# 2250-1 7:30am Lecture Record Week 9 S2010

Last Modified: March 16, 2010, 07:13 MDT.    Today: July 18, 2018, 12:34 MDT.

## 8 Mar: Damped and Undamped Motion. Section 5.4

```Lecture: Applications. Damped and undamped motion.
Last time: Theory of equations and 5.3-32.
Problems discussed in class: 5.4-20,34
Spring-mass equation,
LRC-circuit equation,
Spring-mass DE and RLC-circuit DE derivations.
Electrical-mechanical analogy.
The RLC circuit equation and its physical parameters.
Spring-mass equation mx''+cx'+kx=0 and its physical parameters.
Forced systems.
Forcing terms in mechanical systems. Speed bumps.
Forcing terms in electrical systems. Battery. Generator.
Harmonic oscillations: sine and cosine terms of frequency omega.
Damped and undamped equations. Phase-amplitude form.
Damped oscillations
overdamped, critically damped, underdamped [Chapter 5]
phase-amplitude form of the solution [chapter 5]
Undamped oscillations.
Harmonic oscillator.
Partly solved 5.4-20.
See the FAQ at the web site for answers and details.
Class notes transcribed the wrong coefficients. Corrected in FAQ.
Partly solved 5.4-34.
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, then c and k are known in terms
of symbols a,b. See also the web site FAQ for more details.
```

## 9 Mar: Applications x'' + px' + qx=0. Basic Laplace Theory. Sections 5.4, 10.1

```Slides on Section 5.4
Beats.
Decomposition of x(t) into two harmonic oscillations of different
natural frequencies. Envelope curves. Sound waves.
Pure resonance.
Pendulum.
Cafe door.
Pet door.
Over-damped, Critically-damped and Under-damped behavior.
pseudoperiod.
Shock-less auto.
Rolling wheel on a spring.
Bike trailer.
Physical pendulum.
Washing machine.
Swinging rod.
Mechanical watch.
```
Lecture: Introduction to Laplace theory. Newton and Laplace: portraits of the Two Greats [slides]. Method of quadrature. Comparison of Newton calculus and Laplace calculus. Laplace integral. The Laplace integrator dx=exp(-st)dt. Direct Laplace transform == Laplace integral. Laplace integral table, first three entries.

## 10-12 Mar: Intro to Laplace Theory. Sections 10.1,10.2,10.3.

```Drill
top=x-1, bot=(x+1)(x^2+4)
top/bot = A/(x+1)+(Bx+C)/(x^2+4); find A,B,C.
Sampling in partial fractions.
Method of atoms in partial fractions.
Heaviside's coverup method.
Solution to 4.7-10: Subspace Criterion. Blackboard only.
Lecture: Basic Laplace theory.
Read 5.5, 5.6, ch6, ch7, ch8, ch9 later.
Direct Laplace transform == Laplace integral.
Def: Direct Laplace
transform == Laplace integral
== int(f(t)exp(-st),t=0..infinity)
== L(f(t)).
Introduction to Laplace's method.
The method of quadrature for higher order equations and systems.
Calculus for chapter one quadrature versus the Laplace calculus.
The Laplace integrator dx=exp(-st)dt.
The Laplace integral abbreviation L(f(t)) for the Laplace integral of f(t).
Lerch's cancelation law and the fundamental theorem of calculus.
A Brief Laplace Table
1, t, t^2, t^n, exp(at), cos(bt), sin(bt)
```
``` A brief Laplace table.
Forward table.
Backward table.
Extensions of the Table.
Laplace rules.
Linearity.
The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
Shift theorem.
Parts theorem.
Finding Laplace integrals using Laplace calculus.
Solving differential equations by Laplace's method.
Basic Theorems of Laplace Theory
Lerch's theorem
Linearity.
The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
Shift theorem L(exp(at)f(t)) = L(f(t))|s->(s-a)
Parts theorem L(y')=sL(y)-y(0)

```

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)