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== Model ==
 
== Model ==
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The cantilever system is best modeled by first considering the simplest version, a beam of uniform width, thickness, and height, fixed at one end and free at the other. It's possible to derive a fourth-order differential equation to describe the motion of this system by comparing the shear and torque on each heightwise section of the beam.
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The cantilever system is best modeled by first considering the simplest version, a beam of uniform width, thickness, and height, fixed at one end and free at the other. It's possible to derive a fourth-order differential equation to describe the motion of a beam system by comparing the shear and torque on each height-wise section of the beam.
    
<math>E\frac{d^2}{dx^2}(WT^3\frac{d^2y}{dx^2})=-\rho W T \frac{d^2y}{dx^2}</math>
 
<math>E\frac{d^2}{dx^2}(WT^3\frac{d^2y}{dx^2})=-\rho W T \frac{d^2y}{dx^2}</math>
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By assuming constant thickness, we can simplify this equation to something much more manageable.
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<math>\frac{d^4y}{dx^4}=-(\frac{\rho}{ET^2})\frac{d^2y}{dt^2}</math>
    
[[Media:UniformAnalysis.pdf|Uniform-Width Analytical Model]]
 
[[Media:UniformAnalysis.pdf|Uniform-Width Analytical Model]]
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