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math245lab09

# math245lab09 - Math245 Computer Lab set#9 Fall 2008 Hard...

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Math245 Computer Lab set #9, Fall 2008 Hard spring oscillator (nonlinear 2nd-order ODE) In this lab, we will just glance at the strange effect of nonlinearity by using a hard spring oscillator model as an example. Nonlinear spring model Consider a mechanical spring. When stretching the spring, the force needed to stretch ( F s ) is linearly proportional to the stretch ( x ), if the stretch is sufficiently small. In this case, the spring force is modeled by the Hooke’s law of linear spring . F s = kx where k is called spring constant . If the stretch is larger and larger, this Hooke’s law loses its validity, because some nonlinearity of the spring material comes into play. In some occasions, the spring force is modeled by the following nonlinear expression. F s = kx + sx 3 where s is a constant parameter. If s > 0, the spring called hard spring , otherwise ( s < 0) the spring is called soft spring . F s x k x k x + s x ( s > 0) 3 k x + s x ( s < 0) 3 linear spring hard spring soft spring F s x Spring - mass - damper model Consider a forced, spring-mass-damper model as shown below, where we use a hard spring.

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