7.3. Homogeneous linear second-order ODEs with constant coefficients

Example: Spring and dashpot

We are analyzing how the drag force affects the spring oscilations. This is a mechanics example, but models like this are also used for the rheology of viscoelastic materials.

A mass m is attached to a Hookean spring with spring constant k and experiences a drag force as it moves with drag coefficient \(\gamma\).

If x is the displacement of the spring, the governing equation for x is given by Newton’s equations:

(7.1)\[\begin{equation} m x'' + \gamma x' + k x = 0 \end{equation}\]

The first term represents the acceleration of the mass, the second term is the drag force, and the third term is the spring force.

For what values of \(\gamma\) will the mass oscillate after it is stretched?

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To answer this, first solve the roots of the characteristic polynomial for this second-order ODE:

(7.2)\[\begin{align} &m \lambda^2 + \gamma \lambda + k = 0 \\ \lambda_{1,2} &= \frac{ -\gamma \pm \sqrt{\gamma^2 - 4mk}}{2m} \end{align}\]

You will only get oscillations if \(\lambda_{1,2}\) is complex, which only occurs when \(\gamma^2 < 4mk\). In this case:

(7.3)\[\begin{equation} \lambda_{1,2} = -\frac{\gamma}{2m} \pm \frac{i}{2m}\sqrt{4mk - \gamma^2} \end{equation}\]

so the general solution for x is:

(7.4)\[\begin{equation} x(t) = \exp\left(-\frac{\gamma}{2m} t \right) \left [ c_1 \cos(\omega t)+ c_2 \sin(\omega t) \right] \end{equation}\]

where \(\omega = \sqrt{4mk - \gamma^2}\). What does \(\gamma\) do to the solution?

• \(\gamma\) dampens the oscillations to decay to zero. The decay time \(2m/\gamma\) increases as \(\gamma\) decreases, and without it, oscillations go on forever.

• \(\gamma\) also changes the frequency of the oscillations through \(\omega\).