Class Wiki - User contributions [en]

ProblemCh14-22

2010-01-20T19:52:29Z

Rgratias:

==Problem Statement==
Specify the range over which the the Laplace transform exists if
==Solution==

==References==
<references/>

==Contributors==
*[[Grant, Joshua|Joshua Grant]]
*[[Gratias, Ryan|Ryan Gratias]]
==Reviewed By==

==Read By==

ProblemCh14-22

2010-01-20T19:52:03Z

Rgratias:

==Problem Statement==
Specify the range over which the the Laplace transform exists if
==Solution==

==References==
<references/>
==Helpful Links==

==Contributors==
*[[Grant, Joshua|Joshua Grant]]
*[[Gratias, Ryan|Ryan Gratias]]
==Reviewed By==

==Read By==

ProblemCh14-22

2010-01-20T19:51:05Z

Rgratias: New page: ==Problem Statement== ==Solution== ==References== <references/> ==Helpful Links== ==Contributors== *Joshua Grant *Ryan Gratias ==Reviewed By== ==Re...

==Problem Statement==

==Solution==

==References==
<references/>
==Helpful Links==

==Contributors==
*[[Grant, Joshua|Joshua Grant]]
*[[Gratias, Ryan|Ryan Gratias]]
==Reviewed By==

==Read By==

Basic Laplace Transforms

2010-01-20T19:40:52Z

Rgratias:

'''Basic Laplace Transforms'''

The laplace transform has the standard form of:

:<math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math> (Cited From Fullerton, Colby)

However, in this class applying the standard form exclusively to solve problems is not practical. The use of Laplace transform properties greatly simplifies problems. These properties are listed in the book on page 525. The simple properties are listed below and are imported images from mathcad.

'''Linearity'''

[[Image:P1.jpg]]

[[Image:P2.jpg]]

'''Example:'''

If F(s)=(s+2)/(S+1) and f(t)=0 for t<0, then find the Laplace transform for the following functions identifying each property used to compute answers.

'''(a)''' <math>g_1(t)=5f(t-2)</math> '''(b)'''<math>g_2(t)=5(e^{-2t})f(t)</math>

(a) Linearity:
<math>F(s)=(5(s+2))/(s+1)</math>

Time Shift:
<math>\mathcal{L} \left\{f(t-2)u(t-2)\right\}=e^{-2s}F(s) </math> = '''<math>(5e^{-2s}(s+2))/(s+1)</math>'''

(b) Linearity: <math>a_1F(s) = {5(s+2)}/(s+1) </math>
Frequency Shift:
<math>\mathcal{L} \left\{e^{-at}f(t)\right\}=F(s+a) </math>

<math>(5((s+2)+2)))/((s+2)+1)=(5s+20)/(s+1)</math>

'''Author:'''
Jaymin Joseph

'''Reviewed by:'''
*[[Grant, Joshua|Joshua Grant]]

'''Read By:'''

Class Notes 1-5-2010

2010-01-20T19:37:09Z

Rgratias:

[[Image:January_5_graph_1.jpg|200px|thumb|left|Modeling functions as vectors. Using function approximations, the vector path is described.]]
This article covers the notes given in class on January 5, 2010.
==Subjects Covered==
1) Linear Systems

2) Functions as Vectors

[[Image:Figure_1.jpg|200px|thumb|left|Functions graphed in vector form.]]

The individual component representation of vector <math> \vec{v} </math> in the x-direction.
:<math>v_\mathrm{x} = \vec{v} \cdot \mathbf{\hat{i}}</math>
The two-dimensional example of a vector in its components with the vector designations.
:<math> \vec{v} = v_\mathrm{x} \mathbf{\hat{i}} + v_\mathrm{y} \mathbf{\hat{j}} </math>
This is the summation used to represent the vector <math> \vec{v} </math> as having as many dimensions as needed to express its full value:
:<math> \vec{v} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} </math>
:<math> \langle v_x, v_y\rangle</math>
This is the equation for finding the distance between the two vectors <math> \vec{u} </math> and <math> \vec{v} </math> who are separated by angle <math> \theta </math>.
:<math> \vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos\theta </math>
:<math> \vec{v} \cdot \mathbf{\hat{i}} = v_\mathrm{x} (\mathbf{\hat{i}} \cdot \mathbf{\hat{i}}) + v_\mathrm{y} \mathbf{\hat{j}} \cdot \mathbf{\hat{i}} </math>
:<math> \vec{v} \cdot \mathbf{\hat{i}} = v_\mathrm{x} </math>
:<math> \vec{v} \cdot \mathbf{\hat{a}}_\mathrm{m} = \sum_{i} v_\mathrm{i} \mathbf{\hat{a}}_\mathrm{i} \cdot \mathbf{\hat{a}}_\mathrm{m} = v_\mathrm{m} </math>
:<math> \delta_\mathrm{i,m} \equiv \begin{cases} 1, & \mbox{if } i = m \\ 0, & \mbox{else} \end{cases}</math>

==Example==
[[Image:January_5_graph_2.jpg|200px|thumb|left|Function waves with varying periods based on the function x(t) = x(t+T)]]
Given function: <math> x(t) = x(t+T) \,</math>
:<math> x(t) = \sum^\infty_{n=1} \left[ b_n \sin \left( \left( \frac {2\pi n} {T} \right) t \right) \right] </math>
1) Use vector analogy
:<math> x(t) \cdot \sin \left( \frac {2\pi mt} {T} \right) = \sum^\infty_{n=1} \left[ b_n \sin \left( \left( \frac {2\pi n} {T} \right) t \right) \cdot \sin \left( \frac {2\pi mt} {T} \right) \right] </math>

:<math> \int_{\frac {-T} {2}}^{\frac {T} {2}} x(t) \sin \left( \frac {2\pi mt} {T} \right) \,dt = v_m</math>

==External Links==
*[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR351/2010w/Keystone/index.php Class Notes].

==Authors==
Colby Fullerton

Brian Roath

==Read By==

==Reviewed By==
*[[Gratias, Ryan|Ryan Gratias]]

Gratias, Ryan

2010-01-06T06:16:27Z

Rgratias: New page: 750px == Contact info == '''Email:''' ryan.gratias@hotmail.com '''Phone/Text:''' (253) 212-7410

[[Image:pyramid scheme.jpg|750px]]
== Contact info ==

'''Email:''' ryan.gratias@hotmail.com

'''Phone/Text:''' (253) 212-7410

File:Pyramid scheme.jpg

2010-01-06T06:15:41Z

Rgratias: