Stabilization of Driven Pendulum with Periodic Linear Forces

Table 1

Stability comparison of periodic piecewise linear forces with harmonic force under horizontal fast oscillation.

Force type	Trivial position	Stability condition	Nontrivial position	Stability condition
Sin	0	𝜔^2<2𝑔𝑙	±arccos2𝑔𝑙/𝜔^2	𝜔^2>2𝑔𝑙
Triangular	0	𝜔^2<3.0396𝑔𝑙	±arccos3.0396𝑔𝑙/𝜔^2	𝜔^2>3.0396𝑔𝑙
Linear (sine)	0	𝜔^2<2.172𝑔𝑙	±arccos2.172𝑔𝑙/𝜔^2	𝜔^2>2.172𝑔𝑙
Trapezium	0	𝜔^2<1.4736𝑔𝑙	±arccos1.4736𝑔𝑙/𝜔^2	𝜔^2>1.4736𝑔𝑙
Quadratic	0	𝜔^2<1.2967𝑔𝑙	±arccos1.2967𝑔𝑙/𝜔^2	𝜔^2>1.2967𝑔𝑙
Rectangular	0	𝜔^2<1.2159𝑔𝑙	±arccos1.2159𝑔𝑙/𝜔^2	𝜔^2>1.2159𝑔𝑙