Section 6.3 Delta Functions and Forcing
Exercises Exercises
Supplemental HW Problems.
The Dirac Delta function is defined as a “quick” impact, so that at a time \(t = t_0\) we have an impulse of magnitude \(1,\) but zero everywhere else.
In other words: \(\delta(t-t_0)=0\) for \(t\neq t_0\) but
\begin{equation*}
\int_{-\infty}^\infty\delta(t-t_0)dt=1
\end{equation*}
We can break the above integral up:
\begin{equation*}
\int_{-\infty}^\infty\delta(t-t_0)=\int_{-\infty}^c\delta(t-t_0)+\int_c^\infty\delta(t-t_0)
\end{equation*}
1.
From the right side of the equation above, consider the integral
\begin{equation*}
\int_{-\infty}^c\delta(t-t_0)
\end{equation*}
What is the value of this integral when \(c \lt t_0?\) What is the value of this integral when \(c \gt t_0?\) Write the single integral above as a piecewise function in that way.
2.
The Heaviside function is defined as:
\begin{equation*}
u_c(t) = \begin{cases} 0 \amp \text{if } t \lt c \\
1 \amp \text{if } t \geq c \end{cases}
\end{equation*}
What does that tell us about the integral in problem 1 and the Heaviside function?
3.
Since the Heaviside function and the integral we found in problem 1 are the same, take a derivative of both sides. What does that mean as far as the relationship between the Heaviside function and the Dirac Delta function?
4.
A spring-mass system involves a mass of \(1\) kg, an external friction coefficient of \(2\) kg/sec, and a spring constant of \(10\) kg/s\(^2.\) At time \(t = 0,\) the spring is stretched out by \(2\) meters from rest and released. At \(t = 5,\) a hammer smashes into the mass with a force of \(2\) Newtons.
What do you expect to happen here? Talk about short-run behavior and events, and the long-term behavior of this system.
-
Pick the best equation for the position of the mass from the choices below:
\begin{equation*}
x'' + 2x’ + 10x = 2u_0(t)
\end{equation*}
\begin{equation*}
x'' + 2x’ + 10x = 2u_5(t)
\end{equation*}
\begin{equation*}
x'' + 2x’ + 10x = 2u(t-5)
\end{equation*}
\begin{equation*}
x'' + 2x’ + 10x = 2\delta(t)
\end{equation*}
\begin{equation*}
x'' + 2x’ + 10x = 2\delta(5)
\end{equation*}
\begin{equation*}
x'' + 2x’ + 10x = 2\delta(t-5)
\end{equation*}
Using Laplace transforms, find \(x(t)\text{.}\)
Does your function match up with what you expected from part (a) – does the function for \(x(t)\) match the situation, in the short-run and long-term?
Appendix to Judson's 6.3.
One of the examples in the text refers to \(\sinh\) and \(\cosh.\) These are the Hyperbolic Functions, and we pronounce \(\sinh\) as “sinch” and \(\cosh\) as “cosh.” They are, put very roughly, a way to do periodic function work with exponential functions because of the way they are defined:
\begin{align*}
\sinh(x) \amp = \frac{e^x-e^{-x}}{2}\\
\cosh(x) \amp = \frac{e^x+e^{-x}}{2}
\end{align*}
The derivative of \(\sinh\) is \(\cosh,\) and the derivative of \(\cosh\) is \(\sinh\text{.}\)
5.
Take derivatives of both to verify this!
We have a variety of trig-like identities with these objects, and you can find a summary of these (and more information) at the
Wikipedia page 1 . A few of the important ones are:
\begin{align*}
\sinh(-x) \amp = -\sinh(x)\\
\cosh(-x) \amp = \cosh(x)\\
\cosh(x) + \sinh(x) \amp = e^x\\
\cosh(x) - \sinh(x) \amp = e^{-x}\\
\cosh^2(x) - \sinh^2(x) \amp = 1
\end{align*}
6.
Show these algebraically!
And an interesting analysis of these can be found
on this website 2 from Purdue University.
en.wikipedia.org/wiki/Hyperbolic_functions
www.math.purdue.edu/~pvankoug/math266/sinh_cosh_sin_cos.pdf
