# Targeted energy transfer in nonlinear oscillators

Our goal is the design of nonlinear mechanical oscillators, for targeted (guided) energy transfer applications from broad-band sources, which will go beyond the current state of the art of energy transfers which consists of linear resonance ideas or nonlinear systems excited by just a monochromatic source. These novel mechanical configurations will be able to harvest energy from broad-band sources by mimicking the robust and adaptive nonlinear-energy-transfer mechanism occurring across different scales in turbulent flows resulting in enhanced energy harvesting without the need for resonance conditions. Such properties will allow for the design of robust harvesting devices under stochastic excitations as well as the effective passive protection of structures subjected to random external loads.

*Figure 1: Vorticity and energy spectrum of the flow in a laminar and a turbulent jet – turbulent flow dissipates orders of magnitude larger amounts of energy because energy is nonlinearly distributed along many modes – each one of these dissipates energy.*

In linear systems energy cannot be transferred from one mode to another. Each linear mode interacts independently from the others with the external excitation and there are no modal energy exchanges. This reduces the ability of the designer to intentionally guide energy within the structure. Moreover, the independent operation of linear modes constraints the ability of the system to adapt to different excitations, limiting in this way the robustness of performance.

### Performance limitations for single-degree-of-freedom energy harvesters under stochastic excitation

*Student: Han Kyul Joo*

We develop performance criteria for the objective comparison of different classes of single-degree-of-freedom oscillators under stochastic excitation (H.-K. Joo and T. P. Sapsis, 2014). For each family of oscillators, these objective criteria take into account the maximum possible energy harvested for a given response level, which is a quantity that is directly connected to the size of the harvesting configuration. We prove that the derived criteria are invariant with respect to magnitude or temporal rescaling of the input spectrum and they depend only on the relative distribution of energy across different harmonics of the excitation. We then compare three different classes of linear and nonlinear oscillators and using stochastic analysis methods we illustrate that in all cases of excitation spectra (mono- chromatic, broadband, white-noise) the optimal performance of all designs cannot exceed the performance of the linear design. Subsequently, we study the robustness of this optimal performance to small perturbations of the input spectrum and illustrate the advantages of nonlinear designs relative to linear ones.

#### Objective measures of energy harvesting performance

An objective comparison between harvesters should involve not only the same mass but also the same size. Our goal is to quantify the maximum performance of a harvesting configuration for a given size of the response and for a given form of the input spectrum. We define suitable non-dimensional quantities, which are invariant on linear transformations of the input power spectrum such as the *harvesting power density* that expresses the maximum power that a given class of energy harvesters can collect for a given size of response, normalized by this size. These objective measures are designed so that they do not depend on the specific parameters chosen for each class of harvesters (H.-K. Joo and T. P. Sapsis, 2014).

#### Quantification of performance for SDOF harvesters

The derived criteria are applied on three different classes of nonlinear SDOF energy harvesters in order to examine which one can achieve the best performance under different stochastic excitations (Figure 2a). Our comparisons are presented for three cases of excitation spectra, namely, the monochromatic excitation, the white noise excitation, and an intermediate one characterized by colored noise excitation with Gaussian, stationary probabilistic structure and a power spectrum having the Pierson-Moskowitz form (relevant to water waves).

*In all cases we find that nonlinearity does not increase the effectiveness of the energy harvesting configuration for any given size of the harvesting device. For all the cases studied the performance of the optimal linear oscillator (for a fixed stochastic excitation) defines a barrier that can never be exceeded by any optimal nonlinear oscillator subjected to the same fixed stochastic excitation. We emphasize that this analysis considers only the issue of maximum energy harvesting from a given stochastic source under a given size of response. It does not consider the issue of robustness to excitations of different statistical characteristics (see next section).*

A description of the performance burrier with respect to the response level is given in Figure 2 where the harvested power density curves for the linear oscillator (which is always the best) under three different excitations is presented. We observe that for larger size of the device (B > 1) monochromatic excitation leads generally to increased optimal performance. On the other hand when the excitation is broadband smaller devices (B < 1) tend to have an advantage in terms of their optimal performance.

*Figure 2: (a) Linear and nonlinear SDOF systems considered in this study. (b) Performance burrier shown in terms of the harvested power density vs size of the harvester (B) for three different types of excitation spectra.*

### Robust energy harvesting through nonlinear energy transfers in 2DOF systems

*Student: Jocie Kluger*

Here we study designs that can lead to enhanced robustness under excitation that has variable statistical characteristics over time, i.e. we try to see if and what nonlinear designs can lead to robust energy harvesting under variable stochastic excitation. The applications we have in mind include charging a cell phone from the vibrations of a person walking with different pace, powering small electronics in remote locations under variable environmental conditions, or generating utility scale electricity from ocean waves.

Ambient vibrations are often stochastic, multi-frequency, and time-varying. Traditional linear oscillators are not well-suited for the targeted energy transfer of ambient vibrations because they can only absorb energy near one frequency. For example, if the vibration source is a person, then a properly-tuned linear oscillator can generate a significant amount of power when the person walks, but it generates a negligible amount of power when the person runs, because a person's feet step at different frequencies during the different motions. Our solution to this mistuning problem is to use nonlinear oscillators, which many studies have shown are more robust to vibrations signal changes than linear systems. This is a passive solution to the mistuning problem rather than using control theory. Below are the details on how we constructed a nonlinear oscillator and compared the maximum power and robustness of linear and nonlinear oscillators for the application of a person walking and running.

#### Constructing a nonlinear spring using a cantilever beam and contact surfaces

We study a nonlinear spring mechanism that is comprised of a cantilever wrapping around a curved surface as it deflects (Figure 3). While for the free cantilever, the force acting on the free tip depends linearly with 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 non-linear mechanism has favorable mechanical properties such as low frictional losses, minimal moving parts, and a rugged design that can withstand excessive loads. More details can be found in Kluger et. al.

*Figure 3: (a) A nonlinear spring design with essential nonlinearity, low friction, and one moving part (which increases device lifetime). (b) Nondimensionalized theoretical (black) and experimental (colored) force versus deflection curves for the nonlinear spring.*

#### Enhanced robustness of performance in the 2DOF nonlinear design

We optimize the parameters of single degree-of-freedom (1DOF) and 2DOF linear and nonlinear systems to maximize power during walking and the other forms of excitations with completely different forms of power spectrum (Figure 4). That is, all of the systems (1DOF linear, 1DOF nonlinear, 2DOF linear, 2DOF nonlinear) were optimized for maximum power when the person walks and non-negligible power (defined as >0.01 W) when the person runs. All of the systems used a total of 60 g oscillating mass(es) with an allowable travel height of 6.8 cm. As shown in Figure 5, the 2DOF nonlinear system is the most robust to different excitations and can produce the largest averaged power over the three different excitation signals. More details in Kluger et. al.

*Figure 4: (a) Time series of different hip motions (excitations) while a person walks, walks quickly, and runs. (b) Corresponding power spectra.*

*Figure 5: Comparison of power harvested by optimized systems when a person walks, walks quickly, and run.*

This research is supported by the MIT Energy Initiative. Han Kyul Joo is also supported by a Samsung Fellowship and Jocie Kluger is supported by an NSF Fellowship.

**Publications**

`J. Kluger, T. Sapsis, A. Slocum, `**Robust energy harvesting from walking vibrations by means of nonlinear cantilever beams**, *J. Sound Vib.*, In Press (2014). [pdf]

`H.-K. Joo, T. Sapsis, `**Performance measures for single-degree-of-freedom energy harvesters under stochastic excitation**, *J. Sound Vib.* **313** 4695-4710 (2014). [pdf]

K. Remick, H.-K. Joo, D.M. McFarland, T. Sapsis, L. Bergman, D.D. Quinn, A. Vakakis,Sustained High-Frequency Energy Harvesting Through a Strongly Nonlinear Electromechanical System under Single and Repeated Impulsive Excitations,J. Sound Vib.,3333214-3235 (2014). [pdf]

`K. Remick, A. Vakakis, L. Bergman, D. M. McFarland, D. D. Quinn, T. Sapsis, `**Sustained dynamical instabilities in coupled oscillators with essential nonlinearities,*** ASME J. Vibration & Acoustics,*** ****136 **011013 (2014). [pdf]

`T. Sapsis, D. Quinn, A. Vakakis, & L. Bergman,`** Effective Stiffening and Damping Enhancement of Structures with Strongly Nonlinear Local Attachments**, *ASME J. Vibration & Acoustics*, **134** (2012) 011016. [pdf]

`T. Sapsis, A. Vakakis, & L. Bergman, `**Effect of Stochasticity on Targeted Energy Transfer from a Linear Medium to a Strongly Nonlinear Attachment**, *Probabilistic Engineering Mechanics*, **26** (2011) 119-133. [pdf]

`O. Gendelman, T. Sapsis, A. Vakakis, L. Bergman, `**Radical Enhancement of Passive Targeted Energy Transfer in Strongly Nonlinear Mechanical Oscillators**, *J. Sound Vib.*, **330** (2011) 1. [pdf]

`T. Sapsis & A. Vakakis, `**Subharmonic orbits of a strongly nonlinear oscillator forced by closely spaced harmonics,*** J. Computational and Nonlinear Dynamics*, **6** (2011) 011014. [pdf]

`T. Sapsis, A. Vakakis, O. Gendelman, L. Bergman, G. Kerschen, & D. Quinn. `**Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part II, Analytical study**, *J. Sound Vib.*, **325** (2009) 297-320. [pdf]

`D. Quinn, O. Gendelman, G. Kerschen, T. Sapsis, L. Bergman, & A. Vakakis. `**Efficiency of Targeted Energy Transfers in Coupled Nonlinear Oscillators Associated with 1:1 Resonance Captures: Part I**, *J. Sound Vib.*, **311** (2008) 1228. [pdf]