Multi-input multi-output Fast and Relaxed Vector Fitting for aircraft ground vibration test

Multi-input multi-output Fast and Relaxed Vector Fitting for aircraft ground vibration Beatrice Elsa Bauret Martínez 1Gabriele Dessena 1 Marco Cirvera 2Oscar E. Bonilla-Manrique 3 1Universidad Carlos III de Madrid - Department of Aerospace Engineering 2Politecnico di Torino - Department of Structural, Geotechnical and Building Engineering (DISEG) 3Universidad Carlos III de Madrid - Electronic Technology Department Abstract Fast and Relaxed Vector Fitting (FRVF) is a frequency-domain system identification method widely used in electrical network modeling, but its application to mechanical systems remains limited. This study adapts FRVF to identify modal parameters of aircraft structures from Ground Vibration Test (GVT) data in multi-input multi-output (MIMO) configurations. The methodology consists of a rational approximation of frequency response functions using enhanced input stacking, followed by the extraction of poles from the fitted model. Subsequently, modal parameters are computed based on the pole locations and residues. Numerical validation is performed on 2D MIMO beam model, assessing accuracy and robustness under increasing noise levels. Experimental validation was conducted using the BAE Hawk T1A aircraft dataset. In this experiment, vibrations were induced by five shakers, and responses were recorded across 91 channels, demonstrating performance comparable to that of the improved Loewner Framework (iLF) method. The results demonstrate that the MIMO-extended FRVF approach performs reliably in terms of accuracy, noise resistance, and computational efficiency. These findings highlight the potential of the MIMO-extended FRVF method for practical GVT applications and its future integration into aerospace structural health monitoring systems. Fast and Relaxed Vector Fitting The original VF algorithm approximates measured FRFs with rational functions by iteratively relocating a set of poles until convergence [5] [1]. The identified model takes the following form: H(s) = N X n=1 cn s−an +d+s e where anand cnare the poles and residues of the system, while dand eare constant and proportional terms. Enhancements to Vector Fitting Relaxed VF: Introduces pole relaxation, improving convergence and reducing sensitivity to initial poles, particularly under noisy conditions. Fast and Relaxed VF: Extends RVF with QR-based residue estimation, weak inverse-magnitude weighting, and efficient pole stabilization, yielding faster, more robust, and accurate results with fewer iterations and reduced model order. Enhanced FRVF The enhanced version developed in this study is based on the MIMO extension introduced in [4]. This classical implementation was designed under the assumption of square (same number of input and output channels) FRF matrices, where the number of measured outputs equals the number of excitation inputs. In practice, GVTs rarely satisfy this constraint: the number of sensors is often an order of magnitude larger than the available shakers. To overcome this limitation, an enhanced input stacking strategy is proposed that allows FRVF to process arbitrary nonsquare MIMO configurations efficiently. The enhanced stacking procedure can be summarised as follows: 1. The dimensions of the loops are adapted to handle nonsquare MIMO configurations efficiently. 2. The original three-dimensional FRF tensor, with dimensions [Noutputs ×Ninputs ×Nfreq], is reshaped into a two-dimensional array suitable for vector fitting. The input channels are combined in a superposition framework to construct a single FRF for each output. By stacking multiple excitation contributions, each output signal carries richer dynamic information, improving the signal-to-noise ratio. This procedure preserves the correlation between inputs and outputs, ensuring that the resulting poles and residues can be recombined to extract physically consistent modal shapes. Subsequently, the extraction of the modal parameters is obtained via stabilisation diagrams. Numerical Validation A preliminary numerical example is considered to validate the extended FRVF approach: A 2D MIMO beam model to demonstrate the fundamental modal identification capability under controlled conditions. A hollow rectangular aluminium beam is analysed with multiple excitations in one and two directions. 2D beam model with two excitation forces Results First, the model is tested without noise. Frequency [Hz] Damping [-] MAC Analytical FRVF LSCE Analytical FRVF LSCE FRVF LSCE 1 5.00 5.00 - 0.03 0.031 - 1 - 2 11.36 11.36 - 0.03 0.032 - 1 - 3 31.33 31.33 - 0.03 0.031 - 1 - 4 71.22 71.22 75.32 0.03 0.03 0.355 1 0.10 5 87.87 87.96 - 0.03 0.030 - 1 - 7 199.74 199.76 - 0.03 0.030 - 1 - 8 288.40 288.46 300.07 0.03 0.031 0.081 1 0.57 9 393.21 393.13 - 0.03 0.030 - 1 - 10 431.52 431.39 422.66 0.03 0.030 0.037 1 0.43 11 655.54 656.40 657.34 0.03 0.030 0.031 1 0.21 12 980.85 981.14 981.55 0.03 0.033 0.031 0.99 0.06 Comparison of analytical, FRVF, and LSCE results of modal frequencies, damping ratios, and MAC values for the 2D beam. A noise sensitivity analysis is then conducted, in which noise is incrementally introduced to the input, the output, and both simultaneously. Frequency error with input and output noise 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.02 0.08 0.05 0.03 0.02 0.00 0.00 0.00 0.00 0.00 7.26 7.26 7.26 0.10 0.09 0.01 0.07 0.09 0.10 0.08 0.06 0.01 0.02 0.00 0.01 0.01 17.64 0.27 17.64 0.01 0.01 0.02 0.01 0.05 20.36 20.36 20.36 0.02 0.02 0.08 0.07 0.01 0.07 0.10 29.40 0.02 0.02 0.02 0.03 0.03 40.09 40.09 40.09 0.03 0.03 0.04 0.04 0.05 43.99 43.99 43.99 0.13 0.19 0.21 0.20 0.20 0.28 66.83 66.83 0.03 0.02 0.01 100 100 100 100 100 0 0.1 0.3 0.5 0.7 1 3 5 Noise level [%] 2 4 6 8 10 12 Mode number 0 20 40 60 80 100 Difference [%] Damping error with input and output noise 1.52 1.58 1.76 1.80 1.72 1.67 1.34 0.37 5.42 5.68 4.87 5.63 5.64 5.45 17.86 2.83 2.71 0.32 4.02 31.21 100 100 4.82 24.62 0.05 0.05 1.39 1.96 71.80 100 100 100 9.96 34.73 3.83 11.89 3.88 16.67 18.23 33.91 0.24 2.98 2.23 0.32 1.78 100 28.88 100 0.40 0.41 7.50 2.84 23.03 100 100 100 1.79 1.79 58.75 59.33 3.24 21.84 20.18 100 0.08 0.02 0.10 0.34 0.47 100 100 100 0.17 0.47 0.46 0.28 0.72 100 100 100 0.65 0.31 0.03 8.85 23.36 28.24 100 100 9.29 2.14 8.59 100 100 100 100 100 0 0.1 0.3 0.5 0.7 1 3 5 Noise level [%] 2 4 6 8 10 12 Mode number 0 20 40 60 80 100 Difference [%] MAC changes with input and output noise 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.00 0.99 1.00 1.00 1.00 1.00 1.00 0.00 0.00 0.00 1.00 1.00 1.00 1.00 1.00 0.95 0.09 0.10 1.00 1.00 1.00 1.00 1.00 0.00 0.84 0.00 1.00 1.00 1.00 1.00 1.00 0.00 0.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 0.97 0.00 1.00 1.00 1.00 1.00 1.00 0.00 0.00 0.00 1.00 1.00 1.00 1.00 1.00 0.00 0.00 0.00 1.00 0.90 0.89 0.88 0.88 0.88 0.00 0.00 0.99 0.98 0.98 0.00 0.00 0.00 0.00 0.00 0 0.1 0.3 0.5 0.7 1 3 5 Noise level [%] 2 4 6 8 10 12 Mode number 0 0.2 0.4 0.6 0.8 1 Difference [%] Experimental Validation: HAWK T1A GVT This experimental case study is based on a BAE Systems Hawk T1A (from [6]), commonly referred to as the Hawk. The testing configuration incorporates five excitation sources strategically located across the aircraft structure. To capture the structural response, a total of 85 accelerometers are deployed throughout the airframe. These include both uniaxial and triaxial models, resulting in 91 response channels. Hawk T1A aircraft where experiments are conducted (retrieved from [6] Results FRVF results are compared with those obtained using the iLF method and LSCE for the same dataset. All MAC values are reported relative to the iLF reference modes retrieved from [3]. For brevity, only the first three modes, the most dominant ones, are analyzed. Frequency [Hz] Damping [-] MAC Analytical FRVF LSCE Analytical FRVF LSCE FRVF LSCE 1 6.98 7.02 9.11 0.028 0.010 0.668 0.83 0.04 2 15.44 15.52 13.98 0.008 0.008 0.712 0.90 0.02 3 16.32 16.29 - 0.010 0.007 - 0.74 - Comparison of first three modes identified by iLF, FRVF, and LSCE using the truncated dataset (5–165 Hz). The first three vibration modes are depicted: First mode Second mode Third mode Conclusions The FRVF method with enhanced input stacking showed strong performance in all validation cases. In the 2D MIMO beam example, it accurately estimated modal parameters with MAC values close to 1 for most modes, even under 0.7% synthetic noise, while LSCE degraded much earlier. Frequency and damping errors also remained low under the same noise conditions. For the experimental Hawk T1A GVT, FRVF identified the first 16 global modes in the 5–165 Hz range, showing higher MAC values for the lower modes, which corresponds to the clearer quality of the data in that range. Although accuracy decreases for large experimental datasets due to measurement noise, making stabilization diagrams harder to interpret, FRVF still outperformed LSCE, which missed many modes that iLF could. FRVF is therefore a robust and computationally efficient option for experimental modal analysis, providing reliable frequency and damping estimates with relatively low cost when the frequency range is properly defined. Future improvements may be achieved through adaptive pole initialization, enabling efficient processing of full datasets with higher accuracy. References [1] M. Civera, G. Calamai, and L. Zanotti Fragonara. Experimental modal analysis of structural systems by using the fast relaxed vector fitting method. 2021. [2] G. Dessena and M. Civera. Improved tangential interpolation-based multi-input multi-output modal analysis of a full aircraft. 2023. [3] G. Dessena, M. Civera, A. Marcos, and B. Chiaia. Multi-input multi-output Loewner framework for vibrationbased damage detection on a trainer jet. 2023. [4] B. Gustavsen. Matrix Fitting Toolbox, User’s Guide and Reference. SINTEF Energy Research, 2020. [5] B. Gustavsen and A. Semlyen. Rational approximation of frequency domain responses by vector fitting, volume 14. 1999. [6] J. Wilson, M. D. Champneys, M. Tipuric, R. Mills, D. J. Wagg, and T. J. Rogers. Multiple-input, multipleoutput modal testing of a Hawk T1A aircraft: a new full-scale dataset for structural health monitoring. 2023. This work was supported by the Madrid Government (Comunidad de Madrid – Spain) under the Multiannual Agreement with the Universidad Carlos III de Madrid (IA_aCTRl-CM-UC3M). EASN 15th International Conference [email protected].es