to accompany the paper
J. Galan, W.B. Fraser, D.J. Acheson & A.R. Champneys
in Journal of Sound and Vibration Vol 280 (2005) 359-377.
dx/dt = y
dy/dt = - k*y - (s^2)*x + ( 1 + a*(w^2)*cos(w*t) ) * sin x
dt/dt = 1
and these are integrated by a 4th-order Runge-Kutta method with time step h = .0025.
The parameters in the software are related to those in the published paper as follows:
pivot vibration amplitude a = epsilon
pivot vibration frequency w = omega
damping constant k = gamma
spring constant s = square root of B
The spring constant s can be gradually varied as the simulation proceeds.
While simulation is running press:
s/t to increase/decrease s
h/k to increase/decrease time step (i.e. simulation speed)
p to slightly disturb vertically upward state
c to clear screen
q to start a different demonstration
e to end
In each demonstration the parameters other than s are fixed.
Here a = 0.05 , w = 10 , k = 0.1 . The product a*w is less than 0.707 . Input, say, s = 0.5, and gradually increase s . The flopped-down pendulum gradually moves smoothly upward to the inverted state, which is stabilised when s > 0.935 . Decreasing s again reverses the process; there are no 'jumps'.
Here a = 0.1 , w = 10 , k = 0.1 .
The product a*w is greater than 0.707 but less than 1.414, so the
upside-down state is still unstable without the spring.
Input, say, s =0.5, and gradually increase s until the pendulum suddenly 'jumps' to the upward vertical when s > 0.80 . Then gradually decrease s again, testing for stability of the vertical state by pressing p. There is hysteresis: the pendulum does not 'jump' back to the lower state until s < 0.71 .
Similar to Demonstration 2, but with larger a and smaller w.
In all three demonstrations, rather larger values of s can lead to instability of the upside down state for a range of values around s = w/2 . Thus in Demonstration 2, for instance, having achieved stability of the upside-down state with, say, s = 2 , gradually increasing s to 5.0 will make the upside down state unstable , and large oscillations result, though further increases still in s will give restore stability again.
Here a = 0.7 , w = 10 and k = 0.4 .
Inputting s = 5 , i.e. w/2 , again leads to instability of the
upside-down state, and because the pivot amplitude is quite large, the
pendulum gets flung about in large, chaotic
On a quite different note, a small value of s , such as 0.3 , leads to quite different exotic dynamics in the form of (what appear to be) chaotic 'relaxation oscillations'. The (very weak) spring now gets wound up severely as the pivot vibrations whirl the pendulum repeatedly around the pivot, and eventually the spring 'fights back', unwinding very suddenly to release its pent-up elastic energy.
The software is a compiled DOS-based program
using MS QuickBasic 4.5 . It
should be ready-to-run on most PCs with MS Windows operating systems. If
you wish to download the zipped version, and
need unzipping software, try