laplace table methods
   Dealing with f(t-a)step(t-a)

Theory of Practical Resonance
   The equation for practical resonance is
     mx''+cx'+kx=F_0 cos(omega t)
   THEOREM. The limit of x_h(t) is zero at t=infinity
   THEOREM. The undetermined coefficient trial solution is
                  x(t) = A cos(omega t) + B sin(omega t).
   THEOREM. x_p(t) = C(omega) cos(omega t - phi)
            where C(omega) = F_0/Z, and Z^2 = A^2+B^2. Symbols
            A and B are the undetermined coefficient answers for
            the 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.
   Proof sketches for these theorems. Derivation of formulas in 5.6.

18-19-20 Nov: Fusi and Richins

Review starts for Exam 3, using the 7:30 exam key from S2009.

20 Nov: Sections 7.2,7.3,7.4

Systems of two differential equations
   Solving a system from Chapter 1 methods
   The Laplace resolvent method for systems.
    Cramer's Rule,
    matrix inversion methods.
   Example: Solving a 2x2 dynamical system using Laplace's resolvent method.
     Study of u'=Au, u(0)=vector([2,1]), A=matrix([[2,3],[0,4]]).


Topics from linear systems
    Brine tank models.
    Recirculating brine tanks.
    Pond pollution.
    Home heating.
    All are 2x2 or 3x3 system applications that can be solved by Laplace methods.
Lecture: Fourier's Model. Intro to eigenanalysis, ch6.
  Examples and motivation.
  Fourier's model.
  History.
    J.B.Fourier's 1822 treatise on the theory of heat.
    The rod example.
      Physical Rod: a welding rod of unit length, insulated on the
                    lateral surface and ice packed on the ends.
    Define f(x)=thermometer reading at loc=x along the rod at t=0.
    Define u(x,t)=thermometer reading at loc=x and time=t>0.
    Problem: Find u(x,t).
      Fourier's solution:
      f(x) = 17 sin (pi x) + 29 sin(5 pi x)
           = 17 v1 + 29 v2
      Packages v1, v2 are vectors in a vector space V of functions on [0,1].
      Fourier computes u(x,t) by re-scaling v1, v2 with numbers Lambda_1,
      Lambda_2 that depend on t. This idea is called Fourier's Model.

      u(x,t) = 17 ( exp(-pi^2 t) sin(pi x)) + 29 ( exp(-25 pi^2 t) sin (5 pi x))
             = 17 (Lambda_1 v1) + 29 (Lambda_2 v2)

  Eigenanalysis  on u'=Au
     u(0) = c1 v1 + c2 v2  implies
     u(t) = c1 exp(lambda_1 t) v1 + c2 exp(lambda_2 t) v2
     Fourier's re-scaling idea from 1822, applied to u'=Au.
Main Theorem on Fourier's Model

  THEOREM. Fourier's model
     A(c1 v1 + c2 v2) = c1 (lambda1 v1) + c2 (lambda2 v2)
  with v1, v2 a basis of R^2 holds [for all constants c1, c2]
    if and only if
  the vector matrix system
    A(v1) = lambda1 v1,
    A(v2) = lambda2 v2,
  has a solution with v1, v2 independent
    if and only if
  the diagonal matrix D=diag(lambda1,lambda2) and
  the augmented matrix P=aug(v1,v2) satisfy
     1. det(P) not zero [then v1, v2 are independent]
     2. AP=PD

  THEOREM. The eigenvalues of A are found from the determinant
    equation
                        det(A -lambda I)=0,
    which is called the characteristic equation.
  THEOREM. The eigenvectors of A are found from the frame
    sequence which starts with B=A-lambda I [lambda a root of
    the characteristic equation], ending with last frame rref(B).

    The eigenvectors for lambda are the partial derivatives of
    the general solution obtained by the Last Frame Algorithm,
    with respect to the invented symbols t1, t2, t3, ...
 
    References for Eigenanalysis and Systems of Differential Equations.
    Manuscript: Algebraic eigenanalysis (135.8 K, pdf, 07 Apr 2008)
    Manuscript: What's eigenanalysis 2008 (129.5 K, pdf, 07 Apr 2008)
    Manuscript: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008)
    Manuscript: What's eigenanalysis, draft 2 (124.0 K, pdf, 14 Nov 2007)
    Slides: Cayley-Hamilton method for solving vector-matrix system u'=Au. (111.4 K, pdf, 30 Nov 2009)
    Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)
    Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)
    Manuscript: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008)
    Slides: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)
    Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
    Text: History of telecom companies (1.1 K, txt, 05 Oct 2008)
    Variation of Parameters and Undetermined Coefficients references
    Slides: Basic undetermined coefficients, draft 4 (104.9 K, pdf, 07 Nov 2009)
    Slides: Variation of parameters (109.8 K, pdf, 07 Nov 2009)
    Systems of Differential Equations references
    Manuscript: Systems of DE examples and 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)
    Slides: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)
    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 (87.1 K, pdf, 11 Oct 2009)
    Slides: Forced damped vibrations (235.0 K, pdf, 11 Oct 2009)
    Slides: Forced vibrations and resonance (185.3 K, pdf, 11 Oct 2009)
    Slides: Forced undamped vibrations (174.7 K, pdf, 11 Oct 2009)
    Slides: Resonance and undetermined coefficients (143.3 K, pdf, 07 Nov 2009)
    Slides: Unforced vibrations 2008 (620.4 K, pdf, 11 Oct 2009)