Laplace transforms: Under-damped Mass-Spring System on an Incline: Difference between revisions
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===Laplace Transform=== |
===Laplace Transform=== |
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\mathcal{L} \left\{ \ddot{x} + \frac{\lambda}{m}\,\dot{x}+\frac{k}{m}\,x \right\} = \mathcal{L} \left\{ g \sin \theta \right\} |
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\mathcal{L} \left{ \right} |
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\Rightarrow\ s^2\,X(s) - s\,x(0) - x'(0) + \frac{\lambda}{m}\,s\,X(s) - \frac{\lambda}{m}\,x(0) + \frac{k}{m}\,X(s) = g \sin \theta \, \left(\frac{1}{s}\right) |
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Revision as of 23:45, 25 October 2009
Brandon.plubell 05:44, 26 October 2009 (UTC)
Under-Damped Mass-Spring System on an Incline
Part 1 - Use Laplace Transformations
Problem Statement
Find the equation of motion for the mass in the system subjected to the forces shown in the Free Body Diagram (FBD). The inclined surface is coated in SAE 30 oil.
Initial Conditions and Values
- A is the area of the box in contact with the surface
- g is the gravitational acceleration field constant
- bt is the thickness of the fluid covering the inclined surface
- μ is the viscosity constant of the fluid;
- m is the mass of the box
- k is the spring constant
Let the initial conditions be:
Force Equations
The sum of the moments in the x direction yields the equatiom
Where
To make the algebra easier, let
Then, from the sum of forces equation:
Laplace Transform