The establishment of uncertain single pendulum equation and its solutions ^*

Abstract

The single pendulum equation is commonly used to model the vibration characteristics of a single pendulum subjected to variable forces. A stochastic single pendulum equation driven by Wiener process describes the vibration phenomenon containing a noise term. However, there are also contradictions in some cases. Therefore, in this paper, uncertain single pendulum equation driven by Liu process is proposed to depict noise. Furthermore, analytical solutions as well as the inverse uncertainty distribution (IUD) of solutions for a class of uncertain single pendulum equations are presented. Finally, a paradox related to the stochastic single pendulum equation is given in the appendix.

Stability analysis of uncertain simple pendulum equation *

“…In [32], we use LP to deal with the white noise of the simple pendulum system, and establish the uncertain model of the simple pendulum, which is a second-order uncertain partial equation. In order to discuss the sensitivity of the uncertain simple pendulum equation to the perturbation in the initial state, we give the concept of many kinds of stability of the uncertain simple pendulum equation, including almost deterministic stability, distributional stability and exponential stability.…”