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Mathematics LibreTexts

6.9: Discontinuous Forcing

In the last section we looked at the Heaviside function its Laplace transform. Now we will use this tool to solve differential equations.  

Example

A 1 kg bar is attached to a spring with spring constant 5 and damping constant 2. It is pulled down 3 inches from it equilibrium position and the released. After one second a constant force of 10 Newtons is exerted on the bar. The force remains turned on indefinitely.  Determine the equation of motion of the bar.

Solution

We have the differential equation 

\[ y'' + 2y' + 5y  =  10u_1(t)\]

with

\[ y(0)  =  3 \]

\[  y'(0)  =  0 .\]

We solve this by the method of Laplace transforms.  We have

\[  \begin{align}    L\{y''\}  + 2L\{y'\} + 5L\{y\}  &=  10L\{u_1(t)\}  \\ \implies s^2L\{y\} - 3s - 0 + 2(sL\{y\} - 3) + 5L\{y\}  &=  10e^{-s} /s   \\  \implies  (s^2 + 2s + 5)L\{y\} - 3s - 6  &=  10e^{-s} /s  \\  \implies  L\{y\} &= \dfrac{3s + 6 }{s^2 + 2s + 5} + \dfrac{10 }{s(s^2 + 2s + 5)} . \end{align}  \]

We use partial fractions on the second term

\[ \dfrac{10}{s(s^2 + 2s + 4)} = \dfrac{A}{s} + \dfrac{Bs+C}{s^2+2s+5} \]

\[   A(s^2 + 2s + 5) + (Bs + C)s  =  10 \]

\[    (A + B)s^2 + (2A + C)s + 5A  =  10 \]

The constant term gives

\( A =2 \)

Thus

\( B = -2 \)        \(C = -4 \)

We can complete the square

\[ s^2 + 2s + 5 = (s+1)^2 +4 \]

Putting all the algebra together we get       

\[ \begin{align} L\{y\}  &= \dfrac{3s + 6}{(s + 1)^2 + 4} + e^{-s}\dfrac{2}{s} + e^{-s}\dfrac{-2s - 4}{ (s + 1)^2 + 4 } \\ &= 3\dfrac{s + 1 + 1}{(s + 1)^2 + 4} + e^{-s}\dfrac{2}{s} - 2e^{-s}\dfrac{(s + 1) + 1 }{(s + 1)^2 + 4 } \\ &= 3\dfrac{s + 1}{(s + 1)^2 + 4} + 3/2 \dfrac{2}{(s + 1)^2 + 4} + e^{-s}\dfrac{2}{s} - 2e^{-s}\dfrac{(s + 1)  }{(s + 1)^2 + 4 } - e^{-s}\dfrac{2 }{(s + 1)^2 + 4 } \end{align}\]

Now use the table to take the inverse Laplace transform to get

\[  y  =  3e^{-t}\cos(2t) + 3/2 e^{-t -1}\sin(2t) + 2u_1 - 2u_1(t)e^{-t +1}\cos(2t - 2) - u_1(t)e^{-t +1} \sin(2t - 2). \]

Below is the graph

  

Contributors

  • Integrated by Justin Marshall.