Mathematical Problems in Engineering

Volume 2018, Article ID 7858403, 10 pages

https://doi.org/10.1155/2018/7858403

## Instantaneous Characteristics of Nonlinear Torsion Pendulum and Its Application in Parameter Estimation of Nonlinear System

^{1}Tianjin Key Laboratory for Control Theory and Applications in Complicated Systems, Tianjin University of Technology, Tianjin 300384, China^{2}PetroChina Dagang Oilfield Company, Tianjin 300270, China

Correspondence should be addressed to Junchao Zhu; moc.361@tujt_oahcnujuhz

Received 21 October 2017; Revised 25 February 2018; Accepted 27 March 2018; Published 10 May 2018

Academic Editor: Suzanne M. Shontz

Copyright © 2018 Yan Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The nonlinear model of torsion pendulum is presented by considering the nonlinear damping force and nonlinear restoring force. The analytic solution of the nonlinear model is calculated to analyze the relationship between the characteristics of torsion pendulum and the nonlinear factors. The instantaneous characteristics of nonlinear torsion pendulum are analyzed by instantaneous undamped natural frequency and instantaneous damping coefficient. The instantaneous characteristics can be used for the parameter estimation of nonlinear torsion pendulum system. The nonlinear characteristics of the torsion pendulum are validated by the torsion pendulum based on the air-hovered turntable. The parameter estimation method based on the instantaneous characteristics is validated by the moment of inertia measurement system based on the torsion pendulum.

#### 1. Introduction

The torsion pendulum system has been widely used to measure the moment of inertia of mechanical systems such as missile and engine [1–4].

The torsion pendulum model mainly contains three factors: moment of inertia, damping force, and restoring force. With the increasing complexity of the torsion pendulum, the linear model has not been able to accurately describe the actual torsion pendulum. For the large measured object with complex shape, the air damping is a nonlinear force related to linear terms and square terms of speed [5]. When the torsion bar is alloy or composite material, the restoring force of the torsion bar is a nonlinear force related to linear terms and cubic terms of torsional angle [6].

The linear model cannot accurately describe the complex torsion pendulum system. So it is needed to establish the torsion pendulum model based on the nonlinear dynamic system. Because the nonlinear system and the linear system have significantly different characteristics, it is necessary to analyze the influence of nonlinear factors on the characteristics of the torsion pendulum.

For the quantitative analysis of the nonlinear torsion pendulum, the approximate analytic solution of the nonlinear differential equation of the torsion pendulum is calculated. The solving methods of the nonlinear differential equation mainly include small parameter method, multiple scales method, evolutionary method, and average method [7–10]. The average method regards the amplitude and phase of the solution as slowly changing with time and uses the derivative of amplitude and phase to compute the approximate analytic solution of the nonlinear differential equation. This method is of high computational accuracy and is suitable for solving nonlinear dynamical model.

The logarithmic decrement and vibration period are the important parameters of the torsion pendulum. The logarithmic decrement refers to the ratio of the amplitude of adjacent periods. For the linear torsion pendulum, the logarithmic decrement and vibration period are constant. But for the nonlinear torsion pendulum, the logarithmic decrement and vibration period are time-varying which are associated with nonlinear factors.

The instantaneous characteristics of the nonlinear torsion pendulum can be described by instantaneous undamped natural frequency and instantaneous damping coefficient which are computed by the instantaneous amplitude and instantaneous frequency [11, 12].

To calculate the instantaneous frequency, many methods are developed such as short-time Fourier transformation [13, 14], wavelet transformation [15, 16], Wigner-Vile Distribution [17, 18], and Hilbert transform [19]. Reference [11] presented the construction method of analytic signal based on Hilbert transform, which is used for the calculation of instantaneous frequency.

Considering the influence of nonlinear damping force and nonlinear restoring force, the torsion pendulum model based on the nonlinear dynamic system is established. The analytic solution of the nonlinear model is used to analyze the relationship between the characteristics of torsion pendulum and the nonlinear factors. The instantaneous characteristics of the nonlinear system is analyzed by the instantaneous undamped natural frequency and instantaneous damping coefficient. At last, the nonlinear torsion pendulum experimental system is used to verify the presented model and method.

#### 2. Nonlinear Model of Torsion Pendulum

The modeling methods based on the differential equation include Newton second law, Darren Bell principle method, and Lagrange equation method. Among them, the Lagrange equation method is based on the energy principle and it only needs to analyze the main power of the system. Therefore, the modeling method based on the Lagrange equation is more suitable for the complex system.

As shown in Figure 1, the torsion pendulum system consists of torsion bar, turntable, and test object. The test object is placed on the turntable that is connected to the torsion bar. With impulse excitation, the test object and turntable make torsion pendulum movement together while the restoring force is generated by the torsion bar. The torsion pendulum system in Figure 1 has one freedom degree and the torsion angle is regarded as the generalized coordinate.