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By Patryk Prus and Paul Pearlman |
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Part 1: The Hanging
Chain in Parametrics |
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| Parameterization of the hanging curve by arclength, where theta is the inclination angle (angle of the curve with respect to the horizontal) at any arclength, s, along the curve. | |
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| Parameterization: | Constraints: |
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| Energy: |
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| Modified Energy: | |
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| Euler-Lagrange Equation: | |
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| Transcendental System: | |
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| Solution of Transcendental System: |
For our example: b = 45 cm h = 6 cm L = 61.4 cm So our system becomes: |
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| Given the following initial conditions: | Mathematica computes the constraint coeffecients to be: |
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| Putting it all together: | |
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| Graph of y(x): | Picture of actual chain: |
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Other Trials: |
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Part 2: The Hanging Spring |
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| The potential energy of the hanging spring is comprised of three parts; gravitational potential, bending energy, and the spring's internal energy: | |
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| To simplify the energy, the equation was parametrized by the arclength of the wire (not the core curve): | |
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| Parameterization: | Constraints: |
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| Gravitational Potential Energy: | |
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| Bending Energy: | |
| The bending energy is equal to the square of the curvature (K^2). Because the spring can only be bent to a certain extent, the bending energy must be limited at a critial radius (1/r, where r is the radius of the spring). To accomplish this, the bending energy (K^2) is multiplied by 1/F(K), where F(K) is some function which approaches one until the critical radius is reached, and then quickly approaches 0. A modified version of the inverse tangent function fits these qualifications: | |
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| Spring Potential Energy: | |
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| Creating a System of Energies: | |
| Because the energy equation cotains both x[t] and y[t] (and their respective derivatives) it is necessary to perform two variations, one on x[t] and another on y[t]: | |
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Gravitational Potential Energy: |
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| Bending Energy: | |
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| Spring Potential Energy: | |
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| Constraints: | |
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| Putting it Together: | |
| The energy equations now look as follows: | |
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| Integrating by parts to factor out the eta terms and applying the Fundamental Lemma of Calculus of Variations yields the following two Euler-Lagrange equations: | |
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First Euler-Lagrange Equation |
Second Euler-Lagrange Equation |
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Physical Properties of the Spring: |
The energy equations involve many constant values which
depend on the spring used in the experiment. The following values pertain
to the spring used in this experiment:
radius of coils = r = 1.45 cm number of coils = 131 length of wire = L = 938.393729 cm mass = m = 21.6578 g density = rho = .0230796513 g/cm spring constant = k = 6.25571614 N/cm |
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This was as far as we were able to carry out the calculations of finding a function which describes the hanging curve of a spring. We finished with a system of fourth order Euler-Lagrange equations, which, when solved, should yield the correct shape of a hanging spring. In order to solve this system, the initial conditions for all four derivatives of each function (x and y) need to be established, a task which we were unable to perform. The "shooting method" could be used to test every set of possible conditions. If the necessary conditions could be found, the NDSolve function within Mathematica should be able to solve the system. |
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Our paper in PDF form |
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