Stabilization of Driven Pendulum with Periodic Linear Forces

Table 3

Stability conditions with 𝜖-parametric force.

𝜖; 0<𝜖<1	Sum; 𝑏	Nontrivial position	Stability condition
0.9	0.1320	±arccos(3.7879𝑔𝑙/𝜔^2)	𝜔^2>3.7879𝑔𝑙
0.8	0.1607	±arccos(3.1114𝑔𝑙/𝜔^2)	𝜔^2>3.1114𝑔𝑙
0.75	0.1775	±arccos(2.8169𝑔𝑙/𝜔^2)	𝜔^2>2.8169𝑔𝑙
0.7	0.1956	±arccos(2.5562𝑔𝑙/𝜔^2)	𝜔^2>2.5562𝑔𝑙
0.6	0.2357	±arccos(2.1213𝑔𝑙/𝜔^2)	𝜔^2>2.1213𝑔𝑙
0.5	0.2793	±arccos(1.7902𝑔𝑙/𝜔^2)	𝜔^2>1.7902𝑔𝑙
0.4	0.3239	±arccos(1.5437𝑔𝑙/𝜔^2)	𝜔^2>1.5437𝑔𝑙
0.3	0.3664	±arccos(1.3647𝑔𝑙/𝜔^2)	𝜔^2>1.3647𝑔𝑙
0.25	0.3856	±arccos(1.2967𝑔𝑙/𝜔^2)	𝜔^2>1.2967𝑔𝑙
0.2	0.4029	±arccos(1.241𝑔𝑙/𝜔^2)	𝜔^2>1.241𝑔𝑙
0.1	0.4287	±arccos(1.1663𝑔𝑙/𝜔^2)	𝜔^2>1.1663𝑔𝑙