## 2250 8:05am Lectures Week 9 S2015

Last Modified: March 23, 2015, 05:24 MDT.    Today: December 14, 2018, 01:58 MST.
``` Edwards-Penney, sections 5.4, 5.5, 5.6, 7.1
The textbook topics, definitions and theoremsEdwards-Penney 5.1, 5.2, 5.3, 5.4 (15.6 K, txt, 18 Dec 2013)Edwards-Penney 5.5, 5.6 (15.5 K, txt, 18 Dec 2013)Edwards-Penney 10.1, 10.2, 10.3, 10.4, 10.5 (20.5 K, txt, 18 Dec 2013)```

#### Monday: Applications x'' + px' + qx=0. Section 5.4.

```Slides on Section 5.4
Damped oscillations
overdamped, critically damped, underdamped,
pseudo-period  [Chapter 5]
phase-amplitude form of the solution [chapter 5]
Cafe door.
Pet door.
Undamped oscillations.
Harmonic oscillator. Slides: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014) Slides: phase-amplitude, cafe door, pet door, damping classification (136.0 K, pdf, 08 Mar 2014) Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012) Slides: Forced damped vibrations (264.0 K, pdf, 08 Mar 2014)
Partly solved 5.4-20
See the FAQ at the web site for answers and details.Text: FAQ for section 5.4 (34.5 K, txt, 11 Mar 2015)
The problem breaks into two distinct initial value problems:
(1)   2x'' + 16x' + 40x=0, x(0)=5, x'(0)=4
Characteristic equation  2(r^2+8r+20)=0.
Roots r=-4+2i,r=-4-2i.
Solution Atoms=e^{-4t}cos 2t, e^{-4t}sin 2t.
UNDERDAMPED.

(2)   2x'' + 0x' + 40x=0, x(0)=5, x'(0)=4
Characteristic equation 2(r^2+0+20)=0.
Roots r=sqrt(20)i,r=-sqrt(20)i.
The Euler solution atoms are
cos( sqrt(20)t), sin( sqrt(20)t).
UNDAMPED HARMONIC OSCILLATION.

Each system has general solution a linear combination of Euler
solution atoms. Evaluate the constants in the linear combination, in
each of the two cases, using the initial conditions x(0)=5, x'(0)=4.
There are two linear algebra problems to solve.

Answers: (1)  Coefficients 5, 12  for 2x'' + 16x' + 40x=0
Amplitude = sqrt(5^2 + 12^2) = 13
(2)  Coefficients 5, 2/sqrt(5) for 2x'' + 0x' + 40x=0
Amplitude = sqrt(5^2 + 4/5) = sqrt(129/5)
Plots can be made from these answers directly.
Write each solution in phase-amplitude form, a trig problem. See section
5.4 for specific instructions. The book's answers:
(1) tan(alpha) = 12/5   (2) tan(alpha) = 5 sqrt(5)/2

Partly solved 5.4-34.
See the FAQ at the web site for answers and details.Text: FAQ for section 5.4 (34.5 K, txt, 11 Mar 2015)
The method for finding the unknown coefficients c, k is explained
in Problem 1 of Quiz11, assigned in Week 11. See Sample Quiz 11.

The DE is 3.125 x'' + cx' + kx=0. The characteristic equation
is 3.125r^2 + cr + kr=0 which factors into 3.125(r-a-ib)(r-a+ib)=0
having complex roots a+ib, a-ib.

Problems 32, 33 find the numbers a, b from the given information.
This is an inverse problem, one in which experimental data is used
to discover the differential equation model. The book uses its own
notation for the symbols a,b: a ==> -p and b ==> omega1.

Because the two roots a+ib, a-ib determine the quadratic equation
mr^2+cr+k=0, which in factored form is m[(r-a)^2 + b^2] = 0, then c and k
are known in terms of symbols a,b.

References:  Sections 5.4, 5.6. Forced oscillations.
Slides: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014) Slides: phase-amplitude, cafe door, pet door, damping classification (136.0 K, pdf, 08 Mar 2014) Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012) Slides: Forced damped vibrations (264.0 K, pdf, 08 Mar 2014) Slides: Forced vibrations and resonance (253.0 K, pdf, 08 Mar 2014) Slides: Undetermined coefficients, pure resonance, practical resonance (152.8 K, pdf, 03 Mar 2013) Slides: Electrical circuits (112.9 K, pdf, 08 Mar 2014)```
```    PREVIEW: Undetermined Coefficients
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

Which equations can be solved by undetermined coefficients.
1. Constant coefficients
2. Forcing term a linear combination of Euler solution atoms.

Intro to the basic trial solution method
Solution of x'' + 9x = 30 sin(2t)
x(t)=c1 cos 3t + c2 sin 3t + a cos 2t + b sin 2t
= sum of two harmonics, of frequencies 3 and 2
= BEATS example [see Quiz 9 for more detail]
xh(t) = c1 cos 3t + c2 sin 3t, the first harmonic
xp(t) = a cos 2t + b sin 2t, the second harmonic
TRIAL SOLUTION
The equation y'' + y = 1 + x should have a solution
yp = a + bx for some constants a, b. We find the constants
by substitution of yp into  y'' + y = 1 + x.
Ways to find the Euler solution atoms in y_p(x).
1. Use Laplace theory (a bit slow, even if you already know Laplace theory).
2. Use rules from undetermined coefficient theory. Faster.
Euler solution atoms in y_h(x):  Roots of the  characteristic equation.
```
```Slides: Basic undetermined coefficients, draft 5 (156.3 K, pdf, 28 Mar 2013)Slides: Variation of parameters (164.5 K, pdf, 03 Mar 2012)
Extra Credit Maple Project: Tacoma narrows. Explore an alternative
explanation for what caused the bridge to fail, based on the hanging cables.
```

#### Mon-Wed-Fri: Undetermined Coefficients. Sections 7.1,5.5

```    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_pSlides: Basic undetermined coefficients, draft 5 (156.3 K, pdf, 28 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
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.

EXAMPLES. How to find a shortest trial solution using
Rules I and II.

EXAMPLE 1. 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 2 . 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 3. 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 4. 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 5.6

```Undetermined coefficients
Examples continued from the previous lecture.
EXAMPLES.
y'' = x
y'' + y = x exp(x)
y'' - y = x exp(x)
y'' + y = cos(x)
y''' + y'' = 3x + 4 exp(-x)

Shortest trial solution.
Two Rules to find the shortest trial solution.
1. Compute the atoms in f(x). The number k of atoms found
is the number needed in the shortest trial solution.
2. Correct groups with the same base atom, by
multiplication by x until the group contains no atom
which is a solution of the homogeneous problem
[eliminate homogeneous DE conflicts].
The x^s mystery factor in the book's table. The number s is the
multiplicity of the root in the homogenous DE characteristic
equation. The root is constructed the base atom of the group.
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 LCB math center. All editions of the book have identical
3.7 and 7.6 sections. There have been PDF copies of these
sections available on the WEB. Search for EP3.7.pdf and EP7.6.pdf.
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.

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 oscillationsPDF: 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 ResonanceSlides: 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: For electrical problems, resonance is omega = 1/sqrt(LC).
REVIEW
Reactance S: S=omega*L - 1/(omega*C)
Impedance Z:  Z^2 = S^2 + R^2
Steady-state current (E0/Z)*sin(omega*t - delta)
amplitude = E0/Z
delta = arctan(S/R)
Transfer function explained after sections 10.1, 10.2.
Input and output equation.
Projection: glass-breaking video. Wine glass experiment. Tacoma narrows.
Video: Wine glass breakage (avi) (0.0 K, avi, 31 Dec 1969)       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)

Slides: Basic undetermined coefficients, draft 5 (156.3 K, pdf, 28 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, 28 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 (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)
Oscillations. Mechanical and Electrical.
Slides: Electrical circuits (112.9 K, pdf, 08 Mar 2014)Slides: Forced damped vibrations (264.0 K, pdf, 08 Mar 2014)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)

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