Week 11: 09 Nov, 10 Nov, 11 Nov, 12 Nov, 13 Nov,

# 2250 Lecture Record Week 11 F2009

Last Modified: November 23, 2009, 20:38 MST.    Today: September 24, 2018, 01:55 MDT.

## 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

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.
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.
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
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.
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)