Energy-harvesting series electromagnetic-tuned mass dampers (EMTMDs) have been recently proposed for dual-functional energy harvesting and robust vibration control by integrating the tuned mass damper (TMD) and electromagnetic shunted resonant damping. In this paper, we derive ready-to-use analytical tuning laws for the energy-harvesting series EMTMD system when the primary structure is subjected to force or ground excitations. Both vibration mitigation and energy-harvesting performances are optimized using H_{2} criteria to minimize root-mean-square (RMS) values of the deformation of the primary structure or maximize the average harvestable power. These analytical tuning laws can easily guide the design of series EMTMDs under various external excitations. Later, extensive numerical analysis is presented to show the effectiveness of the series EMTMDs. The numerical analysis shows that the series EMTMD more effectively mitigates the vibration of the primary structure nearly across the whole frequency spectrum, compared to that of classic TMDs. Simultaneously, the series EMTMD can better harvest energy due to its broader bandwidth effect. Beyond simulations, this paper also experimentally verifies the effectiveness of the series EMTMDs in both vibration mitigation and energy harvesting.

## Introduction

Vibration has been a serious concern in many civil structures, such as tall buildings, long-span bridges, and slender towers. For instance, the structures and secondary components of tall buildings can easily be damaged by huge dynamic loadings from winds or earthquakes, which also cause discomfort to its human occupants, whose symptoms range from anxiety, fear to dizziness, headaches, and nausea [1]. In order to mitigate the vibration of civil structures, classic TMDs [2–4], consisting of an auxiliary mass, a spring, and a viscous damper, have been widely used to dissipate vibrational energy into heat. The effectiveness of classic TMDs in vibration mitigation has been demonstrated in many modern buildings, such as the Taipei 101 Tower in Taipei [5] and the Citicorp Center in New York City [6]. Beyond classic TMDs, various alternative TMDs have also been proposed to achieve enhanced vibration mitigation performance, such as three-element TMDs [7], parallel TMDs [8–10], and series TMDs [11,12].

On the other hand, large amounts of associated vibration energy are converted to waste heat by the aforementioned mechanical TMDs and are worth being harvested to provide sustainable energy for many applications in civil structures. In order to enhance vibration mitigation performance and harvest the wasted energy, a new type of TMDs, energy-harvesting EMTMDs, has been proposed in the past decade [12–21]. The idea is to use an electromagnetic transducer to harvest the energy from structural vibration while producing a resisting electromagnetic force to dampen the vibration of the primary structure, acting as an electromagnetic damper as well as an energy harvester. As a result, the energy-harvesting EMTMDs are dual-functional, mitigating the structural vibration of civil structures and harvesting the associated vibrational energy otherwise dissipated by the conventional TMDs.

Inspired by the principles of double-mass series TMDs [11] and resonant shunt damping in piezoelectric structures [22–24], Zuo and Cui [12] recently proposed a series EMTMD in which the original viscous damper of the classic TMD is replaced by an electromagnetic transducer shunt with an RLC circuit. Therefore, the oscillation of the primary structure is mitigated first by the mechanical TMD then by the RLC electrical resonator. This configuration will enhance the effectiveness in vibration mitigation without inducing large additional stroke, as compared to double-mass series TMDs. Moreover, series EMTMDs can also simultaneously harvest the energy that was originally wasted by classic TMDs. However, in Ref. [12], the series EMTMD is only numerically optimized via decentralized control techniques and then verified in simulations. Due to lack of analytical optimizations for the series EMTMD, it is still not clear how the structural design parameters relate to the system performance in terms of both vibration mitigation and energy harvesting.

In this paper, we derive ready-to-use closed-form tuning laws for the series EMTMD system when the primary structure is subjected to force or ground excitations, like wind loads or seismic excitations. Both vibration mitigation and energy-harvesting performances are optimized using $H2$ criteria to minimize RMS values of the deformation of the primary structure, or maximize the average harvestable power. These analytical tuning laws can easily guide the design of series EMTMDs under various ambient loadings. Later, extensive numerical analysis is presented to show the enhanced effectiveness of the series EMTMDs in terms of vibration mitigation and energy harvesting, as compared to that of classic TMDs. Beyond simulations, this paper also experimentally demonstrates the effectiveness of the series EMTMDs in both vibration mitigation and energy-harvesting performances. In our experimental setup, we built a scaled-down series EMTMD system, in which a 3 kg primary structure with a 10% mass ratio mechanical shock absorber is used. The experimental results match the numerical analysis closely.

This paper is organized as follows: Section 2 is a brief introduction to the dynamics of the series EMTMD and optimization problem formulation. In Sec. 3, the exact $H2$ tuning law is derived for the force excitation system and a concise, approximate solution is also provided for practical use. In Sec. 4, the exact $H2$ tuning law is derived for the ground excitation system. In Sec. 5, the numerical analysis of the series EMTMD is presented, in comparison to classic TMDs and structures without TMDs. In Sec. 6, we experimentally verify the dual-functional effectiveness of series EMTMDs. Finally, this paper is concluded in Sec. 7.

## Energy-Harvesting Series EMTMDs and Its Optimization Problem Formulation

### Energy-Harvesting Series EMTMDs.

In many TMD applications like tall buildings and slander towers, the mechanical damping of the primary structure is very small compared to its stiffness [2]. Therefore, we treat these primary structures as lightly damped structures in which the mechanical damping is negligible. Figure 1(b) shows the series EMTMD in which the original energy dissipative damping *c*_{1} in classic TMDs, shown in Fig. 1(a), is replaced by an electromagnetic transducer of coil resistance $Ri$ and inductance *L*. The electromagnetic transducer is then shunted with a circuit that includes a capacitor *C*, an AC–DC converter, a DC–DC converter, energy storage elements, and electric loads. The electromagnetic transducer and the circuit can be modeled as an ideal transducer shunted with an RLC circuit [25–27], as shown in Fig. 1(c). The dynamics of series EMTMDs are summarized as follows. The relative motion between the absorber *m*_{1} and the primary structure *m*_{s} produces an induced voltage, *e*_{EMF}, which is proportional to their relative velocity

where $k1$ is the stiffness of the absorber. $\mu k$ is actually a stiffness ratio (the electromagnetic mechanical coupling stiffness $kvkf/L$ divided by the stiffness of the mechanical shock absorber), which stands for the coupling capability of the mechanical system and the electrical system. For example, if *μ _{k}* is very large, a slow movement between the absorber $m1$ and the primary structure $ms$ will also create a very large equivalent electrical damping force.

where $xs$ and $x1$ are the displacements of the primary structure and the absorber, respectively; $ms$ and $m1$ are the mass of the primary structure and the absorber, respectively; $ks$ and $k1$ are the stiffness of the primary structure and the absorber, respectively; $Fw$ is the external force caused by winds; and $x\xa8g$ is the ground acceleration caused by earthquakes.

### Optimization Problem Formulation.

As mentioned in the “Introduction” section, large-scale civil structures mainly suffer from wind loads and earthquakes. Therefore, for the series EMTMD system, we wish to tune the parameters to minimize building vibration and maximize the average harvestable energy when the primary structure is subjected to force excitations ($x\xa8g=0$ in Fig. 1(c)) or ground excitations ($Fw=0$ in Fig. 1(c)). In detail, the optimization problem of the EMTMD system is how to optimize the parameters of the absorber stiffness $k1$, the inductance $L$, the capacitance $C$, and the total resistance $R$ for a given primary structure $ms$, $ks$ with the absorber mass $m1$ so that the building deformation $xs$ is minimized, and the average harvestable energy $Req\u02d92$ is maximized. Or equivalently, how to optimize the dimensionless parameters of

mechanical tuning ratio $f1=(\omega 1/\omega s)=(k1/m1/ks/ms)$

electromagnetic mechanical coupling coefficient $\mu k=kvkf/Lk1$

electrical tuning ratio $fe=(\omega e/\omega s)=1/LC/ks/ms$

electrical damping ratio $\zeta e=(R/2L\omega e)=(R/2L/C)$

for a given mechanical mass ratio $\mu =(m1/ms)$, where $\omega s=ks/ms$ is the natural frequency of the primary structure.

## H_{2} Optimization for the Force Excitation System

### Vibration Mitigation for the Force Excitation System.

*E*[] stands for the means square value, and <·> stands for the temporal average, respectively. The RMS value of the deformation of the primary mass $xs$ can be obtained as

*d*) with the other three equations in Eq. (15), we can eliminate $\zeta e$ to obtain a set of equations in design variables $f1$, $\mu k$, and $fe$. Then using similar manipulations, we can eliminate $f1$ and $fe$ from the new equation set and obtain Eq. (16) in only variable $\mu k$

where $fopt$ and $\zeta opt$ are the optimal tuning parameter and damping parameter defined in Refs. [31,32].

In addition to broadband excitations, it is also important to optimize the performance of the series EMTMD at a given excitation frequency, such as the resonant frequency of the primary structure. For example, to minimize the vibration of the primary structure subjected to a single-frequency force excitation, we can optimize the design parameters $f1$, $\mu k$, $fe$, and $\zeta e$ to minimize the magnitude of the transfer function in Eq. (11) with a given excitation frequency ratio $\alpha $ and a given mass ratio $\mu $. The corresponding magnitude will be a real-valued function of $f1$, $\mu k$, $fe$, and $\zeta e$. By taking derivative of this magnitude with respect to $f1$, $\mu k$, $fe$, and $\zeta e$ and setting them equal to zero, an optimal set of $f1$, $\mu k$, $fe$, and $\zeta e$ can be obtained to minimize the vibration of the primary structure. Several similar single-frequency optimization problems have been discussed in Refs. [16,33].

### Energy Harvesting for the Force Excitation System.

where $(kv/2kfms)$ and $Ri$ are the fixed parameters in the system. As we can see from Eq. (29), the PI depends on only one of the four design parameters: the external load $Re$. This result agrees with the conclusions in Ref. [35], where the tuning of additional parameters is not necessarily better for energy harvesting under white-noise types of excitation. Therefore, to maximize the power on the external load $Re$, $Re$ needs to be maximized given the physical constraints of the system. Equation (29) also suggests that the energy-harvesting performance is proportional to the mass of the primary structure under unit white-noise force disturbance. It is worth noting that the optimal tuning law for the vibration control given in Eq. (17) and the optimal solution for the energy harvesting obtained from Eq. (29) are different. Therefore, there is a tradeoff between the vibration mitigation and energy-harvesting performance when employing dual-functional EMTMDs. An overall PI can be defined to compromise between vibration control and energy harvesting by assigning respective weights for them, as has been studied in recent researches [17]. According the optimal tuning law in Eqs. (17) and (29), the optimal external load for the overall PI will be slightly larger than that obtained in Eq. (17).

## H_{2} Optimization for the Ground Excitation System

### Vibration Mitigation for the Ground Excitation System.

### Energy Harvesting for the Ground Excitation System.

The same conclusion, the energy harvesting depends on the only tuning variable: the external load $Re$, that drawn from the force excitation system can also be drawn from Eq. (38) for optimizing the energy harvesting in the ground excitation system. It should be noted that the harvestable energy in the ground excitation system under unit ground acceleration is proportional to the mass of the primary structure $ms$ and the mechanical shock absorber $m1$. On the other hand, the harvestable energy in the system under unit wind force excitation is inversely proportional to the mass of the primary structure. This occurs because the heavier the mechanical structure, the smaller the induced motion due to unit force excitation is, which reduces the vibrational energy of the system. However, when the system is subjected to unit ground excitation, a heavier mechanical structure has a larger induced inertia force, therefore increasing the vibrational energy of the system.

Table 1 summarizes the $H2$ tuning laws of the series EMTMD system when it is subjected to the force excitation and the ground acceleration excitation.

Vibration mitigation | Energy harvesting | ||||
---|---|---|---|---|---|

H_{2} tuning laws | PI | H_{2} tuning laws | PI | ||

Under force excitation $Fw$ | Exact solution | ${f1opt=\u2212\mu +r2(1+\mu )\mu kopt=4\mu (4+6\mu +r)16+19\mu feopt=32+58\mu +25\mu 2+\mu r2(16+35\mu +19\mu 2)\zeta eopt=\mu (96+272\mu +247\mu 2+70\mu 3)+(24+44\mu +18\mu 2)r(16+19\mu )(16+23\mu +8\mu 2)$ | $12\pi \u222b\u2212\u221e\u221e|Xs(j\alpha )Fw(j\alpha )/ks|2d\alpha =feopt\zeta eopt(\u22124\u22122\mu +r)\mu \mu kopt$ | $Re\u226bRi$ | $Re2\pi \u222b\u2212\u221e\u221e|Q\u02d9(j\alpha )Fw(j\alpha )/ks|2d\alpha =kv2kfmsReRe+Ri$ |

$r=16+32\mu +17\mu 2$ | |||||

Approx. solution | ${f1opt*=4+3\mu 2(1+\mu )\mu kopt*=32\mu +40\mu 216+19\mu \u224832\mu 16+19\mu feopt*=32+62\mu +29\mu 22(16+35\mu +19\mu 2)\u224816+31\mu 16+35\mu \zeta eopt*=\mu (192+544\mu +495\mu 2+142\mu 3)256+672\mu +565\mu 2+152\mu 3\u2248192\mu 256+672\mu $ | $12\pi \u222b\u2212\u221e\u221e|Xs(j\alpha )Fw(j\alpha )/ks|2d\alpha \u22482feopt*\zeta eopt*\mu kopt*$ | |||

Under ground acceleration excitation $x\xa8g$ (exact solution) | ${f1opt=4\u22123\mu 2(1+\mu )\mu kopt=128\mu 64\u221236\mu \u22129\mu 2feopt=16\u22129\mu 16+19\mu +3\mu 2\zeta eopt=192\mu 256\u221296\mu \u221227\mu 2$ | $12\pi \u222b\u2212\u221e\u221e|Xs(j\alpha )X\xa8g(j\alpha )/\omega s2|2d\alpha $ | $Re2\pi \u222b\u2212\u221e\u221e|Q\u02d9(j\alpha )X\xa8g(j\alpha )/\omega s2|2d\alpha $ | ||

$=(256\u22129\mu 2)(1+\mu )3(48\u221227\mu )32(16+3\mu )\mu (16\u22129\mu )$ | $=(ms+m1)kv2kfReRe+Ri$ |

Vibration mitigation | Energy harvesting | ||||
---|---|---|---|---|---|

H_{2} tuning laws | PI | H_{2} tuning laws | PI | ||

Under force excitation $Fw$ | Exact solution | ${f1opt=\u2212\mu +r2(1+\mu )\mu kopt=4\mu (4+6\mu +r)16+19\mu feopt=32+58\mu +25\mu 2+\mu r2(16+35\mu +19\mu 2)\zeta eopt=\mu (96+272\mu +247\mu 2+70\mu 3)+(24+44\mu +18\mu 2)r(16+19\mu )(16+23\mu +8\mu 2)$ | $12\pi \u222b\u2212\u221e\u221e|Xs(j\alpha )Fw(j\alpha )/ks|2d\alpha =feopt\zeta eopt(\u22124\u22122\mu +r)\mu \mu kopt$ | $Re\u226bRi$ | $Re2\pi \u222b\u2212\u221e\u221e|Q\u02d9(j\alpha )Fw(j\alpha )/ks|2d\alpha =kv2kfmsReRe+Ri$ |

$r=16+32\mu +17\mu 2$ | |||||

Approx. solution | ${f1opt*=4+3\mu 2(1+\mu )\mu kopt*=32\mu +40\mu 216+19\mu \u224832\mu 16+19\mu feopt*=32+62\mu +29\mu 22(16+35\mu +19\mu 2)\u224816+31\mu 16+35\mu \zeta eopt*=\mu (192+544\mu +495\mu 2+142\mu 3)256+672\mu +565\mu 2+152\mu 3\u2248192\mu 256+672\mu $ | $12\pi \u222b\u2212\u221e\u221e|Xs(j\alpha )Fw(j\alpha )/ks|2d\alpha \u22482feopt*\zeta eopt*\mu kopt*$ | |||

Under ground acceleration excitation $x\xa8g$ (exact solution) | ${f1opt=4\u22123\mu 2(1+\mu )\mu kopt=128\mu 64\u221236\mu \u22129\mu 2feopt=16\u22129\mu 16+19\mu +3\mu 2\zeta eopt=192\mu 256\u221296\mu \u221227\mu 2$ | $12\pi \u222b\u2212\u221e\u221e|Xs(j\alpha )X\xa8g(j\alpha )/\omega s2|2d\alpha $ | $Re2\pi \u222b\u2212\u221e\u221e|Q\u02d9(j\alpha )X\xa8g(j\alpha )/\omega s2|2d\alpha $ | ||

$=(256\u22129\mu 2)(1+\mu )3(48\u221227\mu )32(16+3\mu )\mu (16\u22129\mu )$ | $=(ms+m1)kv2kfReRe+Ri$ |

## Numerical Analysis

### Graphical Representations of the H2 Tuning Laws for Vibration Mitigation.

Figure 2 graphically shows the $H2$ tuning laws for the vibration mitigation when the primary structure is disturbed by force and ground accelerations, respectively. It is clear that the error between the exact solution and the approximate solution for the force excitation system is extremely small. Therefore, the approximate solution is a suitable alternative to avoid computational complexities in practice. It is also obvious to see that the $H2$ tuning laws for these two different excitations are close when the mass ratio $\mu $ is small and become more distinct as $\mu $ increases.

### Optimal PI for Vibration Mitigation.

Figure 3 shows the optimal performance index $PIopt$ for the vibration mitigation under force and ground excitations, as obtained in Eqs. (18), (20), and (35). From Fig. 3, we can see that the change in the optimal PI with respect to the mass ratio $\mu $ acts like an exponential decay. It initially decreases very rapidly, but the rate of decrease becomes smaller, as $\mu $ increases from zero. In practice, this exponential-like trend can help make better tradeoffs between the vibration mitigation performance and TMD costs for series EMTMDs.

### Frequency Responses for Vibration Mitigation.

The optimal frequency response of the normalized primary structure displacement under force excitation $(Xs(j\alpha )/(Fw(j\alpha )/ks))$ or ground excitation $(Xs(j\alpha )/(X\xa8g(j\alpha )/\omega s2))$ is shown in Fig. 4 in comparison with that of the classic TMD and the system without TMD, where the mass ratio $\mu =1%$ as a common case. It is noted that the classic TMD is also optimized for vibration mitigation using $H2$ criteria, the corresponding tuning laws can be found in Eq. (23). From Fig. 4, it is clear that the series EMTMD more effectively mitigates the vibration of the primary structure nearly across the whole frequency spectrum for both the force and the ground excitation system, compared to classic TMDs and systems without a TMD. At their own resonant frequencies, the peak value of the normalized displacement in the series EMTMD system is reduced by around 25% compared to that of the classic TMD system.

### Sensitivities of the Tuning Parameters.

In practice, it is difficult to tune perfectly, or perhaps some parameters may change over time. Figure 5 shows how the vibration mitigation performance will change with the uncertainties of the tuning parameters for the force excitation system. It can be concluded that the mechanical tuning ratio, $f1$, namely, the stiffness of the mechanical shock absorber, is the most sensitive design parameter to the vibration performance and that the electrical damping ratio, $\zeta e$, namely, the total resistance of the electrical resonator, is the least sensitive to the vibration performance. It should be noted that Fig. 5 is based on the tuning laws obtained for the force excitation system in Eq. (17). Similar conclusions can be also drawn for the design optimized for the ground excitation system.

### Frequency Responses of Harvestable Power.

Series EMTMDs are capable of mitigating vibration and simultaneously harvesting the vibrational energy of the system. In this subsection, we show the energy-harvesting capability of the series EMTMD when it is optimized for vibration mitigation. The frequency response from force excitation $Fw$ to the square root of the normalized power of the series EMTMD is shown in Fig. 6 in comparison with that of the passive damping power in classic TMD systems. The linear transducer used in the series EMTMD is assumed to be ideal, where $Ri=0$ and $kv=kf$. The mass ratio $\mu $ is 1%. From Fig. 6, it is clear that the series EMTMD outperforms the classic TMD in power harvested due to broader bandwidth.

## Experimental Verification

### Experimental Setup.

Figure 7 shows the experimental setup of the series EMTMD system with adjustable design elements. By using this setup, the motion of the primary structure and the mechanical TMD can be simplified as 1 deg linear motion in the horizontal direction when the motion strokes are small. A voice coil motor is installed between the primary structure and the mechanical TMD, which acts as a linear electromagnetic transducer. A micropositioner is used to align the coil and magnet of the motor to avoid contact friction. To emulate a broadband force excitation, a horizontal impulse force perpendicular to the primary structure was applied to the primary structure by using an impact hammer. An accelerometer is used to measure the frequency response of the acceleration of the primary structure under the force excitation. Later, the frequency response of the normalized displacement of the primary structure can be obtained by

The voltage across on the external resistive load and the impact force are recorded to show the simultaneous energy-harvesting capabilities of series EMTMDs. The parameters of the setup are listed in Table 2.

Description | Symbol | Value |
---|---|---|

Primary structure mass | $ms$ | 3 (kg) |

Mechanical TMD mass | $m1$ | 0.3 (kg) |

Stiffness of the primary structure | $ks$ | 189.5 (kN/m) |

Stiffness of the mechanical TMD | $k1$ | 16.84 (kN/m) |

Total inductance of electrical resonator | $L$ | 41.8 (mH) |

Total capacitance of electrical resonator | $C$ | 387 (μF) |

Internal resistance of linear motor | $Ri$ | 4.35 (Ω) |

External resistance | $Re$ | 1.3 (Ω) |

Force excitation | $Fw$ | Impulse force |

Description | Symbol | Value |
---|---|---|

Primary structure mass | $ms$ | 3 (kg) |

Mechanical TMD mass | $m1$ | 0.3 (kg) |

Stiffness of the primary structure | $ks$ | 189.5 (kN/m) |

Stiffness of the mechanical TMD | $k1$ | 16.84 (kN/m) |

Total inductance of electrical resonator | $L$ | 41.8 (mH) |

Total capacitance of electrical resonator | $C$ | 387 (μF) |

Internal resistance of linear motor | $Ri$ | 4.35 (Ω) |

External resistance | $Re$ | 1.3 (Ω) |

Force excitation | $Fw$ | Impulse force |

### Experimental Results.

Figure 8 shows the theoretical and experimental frequency responses of the normalized displacements of the primary structure when excited by an impulse force. The response of both the series EMTMD system and the system without an electrical resonator is presented to display the influence of the electrical resonator in vibration mitigation performance. The experimental results match the theoretical responses closely. From Fig. 8, it is clear that the series EMTMD system outperforms the system without the electrical resonator in vibration mitigation, reducing the resonant peak by around 58.7%. It should be noted that the system without the electrical resonator has an open RLC circuit, which is not the optimal classic TMD system. The simultaneously generated voltage and the corresponding impulse force are plotted in Fig. 9, in which the peak voltage is around 45 mV, and the peak value of the impulse force is around 43 N. To sum up, Figs. 8 and 9 clearly demonstrated the dual functions of series EMTMDs, namely, simultaneous enhanced vibration mitigation and energy-harvesting functions.

## Conclusion

This paper investigates the energy-harvesting series EMTMD system, which consists of a single degree-of-freedom primary structure, an auxiliary mechanical shock absorber, an electromagnetic transducer, and an electrical RLC resonator. We derive ready-to-use analytical $H2$ tuning laws for the series EMTMD system when the primary structure is subjected to force or ground excitations. Both vibration mitigation and energy-harvesting performance were optimized using $H2$ criteria. These analytical tuning laws can easily guide the design of series EMTMDs under various ambient loadings. Based on the tuning laws of the series EMTMD, we found that an optimal design for vibration mitigation is mainly related to four design parameters, which are the mechanical tuning ratio $f1$, the electromagnetic mechanical coupling coefficient $\mu k$, the electrical tuning ratio $fe$, and the electrical damping ratio $\zeta e$. However, a design that aims to maximize the average harvestable energy is only dependent on one of the four design parameters, the resistance of the electrical resonator. Other than the design parameters, the harvestable energy in the system under a unit ground acceleration excitation is proportional to the mass of the primary structure $ms$ and the mechanical shock absorber $m1$, while the harvestable energy in the system under a unit force excitation is inversely proportional to the mass of the primary structure.

The numerical analysis shows that the series EMTMD can achieve enhanced performance in terms of both vibration and energy harvesting due to tuning both the resonances of the mechanical shock absorber and the electrical resonator, as compared to classic TMDs in which only the mechanical shock absorber is tuned. In experiments, the series EMTMD with the electrical resonator improves the vibration mitigation by reducing the resonant peak by 58.7%, as compared to that without the electrical resonator. The experimental results also show that a large amount of energy is simultaneously generated on the external resistive load by using the series EMTMD under a broad bandwidth force excitation.

## Acknowledgment

The authors gratefully acknowledge the funding support from the National Science Foundation (NSF), NSF #1529380.