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Controlling self-excited vibration using positive position feedback with time-delay

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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.

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Abbreviations

\(A\) :

Non-dimensional amplitude

\(c_{1}\) :

Non-dimensional negative damping coefficient

\(c_{3}\) :

Non-dimensional positive damping coefficient

\(k_{1}\) :

Non-dimensional controller gain

\(k_{2}\) :

Non-dimensional sensitivity of the sensor

\(K_{c} = k_{1} k_{2}\) :

Non-dimensional loop gain

\(x\) :

Non-dimensional displacement

\(x_{f}\) :

Non-dimensional filter variable

\(\zeta_{f}\) :

Non-dimensional damping ratio of filter

\(\zeta_{c}\) :

Damping ratio of closed-loop poles

\(\omega\) :

Non-dimensional frequency

\(\omega_{c}\) :

Frequency of closed-loop poles

\(\omega_{f}\) :

Non-dimensional natural frequency filter

\(\lambda_{1} ,\lambda_{2}\) :

Identical complex conjugate pair of poles used in pole crossover design

\(\tau\) :

Non-dimensional time delay parameter

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Engineering Science and Technology, Shibpur, P.O. Botanic Garden, Howrah, West Bengal, 711103, India

    Akash Sarkar, Joy Mondal & S. Chatterjee

Authors
  1. Akash Sarkar
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  2. Joy Mondal
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  3. S. Chatterjee
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Correspondence to S. Chatterjee.

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Technical Editor: Wallace Moreira Bessa, D.Sc..

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Appendix A

Appendix A

The objective of the design of absorber is to maximize the dissipation of energy so that system converges rapidly to the equilibrium. The rate of dissipation of the absorber was qualitatively measured with the position of the dominant poles in the complex plane. Therefore, the poles should be placed as far from the imaginary axis in the complex plane. To supplement the optimization technique discussed in chapter 3, the movement of the poles with the control gain (Kc) is illustrated in Fig. 12. The optimal absorber parameters were chosen for \(c_{1} = - 0.3\) (from Table 1). The root locus of the controlled system is illustrated with variation control gain (Kc). It is observed from Fig. 12 poles the system merge for Kc = 0.3 with optimized data chosen from Table 1.

Fig. 12
figure 12

Root locus of poles of the controlled system. Arrow indicates the direction of increasing control gain (Kc) \(c_{1} = - 0.3,\zeta_{f} = 0.3783,\omega_{f} = 1.068\)

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Sarkar, A., Mondal, J. & Chatterjee, S. Controlling self-excited vibration using positive position feedback with time-delay. J Braz. Soc. Mech. Sci. Eng. 42, 464 (2020). https://doi.org/10.1007/s40430-020-02544-7

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