Laplace transforms: Critically Damped Spring Mass system

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

Problem Statement

An 98 Newton weight is attached to a spring with a spring constant k of 40 N/m. The spring is stretched 4 m and rests at its equilibrium position. It is then released from rest with an initial upward velocity of 2 m/s. The system contains a damping force of 40 times the initial velocity.

Solution

Given

$m=\frac{98}{9.81}$

$\text{Spring Constant k=40}$

$\text{Damping Constant C=40}$

$\text{x(0)=0}$

$\dot{x}(0)=-4$

$\text{Standard equation: }$

$m\frac{d^{2}x}{d{t}^2}+C\frac{dx}{dt}+khx=0$

Solving the problem

$\text{Therefore the equation representing this system is.}$

$\frac{98}{9.8}\frac{d^{2}x}{d{t}^2}=-40x-40\frac{dx}{dt}$

$\text{Now we put the equation in standard form}$

$\frac{d^{2}x}{d{t}^2}+\frac{40}{10}\frac{dx}{dt}+\frac{40}{10}x=0$

$\text{Now that we have the equation written in standard form we need to send}$ $\text{it through the Laplace Transform.}$

${ℒ}[\frac{d^{2}x}{d{t}^2}+\frac{40}{10}\frac{dx}{dt}+\frac{20}{5}x]$

$\text{And we get the equation (after some substitution and simplification)}.$

${\mathbf{s}}^{2}X(s)+4\mathbf{s}X(s)+4\mathbf{X}(s)=-4$

$\mathbf{X}(s)(s^2+4s+4)=-4$

$\mathbf{X}(s)=-\frac{4}{(s+2{)}^2}$

$\text{Now that we have completed the Laplace Transform}$ $\text{and solved for X(s) we must so an inverse Laplace Transform. }$

${{ℒ}}^{-1}[-\frac{4}{(s+2{)}^2}]$

$\text{and we get}$

$\mathbf{x}(t)=-4t{e}^{-2t}$

$\text{So there you have it the equation of a Critically Damped spring mass system.}$

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

Initial Value Theorem

$\underset{s\to \mathrm{\infty }}{\lim }sF(s)=f(0)$

Final Value Theorem

$\underset{s\to 0}{\lim }sF(s)=f(\mathrm{\infty })$

Applying this to our problem

$\text{The Initial Value Theorem}$

$\underset{s\to \mathrm{\infty }}{\lim }\mathbf{s}\mathbf{X}(s)=-\frac{4}{(s+2{)}^2}$

$\underset{s\to \mathrm{\infty }}{\lim }\mathbf{s}X(s)=-\frac{4}{(\mathrm{\infty }+2{)}^2}=0$

$\text{So as you can see the value for the initial position will be 0. Because the infinity in the denominator always makes the function tend toward zero.}$

$\text{Which makes sense because the system is initially in equilibrium. }$

$\text{The Final Value Theorem}$

$\underset{s\to 0}{\lim }\mathbf{s}X(s)=-\frac{4}{(s+2{)}^2}$

$\underset{s\to 0}{\lim }\mathbf{s}X(s)=-\frac{4}{(0+2{)}^2}=-\frac{4}{4}$

$\text{This shows the final value to be}$ $-\frac{4}{4}ft$

$\text{Which appears to mean the system will be right below equilibrium after a long time. }$

Bode Plot of the transfer function

Transfer Function

$\mathbf{X}(s)=-\frac{4}{(s+2{)}^2}$

Bode Plot

$\text{This plot is done using the control toolbox in MatLab. }$

Fig (1)

Created by Greg Peterson

Checked by Mark Bernet