Laplace transforms: Under-damped Mass-Spring System on an Incline: Difference between revisions
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===Problem Statement=== |
===Problem Statement=== |
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Find the equation of motion for the mass in the system subjected to the forces shown in the |
Find the equation of motion for the mass in the system subjected to the forces shown in the free body diagram. The inclined surface is coated in 1mm of SAE 30 oil. |
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[[Image:BP_Setup-1.jpg|right]] |
[[Image:BP_Setup-1.jpg|right|Problem Setup]] |
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* g is the gravitational acceleration field constant |
* g is the gravitational acceleration field constant |
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* b<sub>t</sub> is the thickness of the fluid covering the inclined surface |
* b<sub>t</sub> is the thickness of the fluid covering the inclined surface |
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* μ is the viscosity constant of the fluid |
* μ is the viscosity constant of the fluid |
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* m is the mass of the box |
* m is the mass of the box |
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* k is the spring constant |
* k is the spring constant |
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</math> |
</math> |
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Let the initial conditions be |
Let the initial conditions be |
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<math> |
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===Force Equations=== |
===Force Equations=== |
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[[Image:BP_FBD-1.jpg|right]] |
[[Image:BP_FBD-1.jpg|right|Free Body Diagram]] |
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The sum of the |
The sum of the forces in the x direction yields the equation |
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<math> |
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</math> |
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Then, from the sum of forces equation |
Then, from the sum of forces equation |
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==Part 4 - Breakpoints and Asymptotes on Bode Plot== |
==Part 4 - Breakpoints and Asymptotes on Bode Plot== |
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From the plot in |
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Revision as of 21:23, 27 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. The inclined surface is coated in 1mm of 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 forces in the x direction yields the equation
Where
To make the algebra easier, let
Then, from the sum of forces equation
Laplace Transform
If we let be 0 and rearrange the equation,
The above is the transfer function that will be used in the Bode plot and can provide valuable information about the system.
Inverse Laplace Transform
Since the Laplace Transform is a linear transform, we need only find three inverse transforms. All of the these have complex roots, since . Because I am not yet comfortable finding the inverse with complex roots by hand, I used a laplace transform program for the TI-89.
Equation of Motion
Putting it all back together again gives,
It is useful to have the equation in the form given above because can be varied and still give accurate results. The Matlab (or Octave) script below can be edited as described. Take note! cannot be altered (else the inverse Laplace is false)!
Part 2 - Final and Initial Value Theorems
Initial Value Theorem
As was derived in class, there are two theorems that relate the initial and final values (in this case positions) of the output functions in the t domain with the output function in the s domain. In a case such as this, in which the initial values are given, the initial value theorem is just a check.
Taking the limit of gives
Final Value Theorem
The Final Value Theorem is a very useful tool that will show what the final value of the output function (as ), which in this case is the final position of the block. Notice that it is not the unstretched length of the spring (else ). It is also of interest to note that only the input function comes into play here, as all the others go to zero, and is not dependent on the initial position or velocity.
Which can be seen in the plot in Equation of Motion section.
Part 3 - Bode Plot
The bode plot shows useful information about the system we are analyzing. It has only to do with the transfer function, which means that it does not change based upon the input. However, it can show what a given frequency of a harmonic input will do to the output. For my example, it can be seen that at about there is a rise in the magnitude of the transfer function. If it were hit with a corresponding frequency by an input function, it could have very larg oscillations.
Part 4 - Breakpoints and Asymptotes on Bode Plot
From the plot in