Mechanisms typically have clearance in their joints due to manufacturing and assembly requirements, and this clearance tends to increase during operation. If the clearance exceeds the allowable limit, it can lead to decreased accuracy, increased vibrations and noise, and reduced system efficiency. Therefore, timely identification of the increase in the amount of clearance in the joints is essential to prevent destructive effects and increased maintenance costs. In recent years, the use of health monitoring methods has gained considerable attention due to their effectiveness and cost efficiency. In this study, the time history of the acceleration of the slider link due to the operation of a slider-crank mechanism with radial clearance in revolute joints is investigated. To identify defects, we present a novel index based on nonlinear monitoring methods, including vibration-acoustic modulation (VAM) and high-order harmonic generation (HHG), which can help predict the presence or absence of clearance in the mechanism. Then, using the collected data, we trained a multilayer perceptron (MLP) neural network to address the study problem and predict the index value. Also, the structure of the neural network was optimized using the particle swarm algorithm (PSO). The results demonstrate that the optimized network can accurately predict the desired index based on the input values.