Prediction of dynamic characteristics for an air spring with an airbag based on physics-data hybrid model under large amplitude excitation

The air spring with an auxiliary airbag (ASAA) is a novel dual-chamber throttling tube air spring featuring a lightweight design, with low cost and flexible installation layout. Different from conventional air springs with constant-volume auxiliary chambers, the ASAA employs an elastic airbag whose volume varies with internal pressure, enabling a wide adjustable range of stiffness and damping. Because of this structural feature, the ASAA exhibits strong adaptability across diverse operating conditions and shows great potential in intelligent air suspension systems. However, under large-amplitude excitation, the coupled nonlinear effects of airbag expansion, the complex viscoelastic behavior of the rubber bladder, and turbulent airflow within the throttling tube lead to highly nonlinear dynamic characteristics. These strong nonlinearities drastically limit the prediction accuracy of traditional physical models. To address this problem, a physics-data hybrid modeling approach is proposed to enhance the prediction accuracy of dynamic stiffness and lag angle under large-amplitude conditions.

A simplified physical model of the ASAA was developed based on the thermodynamic theory, flow resistance analysis of the throttling tube, and viscoelastic modeling of the rubber bladder. For the compressed air part, governing equations were derived under adiabatic assumptions. The nonlinear airbag volume variation term, which is difficult to linearize, was simplified to reduce model complexity, whereas the nonlinear airflow term in the throttling tube was linearized using a first-order Fourier series expansion. The rubber bladder was modeled by combining a Coulomb friction model, which captured amplitude-dependent hysteresis, and a fractional-order Kelvin-Voigt model, which reproduced frequency-dependent viscoelastic effects. Static and dynamic experiments were conducted to identify the key structural parameters. Dynamic experiments were further carried out at excitation frequencies ranging from 0 to 6 Hz and amplitudes ranging from 5 to 40 mm to obtain the experimental data on dynamic stiffness and lag angle. A deep neural network (DNN) was subsequently constructed to learn the residual errors between the experimental measurements and the physical model calculations. Excitation frequency and amplitude were used as input features, and the corresponding errors in dynamic stiffness and lag angle were taken as output targets. The DNN had two hidden layers with rectified linear unit activation and dropout regularization and is trained using the Adam optimization algorithm on normalized datasets to ensure convergence stability and generalization capability.

Validation results demonstrated that the proposed physics-data hybrid model considerably improved the prediction performance, particularly under large-amplitude excitation. In the test cases with excitation amplitudes of 35 mm and 40 mm, which were not included in the training dataset, the hybrid model demonstrated strong generalization ability. For these test cases, the maximum relative errors of dynamic stiffness prediction reduced to approximately 3.22% and 3.98%, respectively, representing an improvement of approximately 6% compared with the conventional physical model. For lag angle prediction, the maximum relative errors were 17.21% and 15.83%, respectively, corresponding to an error reduction of approximately 35% compared with the conventional physical model. Although lag angle prediction remains more sensitive because of its small baseline value and the strong nonlinear stiffness-damping correlation, the hybrid model effectively captures the overall variation trend and achieves substantially improved accuracy.

By integrating physics-based modeling with machine learning-based residual compensation, the proposed hybrid approach effectively overcomes the limitations of simplified physical models in describing strong nonlinear behavior. The method substantially enhances the prediction accuracy of dynamic stiffness and lag angle under highly nonlinear, large-amplitude conditions, providing theoretical support and practical guidance for the design and dynamic modeling of intelligent air suspension systems.