Robust energy harvesting from walking vibrations by means of nonlinear cantilever beams

Abstract

In the present work we examine how mechanical nonlinearity can be appropriately utilized to achieve strong robustness of performance in an energy harvesting setting. More specifically, for energy harvesting applications, a great challenge is the uncertain character of the excitation. The combination of this uncertainty with the narrow range of good performance for linear oscillators creates the need for more robust designs that adapt to a wider range of excitation signals. A typical application of this kind is energy harvesting from walking vibrations. Depending on the particular characteristics of the person that walks as well as on the pace of walking, the excitation signal obtains completely different forms. In the present work we study a nonlinear spring mechanism that is composed of a cantilever wrapping around a curved surface as it deflects. While for the free cantilever, the force acting on the free tip depends linearly on the tip displacement, the utilization of a contact surface with the appropriate distribution of curvature leads to essentially nonlinear dependence between the tip displacement and the acting force. The studied nonlinear mechanism has favorable mechanical properties such as low frictional losses, minimal moving parts, and a rugged design that can withstand excessive loads. Through numerical simulations we illustrate that by utilizing this essentially nonlinear element in a 2 degrees-of-freedom (DOF) system, we obtain strongly nonlinear energy transfers between the modes of the system. We illustrate that this nonlinear behavior is associated with strong robustness over three radically different excitation signals that correspond to different walking paces. To validate the strong robustness properties of the 2DOF nonlinear system, we perform a direct parameter optimization for 1DOF and 2DOF linear systems as well as for a class of 1DOF and 2DOF systems with nonlinear springs similar to that of the cubic spring that are physically realized by the cantilever–surface mechanism. The optimization results show that the 2DOF nonlinear system presents the best average performance when the excitation signals have three possible forms. Moreover, we observe that while for the linear systems the optimal performance is obtained for small values of the electromagnetic damping, for the 2DOF nonlinear system optimal performance is achieved for large values of damping. This feature is of particular importance for the system׳s robustness to parasitic damping.

Introduction

Several technological processes such as energy harvesting from ambient vibrations, shock absorption from external loads, and passive control or suppression of mechanical instabilities involve targeted energy transfer from one component of a structure to another. In particular, energy harvesting is the process of using ambient energy sources to generate useful forms of energy such as electricity. The energy in these ambient sources is usually spread over a range of frequencies. Applications of energy harvesting range from MEMs sensors implanted in the human body to monitor biological signs, von Büren et al. [1], to small electronics such as wireless sensors in remote locations, Paradiso and Starner [2]. Shock absorption is the process of protecting a primary structure from an ambient force or external pressure load. Applications of shock absorption include passive protection of buildings from earthquake excitations, of offshore platforms from water wave impacts, and of delicate instruments from external loads, as studied by Manevitch et al. [3], Vakakis et al. [4], and Sapsis et al. [5]. Passive control of mechanical instabilities is another important area that has recently emerged in the context of targeted energy transfer. Examples include the suppression of aeroelastic instabilities on wings due to fluttering, Lee et al. [6], and the elimination of aeroelastic instabilities in suspension bridges, Vaurigaud et al. [7].

In all these cases, one aims to design elements that are capable of transferring the energy irreversibly and efficiently. In typical applications (especially energy harvesting), the ambient vibration can be described as a stochastic, multi-frequency signal that is often characterized by time-varying features, Stanton et al. [8]. However, traditional single degree of freedom linear vibration harvesters are efficient only close to their design point; that is, when the excitation frequency matches the harvester׳s natural frequency. Therefore, linear harvesters respond inefficiently to vibrations with uncertain characteristics, Tang and Zuo [9]. In order to absorb ambient vibrations effectively, it is essential for an energy harvester to be characterized by performance robustness when the excitation signal has radically different properties over time.

Below, we give an overview of methods for overcoming the excitation mistuning problem. Then, we discuss some mechanical challenges to designing ambient vibration oscillators, and the importance of an oscillator design that can overcome both the excitation mistuning problem and these mechanical challenges.

Methods for overcoming the mistuning problem include designing systems that do not use a spring, control theory of linear spring systems, two-degree-of-freedom linear systems, continuous linear systems, and nonlinear springs.

Devices that oscillate without a spring do not have resonant frequencies and respond similarly to acceleration signals that have similar magnitudes but different frequencies. For example, Mitcheson et al. [10] describe a micro-scale coulomb-force parametric generator (CFPG). Instead of using a spring, the CFPG uses a charged capacitor plate that snaps away from a counter-electrode when excited by large accelerations. The CFPG, however, only functions well when the excitation displacement greatly exceeds the allowable travel length of its sliding plate. Another shock absorption device that functions without a spring is the MEMS-fabricated hydraulic valve that fits inside a shoe, as described by Yaglioglu [11]. In this device, a controller allows hydraulic fluid flowing in between two chambers to pulse on a piezoelectric element. Resulting strain in the piezoelectric element converts the mechanical energy into electric energy. Additionally, Paradiso and Starner [2] discuss a device small enough to fit in a shoe that consists of a clam shell made from two piezoelectric elements that flattens with each heel-strike. Paradiso and Starner also review other energy harvesting devices that absorb ambient energy without vibrating [2].

The performance (i.e. peak power output, adaptivity, and robustness to varied excitations) of energy harvesters with linear springs can be improved by using control strategies to alter the oscillator׳s resonant frequency, as described by Tang and Zuo [9], or creating linear devices with two or more degrees of freedom so that they have multiple resonant frequencies, as described by Trimble in [12]. Refs. [9] and [12] both present devices with better performance than traditional 1DOF linear systems. Some general drawbacks of these devices, however, are that the controlled devices consume some of the collected power, and the multiple degree of freedom systems are bulky and have limited robustness.

Another approach is to use a nonlinear spring. Essentially nonlinear springs (that is nonlinear springs without linear stiffness components) do not have preferential linear frequencies. Therefore, they are more robust to variations in the external excitation and preserve their good performance level for a wide range of conditions, as described in Vakakis et al. [4], Gendelman et al. [13], Sapsis et al. [14], and Quinn et al. [15]. The simplest form of an essentially nonlinear spring is a cubic one. One way to implement a cubic spring is by linear springs supporting the proof mass at various angles to its direction of travel. For example, MacFarland et al. [16] investigate the dynamics of a nonlinear oscillator realized by a thin elastic rod (piano wire) clamped at its ends without pretension that performs transverse vibrations at its center. To leading order approximation, the stretching wire produces a cubic stiffness nonlinearity. The effectiveness of this design has been illustrated for energy harvesting applications from impulsive excitations in Remick et al. [17], [18].

Similarly, Hajati et al. [19] describe an ultra-wide bandwidth resonator made out of a doubly clamped piezoelectric beam. The double-clamps cause the cantilever to stretch as it bends, resulting in a nonlinear stiffness. As described by Freeman [20], nonlinear springs may also be physically implemented by helical coil springs with thickening coil wires or changing overall diameters. Another way to achieve nonlinear behavior is by employing multiple linear components that interact more strongly the further they deflect. For example, in the leaf springs of automobile suspensions, several layers of arc-shaped spring steel are clamped together. As the center of the upper arc deflects, it contacts the arc below it, and both springs further deflect in contact. As more and more arcs deflect, the spring effectively stiffens [20].

A different class of nonlinear springs are those with negative linear stiffnesses, which are usually characterized by bi-stable configurations. Cottone et al. [21] describe a nonlinear spring implemented by an inverted pendulum with a tip magnet that faces an opposing static magnet. For a small enough gap between the magnets, the cantilever has two equilibria. For small base input accelerations, the tip magnet oscillates linearly about one of the equilibria. For sufficiently large accelerations, the tip magnet cycles between the two equilibria. This resonance is insensitive to noise.

Mann and Sims [22] describe an oscillator that is implemented by a magnet sliding in a tube with two opposing magnets as the end caps. This configuration causes the stiffness to be the summation of a linear and cubic component. While linear stiffness hinders the system׳s adaptivity and robustness to different excitation spectra, Carrella et al. [23] describe how the effective linear stiffness component can be reduced by counteracting it with a negative stiffness component using magnets arranged in an attracting configuration. In another study, Kovacic et al. [24] demonstrate that a negative stiffness can be implemented using oblique pre-stressed springs.

The goal of this work is the development of a nonlinear 2DOF system that will be able to maintain its good energy harvesting performance over a wide range of input signals. To achieve this goal we plan to utilize mechanical nonlinearities that will be chosen so that conditions for targeted energy transfers between modes are realized. To implement the desired nonlinearity we propose the use of a cantilever beam that oscillates between two contact surfaces with carefully selected curvature. For the purpose of energy harvesting, the proposed design has many desired properties which we illustrate analytically, numerically, and experimentally. These properties include (i) the resulting nonlinear spring has a negligible linearized component, (ii) the order of its nonlinearity does not remain constant but increases as the amplitude gets larger, and (iii) the spring achieves a theoretically infinite force for a finite displacement. The last property is of crucial importance since it allows the device to act as a typical nonlinear spring (with polynomial nonlinearity) for moderate vibration amplitudes and to effectively behave as a vibro-impact spring for larger amplitudes, which protects the device from excessive accelerations. After we have developed and studied the nonlinear element, we proceed with a thorough numerical study that illustrates clearly the advantages of the 2DOF nonlinear energy harvester compared to 1DOF designs and 2DOF linear systems. More specifically, focusing on the energy harvesting challenge of walking vibrations, we collect three radically different excitation signals that correspond to three different human body motions (walking, walking quickly, and running). For each family of systems we perform an optimization of system parameters, after which we conclude that the 2DOF energy harvester has almost double the average harvested power compared with the 1DOF systems and the 2DOF linear system. In addition, we observe that while the linear system׳s optimal performance is obtained for small values of the electromagnetic damping, the 2DOF nonlinear system optimal performance is achieved for large values of damping. This feature is of particular importance for the system׳s robustness to parasitic damping.