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Analysis of Diamond Cantilever Vibration (view source)
Revision as of 00:08, 4 November 2012
, 00:08, 4 November 2012→Model
== Model ==
== Model ==
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.
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>
By assuming constant thickness, we can simplify this equation to something much more manageable.
<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]]