## 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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## 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