Controlling self-excited vibration using positive position feedback with time-delay
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(2020) 42:464
TECHNICAL PAPER
Controlling self‑excited vibration using positive position feedback with time‑delay Akash Sarkar1 · Joy Mondal1 · S. Chatterjee1 Received: 13 June 2019 / Accepted: 3 August 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract This paper explores the vibration control of Rayleigh oscillator (a self-excited system), by using positive position feedback method. Both linear and nonlinear stability analyses are performed. The stability regions and optimal system parameters are obtained by performing linear stability analysis, and nonlinear analysis is performed using describing function method to get the amplitude and frequency of the system. The effect of time-delay on system performance is also studied in this paper. It is observed that the existence of time-delay can be unfavourable; however, the situation can be improved by increasing the loop gain. However, it is impossible to design a system without time-delay present in the feedback circuit. Therefore, to nullify the effect of uncertain time-delay, authors have intentionally introduced a preselected time-delay in the feedback circuit and re-optimized to stabilize the static equilibrium of the delayed system. Numerical simulations performed in MATLAB SIMULINK confirm the results of the theoretical analysis obtained. Keywords Self-excited vibration · Positive position feedback · Stability · Time-delay List of symbols A Non-dimensional amplitude c1 Non-dimensional negative damping coefficient c3 Non-dimensional positive damping coefficient k1 Non-dimensional controller gain k2 Non-dimensional sensitivity of the sensor Kc = k1 k2 Non-dimensional loop gain x Non-dimensional displacement xf Non-dimensional filter variable 𝜁f Non-dimensional damping ratio of filter 𝜁c Damping ratio of closed-loop poles 𝜔 Non-dimensional frequency 𝜔c Frequency of closed-loop poles 𝜔f Non-dimensional natural frequency filter 𝜆1 , 𝜆2 Identical complex conjugate pair of poles used in pole crossover design 𝜏 Non-dimensional time delay parameter
Technical Editor: Wallace Moreira Bessa, D.Sc.. * S. Chatterjee [email protected] 1
Department of Mechanical Engineering, Indian Institute of Engineering Science and Technology, Shibpur, P.O. Botanic Garden, Howrah, West Bengal 711103, India
1 Introduction Self-excited vibration is a common phenomenon in many physical systems like friction-induced vibration (in vehicle clutches and brakes, vehicle–bridge interaction) and flow-induced vibration (circular wood saws, in machining, fluid-conveying pipelines). Self-excited vibration is caused due to an intrinsic, state-dependent nonlinear force, and its sustenance requires no external excitation. A phenomenological mathematical model of such oscillations includes a second-order oscillator with a nonlinear damping function producing linear negative damping in the phase space around an unstable equilibrium and positive nonlinear damping far away from the equilibrium. Van der Pol and Rayleigh oscillators are the two widely stu
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