# 2250-1 7:30am Lectures Week 10 S2013

Last Modified: March 25, 2013, 05:48 MDT.    Today: September 24, 2018, 01:23 MDT.
``` Edwards-Penney, sections 5.4, 10.1, 10.2, 10.3
The textbook topics, definitions and theoremsEdwards-Penney 10.1, 10.2, 10.3, 10.4, 10.5 (20.5 K, txt, 03 Mar 2013)```

### Week 10: Sections 10.4, 10.5, EPbvp7.6

#### Monday (after the Break): Mechanical oscillators. Resonance. Beats. Sections 10.4, 10.5, 5.4

```
Applications
How to solve differential equations
Solving y''+y=0, y(0)=0, y'(0)=1
Solving y''+y=1, y(0)=y'(0)=0
Solving y''+y=cos(t), y(0)=y'(0)=0 (resonance)
Solving y''+y=cos(2t), y(0)=y'(0)=0 (beats)
LRC Circuit
Specialized mechanical models.
Pure Resonance x''+x=cos(t), frequency mnatching
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 (240.3 K, pdf, 04 Mar 2012)
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.

Chapter 5 referencesSlides: Electrical circuits (124.8 K, pdf, 04 Mar 2012)Slides: Forced damped vibrations (280.7 K, pdf, 04 Mar 2012)Slides: Forced vibrations and resonance (240.3 K, pdf, 04 Mar 2012)Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012)Slides: Resonance and undetermined coefficients (199.4 K, pdf, 03 Mar 2012)Slides: Unforced vibrations 2008 (667.9 K, pdf, 03 Mar 2012)MOVIE: Water glass shattering due to resonant sound waves. (96.8 K, mov, 21 Mar 2013)
```

#### Tuesday: Piecewise functions. More Laplace theory. Sections 10.4, 10.5

```Piecewise Functions
Unit Step: u(t)=1 for t>=0, u(t)=0 for t<0.
Pulse: pulse(t,a,b)=u(t-a)-u(t-b)
Ramp: ramp(t-a)=(t-a)u(t-a)
L(u(t-a)) = (1/s) exp(-as) [for a >= 0 only]
Integral Theorem
L(int(g(x),x=0..t)) = s L(g(t))
Applications to computing ramp(t-a)
L(ramp(t-a)) = (1/s^2) exp(-as) [for a >= 0 only]
Piecewise defined periodic waves
Square wave: f(t)=1 on [0,1), f(t)=-1 on [1,2), 2-periodic
Triangular wave: f(t)=|t| on [-1,1], 2-periodic
Sawtooth wave: f(t)=t on [0,1], 1-periodic
Rectified sine: f(t)=|sin(kt)|
Half-wave rectified sine: f(t)=sin(kt) when positive, else zero.
Parabolic wave
Periodic function theorem
Proof details
Laplace of the square wave. Problem 10.5-25.

Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 04 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012) Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 04 Mar 2012) Slides: Laplace examples (149.1 K, pdf, 04 Mar 2012) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: Maple Lab 7. Laplace applications (156.0 K, pdf, 04 Dec 2012) Manuscript: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008) Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012) Slides: Home heating, attic, main floor, basement (109.8 K, pdf, 04 Mar 2012) 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 (186.8 K, pdf, 20 Oct 2009) Manuscript: Laplace theory 2008 (351.3 K, pdf, 15 Apr 2009) Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010) Text: Laplace theory problem notes F2008 (17.2 K, txt, 03 Dec 2012) Text: Final exam study guide (8.2 K, txt, 05 Dec 2012)```

## Wednesday: Piecewise Functions. Section 10.5 and EPbvp supplement 7.6.

``` Convolution theorem
DEF. Convolution of f and g = f*g(t) = integral of f(x)g(t-x) from x=0 to x=t
THEOREM. L(f(t))L(g(t))=L(convolution of f and g)
Application:   L(cos t)L(sin t) = L(0.5 t sin(t))
Second shifting Theorems
e^{-as}L(f(t))=L(f(t-a)u(t-a)) Backward table
L(g(t)u(t-a))=e^{-as}L(g(t+a)) Forward table
EXAMPLES.
Forward table
L(sin(t)u(t-Pi)) = e^{-Pi s} L(sin(t)|t->t+Pi)
= e^{-Pi s} L(sin(t+Pi))
= e^{-Pi s} L(sin(t)cos(Pi)+sin(Pi)cos(t))
= e^{-Pi s} L(-sin(t))
= e^{-Pi s} ( -1/(s^2+1))
Backward table
L(f(t)) = e^{-2s}/s^2
= e^{-2s} L(t)
= L(t u(t)|t->t-2)
= L((t-2)u(t-2))
Therefore f(t) = (t-2)u(t-2) = ramp at t=2.

Laplace Resolvent Method.
--> This method is a shortcut for solving systems by Laplace's method.
--> It is also a convenient way to solve systems with maple.Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012)
Intro to the Laplace resolvent shortcut for 2x2 systems
Problem: Write a 2x2 dynamical system as a vector-matrix equation u'=Au.
Problem: Solve a 2x2 dynamical system in vector-matrix form u'=Au.
The general vector-matrix DE Model u'=Au
Laplace of u(t) = Resolvent times u(0)
Resolvent = inverse(sI - A)
Chapter 1 methods for solving 2x2 systems
Solve the systems by ch1 methods for x(t), y(t):
x' = 2x, x(0)=100,
y' = 3y, y(0)=50.
Answer: x = 100 exp(2t), y = 50 exp(3t)
x' = 2x+y, x(0)=1,
y' = 3y, y(0)=2.
Answer: y(t) = 2 exp(3t) and x(t) is the solution of the linear
integrating factor problem x'(t)=2x(t)+2 exp(3t).
```

## Friday: Piecewise Functions. Shifting. Section 10.5 and EPbvp supplement 7.6. Delta function and hammer hits.

```Laplace Resolvent Method
Consider problem 10.2-16
x'=x+z, y'=x+y, z'=-2x-z, x(0)=1, y(0)=0, z(0)=0
Write this as a matrix differential equation
u'=Bu, u(0)=u0
Then
u:=vector([x,y,z]);
B:=matrix([[1,0,1],[1,1,0],[-2,0,-1]]); u0:=vector([1,0,0]);
If we think of the matrix differential equation as a scalar equation, then
its Laplace model is
-u(0) + s L(u(t)) = BL(u(t))
or equivalently
sL(u(t)) - B L(u(t)) = u0
Write s = sI where I is the 3x3 identity matrix. Then the Laplace model is
(sI - B) L(u(t)) = u0
which is called the Resolvent Equation.
DEF. The RESOLVENT is the inverse of the matrix multiplier on the left:
Resolvent == inverse(sI - B)
It is so-named because the vector of Laplace answers is
= L(u(t)) = inverse(sI - B) times vector u0
Briefly,
Laplace of VECTOR u(t) = RESOLVENT MATRIX times VECTOR u(0)
ADVICE: Use Cramer's rule or matrix inversion to solve the resolvent
equation for the vector of components L(x), L(y), L(z). Any
linear algebra problem Bu=c where B contains symbols should
be solved this way, unless B is triangular.

Hammer hits and the Delta function
Definition of delta(t)
delta(t) = idealized injection of energy into a system at
time t=0 of impulse=1.
A hammer hit model in mechanics:
Camshaft impulse in a car engine
How delta functions appear in circuit calculations
Start with Q''+Q=E(t) where E is a switch. Then differentiate to get
I''+I=E'(t). Term E'(t) is a Dirac Delta.
Paul Dirac (1905-1985) and impulses
Laurent Schwartz (1915-2002) and distribution theory
Riemann Stieltjes integration theory: making sense of the Dirac delta.
Def: RS-integral equals the limit of RS-sums as N-->infinity and mesh-->zero.
RS-sum = sum of terms  f(x_i)(alpha(x_i)-alpha(x_{i-1})) where alpha(x) is
the monotonic RS integrator.
Why int( f(t) delta(t-a), t=-infinity .. infinity) = f(a)
The symbol delta(t-a) makes sense only under an integral sign.

Engineering models
Short duration impulses: Injection of energy into a mechanical or electrical model.
Definition: The impulse of force f(t) on interval [a,b] equals the
integral of f(t) over [a,b]
An example for f(t) with impulse 5 is defined by
f(t) = (5/(2h))pulse(t,-h,h)
EXAMPLE. The Laplace integral of f(t) and its limit as h --> 0.
EXAMPLE. The delta function model x''(t) + 4x(t) = 5 delta(t-t0),
x(0)=0, x'(0)=0. The model is a mass on a spring with no
damping. It is at rest until time t=t0, when a short duration
impulse of 5 is applied. This starts the mass oscillation.
EXAMPLE. The delta function model from EPbvp 7.6,
x''(t) + 4x(t) = 8 delta(t-2 pi), x(0)=3, x'(0)=0.
The model is a mass on a spring with no damping. The mass is moved
to position x=3 and released (no velocity). The mass oscillates until
time t=2Pi, when a short duration impulse of 8 is applied. This
alters the mass oscillation, producing a piecewise output x(t).
# How to solve it with dsolve in maple.
de:=diff(x(t),t,t)+4*x(t)=f(t);f:=t->8*Dirac(t-2*Pi);
ic:=x(0)=3,D(x)(0)=0; dsolve({de,ic},x(t));
convert(%,piecewise,t);
Details of the Laplace calculus in maple: inttrans package.
with(inttrans): f:=x->cos(omega*t)+8*Dirac(t-2*Pi);
G:=laplace(f(t),t,s); invlaplace(G,s,t);
de:=diff(x(t),t,t)+4*x(t)=f(t);
laplace(de,t,s);
subs(ic,%);
solve(%,laplace(x(t),t,s));
CALCULATION. Phase amplitude conversion [see EP 5.4]
x(t) = 3 cos(2t) until hammer hit at t=2Pi. It has amplitude 3.
x(t) = 3 cos(2t)+4 sin(2t) after the hit. It has amplitude 5.
= 5 cos(2t - arctan(4/3))

An RLC circuit model
Q'' + 110 Q' + 1000 Q = E(t)
Differentiate to get [see EPbvp 3.7]
I'' + 100 I' + 1000 I = E'(t)
When E(t) is a switch, then E'(t) is a Dirac delta.
Resonance examples
x'' + x = cos(t)
Pure resonance, unbounded solution x(t) = 0.5 t sin(t)
mx'' + cx' + kx = F_0 cos(omega t)
Practical resonance, all solutions bounded, but x(t)
can have extremely large amplitude when omega is tuned
to the frequency omega = sqrt(k/m - c^2/(2m^2))
LQ'' + RQ' + (1/C)Q = E_0 sin(omega t)
Practical resonance, all solutions bounded, but the current
I(t)=dQ/dt can have large amplitude when omega is tuned
to the resonant frequency omega = 1/sqrt(LC).
Future lecture, with slides and video:
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.

Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 04 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012) Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 04 Mar 2012) Slides: Laplace examples (149.1 K, pdf, 04 Mar 2012) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: Maple Lab 7. Laplace applications (156.0 K, pdf, 04 Dec 2012) Manuscript: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008) Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012) Slides: Home heating, attic, main floor, basement (109.8 K, pdf, 04 Mar 2012) 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 (186.8 K, pdf, 20 Oct 2009) Manuscript: Laplace theory 2008 (351.3 K, pdf, 15 Apr 2009) Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010) Text: Laplace theory problem notes for Chapter 10 (17.2 K, txt, 03 Dec 2012) Text: Final exam study guide (8.2 K, txt, 05 Dec 2012)
Variation of Parameters and Undetermined Coefficients references
Slides: Basic undetermined coefficients, draft 4 (104.9 K, pdf, 07 Nov 2009)Slides: Variation of parameters (164.5 K, pdf, 03 Mar 2012)
Systems of Differential Equations references
Manuscript: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008) Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012) Slides: Home heating, attic, main floor, basement (109.8 K, pdf, 04 Mar 2012) 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 (124.8 K, pdf, 04 Mar 2012)Slides: Forced damped vibrations (280.7 K, pdf, 04 Mar 2012)Slides: Forced vibrations and resonance (240.3 K, pdf, 04 Mar 2012)Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012)Slides: Resonance and undetermined coefficients (199.4 K, pdf, 03 Mar 2012)Slides: Unforced vibrations 2008 (667.9 K, pdf, 03 Mar 2012)Eigenanalysis and Systems of Differential Equations.
Manuscript: Eigenanalysis 2010, 46 pages (345.3 K, pdf, 31 Mar 2010)Manuscript: Algebraic eigenanalysis 2008 (187.6 K, pdf, 04 Mar 2012)Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)Text: History of telecom companies (1.9 K, txt, 04 Apr 2013)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)Manuscript: What's eigenanalysis 2008 (174.2 K, pdf, 04 Mar 2012)Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (152.9 K, pdf, 04 Mar 2012)```