?Plot a deformed shape given the displacements

1. Sep 24, 2013

TheFerruccio

Plot a deformed shape given the displacements

Preliminary statement: This class has a different convention.

Instead of writing x and y, $x_1$ and $x_2$ are the variables in their respective orthogonal directions.

I need to plot how shapes deform, so I am given an equation for the deformation in the $x_1$ direction, and the deformation in the $x_2$ direction, denoted $u_1$ and $u_2$, respectively.

Problem Statement

Plot how a rectangle with sides a (horizontal) and b (vertical) deforms under the following deformation equations:
$u_1=k({x_1}^2+{x_2}^2)$
$u_2=k(2x_1+x_2)$

Attempt at Solution

As far as I can tell, I have never had to plot something quite like this. The closest I can relate to this is implicit functions and parametric functions. I know that the displacement is simply the original location $x_i$ plus the displacement $u_i$

Thus, I can turn the displacement function into a function for the "final value" as such, denoting $y_1$ and $y_2$ as the final locations in the $x_1$ and $x_2$ directions, respectively.

$y_1=x_1+k({x_1}^2+{x_2}^2)$
$y_2=k(2x_1+x_2)+x_2$

I know what this is doing. These two equations combine to create a mapping from $\mathbb{R^2}\rightarrow\mathbb{R^2}$. I can fix one of the directions to plot a line. In this case, I can take the $x_1$ and $x_2$ coordinates and plot a deformed line from an original line by holding one of my input variables constant, such as defining $x_2=0$ to see how a horizontal line located at $x_2=0$ gets deformed.

This would reduce the equations to:

$y_1=x_1+k({x_1}^2)$
$y_2=k(2x_1)$
Unfortunately, I still have no idea how I would go about doing this considering a and b are left arbitrary. I need some assistance to spark my brain and push me in the right direction with this. Wolfram Alpha has a parametric plotter, but it has no built-in functions to handle constants, and just assumes that everything is a variable. Does anyone know how I can approach this problem in an intuitive manner? My attempts to tackle it have just been failing.

Last edited: Sep 24, 2013
2. Sep 24, 2013

TheFerruccio

4char

Last edited: Sep 24, 2013