Exact H2 Optimal Tuning and Experimental Verification of Energy-Harvesting Series Electromagnetic Tuned-Mass Dampers

Liu, Yilun; Lin, Chi-Chang; Parker, Jason; Zuo, Lei

doi:10.1115/1.4034081

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 H₂ 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 $H_{2}$ 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 $H_{2}$ 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 $H_{2}$ 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 c₁ in classic TMDs, shown in Fig. 1(a), is replaced by an electromagnetic transducer of coil resistance $R_{i}$ 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 m₁ and the primary structure m_s produces an induced voltage, e_EMF, which is proportional to their relative velocity

Fig. 1

View large Download slide

(a) Classic TMD, (b) dual-functional series EMTMD for energy harvesting and vibration control, and (c) simplified model of dual-functional series EMTMD

e_{EMF} = k_{v} ({\dot{x}}_{1} - {\dot{x}}_{s})

(1)

where the proportional gain

k_{v} [V / (m / s)]

is the voltage constant of the electromagnetic transducer. The current in the electromagnetic transducer will produce a force,

f_{EMF}

⁠, proportional to the current,

\dot{q}

⁠,

f_{EMF} = k_{f} \dot{q}

(2)

where the proportional gain

k_{f}

[N/A] is the force constant. The constants

k_{v}

and

k_{f}

are determined by the transducer properties. The relation

k_{v}

=

k_{f}

is held for an ideal transducer without energy loss. In the resonator circuit, the closed-loop voltage drop is zero according to Kirchhoff's voltage law [28]. Thus, we have

e_{EMF} + L \ddot{q} + R \dot{q} + \frac{1}{C} q = 0

(3)

where the total resistance

R = R_{i} + R_{e}

⁠, and

R_{e}

is the equivalent external load. The resonant frequency of the circuit itself is

ω_{e} = 1 / \sqrt{L C}

(4)

The electrical damping ratio of the circuit itself is

ζ_{e} = \frac{R}{2 L ω_{e}}

(5)

And, the electromagnetic mechanical coupling coefficient

μ_{k}

is

μ_{k} = \frac{k_{v} k_{f}}{L k_{1}}

(6)

where $k_{1}$ is the stiffness of the absorber. $μ_{k}$ is actually a stiffness ratio (the electromagnetic mechanical coupling stiffness $k_{v} k_{f} / L$ 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 $m_{1}$ and the primary structure $m_{s}$ will also create a very large equivalent electrical damping force.

The overall dynamic equations of the simplified series EMTMD system, shown in Fig. 1(c), are given by

{\begin{matrix} m_{s} {\ddot{x}}_{s} + k_{f} \dot{q} + (k_{s} + k_{1}) x_{s} - k_{1} x_{1} = F_{w} - m_{s} {\ddot{x}}_{g} \\ m_{1} {\ddot{x}}_{1} - k_{f} \dot{q} - k_{1} x_{s} + k_{1} x_{1} = - m_{1} {\ddot{x}}_{g} \\ L \ddot{q} - k_{v} ({\ddot{x}}_{s} - {\ddot{x}}_{1}) + R \dot{q} + \frac{1}{C} q = 0 \end{matrix}

(7)

where $x_{s}$ and $x_{1}$ are the displacements of the primary structure and the absorber, respectively; $m_{s}$ and $m_{1}$ are the mass of the primary structure and the absorber, respectively; $k_{s}$ and $k_{1}$ are the stiffness of the primary structure and the absorber, respectively; $F_{w}$ is the external force caused by winds; and ${\ddot{x}}_{g}$ 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 (⁠ ${\ddot{x}}_{g} = 0$ in Fig. 1(c)) or ground excitations (⁠ $F_{w} = 0$ in Fig. 1(c)). In detail, the optimization problem of the EMTMD system is how to optimize the parameters of the absorber stiffness $k_{1}$ ⁠, the inductance $L$ ⁠, the capacitance $C$ ⁠, and the total resistance $R$ for a given primary structure $m_{s}$ ⁠, $k_{s}$ with the absorber mass $m_{1}$ so that the building deformation $x_{s}$ is minimized, and the average harvestable energy $R_{e} {\dot{q}}^{2}$ is maximized. Or equivalently, how to optimize the dimensionless parameters of

mechanical tuning ratio $f_{1} = (ω_{1} / ω_{s}) = (\sqrt{k_{1} / m_{1}} / \sqrt{k_{s} / m_{s}})$
electromagnetic mechanical coupling coefficient $μ_{k} = k_{v} k_{f} / L k_{1}$
electrical tuning ratio $f_{e} = (ω_{e} / ω_{s}) = \sqrt{1 / L C} / \sqrt{k_{s} / m_{s}}$
electrical damping ratio $ζ_{e} = (R / 2 L ω_{e}) = (R / 2 \sqrt{L / C})$

for a given mechanical mass ratio $μ = (m_{1} / m_{s})$ ⁠, where $ω_{s} = \sqrt{k_{s} / m_{s}}$ is the natural frequency of the primary structure.

H₂ Optimization for the Force Excitation System

Vibration Mitigation for the Force Excitation System.

Since the wind force that excites civil structures is of broad bandwidth [29],

H_{2}

norm is preferred to evaluate the system performance because it is the RMS value of the performance under unit Gaussian white-noise input [30]. Therefore, to mitigate the vibration of the primary structure, we minimize the

H_{2}

norm from the excitation force

F_{w}

to the deformation of the primary structure

x_{s}

in Fig. 1(c) by optimizing the design parameters of the series EMTMD. The corresponding performance index (PI) is defined as

PI = \frac{k_{s}^{2} E [x_{s}^{2}]}{2 π ω_{s} S_{F}} = \frac{k_{s}^{2} x_{s}^{2}}{2 π ω_{s} S_{F}}

(8)

where

S_{F}

is the uniform power spectrum density of the white-noise input force, E[] stands for the means square value, and <·> stands for the temporal average, respectively. The RMS value of the deformation of the primary mass

x_{s}

can be obtained as

x_{s}^{2} = ω_{s} S_{F} \int_{- \infty}^{\infty} {| \frac{X_{s} (j α)}{F_{w} (j α)} |}^{2} d α

(9)

where

α = ω / ω_{s}

is the excitation frequency ratio.

| (X_{s} (j α) / F_{w} (j α)) |

in Eq. is the norm of the transfer function from the excitation force

F_{w}

to the deformation of the primary structure

x_{s}

⁠, where

j = \sqrt{- 1}

is the unit imaginary number. Substituting Eq. into Eq. , the PI in Eq. can be expressed as

PI = \frac{1}{2 π} \int_{- \infty}^{\infty} {| \frac{X_{s} (j α)}{F_{w} (j α) / k_{s}} |}^{2} d α

(10)

The normalized transfer function from

(F_{w} / k_{s})

to

x_{s}

can be written in the dimensionless form by using the dimensionless parameters defined in Sec. 2.2, which is

\frac{X_{s} (j α)}{F_{w} (j α) / k_{s}} = \frac{{(j α)}^{4} + 2 ζ_{e} f_{e} {(j α)}^{3} + ((1 + μ_{k}) f_{1}^{2} + f_{e}^{2}) {(j α)}^{2} + 2 ζ_{e} f_{e} f_{1}^{2} (j α) + f_{1}^{2} f_{e}^{2}}{{\bar{A}}_{6} {(j α)}^{6} + {\bar{A}}_{5} {(j α)}^{5} + {\bar{A}}_{4} {(j α)}^{4} + {\bar{A}}_{3} {(j α)}^{3} + {\bar{A}}_{2} {(j α)}^{2} + {\bar{A}}_{1} (j α) + {\bar{A}}_{0}}

(11)

where

{\begin{matrix} {\bar{A}}_{6} = 1 \\ {\bar{A}}_{5} = 2 ζ_{e} f_{e} \\ {\bar{A}}_{4} = (1 + μ) (1 + μ_{k}) f_{1}^{2} + f_{e}^{2} \\ {\bar{A}}_{3} = 2 (1 + μ) ζ_{e} f_{e} f_{1}^{2} + 2 ζ_{e} f_{e} \\ {\bar{A}}_{2} = (1 + μ_{k}) f_{1}^{2} + (1 + μ) f_{1}^{2} f_{e}^{2} + f_{e}^{2} \\ {\bar{A}}_{1} = 2 ζ_{e} f_{e} f_{1}^{2} \\ {\bar{A}}_{0} = f_{1}^{2} f_{e}^{2} \end{matrix}

(12)

The integral in Eq. can be solved using the residue theorem, the general formula of which can be found in the Appendix. Hence, the PI in Eq. can be obtained as a function of the four design parameters

f_{1}

⁠,

μ_{k}

⁠,

f_{e}

⁠, and

ζ_{e}

and the given parameter

μ

PI = \frac{1}{4 μ f_{1}^{2} μ_{k} ζ_{e} f_{e}} {4 ζ_{e}^{2} f_{e}^{2} [f_{1}^{4} {(1 + μ)}^{2} - f_{1}^{2} (2 + μ) + 1] + f_{1}^{4} (1 + μ) {[f_{e}^{2} (1 + μ) - (1 + μ_{k})]}^{2} - f_{1}^{2} [2 f_{e}^{4} (1 + μ) - f_{e}^{2} (2 + μ) (2 + μ_{k}) + 2 (1 + μ_{k})] + {(f_{e}^{2} - 1)}^{2}}

(13)

In order to minimize the PI regarding the vibration mitigation performance, the derivatives of PI with respect to all the design parameters should be equal to zero. Thus, we have

\frac{\partial PI}{\partial f_{1}} = 0, \frac{\partial PI}{\partial μ_{k}} = 0, \frac{\partial PI}{\partial f_{e}} = 0, \frac{\partial PI}{\partial ζ_{e}} = 0

(14)

Therefore, the following simultaneous gradients' equations can be obtained from Eq. :

4 ζ_{e}^{2} f_{e}^{2} [f_{1}^{4} {(1 + μ)}^{2} - 1] + f_{1}^{4} (1 + μ) {[f_{e}^{2} (1 + μ) - (1 + μ_{k})]}^{2} - {(f_{e}^{2} - 1)}^{2} = 0

(15a)

4 ζ_{e}^{2} f_{e}^{2} [f_{1}^{4} {(1 + μ)}^{2} - f_{1}^{2} (2 + μ) + 1] + f_{1}^{4} (1 + μ) [f_{e}^{4} {(1 + μ)}^{2} - 2 f_{e}^{2} (1 + μ) + 1 - μ_{k}^{2}] - 2 f_{1}^{2} [f_{e}^{4} (1 + μ) - f_{e}^{2} (2 + μ)] + {(f_{e}^{2} - 1)}^{2} = 0

(15b)

4 ζ_{e}^{2} f_{e}^{2} [f_{1}^{4} {(1 + μ)}^{2} - f_{1}^{2} (2 + μ) + 1] + f_{1}^{4} (1 + μ) (f_{e}^{2} (1 + μ) - (1 + μ_{k})) (3 f_{e}^{2} (1 + μ) + (1 + μ_{k})) - f_{1}^{2} [6 f_{e}^{4} (1 + μ) - f_{e}^{2} (2 + μ) (2 + μ_{k}) - 2 (1 + μ_{k})] + (f_{e}^{2} - 1) (3 f_{e}^{2} + 1) = 0

(15c)

4 ζ_{e}^{2} f_{e}^{2} [f_{1}^{4} {(1 + μ)}^{2} - f_{1}^{2} (2 + μ) + 1] - f_{1}^{4} (1 + μ) {[f_{e}^{2} (1 + μ) - (1 + μ_{k})]}^{2} + f_{1}^{2} [2 f_{e}^{4} (1 + μ) - f_{e}^{2} (2 + μ) (2 + μ_{k}) + 2 (1 + μ_{k})] - {(f_{e}^{2} - 1)}^{2} = 0

(15d)

Solving this set of equations is nontrivial as it involves multiple nonlinear, high-order variables, but in the following we will simply summarize the solving process and then directly present the final results. This will highlight the results and avoid prolixity. By combining Eq. with the other three equations in Eq. (15), we can eliminate

ζ_{e}

to obtain a set of equations in design variables

f_{1}

⁠,

μ_{k}

⁠, and

f_{e}

⁠. Then using similar manipulations, we can eliminate

f_{1}

and

f_{e}

from the new equation set and obtain Eq. in only variable

μ_{k}

(16 + 19 μ) μ_{k}^{2} - 16 μ (2 + 3 μ) μ_{k} + 16 μ^{3} = 0

(16)

Therefore, the

H_{2}

optimal

μ_{k}

can be obtained by Eq. as

μ_{k}^{opt} = \frac{4 μ (4 + 6 μ + \sqrt{16 + 32 μ + 17 μ^{2}})}{16 + 19 μ}

(17a)

Substituting the optimal

μ_{k}

into Eq. (15), the optimal solutions of other design parameters can be similarly obtained, which are

f_{1}^{opt} = \frac{\sqrt{- μ + \sqrt{16 + 32 μ + 17 μ^{2}}}}{2 (1 + μ)}

(17b)

f_{e}^{opt} = \sqrt{\frac{32 + 58 μ + 25 μ^{2} + μ \sqrt{16 + 32 μ + 17 μ^{2}}}{2 (16 + 35 μ + 19 μ^{2})}}

(17c)

ζ_{e}^{opt} = \sqrt{μ \frac{(96 + 272 μ + 247 μ^{2} + 70 μ^{3}) + (24 + 44 μ + 18 μ^{2}) \sqrt{16 + 32 μ + 17 μ^{2}}}{(16 + 19 μ) (16 + 23 μ + 8 μ^{2})}}

(17d)

At the optimal

H_{2}

tuning condition, the performance index

{PI}^{opt}

is

{PI}^{opt} = \frac{f_{e}^{opt} ζ_{e}^{opt} (- 4 - 2 μ + \sqrt{16 + 32 μ + 17 μ^{2}})}{μ μ_{k}^{opt}}

(18)

Since the mass ratio

μ

is a very small number, usually less than 0.1 for building-TMD systems, the square root term

\sqrt{16 + 32 μ + 17 μ^{2}}

in Eq. (17) can be approximated as

4 (1 + μ)

by partially neglecting the terms involving second powers in

μ

⁠. Therefore, a concise, approximate solution set can be obtained as

{\begin{matrix} \begin{matrix} {f_{1}}^{{opt}^{*}} = \frac{\sqrt{4 + 3 μ}}{2 (1 + μ)} \\ {μ_{k}}^{{opt}^{*}} = \frac{32 μ + 40 μ^{2}}{16 + 19 μ} \approx \frac{32 μ}{16 + 19 μ} \end{matrix} \\ \begin{matrix} {f_{e}}^{{opt}^{*}} = \sqrt{\frac{32 + 62 μ + 29 μ^{2}}{2 (16 + 35 μ + 19 μ^{2})}} \approx \sqrt{\frac{16 + 31 μ}{16 + 35 μ}} \\ {ζ_{e}}^{{opt}^{*}} = \sqrt{\frac{μ (192 + 544 μ + 495 μ^{2} + 142 μ^{3})}{256 + 672 μ + 565 μ^{2} + 152 μ^{3}}} \approx \sqrt{\frac{192 μ}{256 + 672 μ}} \end{matrix} \end{matrix}

(19)

with the approximate optimal performance index

{PI}^{opt *}

{PI}^{opt *} = \frac{2 {f_{e}}^{opt *} {ζ_{e}}^{opt *}}{{μ_{k}}^{{opt}^{*}}} = \frac{\sqrt{(32 + 62 μ + 29 μ^{2}) (192 + 544 μ + 495 μ^{2} + 142 μ^{3})}}{4 (4 + 5 μ) \sqrt{2 μ (1 + μ) (16 + 23 μ + 8 μ^{2})}}

(20)

After obtaining optimal dimensionless tuning parameters, the corresponding optimal absorber stiffness

k_{1}^{opt}

⁠, the inductance

L^{opt}

⁠, the capacitance

C^{opt}

⁠, and the total resistance

R^{opt}

are

{\begin{matrix} \begin{matrix} k_{1}^{opt} = m_{1} ω_{s}^{2} f_{1}^{opt 2} \\ L^{opt} = \frac{k_{v} k_{f}}{m_{1} ω_{s}^{2} μ_{k}^{opt} f_{1}^{opt 2}} \end{matrix} \\ \begin{matrix} C^{opt} = \frac{m_{1} μ_{k}^{opt} f_{1}^{opt 2}}{k_{v} k_{f} f_{e}^{o p t 2}} \\ R^{opt} = \frac{k_{v} k_{f} f_{e}^{opt}}{m_{1} ω_{s} μ_{k}^{opt} f_{1}^{opt 2}} \end{matrix} \end{matrix}

(21)

where

R^{opt}

is the optimal total resistance in which the tunable load is the external load

R_{e}^{opt}

⁠, given by

R_{e}^{opt} = R^{opt} - R_{i}

(22)

It should be noted that similar optimization procedures have been employed to obtain the

H_{2}

tuning laws for the classic TMD system, which is [31]

{\begin{matrix} f^{opt} = \frac{1}{1 + μ} \sqrt{\frac{2 + μ}{2}} \\ ζ^{opt} = \sqrt{\frac{μ (4 + 3 μ)}{8 (1 + μ) (2 + μ)}} \end{matrix}

(23)

where $f^{opt}$ and $ζ^{opt}$ 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 $f_{1}$ ⁠, $μ_{k}$ ⁠, $f_{e}$ ⁠, and $ζ_{e}$ to minimize the magnitude of the transfer function in Eq. (11) with a given excitation frequency ratio $α$ and a given mass ratio $μ$ ⁠. The corresponding magnitude will be a real-valued function of $f_{1}$ ⁠, $μ_{k}$ ⁠, $f_{e}$ ⁠, and $ζ_{e}$ ⁠. By taking derivative of this magnitude with respect to $f_{1}$ ⁠, $μ_{k}$ ⁠, $f_{e}$ ⁠, and $ζ_{e}$ and setting them equal to zero, an optimal set of $f_{1}$ ⁠, $μ_{k}$ ⁠, $f_{e}$ ⁠, and $ζ_{e}$ 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.

For optimizing the energy harvesting of the series EMTMD, we wish to maximize the average electrical power on the external load

R_{e}

⁠. The instant power on the external load is

P (t) = R_{e} {\dot{q}}^{2}

(24)

Therefore, when the system is subjected to the force excitation with a uniform power spectrum

S_{F}

⁠, the PI is defined as

PI = \frac{k_{s}^{2} E [P (t)]}{2 π ω_{s} S_{F}} = \frac{k_{s}^{2} R_{e} 〈 {\dot{q}}^{2} 〉}{2 π ω_{s} S_{F}}

(25)

where the RMS value of the current

\dot{q}

can be obtained as

〈 {\dot{q}}^{2} 〉 = ω_{s} S_{F} \int_{- \infty}^{\infty} {| \frac{\dot{Q} (j α)}{F_{w} (j α)} |}^{2} d α

(26)

Substituting Eq. into Eq. , the PI in Eq. can be expressed as

PI = \frac{R_{e}}{2 π} \int_{- \infty}^{\infty} {| \frac{\dot{Q} (j α)}{F_{w} (j α) / k_{s}} |}^{2} d α

(27)

where

(\dot{Q} (j α) / (F_{w} (j α) / k_{s}))

is the normalized transfer function from

(F_{w} / k_{s})

to

\dot{q}

⁠, which is

\frac{\dot{Q} (j α)}{F_{w} (j α) / k_{s}} = \frac{\frac{k_{v}}{L} {(j α)}^{4}}{{\bar{A}}_{6} {(j α)}^{6} + {\bar{A}}_{5} {(j α)}^{5} + {\bar{A}}_{4} {(j α)}^{4} + {\bar{A}}_{3} {(j α)}^{3} + {\bar{A}}_{2} {(j α)}^{2} + {\bar{A}}_{1} (j α) + {\bar{A}}_{0}}

(28)

Similarly, by using the residue theorem [34], the PI in Eq. can be finally obtained as

PI = \frac{k_{v}}{2 k_{f} m_{s}} \frac{R_{e}}{R_{e} + R_{i}}

(29)

where $(k_{v} / 2 k_{f} m_{s})$ and $R_{i}$ 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 $R_{e}$ ⁠. 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 $R_{e}$ ⁠, $R_{e}$ 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).

H₂ Optimization for the Ground Excitation System

Vibration Mitigation for the Ground Excitation System.

In this section, we wish to optimize the design parameters of the series EMTMD shown in Fig. 1(c), when the system is subjected to a ground acceleration

{\ddot{x}}_{g}

⁠. Similar to the definition in Sec. 3.1, the PI regarding the

H_{2}

norm from the ground acceleration

{\ddot{x}}_{g}

to the deformation of the primary structure

x_{s}

is defined as

PI = \frac{1}{2 π} \int_{- \infty}^{\infty} {| \frac{X_{s} (j α)}{{\ddot{X}}_{g} (j α) / ω_{s}^{2}} |}^{2} d α

(30)

where

(X_{s} (j α) / ({\ddot{X}}_{g} (j α) / ω_{s}^{2}))

is the normalized transfer function from

{\ddot{x}}_{g} / ω_{s}^{2}

to

x_{s}

⁠. According to the system dynamics in Eq. , its dimensionless form is given by

\frac{X_{s} (j α)}{{\ddot{X}}_{g} (j α) / ω_{s}^{2}} = - \frac{{(j α)}^{4} + 2 ζ_{e} f_{e} {(j α)}^{3} + ((1 + μ) (1 + μ_{k}) f_{1}^{2} + f_{e}^{2}) {(j α)}^{2} + 2 (1 + μ) ζ_{e} f_{e} f_{1}^{2} (j α) + (1 + μ) f_{1}^{2} f_{e}^{2}}{{\bar{A}}_{6} {(j α)}^{6} + {\bar{A}}_{5} {(j α)}^{5} + {\bar{A}}_{4} {(j α)}^{4} + {\bar{A}}_{3} {(j α)}^{3} + {\bar{A}}_{2} {(j α)}^{2} + {\bar{A}}_{1} (j α) + {\bar{A}}_{0}}

(31)

By using the residue theorem [34],

PI

in Eq. can be similarly obtained as

\begin{matrix} PI = \frac{1}{4 μ f_{1}^{2} μ_{k} ζ_{e} f_{e}} {4 ζ_{e}^{2} f_{e}^{2} [f_{1}^{4} {(1 + μ)}^{4} - f_{1}^{2} (1 + μ) (2 + μ - μ^{2}) + 1] + f_{1}^{4} {(1 + μ)}^{3} \\ [f_{e}^{4} {(1 + μ)}^{2} - 2 f_{e}^{2} (1 + μ) (1 + μ_{k}) + {(1 + μ_{k})}^{2}] - f_{1}^{2} (1 + μ) [2 f_{e}^{4} {(1 + μ)}^{2} (1 - μ) - f_{e}^{2} (2 + μ - μ^{2}) (2 + μ_{k}) + 2 (1 + μ_{k})] \\ + f_{e}^{4} (1 + μ^{3}) - 2 f_{e}^{2} + 1} \end{matrix}

(32)

Similarly, the gradients of the PI in Eq. with respect to the four dimensionless design parameters should be equal to zero, see Eq. . Therefore, we can obtain four simultaneous equations for the ground excitation system, as written below

4 ζ_{e}^{2} f_{e}^{2} [f_{1}^{4} {(1 + μ)}^{4} - 1] + f_{1}^{4} {(1 + μ)}^{3} [f_{e}^{4} {(1 + μ)}^{2} - 2 f_{e}^{2} (1 + μ) (1 + μ_{k}) + {(1 + μ_{k})}^{2}] - f_{e}^{4} (1 + μ^{3}) + 2 f_{e}^{2} - 1 = 0

(33a)

4 ζ_{e}^{2} f_{e}^{2} [f_{1}^{4} {(1 + μ)}^{4} - f_{1}^{2} {(1 + μ)}^{2} (2 - μ) + 1] + f_{1}^{4} {(1 + μ)}^{3} [f_{e}^{4} {(1 + μ)}^{4} - 2 f_{e}^{2} (1 + μ) + 1 - μ_{k}^{2}] - 2 f_{1}^{2} [f_{e}^{4} {(1 + μ)}^{3} (1 - μ) - f_{e}^{2} {(1 + μ)}^{2} (2 - μ) + (1 + μ)] + f_{e}^{4} (1 + μ^{3}) - 2 f_{e}^{2} + 1 = 0

(33b)

\begin{matrix} 4 ζ_{e}^{2} f_{e}^{2} [f_{1}^{4} {(1 + μ)}^{4} - f_{1}^{2} {(1 + μ)}^{2} (2 - μ) + 1] + f_{1}^{4} {(1 + μ)}^{3} [3 f_{e}^{4} {(1 + μ)}^{2} - 2 f_{e}^{2} (1 + μ) (1 + μ_{k}) + {(1 + μ_{k})}^{2}] \\ - f_{1}^{2} (1 + μ) [6 f_{e}^{4} {(1 + μ)}^{2} (1 - μ) - f_{e}^{2} (1 + μ) (2 - μ) (2 + μ_{k}) - 2 (1 + μ_{k})] + 3 f_{e}^{4} (1 + μ^{3}) - 2 f_{e}^{2} - 1 = 0 \end{matrix}

(33c)

\begin{matrix} 4 ζ_{e}^{2} f_{e}^{2} [f_{1}^{4} {(1 + μ)}^{4} - f_{1}^{2} (1 + μ) (2 + μ - μ^{2}) + 1] - f_{1}^{4} {(1 + μ)}^{3} [f_{e}^{4} {(1 + μ)}^{2} - 2 f_{e}^{2} (1 + μ) (1 + μ_{k}) + {(1 + μ_{k})}^{2}] \\ + f_{1}^{2} (1 + μ) [2 f_{e}^{4} {(1 + μ)}^{2} (1 - μ) - f_{e}^{2} (2 + μ - μ^{2}) (2 + μ_{k}) + 2 (1 + μ_{k})] - f_{e}^{4} (1 + μ^{3}) + 2 f_{e}^{2} - 1 = 0 \end{matrix}

(33d)

After manipulations similar to those summarized in Sec. 3.1, the

H_{2}

optimal tuning law for the vibration mitigation under ground accelerations can be obtained as

{\begin{matrix} \begin{matrix} f_{1}^{opt} = \frac{\sqrt{4 - 3 μ}}{2 (1 + μ)} \\ μ_{k}^{opt} = \frac{128 μ}{64 - 36 μ - 9 μ^{2}} \end{matrix} \\ \begin{matrix} f_{e}^{opt} = \sqrt{\frac{16 - 9 μ}{16 + 19 μ + 3 μ^{2}}} \\ ζ_{e}^{opt} = \sqrt{\frac{192 μ}{256 - 96 μ - 27 μ^{2}}} \end{matrix} \end{matrix}

(34)

At the

H_{2}

optimal tuning condition, the performance index

{PI}^{opt}

is

{PI}^{opt} = \frac{(256 - 9 μ^{2}) \sqrt{{(1 + μ)}^{3} (48 - 27 μ)}}{32 (16 + 3 μ) \sqrt{μ (16 - 9 μ)}}

(35)

Then, the corresponding optimal mechanical and electrical elements can be similarly obtained using Eqs. (21) and (22).

Energy Harvesting for the Ground Excitation System.

If the system is subjected to ground accelerations, the energy-harvesting PI can be defined as

PI = \frac{R_{e}}{2 π} \int_{- \infty}^{\infty} {| \frac{\dot{Q} (j α)}{{\ddot{X}}_{g} (j α) / ω_{s}^{2}} |}^{2} d α

(36)

where

(\dot{Q} (j α) / ({\ddot{X}}_{g} (j α) / ω_{s}^{2}))

is the normalized transfer function from

{\ddot{x}}_{g} / ω_{s}^{2}

to

\dot{q}

⁠, the dimensionless form of which is

\frac{\dot{Q} (j α)}{{\ddot{X}}_{g} (j α) / ω_{s}^{2}} = \frac{\frac{k_{v}}{L} (2 ζ_{e} {(j α)}^{3} + {(j α)}^{2})}{{\bar{A}}_{6} {(j α)}^{6} + {\bar{A}}_{5} {(j α)}^{5} + {\bar{A}}_{4} {(j α)}^{4} + {\bar{A}}_{3} {(j α)}^{3} + {\bar{A}}_{2} {(j α)}^{2} + {\bar{A}}_{1} (j α) + {\bar{A}}_{0}}

(37)

Similarly, using the residue theorem [34], the PI in Ref. [36] can be finally obtained as

PI = \frac{(m_{s} + m_{1}) k_{v}}{2 k_{f}} \frac{R_{e}}{R_{e} + R_{i}}

(38)

The same conclusion, the energy harvesting depends on the only tuning variable: the external load $R_{e}$ ⁠, 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 $m_{s}$ and the mechanical shock absorber $m_{1}$ ⁠. 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 $H_{2}$ tuning laws of the series EMTMD system when it is subjected to the force excitation and the ground acceleration excitation.

Table 1

H₂ tuning laws for series EMTMD systems

		Vibration mitigation		Energy harvesting
H₂ tuning laws	PI	H₂ tuning laws	PI
Under force excitation $F_{w}$	Exact solution	${\begin{matrix} f_{1}^{opt} = \frac{\sqrt{- μ + r}}{2 (1 + μ)} \\ μ_{k}^{opt} = \frac{4 μ (4 + 6 μ + r)}{16 + 19 μ} \\ f_{e}^{opt} = \sqrt{\frac{32 + 58 μ + 25 μ^{2} + μ r}{2 (16 + 35 μ + 19 μ^{2})}} \\ ζ_{e}^{opt} = \sqrt{μ \frac{(96 + 272 μ + 247 μ^{2} + 70 μ^{3}) + (24 + 44 μ + 18 μ^{2}) r}{(16 + 19 μ) (16 + 23 μ + 8 μ^{2})}} \end{matrix}$	$\begin{matrix} \frac{1}{2 π} {\int_{- \infty}^{\infty} \| \frac{X_{s} (j α)}{F_{w} (j α) / k_{s}} \|}^{2} d α \\ = \frac{f_{e}^{opt} ζ_{e}^{opt} (- 4 - 2 μ + r)}{μ μ_{k}^{opt}} \end{matrix}$	$R_{e} ≫ R_{i}$	$\begin{matrix} \frac{R_{e}}{2 π} \int_{- \infty}^{\infty} {\| \frac{\dot{Q} (j α)}{F_{w} (j α) / k_{s}} \|}^{2} d α \\ = \frac{k_{v}}{2 k_{f} m_{s}} \frac{R_{e}}{R_{e} + R_{i}} \end{matrix}$
Under force excitation $F_{w}$	$r = \sqrt{16 + 32 μ + 17 μ^{2}}$
Approx. solution	${\begin{matrix} \begin{matrix} {f_{1}}^{{opt}^{}} = \frac{\sqrt{4 + 3 μ}}{2 (1 + μ)} \\ {μ_{k}}^{{opt}^{}} = \frac{32 μ + 40 μ^{2}}{16 + 19 μ} \approx \frac{32 μ}{16 + 19 μ} \end{matrix} \\ \begin{matrix} {f_{e}}^{{opt}^{}} = \sqrt{\frac{32 + 62 μ + 29 μ^{2}}{2 (16 + 35 μ + 19 μ^{2})}} \approx \sqrt{\frac{16 + 31 μ}{16 + 35 μ}} \\ {ζ_{e}}^{{opt}^{}} = \sqrt{\frac{μ (192 + 544 μ + 495 μ^{2} + 142 μ^{3})}{256 + 672 μ + 565 μ^{2} + 152 μ^{3}}} \approx \sqrt{\frac{192 μ}{256 + 672 μ}} \end{matrix} \end{matrix}$	$\begin{matrix} \frac{1}{2 π} \int_{- \infty}^{\infty} {\| \frac{X_{s} (j α)}{F_{w} (j α) / k_{s}} \|}^{2} d α \\ \approx \frac{2 {f_{e}}^{{opt}^{}} {ζ_{e}}^{{opt}^{}}}{{μ_{k}}^{{opt}^{*}}} \end{matrix}$
Under ground acceleration excitation ${\ddot{x}}_{g}$ (exact solution)		${\begin{matrix} \begin{matrix} f_{1}^{opt} = \frac{\sqrt{4 - 3 μ}}{2 (1 + μ)} \\ μ_{k}^{opt} = \frac{128 μ}{64 - 36 μ - 9 μ^{2}} \end{matrix} \\ \begin{matrix} f_{e}^{opt} = \sqrt{\frac{16 - 9 μ}{16 + 19 μ + 3 μ^{2}}} \\ ζ_{e}^{opt} = \sqrt{\frac{192 μ}{256 - 96 μ - 27 μ^{2}}} \end{matrix} \end{matrix}$	$\frac{1}{2 π} \int_{- \infty}^{\infty} {\| \frac{X_{s} (j α)}{{\ddot{X}}_{g} (j α) / ω_{s}^{2}} \|}^{2} d α$		$\frac{R_{e}}{2 π} \int_{- \infty}^{\infty} {\| \frac{\dot{Q} (j α)}{{\ddot{X}}_{g} (j α) / ω_{s}^{2}} \|}^{2} d α$
$= \frac{(256 - 9 μ^{2}) \sqrt{{(1 + μ)}^{3} (48 - 27 μ)}}{32 (16 + 3 μ) \sqrt{μ (16 - 9 μ)}}$	$= \frac{(m_{s} + m_{1}) k_{v}}{2 k_{f}} \frac{R_{e}}{R_{e} + R_{i}}$

		Vibration mitigation		Energy harvesting
H₂ tuning laws	PI	H₂ tuning laws	PI
Under force excitation $F_{w}$	Exact solution	${\begin{matrix} f_{1}^{opt} = \frac{\sqrt{- μ + r}}{2 (1 + μ)} \\ μ_{k}^{opt} = \frac{4 μ (4 + 6 μ + r)}{16 + 19 μ} \\ f_{e}^{opt} = \sqrt{\frac{32 + 58 μ + 25 μ^{2} + μ r}{2 (16 + 35 μ + 19 μ^{2})}} \\ ζ_{e}^{opt} = \sqrt{μ \frac{(96 + 272 μ + 247 μ^{2} + 70 μ^{3}) + (24 + 44 μ + 18 μ^{2}) r}{(16 + 19 μ) (16 + 23 μ + 8 μ^{2})}} \end{matrix}$	$\begin{matrix} \frac{1}{2 π} {\int_{- \infty}^{\infty} \| \frac{X_{s} (j α)}{F_{w} (j α) / k_{s}} \|}^{2} d α \\ = \frac{f_{e}^{opt} ζ_{e}^{opt} (- 4 - 2 μ + r)}{μ μ_{k}^{opt}} \end{matrix}$	$R_{e} ≫ R_{i}$	$\begin{matrix} \frac{R_{e}}{2 π} \int_{- \infty}^{\infty} {\| \frac{\dot{Q} (j α)}{F_{w} (j α) / k_{s}} \|}^{2} d α \\ = \frac{k_{v}}{2 k_{f} m_{s}} \frac{R_{e}}{R_{e} + R_{i}} \end{matrix}$
Under force excitation $F_{w}$	$r = \sqrt{16 + 32 μ + 17 μ^{2}}$
Approx. solution	${\begin{matrix} \begin{matrix} {f_{1}}^{{opt}^{}} = \frac{\sqrt{4 + 3 μ}}{2 (1 + μ)} \\ {μ_{k}}^{{opt}^{}} = \frac{32 μ + 40 μ^{2}}{16 + 19 μ} \approx \frac{32 μ}{16 + 19 μ} \end{matrix} \\ \begin{matrix} {f_{e}}^{{opt}^{}} = \sqrt{\frac{32 + 62 μ + 29 μ^{2}}{2 (16 + 35 μ + 19 μ^{2})}} \approx \sqrt{\frac{16 + 31 μ}{16 + 35 μ}} \\ {ζ_{e}}^{{opt}^{}} = \sqrt{\frac{μ (192 + 544 μ + 495 μ^{2} + 142 μ^{3})}{256 + 672 μ + 565 μ^{2} + 152 μ^{3}}} \approx \sqrt{\frac{192 μ}{256 + 672 μ}} \end{matrix} \end{matrix}$	$\begin{matrix} \frac{1}{2 π} \int_{- \infty}^{\infty} {\| \frac{X_{s} (j α)}{F_{w} (j α) / k_{s}} \|}^{2} d α \\ \approx \frac{2 {f_{e}}^{{opt}^{}} {ζ_{e}}^{{opt}^{}}}{{μ_{k}}^{{opt}^{*}}} \end{matrix}$
Under ground acceleration excitation ${\ddot{x}}_{g}$ (exact solution)		${\begin{matrix} \begin{matrix} f_{1}^{opt} = \frac{\sqrt{4 - 3 μ}}{2 (1 + μ)} \\ μ_{k}^{opt} = \frac{128 μ}{64 - 36 μ - 9 μ^{2}} \end{matrix} \\ \begin{matrix} f_{e}^{opt} = \sqrt{\frac{16 - 9 μ}{16 + 19 μ + 3 μ^{2}}} \\ ζ_{e}^{opt} = \sqrt{\frac{192 μ}{256 - 96 μ - 27 μ^{2}}} \end{matrix} \end{matrix}$	$\frac{1}{2 π} \int_{- \infty}^{\infty} {\| \frac{X_{s} (j α)}{{\ddot{X}}_{g} (j α) / ω_{s}^{2}} \|}^{2} d α$		$\frac{R_{e}}{2 π} \int_{- \infty}^{\infty} {\| \frac{\dot{Q} (j α)}{{\ddot{X}}_{g} (j α) / ω_{s}^{2}} \|}^{2} d α$
$= \frac{(256 - 9 μ^{2}) \sqrt{{(1 + μ)}^{3} (48 - 27 μ)}}{32 (16 + 3 μ) \sqrt{μ (16 - 9 μ)}}$	$= \frac{(m_{s} + m_{1}) k_{v}}{2 k_{f}} \frac{R_{e}}{R_{e} + R_{i}}$

Numerical Analysis

Graphical Representations of the H2 Tuning Laws for Vibration Mitigation.

Figure 2 graphically shows the $H_{2}$ 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 $H_{2}$ tuning laws for these two different excitations are close when the mass ratio $μ$ is small and become more distinct as $μ$ increases.

Fig. 2

View large Download slide

Graphical representations of the $H_{2}$ tuning laws: (a) optimal mechanical tuning ratio $f_{1}$ ⁠, (b) optimal electromagnetic mechanical coupling coefficient $μ_{k}$ ⁠, (c) optimal electrical tuning ratio $f_{e}$ ⁠, and (d) optimal electrical damping ratio $ζ_{e}$

Optimal PI for Vibration Mitigation.

Figure 3 shows the optimal performance index ${PI}^{opt}$ 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 $μ$ acts like an exponential decay. It initially decreases very rapidly, but the rate of decrease becomes smaller, as $μ$ 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.

Fig. 3

View large Download slide

Optimal performance index ${PI}^{opt}$ for vibration mitigation

Frequency Responses for Vibration Mitigation.

The optimal frequency response of the normalized primary structure displacement under force excitation $(X_{s} (j α) / (F_{w} (j α) / k_{s}))$ or ground excitation $(X_{s} (j α) / ({\ddot{X}}_{g} (j α) / ω_{s}^{2}))$ is shown in Fig. 4 in comparison with that of the classic TMD and the system without TMD, where the mass ratio $μ = 1 %$ as a common case. It is noted that the classic TMD is also optimized for vibration mitigation using $H_{2}$ 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.

Fig. 4

View large Download slide

The optimal normalized frequency response for vibration mitigation in comparison with the classic TMD and the system without a TMD, where mass ratio $μ = 0.01$ ⁠. (a) Force excitation system and (b) ground excitation 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, $f_{1}$ ⁠, namely, the stiffness of the mechanical shock absorber, is the most sensitive design parameter to the vibration performance and that the electrical damping ratio, $ζ_{e}$ ⁠, 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.

Fig. 5

View large Download slide

The vibration performance change to the changes of the design parameters in the force excitation system. (a) The changes of the stiffness of mechanical shock absorber $k_{1}$ and the inductance of electrical resonator $L$ and (b) the changes of the capacitance $C$ and total resistance $R$ ⁠.

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 $F_{w}$ 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 $R_{i} = 0$ and $k_{v} = k_{f}$ ⁠. The mass ratio $μ$ is 1%. From Fig. 6, it is clear that the series EMTMD outperforms the classic TMD in power harvested due to broader bandwidth.

Fig. 6

The normalized linear power spectrum density (W/Hz) of harvestable energy in ideal series EMTMDs optimized for vibration mitigation in comparison with that of classic TMDs. In ideal series EMTMDs, Ri=0 and kv=kf.

View large Download slide

The normalized linear power spectrum density (⁠ $\sqrt{W / Hz}$ ⁠) of harvestable energy in ideal series EMTMDs optimized for vibration mitigation in comparison with that of classic TMDs. In ideal series EMTMDs, $R_{i} = 0$ and $k_{v} = k_{f}$ ⁠.

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

Fig. 7

View large Download slide

The experimental setup of series EMTMD with adjustable elements: (a) front view and (b) top view

| \frac{X_{s} (j ω)}{F_{w} (j ω) / k_{s}} | = | \frac{{\ddot{X}}_{s} (j ω)}{F_{w} (j ω)} | \frac{k_{s}}{ω^{2}}

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.

Table 2

The parameters of the experimental setup

Description	Symbol	Value
Primary structure mass	$m_{s}$	3 (kg)
Mechanical TMD mass	$m_{1}$	0.3 (kg)
Stiffness of the primary structure	$k_{s}$	189.5 (kN/m)
Stiffness of the mechanical TMD	$k_{1}$	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	$R_{i}$	4.35 (Ω)
External resistance	$R_{e}$	1.3 (Ω)
Force excitation	$F_{w}$	Impulse force

Description	Symbol	Value
Primary structure mass	$m_{s}$	3 (kg)
Mechanical TMD mass	$m_{1}$	0.3 (kg)
Stiffness of the primary structure	$k_{s}$	189.5 (kN/m)
Stiffness of the mechanical TMD	$k_{1}$	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	$R_{i}$	4.35 (Ω)
External resistance	$R_{e}$	1.3 (Ω)
Force excitation	$F_{w}$	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.

Fig. 8

View large Download slide

Theoretical and experimental frequency response of the series EMTMD system and the system without electrical resonator

Fig. 9

View large Download slide

Experimental output voltage across on external resistive load $R_{e}$ under an impulse force excitation

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 $H_{2}$ 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 $H_{2}$ 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 $f_{1}$ ⁠, the electromagnetic mechanical coupling coefficient $μ_{k}$ ⁠, the electrical tuning ratio $f_{e}$ ⁠, and the electrical damping ratio $ζ_{e}$ ⁠. 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 $m_{s}$ and the mechanical shock absorber $m_{1}$ ⁠, 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.

Appendix

By using the residue theorem or the table in Ref. [34], the MS response integration of a sixth-order system can be obtained as

I_{6} = \int_{- \infty}^{+ \infty} {| \frac{{(j α)}^{5} B_{5} + {(j α)}^{4} B_{4} + {(j α)}^{3} B_{3} + {(j α)}^{2} B_{2} + (j α) B_{1} + B_{0}}{{(j α)}^{6} A_{6} + {(j α)}^{5} A_{5} + {(j α)}^{4} A_{4} + {(j α)}^{3} A_{3} + {(j α)}^{2} A_{2} + (j α) A_{1} + A_{0}} |}^{2} d α = \frac{π}{A_{6}} \frac{{Num}_{6}}{{Den}_{6}}

(A1)

where

\begin{matrix} {Num}_{6} = - B_{5}^{2} (- A_{6} A_{3} A_{1} A_{0} + A_{6} A_{2} A_{1}^{2} - A_{5}^{2} A_{0}^{2} + 2 A_{5} A_{4} A_{1} A_{0} + A_{5} A_{3} A_{2} A_{0} - A_{5} A_{2}^{2} A_{1} - A_{4}^{2} A_{1}^{2} - A_{4} A_{3}^{2} A_{0} + A_{4} A_{3} A_{2} A_{1}) \\ + A_{6} (B_{4}^{2} - 2 B_{3} B_{5}) (A_{4} A_{1}^{2} + A_{3}^{2} A_{0} - A_{1} A_{2} A_{3} - A_{0} A_{1} A_{5}) - A_{6} (B_{3}^{2} - 2 B_{2} B_{4} + 2 B_{1} B_{5}) (- A_{6} A_{1}^{2} - A_{5} A_{3} A_{0} + A_{5} A_{2} A_{1}) \\ + A_{6} (B_{2}^{2} - 2 B_{1} B_{3} + 2 B_{0} B_{4}) (A_{6} A_{3} A_{1} + A_{5}^{2} A_{0} - A_{5} A_{4} A_{1}) + A_{6} (2 B_{0} B_{2} - B_{1}^{2}) (A_{6} A_{5} A_{1} - A_{6} A_{3}^{2} - A_{5}^{2} A_{2} + A_{3} A_{4} A_{5}) \\ + \frac{A_{6}}{A_{1}} B_{0}^{2} (A_{6}^{2} A_{1}^{2} + A_{6} A_{5} A_{3} A_{0} - 2 A_{6} A_{5} A_{2} A_{1} - A_{6} A_{4} A_{3} A_{1} + A_{6} A_{3}^{2} A_{2} - A_{5}^{2} A_{4} A_{0} + A_{5}^{2} A_{2}^{2} + A_{5} A_{4}^{2} A_{1} - A_{2} A_{3} A_{4} A_{5}) \end{matrix}

(A2)

{Den}_{6} = A_{0}^{2} A_{5}^{3} + 3 A_{0} A_{1} A_{3} A_{5} A_{6} - 2 A_{0} A_{1} A_{4} A_{5}^{2} - A_{0} A_{2} A_{3} A_{5}^{2} - A_{0} A_{3}^{3} A_{6} + A_{0} A_{3}^{2} A_{4} A_{5} + A_{1}^{3} A_{6}^{2} - 2 A_{1}^{2} A_{2} A_{5} A_{6} - A_{1}^{2} A_{3} A_{4} A_{6} + A_{1}^{2} A_{4}^{2} A_{5} + A_{1} A_{2}^{2} A_{5}^{2} + A_{1} A_{2} A_{3}^{2} A_{6} - A_{1} A_{2} A_{3} A_{4} A_{5}

(A3)

The integral in Eq. (A1) can also be expressed in a more elegant form using the determinant of certain matrices, which can be derived following the method in Refs. [36,37].

References

1.

Kareem

,

A.

,

Kijewski

,

T.

, and

Tamura

,

Y.

,

1999

, “

Mitigation of Motions of Tall Buildings With Specific Examples of Recent Applications

,”

Wind Struct.

,

2

(

3

), pp.

201

–

251

.

Google Scholar

Crossref

2.

Den Hartog

,

J. P.

,

1928

,

Mechanical Vibrations

,

McGraw-Hill

,

New York

.

3.

Lackner

,

M. A.

, and

Rotea

,

M. A.

,

2011

, “

Passive Structural Control of Offshore Wind Turbines

,”

Wind Energy

,

14

(

3

), pp.

373

–

388

.

Google Scholar

Crossref

4.

Miyata

,

T.

,

2003

, “

Historical View of Long-Span Bridge Aerodynamics

,”

J. Wind Eng. Ind. Aerodyn.

,

91

(

12

), pp.

1393

–

1410

.

Google Scholar

Crossref

5.

Irwin

,

P.

,

Kilpatrick

,

J.

,

Robinson

,

J.

, and

Frisque

,

A.

,

2008

, “

Wind and Tall Buildings: Negatives and Positives

,”

Struct. Des. Tall Spec. Build.

,

17

(

5

), pp.

915

–

928

.

Google Scholar

Crossref

6.

ENR,

1977

, “

Tuned Mass Dampers Steady Sway of Skyscrapers in Wind

,” Engineering News Record, New York, pp.

28

–

29

.

7.

Snowdon

,

J. C.

,

1974

, “

Dynamic Vibration Absorbers That Have Increased Effectiveness

,”

ASME J. Manuf. Sci. Eng.

,

96

(

3

), pp.

940

–

945

.

8.

Xu

,

K.

, and

Igusa

,

T.

,

1992

, “

Dynamic Characteristics of Multiple Substructures With Closely Spaced Frequencies

,”

Earthquake Eng. Struct. Dyn.

,

21

(

12

), pp.

1059

–

1070

.

Google Scholar

Crossref

9.

Yamaguchi

,

H.

, and

Harnpornchai

,

N.

,

1993

, “

Fundamental Characteristics of Multiple Tuned Mass Dampers for Suppressing Harmonically Forced Oscillations

,”

Earthquake Eng. Struct. Dyn.

,

22

(

1

), pp.

51

–

62

.

Google Scholar

Crossref

10.

Zuo

,

L.

, and

Nayfeh

,

S. A.

,

2005

, “

Optimization of the Individual Stiffness and Damping Parameters in Multiple Tuned-Mass-Damper Systems

,”

ASME J. Vib. Acoust.

,

127

(

1

), pp.

77

–

83

.

Google Scholar

Crossref

11.

Zuo

,

L.

,

2009

, “

Effective and Robust Vibration Control Using Series Multiple Tuned-Mass Dampers

,”

ASME J. Vib. Acoust.

,

131

(

3

), p.

031003

.

Google Scholar

Crossref

12.

Zuo

,

L.

, and

Cui

,

W.

,

2013

, “

Dual-Functional Energy-Harvesting and Vibration Control: Electromagnetic Resonant Shunt Series Tuned Mass Dampers

,”

ASME J. Vib. Acoust.

,

135

(

5

), p.

051018

.

Google Scholar

Crossref

13.

Inoue

,

T.

,

Ishida

,

Y.

, and

Sumi

,

M.

,

2008

, “

Vibration Suppression Using Electromagnetic Resonant Shunt Damper

,”

ASME J. Vib. Acoust.

,

130

(

4

), p.

041003

.

Google Scholar

Crossref

14.

Ali

,

S. F.

, and

Adhikari

,

S.

,

2013

, “

Energy Harvesting Dynamic Vibration Absorbers

,”

ASME J. Appl. Mech.

,

80

(

4

), p.

041004

.

Google Scholar

Crossref

15.

Gonzalez-Buelga

,

A.

,

Clare

,

L. R.

,

Cammarano

,

A.

,

Neild

,

S. A.

,

Burrow

,

S. G.

, and

Inman

,

D. J.

,

2014

, “

An Optimized Tuned Mass Damper/Harvester Device

,”

Struct. Control Health Monit.

,

21

(

8

), pp.

1154

–

1169

.

Google Scholar

Crossref

16.

Brennan

,

M. J.

,

Tang

,

B.

,

Melo

,

G. P.

, and

Lopes

,

V.

,

2014

, “

An Investigation Into the Simultaneous Use of a Resonator as an Energy Harvester and a Vibration Absorber

,”

J. Sound Vib.

,

333

(

5

), pp.

1331

–

1343

.

Google Scholar

Crossref

17.

Liu

,

Y.

,

Lin

,

C. C.

, and

Zuo

,

L.

,

2014

, “

Evaluation and Optimum Design of Dual-Functional Electromagnetic Tuned Mass Dampers

,”

International Conference on Motion and Vibration Control

, Hokkaido, Japan, Aug. 4–6.

18.

Gonzalez-Buelga

,

A.

,

Clare

,

L. R.

,

Neild

,

S. A.

,

Jiang

,

J. Z.

, and

Inman

,

D. J.

,

2015

, “

An Electromagnetic Inerter-Based Vibration Suppression Device

,”

Smart Mater. Struct.

,

24

(

5

), p.

055015

.

Google Scholar

Crossref

19.

Gonzalez-Buelga

,

A.

,

Clare

,

L. R.

,

Neild

,

S. A.

,

Burrow

,

S. G.

, and

Inman

,

D. J.

,

2015

, “

An Electromagnetic Vibration Absorber With Harvesting and Tuning Capabilities

,”

Struct. Control Health Monit.

,

22

(

11

), pp.

1359

–

1372

.

Google Scholar

Crossref

20.

Tang

,

X.

,

Liu

,

Y.

,

Cui

,

W.

, and

Zuo

,

L.

,

2016

, “

Analytical Solutions to H2 and H∞ Optimizations of Resonant Shunted Electromagnetic Tuned Mass Damper and Vibration Energy Harvester

,”

ASME J. Vib. Acoust.

,

138

(

1

), p.

011018

.

Google Scholar

Crossref

21.

Liu

,

Y.

,

Zuo

,

L.

,

Lin

,

C. C.

, and

Parker

,

J.

,

2016

, “

Exact H2 Optimal Tuning and Experimental Verification of Energy-Harvesting Series Electromagnetic Tuned Mass Dampers

,”

Proc. SPIE

,

9799

, p.

979918

.

22.

Hagood

,

N. W.

, and

von Flotow

,

A.

,

1991

, “

Damping of Structural Vibrations With Piezoelectric Materials and Passive Electrical Networks

,”

J. Sound Vib.

,

146

(

2

), pp.

243

–

268

.

Google Scholar

Crossref

23.

Lesieutre

,

G. A.

,

1998

, “

Vibration Damping and Control Using Shunted Piezoelectric Materials

,”

Shock Vib.

,

30

(

3

), pp.

187

–

195

.

Google Scholar

Crossref

24.

Moheimani

,

S. O. R.

,

2003

, “

A Survey of Recent Innovations in Vibration Damping and Control Using Shunted Piezoelectric Transducers

,”

IEEE Trans. Control Syst. Technol.

,

11

(

4

), pp.

482

–

494

.

Google Scholar

Crossref

25.

Lefeuvre

,

E.

,

Audigier

,

D.

,

Richard

,

C.

, and

Guyomar

,

D.

,

2007

, “

Buck-Boost Converter for Sensorless Power Optimization of Piezoelectric Energy Harvester

,”

IEEE Trans. Power Electron.

,

22

(

5

), pp.

2018

–

2025

.

Google Scholar

Crossref

26.

Tang

,

X.

, and

Zuo

,

L.

,

2012

, “

Simultaneous Energy Harvesting and Vibration Control of Structures With Tuned Mass Dampers

,”

J. Intell. Mater. Syst. Struct.

,

23

(

18

), pp.

2117

–

2127

.

Google Scholar

Crossref

27.

Liu

,

Y.

,

Zuo

,

L.

, and

Tang

,

X.

,

2013

, “

Regenerative Vibration Control of Tall Buildings Using Model Predictive Control

,”

ASME

Paper No. DSCC2013-3988.

28.

Dow

,

W. G.

,

1937

,

Fundamentals of Engineering Electronics

,

Wiley

,

New York

.

29.

Ni

,

T.

,

Zuo

,

L.

, and

Kareem

,

A.

,

2011

, “

Assessment of Energy Potential and Vibration Mitigation of Regenerative Tuned Mass Dampers on Wind Excited Tall Buildings

,”

ASME

Paper No. DETC2011-48728.

30.

Zhou

,

K.

,

Doyle

,

J. C.

, and

Glover

,

K.

,

1995

,

Robust and Optimal Control

,

Prentice-Hall

,

Englewood Cliffs, NJ

.

31.

Nishihara

,

O.

, and

Asami

,

T.

,

2002

, “

Closed-Form Solutions to the Exact Optimizations of Dynamic Vibration Absorbers (Minimizations of the Maximum Amplitude Magnification Factors)

,”

ASME J. Vib. Acoust.

,

124

(

4

), pp.

576

–

582

.

Google Scholar

Crossref

32.

Asami

,

T.

,

Nishihara

,

O.

, and

Baz

,

A. M.

,

2002

, “

Analytical Solutions to H∞ and H2 Optimization of Dynamic Vibration Absorber Attached to Damped Linear Systems

,”

ASME J. Vib. Acoust.

,

124

(

2

), pp.

67

–

78

.

Google Scholar

Crossref

33.

Tang

,

X.

, and

Zuo

,

L.

,

2011

, “

Enhanced Vibration Energy Harvesting Using Dual-Mass Systems

,”

J. Sound Vib.

,

330

(

21

), pp.

5199

–

5209

.

Google Scholar

Crossref

34.

Gradshtenyn

,

I. S.

, and

Ryzhik

,

I. M.

,

1994

,

Table of Integrals Series, and Products

,

Academic Press

, Burlington, MA.

35.

Tang

,

X.

, and

Zuo

,

L.

,

2012

, “

Vibration Energy Harvesting From Random Force and Motion Excitations

,”

Smart Mater. Struct.

,

21

(

7

), p.

075025

.

Google Scholar

Crossref

36.

Roberts

,

J. B.

, and

Spanos

,

P. D.

,

1990

,

Random Vibration and Statistical Linearization

, Dover Publications,

New York.

37.

Adhikari

,

S.

,

Friswell

,

M. I.

, and

Inman

,

D. J.

,

2009

, “

Piezoelectric Energy Harvesting From Broadband Random Vibrations

,”

Smart Mater. Struct.

,

18

(

11

), p.

115005

.

Google Scholar

Crossref

Exact H₂ Optimal Tuning and Experimental Verification of Energy-Harvesting Series Electromagnetic Tuned-Mass Dampers

Introduction

Energy-Harvesting Series EMTMDs and Its Optimization Problem Formulation

Energy-Harvesting Series EMTMDs.

Optimization Problem Formulation.

H₂ Optimization for the Force Excitation System

Vibration Mitigation for the Force Excitation System.

Energy Harvesting for the Force Excitation System.

H₂ Optimization for the Ground Excitation System

Vibration Mitigation for the Ground Excitation System.

Energy Harvesting for the Ground Excitation System.

Numerical Analysis

Graphical Representations of the H2 Tuning Laws for Vibration Mitigation.

Optimal PI for Vibration Mitigation.

Frequency Responses for Vibration Mitigation.

Sensitivities of the Tuning Parameters.

Frequency Responses of Harvestable Power.

Experimental Verification

Experimental Setup.

Experimental Results.

Conclusion

Acknowledgment

Appendix

References

Get Email Alerts

Cited By

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

Exact H2 Optimal Tuning and Experimental Verification of Energy-Harvesting Series Electromagnetic Tuned-Mass Dampers

Introduction

Energy-Harvesting Series EMTMDs and Its Optimization Problem Formulation

Energy-Harvesting Series EMTMDs.

Optimization Problem Formulation.

H2 Optimization for the Force Excitation System

Vibration Mitigation for the Force Excitation System.

Energy Harvesting for the Force Excitation System.

H2 Optimization for the Ground Excitation System

Vibration Mitigation for the Ground Excitation System.

Energy Harvesting for the Ground Excitation System.

Numerical Analysis

Graphical Representations of the H2 Tuning Laws for Vibration Mitigation.

Optimal PI for Vibration Mitigation.

Frequency Responses for Vibration Mitigation.

Sensitivities of the Tuning Parameters.

Frequency Responses of Harvestable Power.

Experimental Verification

Experimental Setup.

Experimental Results.

Conclusion

Acknowledgment

Appendix

References

Get Email Alerts

Cited By

Related Articles

Related Proceedings Papers

Related Chapters

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

This Feature Is Available To Subscribers Only

Exact H₂ Optimal Tuning and Experimental Verification of Energy-Harvesting Series Electromagnetic Tuned-Mass Dampers

H₂ Optimization for the Force Excitation System

H₂ Optimization for the Ground Excitation System