On the Development of an AI-Based Tool to Assess the Instantaneous Modal Properties of Nonlinear SDOF Systems

Abstract

In this article, a data-driven algorithm is developed to assess the natural frequency and damping ratio of a nonlinear oscillating single-degree-of-freedom (SDOF) system. The algorithm is based on hybrid convolutional–long short-term memory neural networks (CNN-LSTM) that process a short moving window belonging to a free-decay response and provide estimates of both parameters over time. The novelty of the study resides in the fact that the neural network is trained exclusively using synthetic data issued from linear SDOF models. Since the recurrent neural network (RNN) requires relatively small amounts of data to operate effectively, the nonlinear system locally behaves as a quasi-linear model, allowing each data segment to be processed under this assumption. The proposed RecuID tool is experimentally validated on a laboratory-scale nonlinear SDOF system. To demonstrate its effectiveness, it is compared to conventional identification algorithms. The experimental study yields a maximum mean absolute error (MAE) of 0.244 Hz for the natural frequency and 0.015 for the damping ratio. RecuID proves to be a faster and more robust methodology, capable of handling time-varying damping ratios up to 0.2 and a wide range of natural frequencies defined relative to the sampling rate.

1. Introduction

Neural networks have attained considerable attention during the past years due to their ability to solve complex problems and efficiently extract hidden features from data. Nowadays, they are applied to a wide variety of structural engineering problems, such as the following: operational modal analysis (OMA) [1,2,3,4], where they assist in estimating modal parameters or discriminating structural modes from spurious ones; structural health monitoring (SHM) [5,6,7], where neural networks are helpful to foresee damage and its consequences [8,9]; and simulation [10,11,12], where neural networks replace the traditional structural models.

In this context, nonlinear systems emerge as suitable application domains for neural networks. In fact, the inherent nonlinearity of neural networks enables them to identify hidden patterns in data, which, in turn, can lead to better comprehension of a nonlinear system. In the field of nonlinear structural dynamics, neural networks have been applied to a wide range of problems. For instance, in [13], three different approaches—including fully connected neural networks (FCNNs) and long short-term memory (LSTM) networks—are compared in order to predict the dynamic response of structures exhibiting nonlinear behaviour due to environmental effects, ageing, or fatigue. Similarly, ref. [14] employs FCNNs to predict the nonlinear response of a tapered beam, accounting for complex variable dependencies, energy transfer mechanisms, and sensitivity to initial vibration conditions. Supervised learning can be combined with unsupervised learning, as is the case in [15], where seismic inputs are clustered to optimise the training process of an LSTM model for predicting the nonlinear response of building structures. However, response prediction approaches typically model the system as a “black box”, providing limited physical interpretability and preventing direct access to the evolving modal properties of the structure.

In this article, the tracking of the natural frequency and damping ratio of a nonlinear single-degree-of-freedom (SDOF) system is sought. Such estimation, which is intended to be as fast as possible, is performed in this case by a neural network that processes chunks of data as they are registered. Other works have addressed this problem, but none have yet applied a data-driven algorithm to simultaneously estimate both the natural frequency and the damping ratio of the system. For instance, ref. [16] presents a mathematical approach based on the sinusoidal and exponential derivative concepts (introduced in the article), which led to promising results validated on synthetic data. In [17], the authors propose a methodology to identify the properties of nonlinear systems by means of well-designed neural network architectures. However, their estimation relies on frequency response functions, which require large amounts of time domain data and are not suitable for rapid estimation from short data segments. Finally, ref. [18] presents a numerical study using a modified WaveNet approach to estimate nonlinear physical properties of a structural system. Although the data is processed in the time domain, the article focuses on identifying constant physical parameters governing the structural nonlinear behaviour, for which an underlying model of such nonlinearity is required.

Deep learning provides useful tools for estimating structural properties. Modal identification of SDOF systems in the frequency domain using convolutional neural networks (CNNs) has been addressed in [19]. Moreover, hybrid approaches combining CNNs with recurrent neural networks (RNNs), such as LSTM cells, have also been proposed to account for time dependencies [20]. In this work, RecuID is presented. It consists of a hybrid CNN-LSTM architecture trained to estimate the instantaneous natural frequency and damping ratio from the free-decay response of a nonlinear system, enabling the tracking of their evolution over time. The main strength of the proposed approach lies in its generalisation capability to handle arbitrary nonlinear behaviours. Although the system is nonlinear, the free-decay signal can be interpreted as a concatenation of locally quasi-linear free-decays when sufficiently short time segments are considered. This assumption allows the CNN-LSTM model to be trained exclusively using data issued from linear SDOF models. The local linear predictions obtained for each data segment can then be combined to reconstruct the overall nonlinear evolution. This strategy is validated through both numerical and experimental tests, the latter conducted on an SDOF mock-up system with adjustable physical properties.

The manuscript is organised as follows: after this introduction, the theoretical background is presented in Section 2. Section 3 introduces the RecuID algorithm, describing the proposed neural network architecture and its training, validation, and test details. Section 4 is devoted to the nonlinear system analysis; first, RecuID is evaluated over simulated nonlinear responses, and then it is applied to a nonlinear experimental setup. The modal properties are also estimated by means of the logarithmic decrement method (DLG) in order to compare the performance of each approach. Finally, Section 5 summarises the main conclusions of the study.

2. Theoretical Background

The free motion of a linear viscously damped SDOF system follows the following differential equation (Equation (1)):

$$m\ddot{x}\left(t\right)+c\dot{x}\left(t\right)+\kappa x\left(t\right)=0,$$

(1)

where $x\left(t\right)$ represents the displacement of the mass, $\dot{x}\left(t\right)$ and $\ddot{x}\left(t\right)$ its first and second-time derivatives, $m$ stands for the moving mass, $\kappa$ is the stiffness constant of the elastic members, and $c$ accounts for the equivalent viscous damping of the system. A usual alternate form of this equation [21] is obtained by dividing it by $m$ and making the following substitutions (Equation (2)):

$${\displaystyle \frac{c}{m}}=2\zeta {\omega }_0\hspace{1em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{1em}{\displaystyle \frac{\kappa }{m}}={\omega }_0^2,$$

(2)

where ${\omega }_0$ is the natural frequency of the system (in rad/s) and $\zeta$ its damping ratio. Together, both magnitudes form the modal properties of the system. As a result, Equation (1) transforms into the differential equation:

$$\ddot{x}\left(t\right)+2\zeta {\omega }_0\dot{x}\left(t\right)+{\omega }_0^{2}x\left(t\right)=0.$$

(3)

The general solution of Equation (3) can be easily computed, and one of its forms is shown in Equation (4), where $A$ and $\phi$ depend on the initial conditions of the system. Since the underdamped case is studied ($\zeta <1$), ${\omega }_d$ stands for the damped natural frequency ${\omega }_d={\omega }_0\sqrt{1-{\zeta }^2}$ [22]:

$$x\left(t\right)=A\mathit{sin}\left({\omega }_{d}t+\phi \right) e^{-\zeta {\omega }_{0}t}$$

(4)

The motion of a real system is experimentally recorded by means of a data acquisition system and some sensors, as described in Section 4, so the registered data is discretized. The tests are performed at a certain sampling rate ($F_s$) and they last a certain amount of time ($T$), so a total of $N_t=F_s T$ samples are recorded. As a consequence, the recorded signal is composed of a series of uniformly separated samples. The time separation $\mathsf{\Delta }t$ depends on $F_s$:

$$\mathsf{\Delta }t=1/F_s$$

(5)

Each data sample is recorded at a multiple of this value, as follows:

$$t=k \mathsf{\Delta }t, \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} k\in {\mathbb{Z}}^{+}$$

(6)

To perform the discretisation, a change in variable is applied, considering a new non-dimensional frequency, defined in terms of the sampling rate and applicable to both the natural frequency (${\omega }_0$) and the damped natural frequency (${\omega }_d$):

$${\tilde{\omega }}_0={\displaystyle \frac{{\omega }_0}{2\pi F_s}} {\tilde{\omega }}_d={\displaystyle \frac{{\omega }_d}{2\pi F_s}}$$

(7)

Applying the previous changes, Equation (4) turns into Equation (8):

$$x_k\left(k\right)=A\mathit{sin}\left(2\pi {\tilde{\omega }}_{d}k+\phi \right) e^{-\zeta 2\pi {\tilde{\omega }}_{0}k}$$

(8)

Finally, the oscillation frequency of the system can be expressed in terms of the oscillation period (${\tau }_d\in {\mathbb{R}}^{+}$), as shown in Equation (9), where $K_d\in {\mathbb{R}}^{+}$ represents the total amount of time separation $\mathsf{\Delta }t$ that composes the system’s natural period:

$${\tau }_d=2\pi /{\omega }_d=K_d \mathsf{\Delta }t$$

(9)

In a discrete domain, $k_d\in {\mathbb{Z}}^{+}$ and represents the samples within a complete period:

$$k_d=⌊K_d⌋$$

(10)

3. Methodology

This section is devoted to presenting the algorithm for the identification of the modal parameters, including the neural network architecture together with its optimisation procedure, the data generation strategy, and the training hyperparameters. As mentioned in the introduction, the algorithm is intended to detect the instantaneous natural frequency and damping ratio from a free-decay signal, which has several consequences:

• On one hand, the deep-learning algorithm should be prepared to process data in the time domain by means of a moving window that sweeps the free-decay response. Therefore, as mentioned, RNNs are used in this work, specifically LSTM cells, which are specialised in establishing time dependencies between the new and old data. Several architectures are proposed, trained, and validated.
• This way of processing time signals is highly dependent on two parameters: the window size, which is called in this work the “look back” ($n_l$) because it represents the number of past samples that the model processes at each time step; and the “displacement” ($d_l$) of the time window, which stands for the number of samples that it advances before a new estimation is performed.
• Finally, the real-time condition makes it impossible to normalise the time signal as a whole, because future samples are not available in online applications. Section 3.1 explains the solution that is adopted in this work.

The core of the methodology is the RNN, which performs the ${\omega }_0$ and $\zeta$ identification, for it is named RecuID. The following subsections are devoted to explaining its development. The methodology has been implemented in Python 3.11.3, using NumPy 1.24.3 for numerical computations, Tensorflow 2.15.0 for the deep learning framework, Keras 2.15.0 for model definition and training [23], and MATLAB R2023b for the results presentation and data storing.

3.1. Synthetic Data Generation

In a linear free-decay signal, there is a proportional relationship between the amplitudes of two consecutive cycles. Let $k_1$ be the first sample of a given period, $k_2$ be the first sample of the segment one period later, and $k_d$ (Equation (10)) be the number of samples between them (Equation (11)):

$$k_2=k_1+k_d$$

(11)

The responses for these samples are Equations (12) and (13):

$$x_k\left(k_1\right)=A\mathit{sin}\left(2\pi {\tilde{\omega }}_d{k}_1+\phi \right)e^{-\zeta 2\pi {\tilde{\omega }}_0{k}_1}$$

(12)

$$\begin{array}{c}{x}_k\left(k_2\right)=A\mathit{sin}\left(2\pi {\tilde{\omega }}_d{k}_2+\phi \right)e^{-\zeta 2\pi {\tilde{\omega }}_0{k}_2}=\\ =A\mathit{sin}\left(2\pi {\tilde{\omega }}_d{k}_1+2\pi {\tilde{\omega }}_d{k}_d+\phi \right)e^{-\zeta 2\pi {\tilde{\omega }}_0\left(k_1+k_d\right)}\end{array}$$

(13)

Considering that there is a complete period between $k_1$ and $k_2$ ($k_d$), and $\mathit{sin}\left(\theta \right)=\mathit{sin}\left(\theta +2\pi \right)$, a $\lambda$ proportionality coefficient can be defined as Equation (14):

$$\lambda ={\displaystyle \frac{x_k\left(k_1\right)}{x_k\left(k_2\right)}}={\displaystyle \frac{A\mathit{sin}\left(2\pi {\tilde{\omega }}_d{k}_1+\phi \right)e^{-\zeta 2\pi {\tilde{\omega }}_0{k}_1}}{A\mathit{sin}\left(2\pi {\tilde{\omega }}_d{k}_1+2\pi {\tilde{\omega }}_d{k}_d+\phi \right)e^{-\zeta 2\pi {\tilde{\omega }}_0\left(k_1+k_d\right)}}}= {\displaystyle \frac{e^{-\zeta 2\pi {\tilde{\omega }}_0{k}_1}}{e^{-\zeta 2\pi {\tilde{\omega }}_0\left(k_1+k_d\right)}}}$$

(14)

And, regarding Equations (7) and (9), the final $\lambda$ expression is Equation (15):

$$\lambda =e^{-\zeta 2\pi {\tilde{\omega }}_0{k}_1+\zeta 2\pi {\tilde{\omega }}_0\left(k_1+k_d\right)}=e^{\zeta 2\pi {\tilde{\omega }}_0{k}_d}=e^{{\displaystyle \frac{{\omega }_0{k}_d\zeta }{F_s}}}=e^{{\displaystyle \frac{{\omega }_d{\tau }_d\zeta }{\sqrt{1-{\zeta }^2}}}}=e^{{\displaystyle \frac{2\pi \zeta }{\sqrt{1-{\zeta }^2}}}}$$

(15)

Let $k$ be an arbitrary point within each window, $k_1+k$ in the first case and $k_2+k$ one period later. Then, we have the following:

$$\begin{array}{c}{x}_k\left(k_1+k\right)=A\mathit{sin}\left(2\pi {\tilde{\omega }}_d(k_1+k)+\phi \right) e^{-\zeta 2\pi {\tilde{\omega }}_0(k_1+k)}\\ x_k\left(k_2+k\right)=A\mathit{sin}\left(2\pi {\tilde{\omega }}_d\left(k_2+k\right)+\phi \right) e^{-\zeta 2\pi {\tilde{\omega }}_0\left(k_2+k\right)}=\\ =A\mathit{sin}\left(2\pi {\tilde{\omega }}_d\left(k_1+k\right)+2\pi {\tilde{\omega }}_d{k}_d+\phi \right) e^{-\zeta 2\pi {\tilde{\omega }}_0(k_1+k_d+k)}\end{array}$$

(16)

Dividing the expressions in Equation (16), the new proportionality coefficient is as follows:

$$\begin{array}{c}{\lambda }^{\prime }={\displaystyle \frac{x_k\left(k_1+k\right)}{x_k\left(k_2+k\right)}}={\displaystyle \frac{A\mathit{sin}\left(2\pi {\tilde{\omega }}_d\left(k_1+k\right)+\phi \right) e^{-\zeta 2\pi {\tilde{\omega }}_0\left(k_1+k\right)}}{A\mathit{sin}\left(2\pi {\tilde{\omega }}_d\left(k_1+k\right)+2\pi {\tilde{\omega }}_d{k}_d+\phi \right) e^{-\zeta 2\pi {\tilde{\omega }}_0\left(k_1+k_d+k\right)}}}\\ {\lambda }^{\prime }= {\displaystyle \frac{e^{-\zeta 2\pi {\tilde{\omega }}_0\left(k_1+k\right)}}{e^{-\zeta 2\pi {\tilde{\omega }}_0\left(k_1+k_d+k\right)}}}=e^{{\displaystyle \frac{2\pi \zeta }{\sqrt{1-{\zeta }^2}}}}= \lambda \end{array}$$

(17)

Equations (15) and (17) demonstrate that a free-decay response exhibits a periodic proportionality: the amplitude at any sample within the signal is proportional to the amplitude at the same relative position after a shift of “p” periods, which is at the core of the known logarithmic decrement method to estimate the damping ratio of such signals.

As a consequence, a single cycle is sufficient to characterise the free response, since it already contains the oscillation frequency and the damping ratio information, and every subsequent cycle is simply a proportional replica of the first one. Therefore, during the neural network training process, only one cycle is generated to represent each system. Finally, note that the proportionality effects vanish during the normalisation step described below.

The synthetic data generation process is designed according to the required input/output structure of the RNN, and it consists of four steps, depicted in Figure 1:

• Random generation of systems with known properties.
• Calculation of the discrete single-period signals.
• Input data windowing.
• Data normalisation for training.

3.1.1. Random Generation of Systems with Known Properties

The neural network should output the natural frequency and damping ratio. Since the frequency normalisation introduced earlier is considered (Equation (7)), the natural frequency is expressed relative to the sampling rate that is used in the measurement process, and it is dimensionless. Likewise, the damping ratio is also a dimensionless parameter. For this reason, suitable bounds must be defined for both parameters:

$${\tilde{\omega }}_0\in \left(0, 0.1\right) \zeta \in \left(0.001, 0.2\right)$$

(18)

These ranges imply that ${\tilde{\omega }}_0$ is not greater than 10% of $2\pi F_s$, and $\zeta$ remains below 20%. The first limit follows a common rule of thumb, which states that if the frequency of interest of a certain system is $f_0$ (Hz), it is advised to register the corresponding data at a sampling rate 10 times greater ($F_s>10 f_0$). The second range has been defined in order to widely include the common values of structural damping. Please note that these boundaries could be adapted if required. A set of 100,000 random pairs (${\tilde{\omega }}_0$, $\zeta$) are generated following a uniform distribution within the specified ranges.

3.1.2. Calculation of the Discrete Single-Period Signals

In the second step, to generate the corresponding single-period signals, the period (${\tau }_d$) should be calculated through Equation (9). Consequently, depending on ${\tilde{\omega }}_0$ and $\zeta$, each single-period signal contains a different number of samples, $k_d$ (Equation (10)), and the final dataset does not have a predefined size, as discussed below. Equation (8) is evaluated in each case for $k\in {\mathbb{Z}}^{+}$ between (1, $k_d$) to store the 100,000 single-period waves.

3.1.3. Input Data Windowing

The third step, data windowing, requires the parameters $n_l$ and $d_l$. Each generated signal should be windowed and rearranged before training. Since a continuous prediction is targeted, $d_l$ is set to 1, meaning that one prediction is obtained per time step, and the window slides sample by sample along the signal. The parameter $n_l$ controls the window length. In principle, only four points would be sufficient to identify the four parameters of a free response ($A$, ${\omega }_0$, $\zeta$, and $\phi$). However, to initiate the architecture design, $n_l$ is set to 10 samples per window. The result is a matrix ($N_{set}\times n_l$), where $N_{set}$ is an unspecified number of 10-sample rows, each containing a short segment of the waveform, as illustrated in Figure 1.

In Equation (19), $n_i$ represents the number of windows that can be extracted from a single-period wave. Then, assuming $k_d>n_l$, $N_{set}$ can be expressed as follows:

$$N_{set}=\sum _i^{\mathrm{100,000}}{n}_i=\sum _i^{\mathrm{100,000}}⌊{\displaystyle \frac{k_{d,i}-n_l}{d_l}}⌋+1$$

(19)

In this work, $N_{set}$ turns out to be 12,400,834 windowed segments of 10 samples each, forming the training and validation dataset together with their corresponding labels. For each window, a single pair of ${\tilde{\omega }}_0$ and $\zeta$ fully represents the linear behaviour, but many labels are repeated because all the segments from the same period are equally labelled.

3.1.4. Data Normalisation for Training

Before being input into the neural network, each window is rescaled between 0 and 1. This allows the tool to process each window independently of the preceding ones, which is advantageous when approaching a real-time application. Regarding the output, both ${\tilde{\omega }}_0$ and $\zeta$ are also rescaled between 0 and 1. However, predicting the damping ratio independently is particularly challenging for the RNN. Although $\zeta$ also influences the damped natural frequency ${\tilde{\omega }}_d$, it has clearer and more dominant effects on the exponential factor in Equation (8), $e^{-\zeta 2\pi {\tilde{\omega }}_{0}k}$, which governs the amplitude-decay. Since damping is not the only parameter in the exponent, ${\tilde{\omega }}_0$ also plays a fundamental role. For this reason, after several optimisation attempts, it is found that the product ${\tilde{\omega }}_0\zeta$ represents the amplitude-decay phenomenon more consistently, leading to significantly better training performance when used as a label of the data and a value to be estimated by the neural network. This does not pose any issue for the identification, as the RNN predicts both ${\tilde{\omega }}_0$ and ${\tilde{\omega }}_0\zeta$, from which obtaining $\zeta$ is straightforward.

3.2. Neural Network Design and Optimisation

3.2.1. CNN-LSTM Architecture

Figure 2 illustrates the general architecture scheme.

Table 1. Summary of the architectures’ hyperparameters: highlighted in green, yellow and red the first, second and third best-performing architectures respectively.

ID	L1			L2		L3			L4	L5	L6	L7	Weights
	nf	sf	st	ps	st	nf	sf	st	nu	nu	nn	nn
A1	-	-	-	-	-	-	-	-	8	-	-	-	338
A2	-	-	-	-	-	-	-	-	32	-	-	-	4418
A3	-	-	-	-	-	-	-	-	64	-	-	-	17,026
A4	-	-	-	-	-	8	5	1	32	-	-	-	5362
A5	-	-	-	-	-	16	5	1	64	-	-	-	20,962
A6	-	-	-	-	-	16	5	1	64	-	32	-	22,978
A7	-	-	-	-	-	32	5	1	64	-	32	-	27,170
A8	16	3	1	2	1	8	3	1	32	-	32	-	6826
A9	32	3	1	2	1	8	3	1	64	-	32	-	21,738
A10	64	3	1	2	1	32	3	1	64	-	32	-	33,410
A11	32	3	1	2	1	8	3	1	64	-	32	8	21,954
A12	64	3	1	2	1	32	3	1	64	-	32	8	33,626
A13	128	3	1	2	1	64	3	1	128	-	64	16	133,298
A14	128	3	1	2	1	64	3	1	256	-	64	16	371,378
A15	256	3	1	2	1	128	3	1	256	-	128	32	530,786
A16	128	3	1	-	-	-	-	-	128	64	32	-	183,650
A17	128	3	1	-	-	-	-	-	128	64	64	-	185,794
A18	128	3	1	-	-	-	-	-	128	128	64	-	272,066
A19	128	3	1	-	-	-	-	-	128	64	64	32	187,810
A20	128	3	1	2	1	64	3	1	128	32	64	16	147,762

summarises the 20 architectures trained in this study, each defined by a different combination of hyperparameters and denoted by “Ai”. As can be observed, the analysis begins with the input layer (IL) followed by a single LSTM layer (L5) with varying numbers of units (A1 to A3). Architectures A4 and A5 introduce a 1D-convolutional layer (L3) to pre-process the waveform information, while A6 and A7 incorporate an additional dense layer (L6). From A8 to A10, an extra 1D-convolutional layer (L1) is added at the beginning of the architecture, with its corresponding max-pooling layer (L2). Then, from A11 to A15, the hyperparameters are further varied, and an additional dense layer (L7) is included before the two-neuron-dense output layer (OL). Finally, a different configuration is tested by including an additional LSTM layer (L4), with the “return sequences” parameter set to “True”, to concatenate it before L5.

3.2.2. Training-Validation Process

The training-validation process can be summarised in the following points:

• The mean squared error (MSE) is chosen as loss function, and the Adam optimiser with a learning rate of 0.001 is used.
• The dataset is divided into a training set and a validation set with a 70–30% ratio (8,680,583 for training, 3,720,251 for validation).
• The training and validation processes run concurrently. The validation performance serves as the basis for selecting the most suitable architecture.
• A batch size of 2¹⁰ and a maximum of 1000 epochs are considered. An early stopping criterion with a patience of 5 avoids overfitting.

Figure 3 shows the training-validation histories for all architectures, while Figure 4 summarises their final MSE values. Architecture A17 (highlighted in black in Figure 3) achieves the lowest validation MSE, 2.52 × 10⁻⁵, followed by A16 and A19. Consequently, A17 is selected as the optimal model. Table 1 highlights in green its configuration and reveals that incorporating an additional LSTM layer with 64 units is essential to improve the identification performance. The best-performing architecture is not the most complex one (e.g., A20). Instead, A17 provides a balanced compromise between model depth and model size, offering enough representational capacity with manageable computational load. In fact, the three best architectures have approximately 180,000 weights, which reinforces the consistency of the selection. The trained neural network is provided as Supplementary Materials to this manuscript, stored in the ‘.keras’ extension.

3.3. Methodology Test

The test stage is devoted to validating the algorithm with simulated linear free-decay signals to demonstrate its ability to identify the evolution, although constant, of their natural frequency and damping ratio. For that purpose, a set of synthetic free-decay signals is generated with random values of $A$, ${\omega }_0$, $\zeta$, and $\phi$, following a uniform distribution per property within the next limits:

$$\begin{array}{c}A\in \left(0, 1\right) {\tilde{\omega }}_0\in \left(0, 0.1\right)\\ \zeta \in \left(0.001, 0.2\right) \phi \in \left(-\pi , \pi \right)\end{array}$$

(20)

A sampling rate should be selected to perform the test, which is set to $F_s=\mathrm{10,000}$ Hz. This value does not have any influence on the test, because the RNN works with dimensionless signals, so the predictions are independent of $F_s$, and the only consequence is the maximum natural frequency to be detected, which is conditioned by ${\tilde{\omega }}_0=0.1$ (so, in the test, the maximum natural frequency is 1000 Hz).

The settling time, which is the time it takes for the response to obtain and stay within 3% of the steady-state value [24], is calculated for each free-decay through Equation (21):

$${\tau }_s={\displaystyle \frac{3.5}{\zeta {\omega }_0}}$$

(21)

which, in the discrete domain, turns out to be $k_s$ (Equation (22)):

$$k_s={\tau }_s{F}_s=⌊{\displaystyle \frac{3.5{·F}_s}{{\omega }_0\zeta }}⌋=⌊{\displaystyle \frac{7\pi }{{\tilde{\omega }}_0\zeta }}⌋$$

(22)

To test the algorithm, 1000 free-decay signals are generated, with their corresponding 1000 random pairs of ${\tilde{\omega }}_0\left(t\right)$ and $\zeta \left(t\right)$, which are constant. Since each free-decay signal now evolves along a different number of samples depending on its properties, those with the lowest ${\tilde{\omega }}_0\zeta$ are longer. Therefore, the dataset is composed of many more windows with low ${\tilde{\omega }}_0\zeta$ and the error metrics are biased. To avoid that, a different error metric is used in this section, as illustrated in Figure 5. First, $\widehat{\zeta }$ (^{^} denotes the predicted value) is calculated by dividing $\widehat{\left({\tilde{\omega }}_0\zeta \right)}$ by $\widehat{\left({\tilde{\omega }}_0\right)}$ in each sample, and ${\widehat{\omega }}_0$ is calculated through Equation (7). Then, the average of all the predicted values is calculated for each free-decay signal and property through Equation (23):

$$\overline{{\widehat{\omega }}_0}={\displaystyle \frac{1}{n_i}}\sum _1^{n_i}{\widehat{\omega }}_0^i \overline{\widehat{\zeta }}={\displaystyle \frac{1}{n_i}}\sum _1^{n_i}{\widehat{\zeta }}^i$$

(23)

where $n_i$ is defined in Figure 5 and represents the number of predictions in each signal (because of the data windowing). This procedure avoids the bias, since it leads to 1000 pairs of values ($\overline{{\widehat{\omega }}_0}, \overline{\widehat{\zeta }}$), which can be compared to the labelled properties of each system: (${\omega }_0^{truth},{\zeta }^{truth}$) through their relative error (RE, defined in Figure 5). Finally, the mean relative error (MRE) and the standard deviation (STD) of the relative errors are calculated through Equations (24) and (25):

$${MRE}_{{\omega }_0}={\displaystyle \frac{1}{{10}^3}}\sum R{E}_{{\omega }_0} {MRE}_{\zeta }={\displaystyle \frac{1}{{10}^3}}\sum R{E}_{\zeta }$$

(24)

$${STD}_{{\omega }_0}=\sqrt{{\displaystyle \frac{\sum {\left(R{E}_{{\omega }_0}-{MRE}_{{\omega }_0}\right)}^2}{{10}^3-1}}} {STD}_{\zeta }=\sqrt{{\displaystyle \frac{\sum {\left(R{E}_{\zeta }-{MRE}_{\zeta }\right)}^2}{{10}^3-1}}}$$

(25)

The three best architectures (A17, A16, and A19) are tested. The results are summarised in

Table 2. Test results for the three best-performing architectures.

Arch.	${\mathit{\omega }}_0$		$\mathit{\zeta }$
	MRE [%]	STD [%]	MRE [%]	STD [%]
A17	0.54	1.82	9.36	56.76
A16	0.93	2.15	22.83	132.17
A19	0.69	2.08	9.50	53.46

and Figure 6, where the relation of predicted–labelled values is plotted. Note that the test stage confirms A17 as the best-performing architecture.

4. Results and Discussion

To test the described methodology, both simulated and experimental nonlinear free-decay signals are processed in the next subsections.

4.1. Test over Simulated Nonlinear Free-Decay Signals

Firstly, the developed algorithm is tested with synthetically generated nonlinear free-decay signals. For this purpose, different analytical signals with time-varying ${\omega }_0\left(t\right)$ and $\zeta \left(t\right)$ are considered. Accordingly, the expression of a damped chirp is used to compute the signals to be processed [16]. Integrating the time-dependent expression of the instant phase function, and assuming Equation (26),

$$\begin{array}{c}{\omega }_0\left(t\right)={\omega }_0^0+{∆}_{\omega }t\\ \zeta \left(t\right)={\zeta }^0+{∆}_{\zeta }t\end{array}$$

(26)

the damped chirp expression is represented by Equation (27):

$$x\left(t\right)=A\mathit{sin}\left(\left({\omega }_0^0+{\displaystyle \frac{1}{2}}{∆}_{\omega }t\right)t+\phi \right)e^{-\left({\zeta }^0{\omega }_0^0+{\displaystyle \frac{1}{2}}\left({\zeta }^0{∆}_{\omega }+{\omega }_0^0{∆}_{\zeta }\right)t+{\displaystyle \frac{1}{3}}\left({∆}_{\zeta }{∆}_{\omega }\right)t^2\right)t}$$

(27)

Using Equation (27), the signals in

Table 3. Simulated nonlinear systems properties. (a), (b) and (c) stand for the label of each combination of properties.

Properties	$\mathit{A}$ [m]	$\mathit{\phi }$ [rad]	${\mathit{\omega }}_0^0$ [Hz]	${\mathit{\omega }}_0^{\mathit{f}}$ [Hz]	${\mathit{\zeta }}^0$ [-]	${\mathit{\zeta }}^{\mathit{f}}$ [-]
(a)	1	0	2.3	3.0	0.06	0.06
(b)	1	0	3.0	3.0	0.06	0.01
(c)	1	0	2.3	3.0	0.06	0.06

are tested (indicating the initial (⁰) and the final (^f) properties along 15 s), and the results are plotted in Figure 7. Note that three different combinations of ${\omega }_0\left(t\right)$ and $\zeta \left(t\right)$ are tested, and RecuID delivers a precise tracking of both properties, even though it was only trained with linear evolutions. Additionally, since they are analytical and no noise has been artificially added, the algorithm is able to provide good results, even when the signal comes very close to 0 m.

4.2. Experimental System Configurations

Once validated over nonlinear examples, RecuID is experimentally tested through a laboratory-scale SDOF system (Figure 8). It is composed of a moving frame (MF), several steel plates (SP) that confine the movement to the vertical direction, and a rigid frame (RF). As can be seen, the moving frame is hollow, so its mass can be modified as desired. Also, springs can be installed between the moving and the rigid frames in order to modify the stiffness of the ensemble. Finally, in parallel with the springs and the plates, one or two steel cables (H1 and H2, details in Figure 8) can be installed in order to modify the overall damping of the system. The cables are composed of multiple thinner wires (UNE-EN 12385 [25], 7 × 7, 8 mm), and the friction between them is mainly responsible for their equivalent damping, providing the necessary nonlinear effect to test the proposed methodology. As can be seen, they are wrapped helically to improve friction and foster the nonlinearity.

In order to test different scenarios, several configurations are sought. These configurations, summarised in

Table 4. Summary of the experimental configurations.

ID	Added Mass [kg]	Added Springs [N/m]
1	-	-
2	24.2	-
3	-	1820
4	24.2	1820

, have different dynamic properties (the equivalent moving mass and stiffness lead to several natural frequencies and damping ratios). The first one corresponds to the unmodified configuration, which is characterised by a moving mass of 9.1 kg, a stiffness of 6780.5 N/m, and an equivalent viscous damping of 1.12 Ns/m (which is inherent to the steel plates, SP). The physical parameters are subsequently modified by adding discrete mass and stiffness components. All of the configurations in Table 4 are tested without any cable (N) and with one and two cables (H1 and H2, respectively).

The measurement chain is shown in Figure 9 and consists of a laser sensor (Panasonic (Osaka, Japan), HL-G1 A-C5) to measure the displacement of the moving mass and the system SIRIUS as a data acquisition system (DAQ). The signal is registered and stored through the corresponding software, DewesoftX (V23-1).

The SDOF system is separated from its equilibrium position to induce movement manually or with an impact hammer, and it evolves freely until repose. The next subsections present the results of the identification.

4.3. Identification of the Configurations

Before applying the methodology, it is useful to identify the linear modal properties of the tested configurations without the nonlinear dampers. The conventional least-squares algorithm, commonly known as curve-fitting, is applied. Several experiments are carried out following the excitation-response scheme described above, and the resulting free-decay point clouds are fitted to the expression given in Equation (4).

Table 5. Summary of the properties without the nonlinear damper.

ID	${\mathit{\omega }}_0$ [Hz]	$\mathit{\zeta }$ [-]
N1	4.321	0.0019
N2	2.249	0.0012
N3	5.161	0.0015
N4	2.526	0.0013

presents the curve-fitting results for these “Ni” configurations. As expected, ${\omega }_0$ increases with stiffness and decreases with mass. Regarding $\zeta$, every increase in either mass or stiffness leads to a reduction in $\zeta$, which can be physically justified (Equation (2)).

4.4. Nonlinear Identifications

Two methodologies are compared in this section in order to evaluate the changing properties of the nonlinear system when helical dampers are added. The experimental free-decay signals are available as Supplementary Materials to this manuscript (‘.mat’ file).

Firstly, RecuID is applied to each one of the experimental setups listed in Table 4. Since the identification process of the proposed algorithm is strongly dependent on the sampling rate, an iterative decision procedure is carried out regarding the order of magnitude of the ${\omega }_0$ detected in the configuration without steel cables (described in Table 5). Figure 10 and Figure 11 describe in black the tracked values of ${\omega }_0$ and $\zeta$ for each free-decay during the first seconds of response, indicating the corresponding final $F_s$ used for the identification. Figure 10 corresponds to the H1 case (only one helical steel cable), and Figure 11 corresponds to the H2 case (two helical steel cables). Note that the system nonlinearity leads to time-varying modal properties. To simplify the analysis, only one experimental free-decay signal per setup is presented.

As can be observed, in all cases, RecuID is able to sweep the free-decay and provide an estimate per time step. The smoothness of the properties’ evolution is coherent with the initial assumption of local linear behaviour for short segments. The identified ${\omega }_0\left(t\right)$ exhibits an inverse relationship with the amplitude of response, with lower values predicted at higher displacement levels. Moreover, $\zeta \left(t\right)$ systematically decreases as the amplitude decreases, showing a direct dependence, which is consistent with the behaviour commonly reported for continuous beams and steel plates [26]. The selected $F_s$ is always between 1, 5, and 2 times the highest value of ${\omega }_0$ in the evolutions.

Table 6. Summary of the experimental results describing the overall evolutions of the modal properties. Superscripts “i” and “f” denote initial and final, respectively.

ID	${\mathit{\omega }}_0^{\mathit{i}}$ [Hz]		${\mathit{\omega }}_0^{\mathit{f}}$ [Hz]		${\mathit{\zeta }}^{\mathit{i}}$ [-]		${\mathit{\zeta }}^{\mathit{f}}$ [-]
	RecuID	DLG	RecuID	DLG	RecuID	DLG	RecuID	DLG
H1.1	5.507	5.563	6.566	6.667	0.063	0.067	0.014	0.010
H2.1	6.236	6.695	7.860	7.693	0.128	0.091	0.015	0.007
H1.2	3.380	3.343	3.810	3.847	0.077	0.076	0.016	0.019
H2.2	3.839	4.201	5.108	5.007	0.165	0.128	0.027	0.054
H1.3	6.620	6.674	7.502	7.693	0.047	0.049	0.008	0.009
H2.3	7.863	7.717	9.177	9.094	0.088	0.080	0.007	0.025
H1.4	3.359	3.335	3.636	3.572	0.061	0.047	0.007	0.006
H2.4	3.639	3.760	4.323	4.412	0.128	0.073	0.007	0.013

summarises the results, reporting the initial and final ${\omega }_0$ and $\zeta$ levels.

To benchmark the RecuID results, the conventional approach known as the logarithmic decrement (DLG) method is applied [27]. It is based on a local maxima detection, by looking for the time samples ($t_1,t_2$) at which the amplitude peaks of the free response occur ($x_1$,$x_2$). From these peaks, the damped natural frequency (${\omega }_d$) is first estimated (Equation (28)). Subsequently, the natural logarithm of the ratio between successive peak amplitudes ($\delta$) is computed and used in Equation (28):

$${\omega }_d={\displaystyle \frac{2\pi }{t_2-t_1}} \delta =ln\left({\displaystyle \frac{x_1}{x_2}}\right)={\displaystyle \frac{2\pi \zeta }{\sqrt{1-{\zeta }^2}}}$$

(28)

from which $\zeta$ is estimated. This method progressively loses accuracy for high $\zeta$, and it is inherently limited to estimating ${\omega }_d$. Thus, only one value per oscillation cycle can be estimated. Finally, ${\omega }_0$ is calculated through the relation ${\omega }_d={\omega }_0\sqrt{1-{\zeta }^2}$.

The results obtained with the DLG method are overlapped in orange in Figure 10 and Figure 11. The same time evolutions, considering the sampling rate selected for RecuID, are processed. As can be observed, the estimations provided by the DLG are consistent with those obtained using RecuID, which further supports the reliability of the proposed approach. Table 6 summarises the results, reporting the initial and final ${\omega }_0$ and $\zeta$ levels. Moreover, to deeply compare both methods, the mean absolute error (MAE) between the RecuID and DLG estimations is calculated, using Equation (29):

$${MAE}_{{\omega }_0}={\displaystyle \frac{1}{N_t}}\sum \left|{\omega }_0^{RecuID}-{\omega }_0^{DLG}\right| {MAE}_{\zeta }={\displaystyle \frac{1}{N_t}}\sum \left|{\zeta }^{RecuID}-{\zeta }^{DLG}\right|$$

(29)

where $N_t$ was previously defined as the number of time samples in each free-decay. Since RecuID has a higher estimation rate, the DLG estimations are linearly interpolated to enable the comparison. The MAE results are presented in

Table 7. Comparison of the RecuID and DLG methods by means of the MAE for each modal property.

ID	${\mathit{M}\mathit{A}\mathit{E}}_{{\mathit{\omega }}_0}$ [Hz]	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{\zeta }}$ [-]
H1.1	0.223	0.007
H2.1	0.163	0.007
H1.2	0.071	0.010
H2.2	0.010	0.015
H1.3	0.154	0.004
H2.3	0.244	0.013
H1.4	0.139	0.004
H2.4	0.067	0.007

.

Table 6 numerically reveals the similarity of the predictions between the two methods. Furthermore, the maximum MAE values presented in Table 7 are 0.244 Hz for ${\omega }_0$ and 0.015 for $\zeta$, which are sustainably low values with respect to the general evolutions. As a conclusion, DLG is simpler and is able to reproduce the evolving behaviour of the modal properties; however, it presents two main drawbacks. First, the estimations are noticeably noisier and exhibit lower accuracy. Second, and more importantly, although the computational cost of the method is low, it is not possible to obtain an estimation at each time sample, as explained previously. Consequently, the identification speed is inherently limited by the natural period of the system. While the modal properties that RecuID can identify are also bounded, the limits for ${\omega }_0$ are defined relative to $F_s$, so the accuracy can be improved through an appropriate sampling rate selection. Regarding $\zeta$, the admissible range is sufficiently wide to ensure applicability to conventional structures.

Modal Parameters’ Evolution

As a final analysis, the effects of adding the steel cables can be studied regarding the identification of the linear system in Table 5. Figure 12 and Figure 13 show the changes in ${\omega }_0$ and $\zeta$, taking as reference the RecuID estimations in Table 6.

Predictably, Figure 12 shows how the steel cables increase the stiffness of the SDOF device; therefore, ${\omega }_0$ consistently increases with respect to the “N” setup. The system could reach a maximum value of 9.18 Hz during its evolution in configuration 3. Bars with the same colour represent the nonlinear evolution of ${\omega }_0$ for a given steel cable case, from the initial instant (“i”) to the final one (“f”). For each configuration, the maximum variability of ${\omega }_0$ relative to the corresponding linear case is highlighted (in %). The results indicate that configuration 2 (mass addition only) exhibits the largest frequency change induced by the helical nonlinear dampers with respect to the linear case, reaching a difference of 127.1%. Moreover, the maximum nonlinear variation of ${\omega }_0$ within a free-decay response is indicated in red (“max”); it occurs at the H2.2 case (added mass and two helical dampers), and the natural frequency increases 33.1% between “i” and “f”.

Finally, Figure 13 compares the estimated $\zeta$. The highest value of $\zeta$ (0.165) remains below 0.2, which is the upper bound defined for the proposed methodology. Since the $\zeta$ values obtained for the “N” case were not meaningful (note that the intrinsic damping is nearly negligible), the percentage comparison is now performed between the H1 and H2 cases, taking as reference the initial values in each nonlinear evolution. As observed, for high amplitudes of vibration (“i” instant), the addition of one helical damper (H1) produces a significant increase in $\zeta$, which is approximately doubled when a second helical damper is added (H2 case). Note that the final $\zeta$ value in each evolution (“f” instant) decreases markedly, revealing a maximum reduction between “i” and “f” of 94.1% for configuration H2.4 (highlighted in red as “max”).

5. Conclusions

A novel methodology has been successfully developed to estimate the instantaneous evolution of the natural frequency and damping ratio in nonlinear SDOF systems. The proposed RecuID algorithm proves to be a general approach capable of performing accurate, sample-by-sample identification by means of a windowing technique that normalises the magnitude of the response over a 10-sample segment of a free-decay signal. The main strength of the methodology lies in the fact that the neural network is trained exclusively using synthetic responses from linear SDOF systems. The assumption of local quasi-linear behaviour along the nonlinear response has been shown to be valid, enabling RecuID to reliably track modal properties in both linear and nonlinear regimes.

Although large amounts of data could be efficiently generated, it has been proven that a single oscillation cycle was sufficient to represent the complete free-decay evolution per system. These synthetic cycles were windowed and labelled with their corresponding modal properties. In this process, the discretisation led to a natural frequency normalisation with respect to the sampling rate ($F_s$), which generalised the methodology to any system regardless of its properties. The damping ratio was bounded within a wide interval, and its label was defined as the product ${\tilde{\omega }}_0\zeta$, which improved the learning efficiency of the neural network.

A total of 20 CNN-LSTM architectures were trained to simultaneously estimate both modal properties. The validation process enabled the selection of the best-performing architecture, A17, which provided a balanced trade-off between model depth and size, achieving a validation MSE of 2.52 × 10⁻⁵ and 185,794 trainable weights. The test stage proved the methodology to be effective over complete linear free-decay signals, with a maximum MRE of 0.93% for ${\omega }_0$ and 22.83% for $\zeta$.

Section 4 addressed the nonlinear application, demonstrating the performance of RecuID on both simulated and experimental nonlinear responses. The experimental validation was performed on an SDOF device with four possible configurations of different mass and stiffness. RecuID was compared with the DLG method, reaching a comparable performance: the maximum MAE between them (regarding Table 7) has been 0.244 Hz for ${\omega }_0$ and 0.015 for $\zeta$. The helical dampers have been demonstrated to increase $\zeta$ up to 0.17 in the H2 case (Figure 13), but with a strong dependency on the amplitude of vibration, revealing a pronounced nonlinear behaviour which has been successfully captured.

The identification results exhibit more accuracy for ${\omega }_0$ than for $\zeta$, due to the inherent uncertainty associated with damping estimation. RecuID outperforms other conventional methods because, although the DLG method is computationally simpler, it cannot provide the modal parameter estimates within a fraction of an oscillation cycle. In contrast, the proposed algorithm is trained to operate on a generic segment of 10 consecutive samples, which, depending on the selected $F_s$, can represent a very short fraction of the cycle, regardless of whether they include the amplitude peak or not. Thus, the identification strategy is significantly faster and leads to a considerably more robust approach.

The direct application of this work is to approach the study of nonlinear systems in order to fully characterise their dynamic behaviour and identify the sources of nonlinearity, such as amplitude-dependent effects. There is a variety of applications in which steel cables are used as dampers, such as the Stockbridge damper [28], which behaves nonlinearly and is used in overhead power lines to suppress vibration caused by wind. In addition, a robust and fast algorithm to detect modal properties from free-decay responses is particularly valuable in adaptive control applications, where active or passive controllers should be tuned based on immediate system-state estimates. In a broader structural engineering context, SHM is evolving towards continuous monitoring strategies, for which algorithms such as RecuID can address large displacements analysis, dry friction phenomena, instability detection, and time-varying boundary conditions, such as mechanical end-stops and clearance take-up phenomena.

Future research will focus on incorporating the mode-shape identification and on extending the methodology to multi-degree-of-freedom (MDOF) systems, in which additional types of nonlinear dependencies may arise. Such an extension will require addressing the synchronisation of multiple measurement signals, as well as evaluating the robustness of the approach against time delays and other common experimental issues. Unlike the natural frequency and damping ratio, mode shapes depend explicitly on the number of degrees-of-freedom; therefore, the proposed methodology will need to be adapted to properly handle these challenges in future developments.