Laplace transforms: Critically Damped Motion: Difference between revisions

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<math>\begin{bmatrix} \dot{x} \\ \ddot{x} \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ -k/m & -C/m \end{bmatrix} \begin{bmatrix} x \\ dot{x} \end{bmatrix} + \begin{bmatrix} 0 \\ 1/m \end{bmatrix}u(t)</math>
<math>\begin{bmatrix} \dot{x} \\ \ddot{x} \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ -k/m & -C/m \end{bmatrix} \begin{bmatrix} x \\ \dot{x} \end{bmatrix} + \begin{bmatrix} 0 \\ 1/m \end{bmatrix}u(t)</math>





Latest revision as of 16:58, 18 November 2009

Using the Laplace Transform to solve a spring mass system that is critically damped

Problem Statement

An 8 pound weight is attached to a spring with a spring constant k of 4 lb/ft. The spring is stretched 2 ft and rests at its equilibrium position. It is then released from rest with an initial upward velocity of 3 ft/s. The system contains a damping force of 2 times the initial velocity.

Solution

Things we know


Solving the problem















Apply the Initial and Final Value Theorems to find the initial and final values

Initial Value Theorem
Final Value Theorem


Applying this to our problem




Bode Plot of the transfer function

Transfer Function




Bode Plot

Fig (1)



     ==Break Points and Asymptotes==                                                                                                                                                      






Convolution











State Space






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Written By: Mark Bernet


Error Checked By: Greg Peterson