article · Scientific Reports
A non-perturbative approach is established to convert nonlinear structural vibrations into an equivalent linear differential equation, allowing for precise structural analysis. Numerical computations align closely with precise frequencies and confirm the accuracy of the mathematical solutions across various nonlinear oscillations. To eliminate damaging vibrations in the system, four distinct vibration control strategies are assessed: positive position feedback, integral resonant control, nonlinear integral positive position feedback, and negative derivative feedback. Among these techniques, negative derivative feedback proves the most effective at suppressing vibrations. The stability and parametric behaviour of the structure are examined under severe resonance conditions, specifically primary and one-to-one internal resonance, using the averaging method. Close agreement is achieved between the analytical models and computational simulations, confirming the stability of the proposed structural configuration.
Unwanted vibrations can cause severe structural damage and mechanical failure, especially under resonance conditions. Identifying optimal control techniques such as negative derivative feedback helps engineers safeguard physical systems against extreme vibrational stress, while simplified mathematical modelling reduces the computational complexity needed to predict and manage nonlinear dynamic behaviours.
This is early-stage theoretical and computational research focused on mathematical modelling and control simulation. The findings could eventually assist mechanical and structural control engineers in designing active damping mechanisms to mitigate resonance in vibrating machinery. However, because the abstract reports only mathematical solutions and numerical simulations of a prototype model, practical deployment remains far from real-world commercial use.
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To arrive at an equivalent linear differential equation, the non-perturbative approach (NPA) is established. The corresponding linear equation is employed for performing the structural analysis. A numerical computation demonstrates a high consistency with the precise frequency. The correlation with the numerical solution explains the reasonableness of the obtained solutions. For additional nonlinear kinds of oscillation, the methodology gives an exact simulation. The stable construction of the prototype is shown in a series of diagrams. Positive position feedback (PPF), integral resonant control (IRC), nonlinear integral positive position feedback (NIPPF), and negative derivative feedback (NDF) are proposed to get rid of the damaging vibration in the system. It is found that the NDF control is more efficient than other controllers for vibration suppression. The theoretical methodology is applied by using the averaging method for getting a perturbed solution. The stability and influence of various parameters of the structure are established at main and 1:1 internal resonance, which is presented as one of the worst resonance cases. Association concerning mathematical solution and computational simulation is achieved.
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DOI: 10.1038/s41598-023-50750-9
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