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?Plot a deformed shape given the displacements

  1. Sep 24, 2013 #1
    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. jcsd
  3. Sep 24, 2013 #2
    4char
     
    Last edited: Sep 24, 2013
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