^{1}, Antonio Carcaterra

^{2}, Zhaoshun Xu

^{3}and Adnan Akay

^{3,a)}

### Abstract

This paper describes a new concept referred to here as “energy sinks” as an alternative to conventional methods of vibration absorption and damping. A prototypical energy sink envisioned here consists of a set of oscillators attached to, or an integral part of, a vibrating structure. The oscillators that make up an energy sink absorb vibratory energy from a structure and retain it in their phase space. In principle, energy sinks do not dissipate vibratory energy as heat in the classical sense. The absorbed energy remains in an energy sink permanently (or for sufficiently long durations) so that the flow of energy from the primary structure appears to it as damping. This paper demonstrates that a set of linear oscillators can collectively absorb and retain vibratory energy with near irreversibility when they have a particular distribution of natural frequencies. The approach to obtain such a frequency distribution is based on an optimization that minimizes the energy retained by the structure as a function of frequency distribution of the oscillators in the set. The paper offers verification of such optimal frequency spectra with numerical simulations and physical demonstrations.

I. INTRODUCTION

II. MODEL

III. OPTIMIZATION

IV. NUMERICAL RESULTS

A. Influence of initial frequency estimates

B. Role of loss factor used in optimization of

C. Simulations

D. Role of damping in an energy sink

E. Multiple-degree-of-freedom systems

F. An analytical expression for optimal frequency distribution

V. EXPERIMENTS

A. A set of thin beams attached to a -configuration

B. A set of flexible beams attached to a rigid oscillator

VI. CONCLUDING REMARKS

### Key Topics

- Oscillators
- 82.0
- Optimization
- 5.0
- Coupled oscillators
- 3.0
- Energy transfer
- 3.0
- Excitation energies
- 3.0

## Figures

Schematic description of a primary structure and attached set of oscillators.

Schematic description of a primary structure and attached set of oscillators.

(Color online) Simulation results using oscillators: (a) linear frequency distribution of the attached oscillators, (b) displacement response, and (c) total energy of the primary structure.

(Color online) Simulation results using oscillators: (a) linear frequency distribution of the attached oscillators, (b) displacement response, and (c) total energy of the primary structure.

(Color online) Optimized frequency distributions (dotted lines) for a system of oscillators each with a loss factor . The change of the integral of the energy of the primary structure, Eq. (5), (right-hand column) at each iteration illustrates the optimization process for different initial estimates for the frequency distribution from which optimization starts: (a) all oscillators have the same frequency as the primary structure , (b) linear distribution, and (c) polynomial distribution.

(Color online) Optimized frequency distributions (dotted lines) for a system of oscillators each with a loss factor . The change of the integral of the energy of the primary structure, Eq. (5), (right-hand column) at each iteration illustrates the optimization process for different initial estimates for the frequency distribution from which optimization starts: (a) all oscillators have the same frequency as the primary structure , (b) linear distribution, and (c) polynomial distribution.

(Color online) Optimized frequency distributions for oscillators that result from different initial frequency distributions for two different loss factors : (a) all oscillators have the same frequency as the primary structure , (b) linear distribution, and (c) polynomial distribution.

(Color online) Optimized frequency distributions for oscillators that result from different initial frequency distributions for two different loss factors : (a) all oscillators have the same frequency as the primary structure , (b) linear distribution, and (c) polynomial distribution.

(Color online) Influence of loss factors on the resulting optimized frequency distributions for a system with oscillators, which initially have the same frequencies as that of the primary . (a) , (b) , and (c) .

(Color online) Influence of loss factors on the resulting optimized frequency distributions for a system with oscillators, which initially have the same frequencies as that of the primary . (a) , (b) , and (c) .

(Color online) Effect of loss factor on the optimized frequency distribution for . Optimization begins with the same initial frequencies for all oscillators and the primary structure. (a) , (b) , and (c) .

(Color online) Effect of loss factor on the optimized frequency distribution for . Optimization begins with the same initial frequencies for all oscillators and the primary structure. (a) , (b) , and (c) .

(Color online) Simulation results for a set of attached oscillators each with a loss factor of . The top row shows the response of the primary when all attached oscillators and the primary have the same frequency . The bottom row shows the results obtained using the optimum frequency distribution: (a) displacement time, (b) displacement-frequency response, and (c) total energy of the primary structure.

(Color online) Simulation results for a set of attached oscillators each with a loss factor of . The top row shows the response of the primary when all attached oscillators and the primary have the same frequency . The bottom row shows the results obtained using the optimum frequency distribution: (a) displacement time, (b) displacement-frequency response, and (c) total energy of the primary structure.

(Color online) Simulation results for the optimum frequency distributions obtained starting with initial frequencies that correspond to those in Fig. 4 show negligible difference in the displacement response of the primary in time and frequency domains as well as the total energy it retains. . Optimum frequency used in each row corresponds to a different set of initial frequency estimates used in optimization: (a) all oscillators have the same frequency as the primary structure , (b) linear distribution, and (c) polynomial distribution.

(Color online) Simulation results for the optimum frequency distributions obtained starting with initial frequencies that correspond to those in Fig. 4 show negligible difference in the displacement response of the primary in time and frequency domains as well as the total energy it retains. . Optimum frequency used in each row corresponds to a different set of initial frequency estimates used in optimization: (a) all oscillators have the same frequency as the primary structure , (b) linear distribution, and (c) polynomial distribution.

(Color online) Simulation results for the optimum frequency distributions obtained starting with initial frequencies that correspond to those in Fig. 4: (a) all oscillators have the same frequency as the primary structure , (b) linear distribution, and (c) polynomial distribution. Results show negligible difference in the displacement response of the primary in time and frequency domains as well as the total energy it retains. .

(Color online) Simulation results for the optimum frequency distributions obtained starting with initial frequencies that correspond to those in Fig. 4: (a) all oscillators have the same frequency as the primary structure , (b) linear distribution, and (c) polynomial distribution. Results show negligible difference in the displacement response of the primary in time and frequency domains as well as the total energy it retains. .

(Color online) Simulation results for an energy sink with oscillators. The optimum frequency distribution, shown in Fig. 6(a), is obtained using a rather large loss factor for each of the oscillators: (a) linear frequency distribution of the attached oscillators, (b) displacement response, and (c) energy of the primary structure.

(Color online) Simulation results for an energy sink with oscillators. The optimum frequency distribution, shown in Fig. 6(a), is obtained using a rather large loss factor for each of the oscillators: (a) linear frequency distribution of the attached oscillators, (b) displacement response, and (c) energy of the primary structure.

(Color online) Simulated response of the primary with oscillators corresponding to Fig. 9(a) but each oscillator has a loss factor .

(Color online) Simulated response of the primary with oscillators corresponding to Fig. 9(a) but each oscillator has a loss factor .

(Color online) Optimized frequency distribution of oscillators attached to a two-degree-of-freedom primary structure and the responses of the platform with the attached oscillators and platform attached to ground (bottom). The natural frequencies of the primary structure are and .

(Color online) Optimized frequency distribution of oscillators attached to a two-degree-of-freedom primary structure and the responses of the platform with the attached oscillators and platform attached to ground (bottom). The natural frequencies of the primary structure are and .

(Color online) Examples of frequency distribution for different values of in Eq. (8).

(Color online) Examples of frequency distribution for different values of in Eq. (8).

(Color online) Simulation results for displacement response in time and frequency domains and the energy retained by the primary mass using values of in Eq. (8): (a) , (b) , (c) , and (d) .

(Color online) Simulation results for displacement response in time and frequency domains and the energy retained by the primary mass using values of in Eq. (8): (a) , (b) , (c) , and (d) .

(Color online) A comparison of the frequency distribution for oscillators obtained by two approaches. Dotted line represents result by direct optimization and the solid line represents interpolation for using the optimum distribution obtained for oscillators.

(Color online) A comparison of the frequency distribution for oscillators obtained by two approaches. Dotted line represents result by direct optimization and the solid line represents interpolation for using the optimum distribution obtained for oscillators.

(Color online) Response of a structure (top) shown with and without the oscillators that make up an energy sink. The first bending frequency of the thin beams that act as oscillators follow the analytical optimum frequency distribution given in Eq. (8).

(Color online) Response of a structure (top) shown with and without the oscillators that make up an energy sink. The first bending frequency of the thin beams that act as oscillators follow the analytical optimum frequency distribution given in Eq. (8).

(Color online) Response of the primary structure with an interpolated frequency distribution. Optimized frequency distribution obtained using oscillators interpolated for oscillators. .

(Color online) Response of the primary structure with an interpolated frequency distribution. Optimized frequency distribution obtained using oscillators interpolated for oscillators. .

(Color online) A set of 40 oscillators attached to a primary structure.

(Color online) A set of 40 oscillators attached to a primary structure.

(Color online) Response of the block with 40 oscillators to an impulse excitation shows that oscillators with an optimum frequency distribution (right) spread the return energy over both time and frequency and reduce its amplitude. Inherent damping in the system also reduces the amplitude of recurrence for the linear distribution, at .

(Color online) Response of the block with 40 oscillators to an impulse excitation shows that oscillators with an optimum frequency distribution (right) spread the return energy over both time and frequency and reduce its amplitude. Inherent damping in the system also reduces the amplitude of recurrence for the linear distribution, at .

(Color online) Simulated response of the block corresponding to conditions in Fig. 19 using loss factors measured from the experimental setup as and .

(Color online) Simulated response of the block corresponding to conditions in Fig. 19 using loss factors measured from the experimental setup as and .

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