Periodic Solutions - Soft and Hard Springs
Previously we have always assumed that the force F(x) exerted by the spring on the mass is a linear function of x: (Hooke’s law). In reality, however, every spring in nature actually is nonlinear (even if only slightly so). Moreover, springs in some automobile suspension systems deliberately are designed to be nonlinear. Here, then, we are interested specifically in the effects of nonlinearity. If we assume also that the reaction of the spring is symmetric with respect to positive and negative displacements by the same distance, then F(-x)=-F(x), so F(x) is an odd function.
For a simple mathematical model of a nonlinear spring we therefore take F(x)=-kx+bx3 and the equation of motion becomes
mx''=-kx+bx3
If we introduce the velocity y(t)=x'(t)
of the mass with position x(t), then we get the equivalent first-order system
x'=y
my'=-kx+bx3
Dividing the above equations we obtain mydy+(kx-bx3)dx=0
Integration then yields (1/2)my2+(1/2)kx2-(1/4)bx4=E
for the equation of a typical trajectory. If write E for the arbitrary constant of integration because, then this expresses conservation of energy for the undamped motion of a mass on a spring.
The behavior of the mass depends on the sign of coefficient b. The spring is called
- hard if b<0
- soft if b>0
Soft Spring Oscillations: If b>0, then the system has the two critical points in addition to the critical point (0, 0). These three critical points yield the only solutions for which the mass can remain at rest. The following example illustrates the greater range of possible behaviors of a mass on a soft spring.
Graphs of x(t) and y(t)
Now let's go interactive 
Soft Springs
Hard Spring Oscillations: If b<0, then the only critical point of the system is the origin (0, 0). Each trajectory is an oval closed curve like those shown below, and thus (0, 0) is a stable center. As the point traverses a trajectory in the clockwise direction, the position and velocity of the mass oscillate alternately. The mass is moving to the right (with x increasing) when y>0, to the left when y<0. Thus the behavior of a mass on a nonlinear hard spring resembles qualitatively that of a mass on a linear spring with b=0. But one difference between the linear and nonlinear situations is that, whereas the period of oscillation of a mass on a linear spring is independent of the initial conditions, the period of a mass on a nonlinear spring depends on its initial position and initial velocity.
Graphs of x(t) and y(t)
Now let's go interactive Hard Springs
Comparison The phase plots illustrate a significant qualitative difference between hard springs with and soft springs. Whereas the phase plane trajectories for a hard spring are all bounded, a soft spring has unbounded phase plane trajectories (as well as bounded ones). However, we should realize that the unbounded soft-spring trajectories cease to represent physically realistic motions faithfully when they exceed the spring’s capability of expansion without breaking.
Comparison