Abstract
This paper presents an active control method for a quasi-zero stiffness (QZS) isolator using flexures based on a Lyapunov function. First, shown is a dynamic model of an active QZS isolator having indirect horizontal actuation. In the model, the control force is applied along the horizontal direction to compensate for vertical vibrations. Next, a nonlinear control algorithm for the active isolator is developed based on a Lyapunov function. Simulation of the active isolator model which consists of passive QZS isolators, sensors and actuator dynamic models is done to study the effects of control tuning gain on the system performances. In order to verify the active isolation performances, developed is an experimental model including an active QZS isolator, an exciter device and various sensors. Finally, experiments for such as impulse disturbance rejection and transmissibility are performed and the results show that the indirect horizontal actuation by the active QZS isolator using flexure attenuates impulse disturbance as well as isolates the base vibration effectively.
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Abbreviations
- A 1–3 :
-
Coefficients of nonlinear model of notched flexure
- a 1–3 :
-
Control gains
- B 1–3 :
-
Coefficients of Lyapunov functions
- C,c :
-
Dimensional and non-dimensional damping coefficient of the isolator system
- f :
-
Non-dimensional vertical restoring force of the flexures
- F ch , f ch :
-
Dimensional and non-dimensional control force produced by the horizontal actuator
- f cv :
-
Non-dimensional control force produced by the horizontal actuator
- f mg :
-
Non-dimensional weight of the payload
- G s :
-
The transfer function of a sensor
- k h , k v :
-
Non-dimensional stiffness of horizontal and vertical spring
- K l , k l :
-
Dimensional and non-dimensional linear stiffness of the isolator
- K n , k n :
-
Dimensional and non-dimensional nonlinear stiffness of the isolator
- L, \(\dot L\) :
-
The Lyapunov function and its derivative
- L a , L b :
-
Length of the notched and thick parts of the flexure
- M, m :
-
Dimensional and non-dimensional supported mass
- M B :
-
Base mass
- n :
-
Flexure shape ratio (L b /L a )
- g :
-
Number of flexures
- p :
-
Non-dimensional compression force
- p 0 :
-
Non-dimensional initial compression force
- s :
-
The number of flexures in one lateral side of the isolator
- T 1, T 2 :
-
Coefficients for flexure stiffness
- x l :
-
Non-dimensional horizontal parasitic motion of flexure at right end of flexure
- y :
-
Non-dimensional vertical coordinate
- y 0 :
-
Non-dimensional initial deflection of vertical spring for gravity compensation
- Y B :
-
Base excitation
- Y m , y m :
-
Dimensional and non-dimensional vertical displacement of payload mass
- z 1, z 2 :
-
System states for Lyapunov function
- W(t), w(t) :
-
Dimensional and non-dimensional external disturbance force
- ω c :
-
Cut-off frequency of the sensor
- ξ :
-
Damping ratio of the sensor model
References
Ibrahim, R. A., “Recent advances in nonlinear passive vibration isolators,” J. of Sound and Vibration, Vol. 314, No. 3–5, pp. 371–452, 2008.
Harris, C. M. and Piersol, A. G., “Shock and Vibration Handbook,” fifth ed., McGraw-Hill, 2002.
Rivin, E. I., “Passive Vibration Isolation,” ASME Press, 2001.
Platus, D. L., “Negative-stiffness-mechanism vibration isolation systems,” Proc. of SPIE, Vol. 1619, Vibration Control in Microelectronics, Optics and Metrology, pp. 44–54, 1991.
Minus K Technology, www.minusk.com
Zhang, J., Li, D., and Dong, S., “An ultra-low frequency parallel connection nonlinear isolator for precision instruments,” Key Engineering Materials, Vol. 257–258, pp. 231–236, 2004.
Schenka, M., Guestb, S. D., and Herdera, J. L., “Zero stiffness tensegrity structures,” Int. J. of Solids and Structures, Vol. 44, No. 20, pp. 6569–6583, 2007.
Tarnai, T., “Zero stiffness elastic structures,” Int. J. Mech. Sci., Vol. 45, No.3, pp. 425–431, 2003.
Park, S. T. and Luu, T. T., “Techniques for optimizing parameters of negative stiffness,” Proc. IMechE part C: J. Mec. Eng. Sci., Vol. 221, No.5, pp. 505–511, 2007.
Kovacic, I., Brennan, M. J., and Waters, T. P., “A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic,” J. of Sound and Vibration, Vol. 315, No. 3, pp. 700–711, 2008.
Ahn, H. J., “Performance limit of a passive vertical isolator,” J. of Mec. Science and Technology, Vol. 22, No. 12, pp. 2357–2364, 2008.
Carrella, A., Brennan, M. J., and Waters, T. P., “Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic,” J. of Sound and Vibration, Vol. 301, No. 3–5, pp. 678–689, 2007.
Trung, P. V., Kim, K. R., and Ahn, H. J., “Active Control of a Quasi-Zero Stiffness Isolator Using Flexures,” 12th International Symposium on Advanced Intelligent Systems, 2011.
Danh, L. T. and Ahn, K. K.., “Experimental investigation of a vibration isolation system using negative stiffness structure,” 11th IEEE Conference on Control, Automation and Systems, pp. 1582–1587, 2011.
Coppola, G. and Liu, K., “Control of a unique active vibration isolator with a phase compensation technique and automatic on/off switching,” J. of Sound and Vibration, Vol. 329, No. 25, pp. 5233–5248, 2010.
Kim, K. R., You, Y. H., and Ahn, H. J., “Optimal design of a QZS isolator using flexures for a wide range of payload,” Int. J. Precis. Eng. Manuf., Vol. 14, No.6, pp. 911–917, 2013.
Ahn, H. J., “A Unified Model for Quasi-Zero-Stiffness Passive Vibration Isolators With Symmetric Nonlinearity,” Proc. of the ASME/IDETC, Paper No. DETC2010-28604, 2010.
Ahn, H. J. and Kim, K. R., “Active Control of Quazi-Zero Stiffness Isolators,” 4th Annual Dynamic Systems and Control Conference, 2011.
Carrella, A., Brennan, M. J., Waters, T. P., and Lopes, Jr. V., “Force and displacement transmissibility of a nonlinear isolator with high-static-low-dynamic-stiffness,” International Journal of Mechanical Sciences, Vol. 55, No. 1, pp. 22–29, 2012.
Kim, M. S. and Kim, J. H., “Design of a gain scheduled PID controller for the precision stage in lithography,” Int. J. Precis. Eng. Manuf., Vol. 12, No.6, pp. 993–1000, 2011.
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Trung, P.V., Kim, KR. & Ahn, HJ. A nonlinear control of an QZS isolator with flexures based on a lyapunov function. Int. J. Precis. Eng. Manuf. 14, 919–924 (2013). https://doi.org/10.1007/s12541-013-0121-z
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DOI: https://doi.org/10.1007/s12541-013-0121-z