Iterative method for real-time Hybrid testing: application to a cantilever beam with two interface degrees of freedom

TL;DR

This paper presents an iterative, Fourier-domain RTHT method tailored for periodic, resonant responses and demonstrates it on a two-DOF cantilever beam with a numerical root at the clamped end and a physical tip. By updating actuator demands every few periods, the method decouples NS and PS dynamics in time and eliminates the need for fast real-time control, with accuracy limited primarily by interface sensing noise. The authors provide a rigorous theory for dynamic substructuring, substructurability, and a Newton/Broyden-based residual solver, and validate the approach experimentally using strain-gauge-derived interface forces to achieve high-fidelity interface synchronization even in low-damping, two-DOF configurations. The results show robust convergence and excellent synchronization at resonance, suggesting practical viability for wing-scale hybrid testing where traditional RTHT faces stability challenges due to delays.

Abstract

In this paper, an iterative method for real-time hybrid testing (RTHT) is proposed. The method seeks to iteratively balance the interface conditions between the physical and numerical substructures by controlling the periodic demand of the actuators. It is then suitable for RTHT of structures undergoing a periodic response, e.g. structures excited at resonance. We demonstrate the capabilities of the method on a cantilever beam in bending motion with two degrees of freedom at the interface, which we use as a prototype for future testing of aircraft wings. We show that a number of challenges arise in these settings, such as the difficulty in measuring interface forces while controlling a continuous structure and the instability of the hybrid test for small time delays. Classical RTHT strategies could produce inaccurate or unstable outcomes, whereas the proposed method is able to attain very good interface synchronisation in a wide range of tested scenarios.

Figures (13)

• Figure 1: Generic scheme of a RTHT: $N$ numerical substructure, $C$ control algorithm, $A$ actuators, $T$ transfer system, $P$ physical substructure. The grey block represents the whole control system.
• Figure 2: RTHT scheme under the pure delay assumption.
• Figure 3: Schematic of the original (a) and hybrid (b) cantilever beam in bending vibration. The numerical substructure ($N$) is denoted in green, the physical one ($P$) in cyan, and the interface in orange. The three blocks in the right plot represent the numerical structure, the displacement controller ($C$), and the test rig, which in turn comprises the actuators ($A$), the transfer system ($T$), and physical substructure.
• Figure 4: Eigenvalues of the characteristic equation of the hybrid cantilever with interface located in the middle of the beam ($\alpha = 0.5$) computed for four different values of the delay.
• Figure 5: Root locus of unstable eigenvalues of a hybrid cantilever beam for varying delay $\tau$, ratio of physical length over numerical length $\alpha$, and frequency $f$ with a cut off frequency of $500$ Hz. Four different values of the real part $\delta$ [1/ms] are reported in red scale. In the left plot the $\alpha-\tau$ view and in the right plot the $f-\tau$ view are given.