energy harvesting and vibration suppression

Metamaterial-inspired piezoelectric system with dual functionalities: energy harvesting and vibration suppression Guobiao Hua, Lihua Tang*a, Raj Dasb a Department of Mechanical Engineering, University of Auckland, Auckland 1010, New Zealand; b School of Engineering, RMIT University, GPO Box 2476, Melbourne, VIC 3001, Australia ABSTRACT Elastic metamaterials can be used for vibration control where environmental vibrations exist. While, vibration energy harvesters can be designed to harness the environmental vibrations and convert them into useful electricity. These facts inspire us to develop a system with simultaneous vibration suppression and energy harvesting ability by combining them together. A piezoelectric metamaterial beam is presented in this paper to achieve dual functionalities. First, an analytical model of this system is developed and analyzed. Regarding the location of the metamaterial section on the beam, two configurations are proposed and studied. In order to achieve good dual functionalities, covering the beam with the metamaterial section from the free end should be given the priority. A parametric study is then performed to investigate the effect of the number of piezoelectric oscillators on the performance of the system. The result shows that by adding more oscillators, the system performance in terms of both vibration suppression and energy harvesting can be enhanced. Finally, a finite element model is developed with the consideration of implementing a realistic structure. The finite element results are in good agreement with the analytical results. Keywords: Metamaterial, piezoelectric, vibration suppression, energy harvesting,

1. INTRODUCTION Numerous research interests have been attracted to the band gap phenomenon of metamaterials [1, 2] recently. The mechanism of the band gap generation in metamaterials opens the possibility to achieve the low-frequency vibration attenuation [3, 4]. Theoretical studies often model metamaterials as mass-spring systems [5-8] which are lumped parameter models. Sugino et al. [9] developed a distributed parameter model to study the band gap formation mechanism in a metamaterial beam. However, their analytical model was based on the assumption of infinite long beam and excluded the damping effect. Zhu et al. [10] also presented a metamaterial beam and developed a distributed parameter model with lumped local resonators. Both infinite model and finite model results were presented in their work. Zhang et al. [3] studied a realistic locally resonant phononic plate through finite element simulations. Other distributed metamaterial models could be found in [4, 11]. In recent years, the application of metamaterials has been extended into the field of vibration energy harvesting [12] which is the technology [13-15] to convert kinetic energy into electric energy. Shen et al. [16] developed a phononic crystal plate with an array of spiral beams as conversion medium for energy harvesting and reported that the output power was enhanced at a dozen of resonant frequencies within a low-frequency range. Mikoshiba et al. [17] proposed an energy harvesting system with periodic structure embedded with multiple local resonators made of spring-suspended magnets. The magnetic fields of the magnets induced the current in the coils fixed outside. They claimed that their energy harvester possessed the functionality of vibration suppression as well, but they only investigated the energy harvesting performance in their experiment. Ahmed et al. [18] proposed and experimentally tested a single unit-cell acoustic elastic metamaterial (AEMM) based energy harvester. Though their system intrinsically owned the functionality of vibration suppression derived from the properties of AEMM, they didn’t analyze and evaluate the system in terms of this functionality. A recent paper by Hu et al. [19] is a first attempt to theoretically investigate the dual-functionalities of metamaterial-inspired energy harvesting system. However, that work was based on lumped parameter models. In this paper, a piezoelectric metamaterial beam system for simultaneous energy harvesting and vibration suppression is proposed. The analytical model of this electromechanical system is developed. The host beam is modelled with * [email protected], phone: +64 9 373 7599 ext 89535

Active and Passive Smart Structures and Integrated Systems 2017, edited by Gyuhae Park, Alper Erturk, Jae-Hung Han, Proc. of SPIE Vol. 10164, 101641X © 2017 SPIE · CCC code: 0277-786X/17/$18 · doi: 10.1117/12.2260396 Proc. of SPIE Vol. 10164 101641X-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 04/11/2017 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx

distributed parameters while the piezoelectric oscillator is modelled with lumped parameters. The section of the host beam with piezoelectric oscillators is regarded as piezoelectric metamaterial section. The effect of the location of the piezoelectric metamaterial section on the system performance is analyzed. The better configuration is to cover the host beam with the metamaterial section from the free end, which achieves good dual functionalities. A parametric study then demonstrates the effect of the number of piezoelectric oscillators (the size of the metamaterial section) on the system performance in terms of both energy harvesting and vibration suppression. To consider the implementation, a realistic design is conceptualized and analyzed by finite element analysis and its performance is evaluated and compared with the analytical results.

2. ANALYTICAL MODELLING The piezoelectric metamaterial beam system is as shown in Figure 1, an array of oscillators (assume there are S oscillators) with mass of m1, stiffness of k1 and damping of c1 are attached onto the host beam of length L at a uniform spacing of d. One piezoelectric element is placed at the position of each oscillator. The left-hand side of the beam is clamped on a base that experiences a harmonic exicitation wb  t   Wb eit transversely. The acceleration of the base excitation is controlled at a constant acc   2Wb . d

Y

Y L

Figure 1. Schematic of the piezoelectric metamaterial beam

By adopting the Euler-Bernoulli beam theory, the governing equation of the piezoelectric metamaterial beam can be written as: EI

S  4 wrel  x, t   5 wrel  x, t   2 wrel  x, t  i t c I  A  Aa e Fbj   x  x j  eit      s cc x 4 x 4 t t 2 j 1

(1)

where wrel  x, t  is the relative transverse displacement between the beam w  x, t  and the base wb  t  i.e., wrel  x, t   w  x, t   wb  t  ; EI is the bending stiffness of the beam; cs is the equivalent strain rate damping constant; 

is the material density of the beam; A is the cross-section area of the beam; Fbj is the total reacting force exerted by the jth oscillator and piezoelectric element onto the beam during the vibration;   x  is the Dirac delta function. The mechanical governing equation for each oscillator is:   x j , t  m1uj  t   c1u j  t   k1u j  t    v j  t    m1 w

(2)

in which u j  t  is the relative displacement between the j-th oscillator mass and the beam i.e., the absolute displacement of the j-th oscillator mass equals to u j  t   w  x j , t  . The circuit governing equation for each piezoelectric element is v j t  R

 C S v j  t    u j  t   0

Proc. of SPIE Vol. 10164 101641X-2 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 04/11/2017 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx

(3)

in which  is the electromechanical coupling coefficient; C S is the clamped capacitance of the piezoelectric transducer; R is the electric resistor connected to the piezoelectric element; and v j  t  is the voltage across R connected to the j-th piezoelectric element. Using the modal superposition method, the relative displacement along the beam can be written as the sum of the normal modes 

wrel  x, t    k  x k  t 

 k  1, 2,3...

(4)

k 1

in which k  x  are normalized mode shape functions of the plain beam (i.e., without oscillators) and  k  t  are modal coordinates. Substituting Eq.(4) into Eq.(1), multiplying by n  x  and integrating over the beam length i.e., from 0 to L, then using the orthogonal conditions, we obtain the modal governing equation as S

n  t   2 nnn  t   n2 n  t    Aacc eit  n  x  dx   Fbjn  x j  eit L

0

(5)

j 1

cs I  n . The expression of  n  t  can be derived from Eq.(5). Substituting the derived  n  t  back into Eq.(4) 2E gives the closed-form solution of the relative displacement

where  n 

S



 Aacc  k  x  dx   Fbjk  x j 

k 1

k2   2  2i k k 

L

wrel  x, t    k  x 

0

j 1

eit

(6)

From Eqs.(2) and (3), we can obtain:  m1 2  wrel  x j , t   wb  t   u j  t    i 2 R   2    k  m i  c   1 1 1  1  i RC S    m1 2  wrel  x j , t   wb  t    i R    v t  j 1  i RC S  i 2 R   k1   2 m1  i c1     1  i RC S  

(7)

The reacting force exerted by the j-th oscillator and piezoelectric element onto the beam at the position of xj can be expressed as f bj    c1u j  t   k1u j  t    v j  t   2   k1  i c1  m1 2  wrel  x j , t   wb  t   m1 2  wrel  x j , t   wb  t      i R   1  i RC S  i 2 R  i 2 R   2 2  k1   m1  i c1    k1   m1  i c1  1  i RC S  1  i RC S  

    

Substituting Eq.(6) into Eq.(8) gives the expression of the force amplitude as


(8)

   k1  i c1  m1 Fbj    2   k   2 m  i c  i R  1 1   1 1  i RC S

i 2 R m1 1  i RC S    i 2 R 2   k1   m1  i c1  1  i RC S  

     

(9)

   Aacc  k  x  dx   Fbhk  xh     0 2 2 h 1    k  x j    Wb  2 2  k    2i k k   k 1  acc  S

L

Rearrange the S Fbj (i.e., Fb1, Fb2,… FbS ) equations a1,1 Fb1  a1,2 Fb 2 ...  a1, S FbS  b1  ...  a j ,1 Fb1  a j ,2 Fb 2 ...  a j , S FbS  b j  ... aS ,1 Fb1  aS ,2 Fb 2 ...  aS , S FbS  bS

in which 

k  x j  k  xh 

k 1

    2i k k 

a j , h   2  

2 k

2



k  x j  k  x j 

k 1

   2  2i k k 

, a j , j   2  

2 k

(10)

1 ，

 i 2 R  L   k1  i c1    m1     Aa  x dx cc 0 k 1  i RC S  ，   b j     2   k  x j  2 a  . cc  k   2  2i k k   i 2 R  k 1 2    k1   m1  i c1   1  i RC S  

By solving Eq.(10), the S Fbj can be derived. Then substituting them back into Eq.(6), the relative displacement amplitude Wrel  x  can be obtained. The transmittance of the system is then calculated:



Wrel  L   Wb Wb

(11)

The voltage response and the power response of the j-th piezoelectric element are derived as  i R V j  1  i RC S     2 P  Vj  j R

m1 2W  x j   i 2 R  2  k1   m1  i c1   1  i RC S  

(12)

3. PARAMETRIC STUDY System parameters used for case studies in this section are listed in Table 1. Assume that the base excitation is kept at a constant acceleration of 2m/s2. For simplicity, in the following studies, we assume that each resistor R connected to the piezoelectric element has the same value all the time and changes simultaneously, and the power outputs from the resistors are summed up to evaluate the energy harvesting performance.


Table 1. System parameters used in case studies

Parameters Host beam Beam length Beam width Beam thickness Mass density Young’s modulus Material damping ratio

Values

Parameters Values Local piezoelectric oscillator d 30 mm m1 4.5 g k1 187 N/m c1 0.0128 Nm/s  6.6997˟10-4 N/V CS 15 nF

300 mm 25 mm 3 mm 2700 kg/m3 69.5 Gpa 0.01

3.1 Effects of the location of the metamaterial covered section

In this subsection, performances of two configurations are investigated and compared: Configuration A – the metamaterial section begins from the clamped end of the beam (similar to the configuration presented in [10]); Configuration B – metamaterial section begins from the free end of the beam. For the case that the metamaterial section covers the whole beam, both configurations are exactly the same. 101

Configuration A Configuration B

10°

t 10x

a0 10_

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Frequency (Hz)

Frequency (Hz)

(a)

(b) 101


10°

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a0 10_

Plain Beam Configuration A Configuration B

q 10

fon fipartlIon

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Frequency (Hz)

Frequency (Hz)

(c)

(d)


60

101


10°

10'

t 10 x 0

a

10_

Plain Beam Configuration A

PI -

oo

Q 10

m.

Configuration B 10

108 15

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.

0

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Frequency (Hz)

Frequency (Hz)

(e)

(f)

Figure 2. Transmittance patterns of different configurations with different metamaterial coverage: (a) one oscillator, (c) three oscillators, (e) five oscillators; power responses (log scaled) of different configurations with different metamaterial coverage: (b) one oscillator, (d) three oscillators, (f) five oscillators

Investigation of the vibration suppression performance: Figure 2(a), (c) and (e) show that the transmittance patterns of different configurations containing different metamaterial coverage. It is worth mentioning that the one oscillator case is exactly a classical vibration absorber structure, the anti-resonance happens around the natural frequency of the oscillator (32.4Hz for the selected oscillator). The solid black line denotes the plain beam case in which there isn’t any local oscillator attached, thus no band gap phenomenon is observed in this case. It can be speculated that with the increase of the number of oscillators, the difference between these two configurations will be smaller, as it approaches the case that the whole beam is covered with metamaterial section where Configuration A and B become exactly the same. However, for limited number of oscillators, by comparing the results in Figure 2(a), (c) and (e), we can find that Configuration B provides a broader band gap than Configuration A. Therefore, if the metamaterial doesn’t cover the whole beam, Configuration B shows an obvious competitive edge in terms of broadband vibration filtering ability. Investigation of the energy harvesting performance: Similarly, the corresponding energy harvesting performances of both configurations are compared. As shown in Figure 2(b) and (d), it is noted that Configuration B gives much higher power output when there are less number of local oscillators. While when the number of local oscillators increases, Configuration A may overtake Configuration B (Figure 2(f)).

From above comparison results, in order to take account of both functionalities (i.e., energy harvesting and vibration suppression), Configuration B should be given the priority in design. Therefore, in the following study, we will focus only on investigating the performance of Configuration B. 3.2

Effects of the number of local oscillators

Investigation of the vibration suppression performance: Figure 3(a) shows the transmittance patterns of systems with different number of oscillators. By comparing all the cases illustrated in Figure 3(a), it can be noted that with the increase of the number of oscillators, the band gap becomes broader, which indicates a better vibration suppression performance. Therefore, from the perspective of vibration suppression purpose, more oscillators are suggested to be used to form a metamaterial section with more unit cells (a portion of the host beam plus a local piezoelectric oscillator) to achieve better vibration filtering ability. Investigation of the energy harvesting performance: Under the same conditions, Figure 3(b) compares the energy harvesting performance of different cases. With the increase of the number of oscillators, the amplitudes of the first peak and the second peak in power response increase; while the third peak decreases. Considering that amplitudes of first two peaks are much higher than that of the third peak i.e., the contribution to power output will be mainly from the first two peaks, therefore, we can conclude that more local oscillators will benefit the energy harvesting performance.


lo' With 1- Oscillator With 2- Oscillators With 3- Oscillators With 4- Oscillators With 5- Oscillators

10°

10 E

0 10

W

Plain Beam

With 1- Oscillator With 2- Oscillators

-With 3- Oscillators

103

-- With 4- Oscillators With 5- Oscillators

10

20

104 30

40

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60

0

10

Frequency (Hz)

20

30

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Frequency (Hz)

(a)

(b)

Figure 3. Effects of number of local oscillators on: (a) transmittance pattern; (b) power output response (log scaled)

4. FINITE ELEMENT MODELLING In order to validate the analytical results related to Configuration B and achieve the simultaneous vibration suppression and energy harvesting, a realistic design of the proposed piezoelectric metamaterial beam has been conceptualized and the corresponding finite element model has been developed as shown in Figure 4.

Figure 4. Finite element model of proposed piezoelectric metamaterial beam

The implementation of those local one-degree-of-freedom (1DOF) oscillators is realized by using cantilever beams with tip masses. Piezoelectric elements are bonded on these parasitic cantilever beams. The commercial software ANSYS was used to build and analyze this model. The material of the beams is assumed to be aluminum and the tip masses are assumed to be made of steel. The 3-D 20-node structural solid element SOLID186 is used for modelling the solid structures of the beams and tip masses. The 3D 20-node coupled-field solid element SOLID226 is applied for the piezoelectric sheets boned on the parasitic beams in order to include the piezoelectric effect in the dynamic analysis. To emulate the electrodes of the piezoelectric sheet, the voltage degrees of freedom (DOFs) on the top and bottom surfaces were coupled together respectively to guarantee uniform electrical potentials. A harmonic analysis is then performed to obtain the frequency-domain displacement response and the steady-state power output of the system. In order to keep the excitation acceleration constant during the harmonic analysis, instead of applying the displacement constraint at the clamped side of the main beam, a corresponding acceleration field acc   2Wb  2m / s 2 due to base excitation was applied to the whole system.


5. FE SIMULATION VALIDATION According to the previous analytical results, the parameters of the FE model are selected as listed in Table 2. Based on the finite element model, we choose to repeat the same investigation of the effect of the number of attached oscillators on the performance of the system. Figure 5(a) shows the effect on the transmittance pattern. By comparing the results presented in Figure 5(a) and Figure 3(a), it can be found that the simulation results qualitatively agree well with the analytical results: with the increase of the number of parasitic beams (oscillators), the band gap becomes broader. Figure 5(b) shows the effect on the power response. As compared to Figure 3(b), though the power responses obtained from FE simulations are different from their analytical counterparts, the overall trend is similar: with the increase of the number of parasitic beams, the power output increases. By sweeping the frequency and the resistance much more finely in FE simulations, the FE simulation may be in better agreement with analytical results. But the difference between FE simulation results and analytical results may still exist, because in analytical study the piezoelectric oscillator is a lumped parameter model; while in FE analysis, both the parasitic beam and the piezoelectric element are modelled as distributed parameter models. In addition, the coupling coefficient in the distributed FE model is actually frequency-dependent; however, in analytical studies we assumed a constant coupling coefficient. Table 2. Properties of the FE model under investigation

Parameters Main beam length Main beam width Main beam thickness Parasitic beam length Parasitic beam width Parasitic beam thickness Distance between parasitic beams Tip mass length Tip mass width Tip mass thickness Piezoelectric sheet length

Values 300 mm 25 mm 3 mm 66 mm 8 mm 0.6 mm 30 mm 10 mm 8 mm 5.8 mm 28 mm

Piezoelectric sheet width Piezoelectric sheet thickness

7 mm 0.2 mm

Material

Parameters Mass density

Values 2700 kg/m3

Young’s Modulus

69.5 Gpa

Mass density

7800 kg/m3

Young’s Modulus Mass density

180 Gpa 7750 kg/m3

Young’s Modulus d31

120 Gpa 171˟10-12 C/N 1.531˟10-08 F/m

Aluminium

Steel

PZT-5A

T

10

100

- 6.-1 Parasitic Beam

Plain Beam

- 6-3

1 Parasitic Beam

3 Parasitic Beams 5 Paras@IC Beams

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- 4P-5 Parasitic Beams

-0

30

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SO

60

i 0.01

0.001

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Frequency (Hz)

(a)

(b)

Figure 5. FE simulation results ─ effects of the number of local oscillators on the: (a) transmittance pattern; (b) power output response (log scaled)


6. CONCLUSIONS This paper presents a piezoelectric metamaterial beam for simultaneous energy harvesting and vibration suppression. The analytical model of this electromechanical system is developed and the solution is derived. Two configurations are proposed and their performances are analytically evaluated and compared. The configuration of placing the metamaterial section from the free end of the beam is recommended to achieve better dual-functionalities. A parametric study then demonstrates that with the increase of the number of local oscillators, both functionalities of this system could be enhanced. Other than the analytical model, a finite element model is built to give an attempt to realize the dual-functional design in a realistic structure. The finite element results qualitatively verify our analytical predictions.

ACKNOWLEDGEMENT This work is financially supported by the Energy Education Trust of New Zealand (no. 3708242) and the PhD scholarship from China Scholarship Council (no. 201608250001).

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