Discovering governing equation in structural dynamics from acceleration-only measurements

TL;DR

This work tackles the problem of discovering governing equations for dynamical systems from acceleration-only measurements, a common constraint in structural dynamics. It introduces sparse ABC (s-ABC), a library-based equation-discovery framework that uses spike-and-slab priors and a Gaussian-mixture sampling strategy to perform model selection and parameter estimation without requiring displacement or velocity data; the distance-based ABC acceptance mechanism guides parsimonious inference. Across four benchmark SDOF systems (damped pendulum, linear oscillator, Duffing oscillator, and quadratic viscous damping), the method accurately identifies the active terms in the candidate library and estimates parameters with high fidelity, while producing accurate predictive trajectories despite measurement noise. The approach holds practical potential for accelerometer-based system identification, enabling interpretable, parsimonious models when traditional state measurements are unavailable, though it relies on a well-chosen dictionary and can be computationally intensive due to ABC iterations.

Abstract

Over the past few years, equation discovery has gained popularity in different fields of science and engineering. However, existing equation discovery algorithms rely on the availability of noisy measurements of the state variables (i.e., displacement {and velocity}). This is a major bottleneck in structural dynamics, where we often only have access to acceleration measurements. To that end, this paper introduces a novel equation discovery algorithm for discovering governing equations of dynamical systems from acceleration-only measurements. The proposed algorithm employs a library-based approach for equation discovery. To enable equation discovery from acceleration-only measurements, we propose a novel Approximate Bayesian Computation (ABC) model that prioritizes parsimonious models. The efficacy of the proposed algorithm is illustrated using {four} structural dynamics examples that include both linear and nonlinear dynamical systems. The case studies presented illustrate the possible application of the proposed approach for equation discovery of dynamical systems from acceleration-only measurements.

Figures (9)

• Figure 1: Schematic illustration of the proposed s-ABC algorithm. First, acceleration data from the system is collected. Then, using predefined priors, an initial population of particles is generated. Using the active particles from this population, a Gaussian mixture model for sampling is constructed. After this, a new population is formed using the active particles and newly sampled particles from this Gaussian mixture. Then, a Gaussian mixture using the active particles of the current population is constructed. The last two steps are repeated to generate new populations successively. This is done until the stopping criterion is satisfied. After this, if fine-tuning of the current parameters is necessary, then the process is reinitialization while encouraging exploration, and the process of constructing a Gaussian mixture model and sampling from it is repeated. If the estimated parameters are acceptable and reinitialization is not needed, then the parameters with the lowest loss value are selected to be the parameters of the identified model.
• Figure 2: Inclusion probability (IP) of each candidate function in the final s-ABC population for the damped pendulum.
• Figure 3: Simulated response of the actual model (black dashed line) and the discovered model (red line) for the damped pendulum.
• Figure 4: Inclusion probability (IP) of each basis function in the final s-ABC population for the linear spring-dashpot system.
• Figure 5: Simulated response of the actual model (black dashed line) and the discovered model (red line) for the linear spring-dashpot system.