SLIDE: A machine-learning based method for forced dynamic response estimation of multibody systems

TL;DR

The SLiding-window Initially-truncated Dynamic-response Estimator (SLIDE), a deep learning-based method designed to estimate output sequences of mechanical or multibody systems with primarily, but not exclusively, forced excitation, is presented.

Abstract

In computational engineering, enhancing the simulation speed and efficiency is a perpetual goal. To fully take advantage of neural network techniques and hardware, we present the SLiding-window Initially-truncated Dynamic-response Estimator (SLIDE), a deep learning-based method designed to estimate output sequences of mechanical or multibody systems with primarily, but not exclusively, forced excitation. A key advantage of SLIDE is its ability to estimate the dynamic response of damped systems without requiring the full system state, making it particularly effective for flexible multibody systems. The method truncates the output window based on the decay of initial effects, such as damping, which is approximated by the complex eigenvalues of the systems linearized equations. In addition, a second neural network is trained to provide an error estimation, further enhancing the methods applicability. The method is applied to a diverse selection of systems, including the Duffing oscillator, a flexible slider-crank system, and an industrial 6R manipulator, mounted on a flexible socket. Our results demonstrate significant speedups from the simulation up to several millions, exceeding real-time performance substantially.

Figures (18)

• Figure 1: The components of the explored surrogate models: using an original model the dataset is created. The neural network surrogate model is trained and evaluated using this dataset and reproduces the input-output behavior of the system.
• Figure 2: Structure of a general feedforward network. The layers are connected with weights $\mathbf{W}$The hidden layers typically utilize a nonlinear function $a(z)$ as activation function.
• Figure 3: The SLIDE method uses an input window of length $n_\mathrm{in}$ and an output window of length $n_\mathrm{out}$. Because of the damping in the mechanical system, initial conditions and oscillations vanish. The Surrogate Neural Network (S-NN) is trained to map the input $\hat{\mathbf{x}}$ to the output $\hat{\mathbf{y}}$. After training the S-NN, the Error Estimator Network (EE-N) is trained to predict the RMSE between the dataset and the neural network's output from the system's input.
• Figure 4: The structure of the proposed error estimator. For better training performance, the output from the simulation model is scaled to normalize $\mathbf{y}_s$. Before calculating the error of the S-NN, the output is rescaled to its original units. The root mean squared error $e$ is transformed to $\epsilon$ by a logarithmic function Eq. (\ref{['eq:logmapping']}) to \ref{['eq:logmapping2']}, denoted by $f()$. The inverse mapping $f^{-1}()$ is used to obtain $\hat{e}$ from the error estimator.
• Figure 5: The spring damper model consists of the mass $m$, stiffness $k$ and damping $d$. The natural frequency $\omega_0$ and the dimensionless damping $D$ are derived from these parameters. The factor $\alpha$ is used to describe the nonlinearity of the Duffing oscillator.