Stochastic Response Spectrum Study of Nonlinear Systems: Introduction and Analysis Methods of Fractional Derivative Elements

Academic Background

In the fields of engineering and physics, nonlinear dynamic systems are widely used to model complex phenomena. However, predicting the response of these systems under stochastic excitation becomes highly challenging, especially when fractional derivative elements are introduced. Fractional derivatives can more accurately describe memory effects and hereditary phenomena, but their introduction also brings additional analytical and computational difficulties. Traditional linear system analysis methods cannot be directly applied to nonlinear systems, particularly when the system includes fractional derivatives, making the determination of the response power spectral density (PSD) even more complex.

The core issue of this study is how to accurately estimate the power spectral density of the stochastic response in nonlinear systems containing fractional derivatives. The introduction of fractional derivatives makes the dynamic behavior of the system non-local, and traditional statistical linearization (SL) methods have limitations in dealing with such problems. Therefore, the authors propose a new method based on the Conditional Spectrum to improve the estimation accuracy of the response spectrum of nonlinear systems.

Source of the Paper

This paper is co-authored by Pol D. Spanos and Beatrice Pomaro. Pol D. Spanos is affiliated with the Department of Mechanical and Civil Engineering at Rice University, USA, while Beatrice Pomaro is from the Department of Civil, Environmental, and Architectural Engineering at the University of Padova, Italy. The paper was accepted on February 18, 2025, and published in the journal Nonlinear Dynamics.

Research Process

1. Research Objectives and Method Overview

The main objective of this study is to develop a reliable method for estimating the power spectral density of nonlinear systems containing fractional derivatives under stochastic excitation. To this end, the authors propose an improved method based on the Conditional Spectrum, which estimates the total response spectrum by decomposing the response of the nonlinear system into the responses of multiple linearized oscillators and performing a weighted average of these responses.

2. System Model and Equation Derivation

The starting point of the research is a single-degree-of-freedom nonlinear oscillator whose dynamic equation includes a fractional derivative term. The equation of motion for the system is as follows:

[ m\ddot{x}(t) + cD^\beta_{0,t}x(t) + f(x, \dot{x}) = w(t) ]

where ( D^\beta_{0,t} ) is the fractional derivative operator, and ( f(x, \dot{x}) ) represents the nonlinear force of the system. The introduction of the fractional derivative makes the dynamic behavior of the system more complex, and traditional linearization methods are difficult to apply directly.

3. Statistical Linearization and Conditional Spectrum Method

To handle the nonlinear system, the authors employ statistical linearization techniques to approximate the nonlinear system as a linear system. However, traditional statistical linearization methods have limitations when dealing with fractional derivatives. Therefore, the authors propose the Conditional Spectrum method, which estimates the total response spectrum by decomposing the system’s response into the responses of multiple linearized oscillators and performing a weighted average of these responses.

Specifically, the formula for the Conditional Spectrum method is as follows:

[ S_x(\omega) = \int_0^\infty S_x(\omega|a) p_s(a) da ]

where ( S_x(\omega|a) ) is the response power spectral density given amplitude ( a ), and ( p_s(a) ) is the probability density function of the amplitude. By introducing a corrective term, the authors further improve the Conditional Spectrum method to more accurately capture the response characteristics of the system.

4. Numerical Simulation and Result Validation

To validate the effectiveness of the proposed method, the authors conducted Monte Carlo (MC) simulations. The simulations considered different orders of fractional derivatives (( \beta = 0.5, 1.0, 1.3 )) and compared the results of the improved Conditional Spectrum method with those of the traditional statistical linearization method.

5. Result Analysis and Conclusion

Through MC simulations, the authors found that the improved Conditional Spectrum method demonstrates higher accuracy in estimating the response spectrum of nonlinear systems, especially in capturing the spectral broadening near resonance frequencies. In contrast, the traditional statistical linearization method shows significant limitations when dealing with fractional derivatives.

Main Results

1. Superiority of the Improved Conditional Spectrum Method: The improved Conditional Spectrum method can more accurately estimate the response spectrum of nonlinear systems, especially in terms of spectral broadening near resonance frequencies.
2. Impact of Fractional Derivatives: The introduction of fractional derivatives significantly affects the dynamic behavior of the system, and the improved Conditional Spectrum method effectively captures these effects.
3. Validation by Monte Carlo Simulations: Through MC simulations, the authors validated the effectiveness of the improved Conditional Spectrum method and demonstrated its advantages in handling fractional derivatives.

Research Conclusions and Significance

The conclusion of this study is that the improved Conditional Spectrum method provides a reliable tool for estimating the stochastic response spectrum of nonlinear systems containing fractional derivatives. This method not only improves the estimation accuracy of the response spectrum of nonlinear systems but also offers a new approach to handling fractional derivatives. Its scientific value lies in providing a new theoretical framework for the analysis of complex nonlinear systems, while its application value lies in offering strong support for system design and optimization in engineering practice.

Research Highlights

1. Innovation in the Conditional Spectrum Method: The authors propose an improved method based on the Conditional Spectrum, effectively addressing the limitations of traditional statistical linearization methods in handling fractional derivatives.
2. Handling of Fractional Derivatives: The study introduces fractional derivatives into the stochastic response spectrum analysis of nonlinear systems for the first time, providing a new perspective for modeling complex systems.
3. Validation by Monte Carlo Simulations: Through extensive MC simulations, the authors validate the effectiveness of the proposed method and demonstrate its potential in engineering applications.

Other Valuable Information

Another significant contribution of this study is that the improved Conditional Spectrum method can be widely applied to other types of nonlinear systems, such as those with hysteretic characteristics. Additionally, the authors point out that although the method performs well in capturing the response spectrum in the low-frequency range, its accuracy in the high-frequency range still needs further improvement. Future research could explore how to extend this method to capture higher-order effects, such as super-harmonic generation.

This study provides new theoretical tools for the stochastic response analysis of nonlinear systems and offers important references for system design and optimization in engineering practice.