A Verifiable Steady-State Frequency–Velocity Mapping for Desktop FDM Printers Based on an Electromechanical Coupling Framework

To monitor online the operational condition and quality of a desktop fused deposition modeling (FDM) printer, the dynamics of vibro-acoustics must be accurately understood. In this paper, an electromechanical coupling (EMT) framework is established to relate the dynamics of stepper actuation, the transmission chain, and machine motion, deriving a steady-state frequency–velocity mapping for steady or near steady printing segments. The mapping is evaluated by numerical calculation to obtain a theoretical drive frequency for different toolpath directions and commanded printing velocities. Validation is performed on the experiment platform I. Drive-side vibration is measured by an accelerometer mounted on the x-axis beam near the motor end. An acoustic channel is recorded as an auxiliary qualitative cross-check rather than for quantitative error evaluation. For steady printing segments, the dominant frequency in drive-side vibration is compared with the theoretical drive frequency. In the tested steady segments and toolpath directions, the relative error remained below 3%. In a further case study, the G-code is modified to introduce two constant printing velocity segments (40 mm/s and 80 mm/s) within the same continuous record, enabling a direct comparison of dominant frequencies between two steady segments. The results show that, under open-loop stepper drive and within the steady/near steady scope adopted here, a drive-related dominant frequency can be observed stably in the x-axis beam vibration response and matches the theoretical drive frequency. When the commanded constant printing velocity is doubled, the dominant frequency in drive-side vibration in the corresponding steady segment changes by approximately a proportional factor. This study provides a verifiable drive referenced frequency–velocity mapping for steady segments under the tested configuration and a traceable frequency reference for steady segment comparisons within the same print record in subsequent case studies.