Stability diagrams for the inverted pendulum with varying amounts of damping. The stable regions are shaded gray and the unstable regions are not shaded. (a) , (b) , (c) , (d) , (e) zoomed view, (f) zoomed view, (g) extended view, (h) extended view. (a)–(d) all have the same scales; scales for figures (e) and (f) are restricted in order to display further detail and the scales for (g) and (h) are greatly expanded such that the lower boundary can be observed. Damping increases, or equivalently the quality decreases, from left to right. The solid black line (–) corresponds to the modified Blackburn approximation Eq. (27), the dashed line (– –) to the averaging method Eq. (24), and dash-dot (- • -) line to the Wolf approximation Eq. (26).
Mathieu solutions for and . (a) Solution slice, (b) perturbation instability region waterfall plot, (c) stable region waterfall plot, (d) parametric resonance region waterfall plot. The large Floquet exponents associated with the solution in (d) prevents oscillations in the first few periods.
Undamped hanging pendulum stability diagram with inverted pendulum boundary. The shaded patches for are partially the result of the minor band density increasing and partially poor numerical accuracy.
Real part of the Floquet exponent across the parameter space. Positive values indicate unstable regions. The Floquet exponents in the parametric resonance region are much greater than those in the perturbation region.
Experimental apparatus, (a) paddles used in the experiment to produce varying amounts of damping; the numbers correspond to those in Table I, (b) physical pendulum with mounted permanent magnets.
Experimental data and model predictions for the four paddles. The five lines indicate the stability boundaries as calculated from the numerical routine and the symbols represent experimental data points. The region to the right of each line is stable and the region to the left of each line is unstable.
Characteristics of physical pendulums used in the experiment.
Solution forms of Eq. (3) based on multipliers. The characteristic multipliers for cases 1 and 2 are complex and cases 3–10 are real. The periodicity of the solutions has been captured in the last two columns. Period 1 is the periodicity of the solution associated with and Period 2 is the periodicity of the solution associated with . The abbreviation AP indicates an aperiodic solution. Stability is uniquely determined from the real part of the Floquet exponent.
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