article · Mathematics
An analytical framework has been established to evaluate harmonically excited, self-excited nonlinear oscillators governed by fractional-order state feedback control. The approach uses a Detuned Multiple Scale Method (DMSM), which perturbs the system around its amplitude-dependent response frequency rather than its fixed linear natural frequency as seen in traditional multiple scales techniques. This formulation produces reduced-order amplitude-phase modulation equations that capture identical backbone curves to the first-order Harmonic Balance Method. It also demonstrates that effective damping and stiffness coefficients vary with the amplitude-dependent response frequency. The analytical findings were verified against numerical simulations using a dedicated Runge-Kutta Grünwald-Letnikov algorithm. The results confirm that the detuned method accurately predicts frequency response curves, stability, and bifurcations across weak, moderate, and strong feedback gains, avoiding the misleading predictions produced by traditional methods under large detuning and strong feedback.
Nonlinear oscillations can cause severe instability and damage in dynamic systems. Accurate mathematical models are essential for designing control strategies to suppress these vibrations. By capturing frequency-dependent damping and stiffness effects that traditional analytical methods miss, this formulation gives control engineers more dependable tools to predict system stability and avoid erroneous designs when applying fractional-order feedback controls.
This work represents early-stage theoretical and computational research aimed at vibration control and advanced control systems design. It could eventually inform the design of control algorithms used by mechanical and aerospace engineers to stabilise oscillating machinery or flexible structures. However, the abstract indicates no physical prototyping or experimental validation, placing the research at an early analytical stage with no immediate market-ready application pathway.
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In this work, an accurate analytical solution is developed for a harmonically excited self-excited nonlinear oscillator under fractional-order state feedback control using a modified version of the Traditional Multiple Scales Method (TMSM), termed the Detuned Multiple Scale Method (DMSM). In contrast to the TMSM, which perturbs the nonlinear system about its linear natural frequency, the DMSM perturbs the system implicitly about its amplitude-dependent response frequency. Based on this formulation, reduced-order amplitude–phase modulation equations for the considered fractional-order system are derived and compared with those obtained by TMSM. It is found that the DMSM yields exactly the same backbone curve as that obtained using the first-order Harmonic Balance Method (HBM), whereas the TMSM provides only its leading-order approximation. In addition, the DMSM reveals that the effective linear and nonlinear damping and stiffness coefficients depend on the amplitude-dependent response frequency, unlike the TMSM, where they depend on the fixed linear natural frequency. Furthermore, it is shown that the TMSM detuning term represents only the first-order approximation of the exact detuning term obtained by the DMSM. Accordingly, the considered fractional-order system is analyzed using the DMSM in comparison with the TMSM through Frequency Response Curves (FRCs), bifurcation diagrams, and stability charts. The system dynamics are investigated for different fractional-order derivatives under weak, moderate, and strong feedback gains. Moreover, a Runge–Kutta Grünwald–Letnikov (RK–GL) algorithm is developed and validated for fractional-order simulations, and all obtained FRCs are numerically verified. The numerical results clearly demonstrate that the proposed DMSM maintains excellent agreement with the numerical results not only near the primary linear resonance condition, but also over a wide range of excitation frequencies under weak, moderate, and strong feedback gains. In contrast, the TMSM fails to preserve this level of accuracy and may produce misleading predictions, especially under strong feedback and large detuning conditions. Extracting reduced-order amplitude–phase equations using the DMSM provides highly accurate analytical predictions along with deep physical insight into system dynamics, which, despite their accuracy in steady-state solutions, offer limited insight into transient behavior and the overall evolution of the response.
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DOI: 10.3390/math14173106
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