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Closed-Form Solution to H-Infinity Optimization of Pre-Tensioned Tuned Mass Damper

Okba Abid Charef(1*), Salah Khalfallah(2)

(1) Polytechnic National School Of Constantine, Algeria
(2) Polytechnic National School Of Constantine, Algeria
(*) Corresponding author



The H-infinity optimal parameters of a pre-tensioned tuned mass damper are derived to overcome the great amplitude of un-damped primary structure vibrations under harmonic excitation. In this case, damping, mass, and stiffness elements are connected directly to the ground. The two fixed points approach is used to optimize analytically frequency response function, damping ratio and pre-tensioned stiffness coefficient. The optimum pre-tensioned spring is used to keep more the stability of structure according to the principle of preload elastic device properties. Finally, the mitigation of resonance oscillation amplitude of the primary structure using the proposed dynamic absorber is compared with the traditional one. Under conditions of optimal control performance, it is proved analytically that the proposed vibration absorber with pre-tensioned stiffness provides well attenuation in the resonant vibration range. Adding, this device can be also broadening the efficient frequency range of vibration.
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Tuned Mass Damper; Pre-Tensioned Stiffness; H-Infinity Optimization; Mitigation of the Resonant Vibration Amplitude; Control Performance; Primary Structure

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H. Frahm, E. H. Amara, Device for damping vibrations of bodies, U.S. Patent, pp. 989-958, 1909.

J. Ormondroyd, JP. Den Hartog, The theory of the dynamic vibration absorber, Appl Mech, Vol. 50, pp. 9-22, 1928.

E. Hahnkamm, The damping of the foundation vibrations at varying excitation frequency, Master of Architecture, Vol. 4, pp.192–201, 1932.

J. Dayou, S. Wang, Derivation of the fixed-points theory with some numerical simulations for global vibration control of structure with closely spaced natural frequencies, Mechanics Based Design of Structures and Machines, Vol. 34, n. 1, pp. 49-68, 2006.

J. Dayou, Fixed-points theory for global vibration control using vibration neutralizer, Journal of Sound and Vibration, Vol. 292, n. 3-5, pp. 765-776, 2006.

Y.L. Cheung, W.O. Wong, H∞ and H2 optimizations of a dynamic vibration absorber for suppressing vibrations in plates, Journal of Sound and Vibration, Vol. 320, n. 1-2, pp. 29-42, 2009.

M. B. Ozer, T.J. Royston, Extending Den Hartog’s vibration absorber technique to multi-degree-of-freedom systems, Journal of Vibration and Acoustic, Vol. 127, n. 4, pp. 341-350, 2005.

J.E. Brock, A note on the damped vibration absorber, Transactions of the ASME, Journal of Applied Mechanics, Vol. 13, n. 4, pp. A-284, 1946.

O. Nishihara, T. Asami, Close-form solutions to the exact optimizations of dynamic vibration absorber (minimizations of the maximum amplitude magnification factors), Journal of Vibration and Acoustics, Vol. 124, n. 4, pp. 576-582, 2002.

J.P. Den Hartog, Mechanical Vibration (New York: McGraw-Hill Publishers. 1956).

M.Z. Ren, A variant design of the dynamic vibration absorber, Journal of Sound and Vibration, Vol. 245, n. 4, pp. 762-770, 2001.

K.F. Liu, J. Liu, The damped dynamic vibration absorbers: revisited and new result, Journal of Sound and Vibration, Vol. 284, n. 3, pp. 1181-1189, 2005.

O. Araz, Effect of detuning conditions on the performance of non-traditional tuned mass dampers under external excitation, Archive of Applied Mechanics, Vol. 90, n. 3, pp 523–532, 2020.

E. Barredo, J.G. Mendoza Larios, J. Colín, J. Mayén, A.A. Flores-Hernández, M. Arias-Montiel, A novel high-performance passive non-traditional inerter-based dynamic vibration absorber, Journal of Sound and Vibration, Vol. 485, pp. 115583, 2020.

M. Yuan, K. Liu, Vibration Suppression and Energy Harvesting with a Non-traditional Vibration Absorber: Transient Responses, Vibration, Vol. 1, n. 1, pp. 105-122, 2018.

H. Naderpour, N. Naji, D. Burkacki, R. Jankowski, Seismic Response of High-Rise Buildings Equipped with Base Isolation and Non-Traditional Tuned Mass Dampers, Applied Sciences, Vol. 9, n. 6, pp 1201, 2019.

M. Yuan, K. Liu, A. Sadhu, Simultaneous vibration suppression and energy harvesting with a non-traditional vibration absorber, Journal of Intelligent Material Systems and Structures, Vol. 29, n. 8, pp 1748–1763, 2018.

E. Pennestri, An application of Chebyshev's min- max criterion to the optimal design of a damped dynamic vibration absorber, Journal of Sound and Vibration, Vol. 217, n. 4, pp. 757-765, 1998.

V. Piccirillo, A.M. Tusset, J.M. Balthazar, Optimization of dynamic vibration absorbers based on equal-peak theory, Latin American Journal of Solids and Structures, Vol. 16, n. 4, 2019.

V.D. Phuc, T.V. Canh, P.V. Lieu, Optimal parameters of dynamic vibration absorber for linear damped rotary systems subjected to harmonic excitation, Vietnam Journal of Mechanics, Vol. 42, n. 4, 2020.

K.K. Dudek, R. Gatt, M.R. Dudek, J.N. Grima, Negative and positive stiffness in auxetic magneto-mechanical metamaterials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 474, n. 2215, 2018.

D.J. Braun, V. Chalvet, A. Dahiya, Positive–negative stiffness actuators, IEEE Transactions on Robotics, Vol. 35, n. 1, pp. 162-173, 2019.

T.C. Wang, C.C. Ko, K.W. Chang, T.W. Ko, Negative-stiffness composite systems and their coupled-field properties, Continuum Mechanics and Thermodynamics, 2021.

Chiroiu, V., Munteanu, L., Dumitriu, D., The Relationship between Auxetic and Negative Stiffness Materials Behavior. Part I: Theory, (2015) International Journal of Earthquake Engineering and Hazard Mitigation (IREHM), 3 (2), pp. 50-60.

B. Chen, L. Chen, B. Du, H. Liu, W. Li, D. Fang, Novel multifunctional negative stiffness mechanical metamaterial structure: Tailored functions of multi-stable and compressive mono-stable, Composites Part B: Engineering, Vol. 204, pp. 108501, 2021.

G.F. Giaccu, An equivalent frequency approach for determining non-linear effects on pre-tensioned-cable cross-braced structures, Journal of Sound and Vibration, Vol. 422, pp. 62–78, 2018.

M. Baduidana, X. Wang, A. Kenfack-Jiotsa, Parameters optimization of series–parallel inerter system with negative stiffness in controlling a single–degree–of–freedom system under base excitation. Journal of Vibration and Control, 2021.

F.S. Li, Q. Chen, J.H. Zhou, Dynamic Properties of a Novel Vibration Isolator with Negative Stiffness, Journal of Vibration Engineering & Technologies Vol. 6, pp. 239-247, 2018.

G. Hu, L. Tang, J. Xu, C. Lan, R. Das, Metamaterial with Local Resonators Coupled by Negative Stiffness Springs for Enhanced Vibration Suppression, Journal of Applied Mechanics, Transactions ASME, Vol. 86, n. 8, 2019.

M. Li, W. Cheng, R. Xie, Design and experiments of a quasi–zero-stiffness isolator with a noncircular cam-based negative-stiffness mechanism, Journal of Vibration and Control, Vol. 26, n. 21-22, pp. 1935-1947, 2020.

Y.L. Cheung, W.O. Wong, H-infinity optimization of a variant design of the dynamic vibration absorber—Revisited and new results, Journal of Sound and Vibration, Vol. 330, n. 16, pp. 3901-3912, 2011.

Y. Hao, Y. Shen, X. Li, J. Wang, S. Yang, H-∞ optimization of Maxwell dynamic vibration absorber with multiple negative stiffness springs, Journal of Low Frequency Noise, Vibration and Active Control, pp. 1-13, 2020.

B.C. Christopher, W.S. David, P.S. Sloan, C.K. Andrew, P.M. Geoffrey, Dynamically variable negative stiffness structures, Science Advances, Vol. 2, n. 2, pp. e1500778, 2016.

X. Huang, Z. Su, H. Hua, Application of a dynamic vibration absorber with negative stiffness for control of a marine shafting system, Ocean Engineering, Vol. 155, pp. 131-143, 2018.

Salem, M., Anany, M., El-Habrouk, M., Rezeka, S., Control of a Dynamic Vibration Absorber Using a Magneto-Rheological Damper, (2013) International Review of Mechanical Engineering (IREME), 7 (1), pp. 81-90.

G.P. Cimellaro, S. Marasco, Tuned-Mass Dampers, In. Introduction to Dynamics of Structures and Earthquake Engineering. Geotechnical, Geological and Earthquake Engineering, 45 (Springer, Cham, 2018, 421-438).

Pahlevan, L., Rezaeepazhand, J., Semi-Active Cabin Suspension of Agricultural Vehicles Using ER Mounts, (2018) International Journal on Engineering Applications (IREA), 6 (6), pp. 196-201.

Sanposh, P., Chinthaned, N., Handling Torque Input Constraints Under Robust Nonlinear Regulation Control of Robotic Systems with Parametric Uncertainties, (2020) International Review of Automatic Control (IREACO), 13 (3), pp. 117-127.

Ali, H., Mhmood, A., Nonlinear H-Infinity Model Reference Controller Design, (2021) International Review of Automatic Control (IREACO), 14 (1), pp. 39-50.

Mekki, I., Bouhamida, M., Saad, M., Robust Control of a Chemical Multivariable System in the Presence of Strong Uncertainties in the Model Parameters, (2018) International Review of Automatic Control (IREACO), 11 (4), pp. 166-173.


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