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Math Help - Mathematical Model of a Catenary

  1. #1
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    Mathematical Model of a Catenary

    Hi everyone,
    I have a Yr 11 maths assignment which investigates the geometry of a suspension bridge (mainly catenarys). I used a piece of 5m rope to create the shape of one and recorded the following measurements:
    X Axis (cm)
    0
    39.1
    78.2
    117.3
    156.4
    195.5
    234.6
    273.7
    312.8
    351.9
    391
    430.1
    469.2


    Y Axis (cm)
    90
    68.5
    51
    37.2
    27.4
    19.7
    17.8
    19.7
    27.4
    37.2
    51
    68.5
    90





    One of the questions asks to generate a mathematical model using either y=ax^2, y=ax^3, y=a/x or y = the square root of x.The textbook also states
    sometimes non-linear relationships can be reduced to linear relationships. For instance, y =m/x + c is a hyperbola. However, if we substiture p for 1/x, the rule becomes linear: y=mp+c. The graph of y versus p will be the straight line with the gradient of m and the y intercept of c. These values (m and c) can then be established from the graph and thus the hyperbolic model can be determined.
    What do I use? How can I generate a model? I have tried but none have worked for more than one value .
    Thanks in advance,
    Daniel
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  2. #2
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    Hyperbolas can be rotated so that the orientation is closer.

    You will have to keep modelling until you settle on the Hyperbolic Cosine, cosh(x) = \frac{e^{x}+e^{-x}}{2}, but that's a different animal than those you have suggested.
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by daewoo_lowrider View Post
    Hi everyone,
    I have a Yr 11 maths assignment which investigates the geometry of a suspension bridge (mainly catenarys). I used a piece of 5m rope to create the shape of one and recorded the following measurements:

    X Axis (cm)


    0


    39.1


    78.2


    117.3


    156.4


    195.5


    234.6


    273.7


    312.8


    351.9


    391


    430.1


    469.2




    Y Axis (cm)


    90


    68.5


    51


    37.2


    27.4


    19.7


    17.8


    19.7


    27.4


    37.2


    51


    68.5


    90






    One of the questions asks to generate a mathematical model using either y=ax^2, y=ax^3, y=a/x or y = the square root of x.The textbook also states
    sometimes non-linear relationships can be reduced to linear relationships. For instance, y =m/x + c is a hyperbola. However, if we substiture p for 1/x, the rule becomes linear: y=mp+c. The graph of y versus p will be the straight line with the gradient of m and the y intercept of c. These values (m and c) can then be established from the graph and thus the hyperbolic model can be determined.
    What do I use? How can I generate a model? I have tried but none have worked for more than one value .
    Thanks in advance,
    Daniel
    For instance, take y = ax^3. Define a new variable p = x^3 and calculate all the p values. Then do a linear regression using p as your "x" value and y as your "y" value.

    -Dan
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