A spring is nonlinear if the force exerted by the spring is a nonlinear function of the displacement. The simplest nonlinear spring has
F(x) = -k x + b x3where k and b are spring constants.
If b < 0 the spring is called hard. The behaviour of a mass on a hard spring is similar to that of a mass on a linear spring.
If b > 0 the spring is called soft. How does a mass on a soft spring behave?
Non-linear mechanical systems are described in Section 6.4 of Edwards & Penney (1996). This problem is Example 1 on page 371, with spring constants k = 4 and b = 1.
The energy in the system is
E = m v2/2 + k x2/2 - b x4/4.
The behavior of the system depends on the energy. When the energy is low, the motion is periodic. The first animation shows this behaviour when x(0) = -1, v(0) = 0 and E = 1.75. The motion of the mass was calculated using a Runge-Kutta method.
The period increases as E approaches 4. The next animation shows the behaviour of the system when x(0) = -1.9, v(0) = 0 and E = 3.962.
When E > 4 the motion is unbounded. The next animation shows the behaviour of the system when x(0) = -3, v(0) = 5.536 and E = 4.00165. (If E is much higher the displacement and velocity increase too rapidly to see what is happening.)
The final animation demonstrates this behaviour for x(0) = 3, v(0) = 5.535 and E = 3.99811. The description of the motion in Edwards & Penney does not agree with their phase portrait!
