Motivation - mass-spring systems.

Section 3.1 Motivation - mass-spring systems.

This chapter is concerned with second order differential equations, and in particular those with constant coefficients. That is, we're going to be spending quite a bit of time thinking about equations of the form

$\begin{equation*} a y'' + b y' + c y = 0 \end{equation*}$

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and

$\begin{equation*} a y'' + b y' + c y = f(t). \end{equation*}$

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At first glance, these equations seem artificially simple in structure. However, some of the most useful differential equations in the physical sciences and mathematics have this form, which motivates our close attention to second order linear equations with constant coefficients.

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Subsection 3.1.1 Undamped mass-spring systems

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The pictures above are a derivation of the differential equation for a simple mass-spring system. Notice that the resulting equation has a second derivative

$x''$ in it - thus, this is a second order equation where the position of the mass

$x(t)$ relative to the equilibrium position is a function of time

$t\text{:}$

$\begin{equation*} x'' + \frac{k}{m} x = 0. \end{equation*}$

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Also, note the important fact that the equation is linear and has constant coefficients. If we want to describe how the solutions to this equation behave, we should study second order equations with constant coefficients.

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Subsection 3.1.2 Damped mass-spring systems

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One way to model more complicated situations with a mass-spring system is to include a damper that applies force against the direction of motion. The spring/shock absorber system in a car wheel is an example of a damped mass-spring system. The pictures below derive the equation for this in the case that we assume that the damper exerts a force proportional to and in the opposite direction from the velocity of the mass.

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Thus, the equation for a damped mass-spring system is

$\begin{equation*} x'' + \frac{b}{m} x' + \frac{k}{m} x = 0 \end{equation*}$

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which is also a second order linear equation with constant coefficients.

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The mass-spring system is one of the most useful models in all of science. For example, RLC circuits (resistor/inductor/capacitor) are typically modeled as mass-spring systems.

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This motivates the study of second order linear differential equations with constant coefficients, even though that might seem like an extremely restricted family of problems to think about. In later sections, we will extend our ideas to consider what happens if an external driving or forcing function is applied to the system.

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We should already have an idea about the sorts of functions that solve these systems.

Physical intuition tells us that when we release a mass attached to a spring from a non-equilibrium starting position, the mass oscillates up and down around the equilibrium position. This suggests that sine or cosine waves might be involved in the solution.
Mathematical intuition tells us that functions that are related to their second derivatives by constant factors are also sines and cosines. That is, $(\sin x)'' = - \sin x\text{.}$

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We will see that, indeed, this intuition is correct for undamped systems with no external forces.