Trapping of vibration energy into a set of resonators: Theory and application to aerospace structures

Abstract

This paper presents the theory of a novel mechanism of energy absorption and induced damping in structural systems and its application to aerospace industry. The underlying principles of the physical phenomena have been addressed in several earlier publications, which focused on prototypical systems of absorbers that consist of a set of single-degree-of-freedom resonators. This paper generalizes those theoretical developments to the case of a cluster of beams attached to a continuous primary structure, to develop predictive methods for the expected performance of this new type of absorber, with particular emphasis on its optimal design. An embodiment of the conceived device is illustrated for an aerospace structure, a satellite, with the purpose of reducing the vibration of the electronic components on board during lift-off. Experimental results illustrate the feasibility and the attractiveness of this new absorption technique.

Introduction

The effect of complex attachments on the dynamic response of built-up structures and their use as vibration absorbers have been the subject of numerous investigations in recent years. These studies examined different aspects of energy exchange between a so-called master structure that in the simplest case represented as a single-degree-of-freedom oscillator and a set of linear oscillators attached to it [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Several investigations (viz., [1], [2], [3], [4]) showed that the numerous oscillators attached to a master structure collectively act like a viscous damper and increase its effective damping significantly. These results further suggested that damping induced by the attached oscillators on the master structure does not depend on the details of loss factor in each oscillator and that such a set of oscillators absorbs energy even for vanishing values of loss factor in each oscillator so long as the number of oscillators remain large, approaching infinity [1].

Weaver in a series of investigations [6], [7], [8], [9], [10] generally corroborated these results, providing alternative approaches and results, and validated the expression developed by Pierce et al. [1] for vanishing loss factors in the oscillators for early times, pointing out that, in the case of a finite number of oscillators, the assertions for vanishing loss factors hold true only for transient exchange of energy. At later times, energy returns to the master [6], behaving as a series of exponentially decaying pulses in the impulse response [3].

This subtle but significant difference relates to the issues surrounding modal overlap and the conditions under which it is permissible to represent the summation in the equations of motion by an integration as discussed previously by, for example, [3], [4], [14]. At a more fundamental level, however, the issue centers around the definitions of “apparent damping” that a master experiences as energy flows to the attached oscillators vs. true dissipation that converts vibratory energy into thermal energy, through loss mechanisms in the attached oscillators. (The one exception that crosses over these two cases would be a nonlinear system, such as a large number of atoms in a solid, that undergoes oscillations, which themselves represent thermal energy or true dissipation [16].)

Consequently, as Maidanik [14] pointed out, the actual dissipation of energy in a linear system requires a physical loss mechanism for the entire system. For a large number of oscillators, loss factor in each can be small so long as their sum is finite [14]. On the other hand, if the attached oscillators have zero loss, even for very large number of oscillators, their presence offers an apparent damping source. Weaver [6] pointed out that apparent damping in linear systems is a transient event followed by a periodic return of energy with a return time that depends on the number of oscillators. For a very large number of oscillators, the transient event has a very long duration.

These studies have shown that achieving irreversible energy transfer in conservative linear systems requires infinite degrees of freedom or infinite number of attached oscillators. However, when their number is finite, the mechanism of energy trapping in the attachment becomes more complicated [17], [18]. For practical cases where the primary structure has a finite number of oscillators attached to it, the concept of irreversibility applies only during a transient period described as the return time during which energy flows into the satellite oscillators before returning to the primary structure.

The conditions that influence the return time were investigated to obtain a lower bound for it by an analysis of the reaction force on a rigid base by a finite number of oscillators demonstrating the influence of natural frequency distribution [17]. Based on these findings, the concept of linear energy sink was developed by optimizing the natural frequency distribution of the attached oscillators to minimize the energy retained by the structure to which they are attached [19]. The concept of linear energy sink was demonstrated through simple experiments that demonstrated how attached oscillators with the optimum frequency distributions can collectively absorb and retain the vibratory energy from an impulsively excited structure [20]. Analyses inspired by probably density distributions showed the existence of a family of frequency distributions for attached oscillators that lead to near-irreversibility of vibratory energy transfer [18], [19], [20], [21]. These studies demonstrated the key characteristic of the frequency distributions as having condensation points, which lead to decaying impulse response function even in lossless systems, a phenomenon that can be observed in preloaded plates on elastic foundations and shells [22]. The loss-free linear oscillators with the same frequency distributions showed comparable performance in energy absorption with the cases when the oscillators in the complex attachment had nonlinearities resulting from impacts among them or having parametric stiffness [23]. The near-irreversibility observed in the studies cited above has close connections with energy flow between mechanical resonators, energy equipartitioning and entropy [24], [25], [26], [27].

It is worth noting that the concept of damping in lossless systems has also been explored in control systems (viz., [28], [29], [30]), and also used as a basis for macroscopic irreversibility in thermodynamics [31] and for energy interaction in complex hybrid systems [32]. The idea of using complex attachments can be realized using different physical configurations and with mechanical as well as a set of electrical resonators. An electrical analog of the systems reported in [18], [19], [21] is described in [33] that is based on the use of piezoelectric coupling. Another physical configuration assumes the master is a floating body in waves subjected to shock excitations as described in [34], [35] and in this case the attachment, as reported in [33], has the twofold effect of mitigating the floating platform response and that of producing energy.

Many of these studies focused on time-domain response of the system and considered impulse response and short-time behavior. Building on these findings, the present paper extends the concept of apparent damping to a cluster of parallel beams and develops design rules for its application to a class of systems to absorb vibratory energy from primary structure. In particular, design and construct of an absorber and its application to a satellite is described.

The theoretical part, developed in 2 Apparent damping effect of a cluster of beams, 3 Physical considerations and properties of, synthesizes two basic results developed in time and frequency domains, together with physical interpretation of the induced equivalent damping. These results are then used to develop design rules for an optimal device and applied to a satellite, as illustrated in 4 Cluster design and performances of the built-up device, 5 Conclusions.