From differential equations to convolution: An active-learning framework for mechanical system modelling and physical insight

Modelling dynamical systems and predicting their responses are fundamental skills in mechanical engineering, yet they remain challenging due to the abstract nature of the underlying mathematics. In particular, convolution, central to the analysis of linear time-invariant (LTI) systems, is often introduced theoretically, limiting students' ability to develop physical intuition and practical understanding. This work presents an active learning framework for teaching continuous- and discrete-time convolution by linking differential equation-based modelling to system response prediction. The approach integrates visual animations, MATLAB-based computational activities, and a structured hands-on computational laboratory to support progressive learning. Students progress from interpreting basic signal interactions to applying convolution to predict dynamical system responses. A key contribution is the comparison between convolution-based solutions and numerical time integration using the Runge–Kutta method for a single-degree-of-freedom (SDOF) mechanical system. This dual perspective reinforces impulse response, superposition, and system modelling while demonstrating the equivalence between different solution strategies. Student assessment results indicate improved engagement and conceptual understanding. The proposed framework provides a coherent methodology connecting mathematical formulation, physical interpretation, and computational implementation, with its novelty lying in the structured integration of these complementary elements within a unified learning framework for engineering education.