FDEs

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WebPilot
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FDEs

Post by WebPilot » Mon Jan 28, 2008 2:55 am

... is an acronymn for:

1. Finite Difference Equations or

2. my initials, whichever.

There was a time when the following intimidated me ... those days are
over.

Image (c.t.)
Image

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FDEs

Post by WebPilot » Mon Jan 28, 2008 10:32 pm

This is a test.

Notice the courant number is only 0.4 .

Image (c.t.)
Last edited by WebPilot on Tue Jan 29, 2008 4:22 pm, edited 1 time in total.
Image

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Post by Mike Everman » Tue Jan 29, 2008 12:12 am

I remain intimidated. ha
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WebPilot
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FDEs

Post by WebPilot » Tue Jan 29, 2008 4:26 pm

Hmmm, I can identify.

Here, a Courant number of 0.8 was used to generate the plots.

Comparing this with the last plot, notice an overall shift to the
right and the 'smearing' of the numerical solution is diminishing.

Image (c.t.)
Image

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FDEs

Post by WebPilot » Wed Jan 30, 2008 3:37 pm

Notice here that with a value of Cn=1.0, the numerical solution
converges with the exact, analytical solution. This is the value
I shall use in all subsequent calculations for this problem.

Image (c.t.)
Image

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the Makings of a Right Moving Pulse (re: FDEs)

Post by WebPilot » Wed Jan 30, 2008 9:26 pm

Heretofore, I have been showing images after 20 time steps. Now,
this is a plot of the first 2 timesteps, using CN=1.0.

The pulse is starting to move to the right ...

Image (c.t.)

NOTE: 'x' is a dimensionless parameter which signifies the
station along the tube. For example, a point 10" from the left end of
a 100" tube would have an 'x' value of 10/100 = 0.10 .
Image

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... Right Moving Pulse (re: FDEs)

Post by WebPilot » Thu Jan 31, 2008 4:27 am

Here, all 20 time steps are plotted. The pulse is definitely moving to
the right.

Image (c.t.)
Image

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... linear wave equation (re: FDEs)

Post by WebPilot » Fri Feb 01, 2008 4:22 am

Now, some of you are probably wondering why study the
linear wave equation.

Well, ...
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... advective transport (re: FDEs)

Post by WebPilot » Fri Feb 01, 2008 2:59 pm

Advective transport refers to a substance being carried along
with fluid motion. For example, consider a contaminant being
advected downstream with some fluid flowing through a one-
dimensional pipe at a constant velocity, u.

The concentration profile (or waveform) propagates with con-
stant speed, u and unchanged shape. In this context, the
linear wave equation is generally called the advection equation.
Image

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... one-way wave equation (re: FDEs)

Post by WebPilot » Sun Feb 03, 2008 4:02 am

The phenomenon of wave motion is observed in its most basic form
if we model a sound wave traveling down a tube of gas or through an
elastic solid. In this case the molecules of the gas or solid barely move,
and yet a distinct wave can propagate through the material with its
shape essentially unchanged over long distances, and at a speed c
(the speed of sound in the material) that is much larger than the velocity
of material particles.

A sound wave propagating in one direction (to the right with speed c > 0)
can be modeled by the equation

wt(x, t) + c wx(x, t) = 0,

where w(x, t) is an appropriate combination of the pressure and particle
velocity.

This again has the form of a scalar first-order hyperbolic equation. In this
context the equation is some-times called the one-way wave equation
because it models waves propagating in one particular direction.
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... DiffEQ (re: FDEs)

Post by WebPilot » Sun Feb 03, 2008 3:33 pm

Let's try to solve an ordinary DiffEQ, first.
Image

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... spreadsheet calc (re: FDEs)

Post by WebPilot » Sun Feb 03, 2008 4:17 pm

It is my intention to use a spreadsheet for this calculation.

Solving the ordinary diffEQ first, will prepare the interested
reader to solve the linear wave equation, next.

I am going to use Excel, but any other spreadsheet may
do. So, follow along and see if you can get it to work in
the one you use, too.
Image

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Post by GRIM » Sun Feb 03, 2008 6:45 pm

Hey Web Pilot
I see you dont get many replies to your posts , rest assured I for one am eagerly awaiting the next instalment , fascinating stuff , and the car and bike photos are excellent.

keep em coming and thanks.

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... DiffEQ (re: FDEs)

Post by WebPilot » Tue Feb 05, 2008 6:19 am

You are quite welcome, GRIM.

~~~~~~~~~~~~

As a start, I've decided to use the differential equation describing
the deflection of a simply supported beam, due its own weight.

The derivation is in any engineering mechanics book.

Image (c.t.)

NOTE:

The beam's length is >> greater than its cross-sectional dimensions.

The deflection, y, is measured from the central axis; +tive down.

The station, x, is measured from the left support.
Image

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Bending in a ULB (re: FDEs)

Post by WebPilot » Tue Feb 05, 2008 5:45 pm

Uniformly Loaded Beam ...

Image (c.t.)
Image

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