## Abstract

An active mass damper for vibration control of an oscillator with two translational degrees of freedom is presented along with the corresponding closed-loop control algorithm. The damper consists of two eccentrically rotating masses. In a preferred mode of operation, the masses rotate in opposite directions with a mostly constant angular velocity about a single axis. The resulting force is a harmonic control force. Its direction is determined by the relative angular position of the masses. In previous research, a similar control algorithm for vibration control of a single degree of freedom oscillator has been proven to be effective. In this paper, the control algorithm is augmented such that it can be applied to an oscillator with two translational degrees of freedom. Various state variables are introduced and a feedback control algorithm is developed. The presented algorithm ensures that the rotational motion of the masses is smooth and that the control force has the required orientation. The algorithm is verified experimentally with a test setup for the damping of free vibrations and numerically for stochastically forced vibrations. Finally, the device is compared with a conventional active mass damper. It is shown that the power demand and the energy consumption of the presented device are smaller than those of the conventional active mass damper.

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## References

- 1.
Inman D (1994) Engineering vibration. Prentice-Hall, Englewood Cliffs

- 2.
Korenev B, Reznikov L (1993) Dynamic vibration absorbers. Wiley, Chichester

- 3.
Den Hartog CP (1956) Mechanical vibrations. McGraw-Hill, New York

- 4.
Peterson C (2000) Dynamik der Baukonstruktionen. Vieweg, Braunschweig

- 5.
Sun J, Jolly M, Norris M (1995) Passive, adaptive and active tuned vibration absorbers—a survey. J Mech Des ASME 117((b)):234–242. doi:10.1115/1.2836462

- 6.
Benicke O, Butz C (2015) Volgograd-bridge: efficiency of passive and adaptive tuned mass damper. In: IABSE symposium report, IABSE conference Geneva 2015: structural engineering: providing solutions to global challenges, pp. 1-8(8). doi:10.2749/222137815818358826

- 7.
Kobori T, Koshika N, Ikeda Y, Yamada K (1991) Seismic-response controlled structure with active mass driver system. Part 1: design. Earthq Eng Struct Dyn 20(2):133–149. doi:10.1002/eqe.4290200204

- 8.
Kobori T, Koshika N, Ikeda Y, Yamada K (1991) Seismic-response controlled structure with active mass driver system. Part 2: verification. Earthq Eng Struct Dyn 20(2):151–166. doi:10.1002/eqe.4290200205

- 9.
Manabu I, Oguz B, Ryuki I, Satoshi O, Takeshi K, Yusuke T (2015) Izmit bay suspension bridge—finding and consideration for vibration control of tower by active mass damper. In: IABSE symposium report, IABSE conference Geneva 2015: structural engineering: providing solutions to global challenges, pp. 1-8(8). doi:10.2749/222137815818359645

- 10.
Scheller J (2013) Power-efficient active structural vibration control by twin rotor dampers. Dissertation, Hamburg University of Technology

- 11.
Bäumer R, Starossek U, (2016) Active vibration control using centrifugal forces created by eccentrically rotating masses. J Vib Acoust ASME 138(4) 041018:1–14. doi:10.1115/1.4033358

- 12.
Ekelund T (2000) Yaw control for reduction of structural dynamic loads in wind turbines. J Wind Eng Ind Aerodyn 85(3):241–262. doi:10.1016/S0167-6105(99)00128-2

- 13.
Ming-Yi L, Wei-Ling C, Chia-Ren C, Shih-Sheng L (2003) Analytical and experimental research on wind-induced vibration in high-rise buildings with tuned liquid column dampers. Wind Struct Int J 6(1):71–90. doi:10.12989/was.2003.6.1.071

- 14.
Gendelman OV, Sigalov G, Manevitch LI, Mane M, Vakakis AF, Bergman LA (2012) Dynamics of an eccentric rotational nonlinear energy sink. J Appl Mech ASME 79(1) 011012:1–9. doi:10.1115/1.4005402

- 15.
Vorotnikov K, Starosvetsky (2015) Nonlinear energy channeling in the two-dimensional, locally resonant, unit-cell model. I. High energy pulsations and routes to energy localization. Chaos Interdiscip J Nonlinear Sci 25:073106. doi:10.1063/4922964

- 16.
Vorotnikov K, Starosvetsky (2015) Nonlinear energy channeling in the two-dimensional, locally resonant, unit-cell model. II. Low energy excitations and unidirectional energy transport. Chaos Interdiscip J Nonlinear Sci 25:073107. doi:10.1063/4922965

- 17.
Clough RW, Penzien J (1993) Dynamics of structures. McGraw-Hill, New York

- 18.
Bronstein IN, Semendyayev KA (1979) Handbook of mathematics. Verlag Harri Deutsch, Thun and Frankfurt/Main

- 19.
Franklin GF (2010) Feedback control of dynamic systems. Pearson, New York

- 20.
Lunze J (1996) Regelungstechnik 1. Springer, Berlin

- 21.
Ljüng L (1999) System identification: theory for the user. Prentice-Hall, Englewood Cliffs

- 22.
Preumont A (2008) Active control of structures. Wiley, Chichester

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## Appendix

### Appendix

### Appendix A

To make the velocities available, two identical observers for both directions were designed, see Fig. 13 [16, 17]. The properties of Table 1 have been used. By choosing the vector gain \({\varvec{L}}=\left[ {\begin{array}{*{20}c} L_{1} &{} L_{2}\\ \end{array} } \right] ^{T}\), the poles of the transfer function from the measured displacement \(x_{m}\) to the observed displacement \(x_{a}\) can be set to any position; *T* indicates the transpose of a vector. The placement of the poles has been done with help of the *place* command in *MATLAB* such that the damped (circular) frequency is \(15(\mathrm {rad})/\mathrm {s}\) and the damping ratio is .95 of the loop from \(x_{m}\) to \(x_{a}\). These requirements result in a vector gain of \({\varvec{L}}=\left[ {\begin{array}{*{20}c} -2263.3 &{} -92.2\\ \end{array} } \right] ^{T}\). Outputs of the observer are the displacement \(x_{a}\) and the velocity \(\dot{x}_{a}\), which are used for the control algorithm of Sect. 3. For the *Y*-direction, the same observer is used.

### Appendix B

See Fig. 14.

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Bäumer, R., Starossek, U. Active vibration control of an oscillator with two translational degrees of freedom using centrifugal forces created by two eccentrically rotating masses.
*Int. J. Dynam. Control* **6, **284–299 (2018). https://doi.org/10.1007/s40435-016-0280-8

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### Keywords

- Two degree of freedom oscillator
- Active damping
- Twin rotor damper
- Centrifugal force
- Loop-shaping
- Closed-loop control