Physics Involved

Understanding a system of coupled springs, such as modeled by Spring Chain, requires a small knowledge of single spring physics and differential equations. The motion of a mass attached to a spring can be derived from Hooks law, which relates the force exerted by the spring to the spring’s displacement from equilibrium. Simply stated, Hooks law says:


where k, known as the spring constant, is a measurable characteristic of the spring. According to this relationship, the acceleration of the mass attached to the spring will be directly proportional to the displacement from equilibrium. From this equation, it can be shown that for an undamped and undriven spring-mass system, the position of the mass as a function of time is given by the equation

x=A cos(wt+d)

where w is equal to the square root of the spring constant divided by the mass, and d is related to the distance the mass is from equilibrium at time t=0.

For a system of coupled oscillators, the system becomes slightly more complicated. The force on any single mass now will depend on that mass’s separation distance from its two neighboring masses on either side. In such a situation, Hooks law becomes:


where xn+1 and xn-1 are the positions of the particle before and after the particle for which the position is being calculated. As you will note, this makes for a slightly more difficult differential equation, because the acceleration of one mass depends not only on its position, but the position of the two particles next to it. Consequently, this is a “coupled” differential equation which will be much easier to solve using numerical methods and a computer than using pencil and paper.

00000002.gifSpring Chain will solve this differential equation for you.