Spring Mass Systems

This was a fun one.

A few weeks back we started second order differential equations in my DE class. After working through homogenous DE’s with constant coefficients, Cauchy-Euler DE’s, Undetermined Coefficients, and Variation of Parameters, we finally started looking at applications of second order DE’s. The big one that comes to mind are Spring-Mass Systems.

If the set up escapes you, it goes a little something like this..

Imagine you have a mass of m kg, the force a spring applies to that mass is proportional to the displacement of that mass from equilibrium, x m, in the direction opposite the displacement. This is known as Hooke’s Law, and is written as.

F=ma=mx”=-kx

Hooke’s law makes the assumption that the mass is sliding across a frictionless surface, however if you think about how friction will impact the sliding of the mass, the force due to friction will act on the mass proportional to the velocity of the mass, in the opposite direction, thus the sum of forces acting on the mass is given by

F=ma=mx”=-kx-bx’

or

mx”+bx’+kx=0

A second order, linear, homogeneous differential equation.

Now, remembering how these work we have three cases for solutions.

1. b^2-4mk>0 – In this case, the system is said to be over damped. The mass will slowly return to its equilibrium position.
2. b^2-4mk<0 – In this case, the system is said to be under damped. In this case the mass will oscillate, and if b is not equal to zero, then the mass will oscillate to its equilibrium position.
3. b^2-4mk=0 – In this case, the system is said to be critically damped. This is a sensitive case in the sense that decreasing friction / increasing mass or spring constant will cause the system to become under damped and increasing friction / decreasing mass or spring constant will cause the system to become over damped. The other way to think about the behavior of a critically damped system is that the mass will return to equilibrium as fast as possible without oscillating. 

You can experiment with each case by playing with the values of m, b, and k in the applet here.