Active vibration control of nonlinear flexible structures via reduction on spectral submanifolds

TL;DR

The paper tackles the challenge of real-time active vibration control for high-dimensional nonlinear flexible structures. It develops a control framework based on spectral submanifolds (SSMs) to produce low-dimensional, invariant ROMs that transform the original high-dimensional nonlinear optimal control problem into a low-dimensional linear optimal control problem, solvable via an extended linear-quadratic regulator (LQR). The authors extend SSM theory to non-autonomous forcing and control, introduce two reduction strategies and an extended-LQR solution, and demonstrate the approach on diverse FE models, including an aircraft wing with over 130,000 DOF. The results indicate accurate ROM predictions and real-time viability, with broad applicability to complex structural dynamics and potential integration with digital twins for advanced vibration control.

Abstract

Large amplitude vibrations can cause hazards and failure to engineering structures. Active control has been an effective strategy to suppress vibrations, but it faces great challenges in the real-time control of nonlinear flexible structures. Here, we present a control design framework using reductions on aperiodic spectral submanifolds (SSMs) to address the challenges. We formulate high-dimensional nonlinear optimal control problems to suppress the vibrations and then use the SSM-based reductions to transform the original optimal control problems into low-dimensional linear optimal control problems. We further establish extended linear quadratic regulators to solve the reduced optimal control problems, paving the road for real-time active control of nonlinear flexible structures. We demonstrate the effectiveness of our control design framework via a suite of examples with increasing complexity, including a finite element model of an aircraft wing with more than 130,000 degrees of freedom.

Figures (20)

• Figure 1: An SSM-based control design framework for the active control of nonlinear flexible structures. Here, HNOCP stands for high-dimensional nonlinear optimal control problem \ref{['P1']}, HLOCP denotes high-dimensional linear optimal control problem \ref{['P2']}, LLOCP represents low-dimensional linear optimal control problem \ref{['P3']}, and LQR stands for a linear quadratic regulator that solves problem \ref{['P4']}.
• Figure 2: A flowchart depicting the implementation of our control design framework via SSM-based model reduction. Here, the online computation is minimal because only the simulation of the linear dynamics for the reduced coordinates $\boldsymbol{q}$ is involved in the online stage. In some of our examples, the dimension of the vector $\boldsymbol{q}$ is just two. The offline computation cost is also small because the dimension of vector $\boldsymbol{p}$ often equals to two or four.
• Figure 3: The schematic of an $n$-mass oscillator chain connected via cubic nonlinear springs. In this example, we set $n=10$.
• Figure 4: Normalized DCgain and MHSV for a system of oscillator chain. Here and throughout this paper, the 'number of mode pair' in horizontal axis denote the index of mode pair instead of the total number of mode pairs.
• Figure 5: Free oscillation of full model for the oscillator chain. The black solid line gives the nonlinear free oscillation and the blue dashed line represents the linear free oscillation.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3