A multi-layered polydimethylsiloxane structure for application in low-excitation, broadband and low frequency energy harvesting

Sensors and Actuators A 222 (2015) 140–148

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

A multi-layered polydimethylsiloxane structure for application in low-excitation, broadband and low frequency energy harvesting Mengying Xie a , Kean C. Aw a,∗ , Wei Gao b a b

Department of Mechanical Engineering, University of Auckland, Auckland, New Zealand Department of Chemical and Materials Engineering, University of Auckland, Auckland, New Zealand

a r t i c l e

i n f o

Article history: Received 23 September 2014 Received in revised form 7 December 2014 Accepted 8 December 2014 Available online 16 December 2014 Keywords: Polydimethylsiloxane Bistable Nonlinear energy harvester Broad bandwidth

a b s t r a c t Vibrational kinetic energy is a promising energy source that can be harvested due to its abundance in daily life, especially from human motion. This low frequency vibration poses a challenge in achieving a small device that is practical and wearable. Further, the effective energy harvesting of a linear structure is very limited to a narrow range of frequencies around the resonance. These two limitations call for the development of wide bandwidth energy harvesters that can work at a wider low frequency range. Using low Young’s modulus material is a common technique to achieve a low resonant frequency energy harvester. Nonlinear bistability is a potential solution for bandwidth broadening. In this paper, a broadband low frequency vibrational energy harvester using multi-layered soft polydimethylsiloxane (PDMS) will be presented. First, a multi-layered PDMS structure was created by sandwiching a thicker PDMS with two thinner pre-stressed PDMS films. After releasing the prestresses, the multi-layered PDMS structure can settle into two cylindrical configurations. An analytical model that can predict the shapes of this multi-layered structure has been developed using classical lamination theory along with Rayleigh–Ritz approximation technique. Through this model, PDMS shapes can be easily predicted by changing various parameters such as the ratio of side length to thickness, prestrain levels, and Young’s modulus. A multi-layered PDMS structure has been further proposed to be used as a bistable energy harvester. The soft PDMS allows working frequencies of lower than 15 Hz. The dynamic response of this harvester under small input excitation shows a softening spring system, before the strong nonlinear ‘snap through’ effect occurs. This softening spring system is able to broaden the bandwidth of the energy harvesting device. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Vibration kinetic energy is one promising source for energy that can be harvested due to its ready availability in daily life [1], especially if suitable devices can be developed to harvest energy from the low-frequency excitation produced by human motion. This low frequency vibration poses a challenge in achieving a small device that is practical and wearable. The use of a low Young’s modulus material can easily realize an energy harvester resonating at low frequency. Polydimethylsiloxane (PDMS) has been successfully applied in previous research [2,3] due to its low Young’s modulus. However, most vibrational energy generators are designed based on the principle of linear oscillation where an inertial mass is mounted on a spring damper and excited at the

∗ Corresponding author at: 20 Symonds Street, Auckland, New Zealand. Tel.: +64 9 923 9767; fax: +64 9 3737479. E-mail addresses: [email protected] (M. Xie), [email protected] (K.C. Aw), [email protected] (W. Gao). http://dx.doi.org/10.1016/j.sna.2014.12.009 0924-4247/© 2014 Elsevier B.V. All rights reserved.

resonant frequency [4]. They can generate maximum power only when the natural frequency of the generator matches the frequency of ambient vibration. Even a small difference between the two frequencies can lead to a significant decrease in generated power. In order to avoid this limitation of linear oscillators, many approaches have been investigated to broaden the bandwidth of the harvesters such as using an array of linear oscillators [5], linear oscillators with an amplitude stopper [6], nonlinear oscillators [7–17]. Among these techniques, nonlinear oscillators can provide better performance in terms of the energy extracted from a generic wide spectrum vibration. Nonlinearity can be easily obtained by utilizing large deflection or large strain at low frequency [12], impact (or contact) or non-impact driven frequency up-conversion mechanism [9,18,19], magnetic levitation [13], etc. Mono-stable nonlinear energy harvesters utilize nonlinear stiffness and act as hardening or softening spring [13,20]. They can broaden frequency range with larger amplitude oscillations and provide bigger output compare to linear configuration. However, most of mono-stable generators can only broaden the frequency response in one direction. A bistable oscillator is another popular nonlinear oscillator

M. Xie et al. / Sensors and Actuators A 222 (2015) 140–148

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bistable energy harvester. Frequency sweep experiments with small excitation vibration have been carried out to characterize the device. The device demonstrates broadening of the useful energy harvesting bandwidth before ‘snap through’ occurs.

Potential energy

2. Experimental Displacement

Fig. 1. Potential energy of bistable oscillator. (Solid line shows the symmetric potential energy double-well while the dashed line shows the asymmetric potential energy well.)

design [7,8,21,22], which exploits a unique symmetric or asymmetric potential energy double-well, as shown in Fig. 1. It is well known that a system with double well potential can exhibit chaotic behavior. If the input excitation is small, the state of the device remains within the potential well related to one side of the neutral position. In this case, the device behaves as a regular linear or nonlinear resonant system [23]. However, under sufficiently large excitation, the device will have enough energy input to overcome the potential well and switch state. This condition is called ‘snap though’ [24]. The complex dynamic response of such a bistable system, if correctly designed, can be favorable to the performance of vibrational energy harvesters. Arrieta et al. [8] studied a piezoelectric bistable nonlinear energy harvester that was fabricated by bonding a piezoelectric patch to a bistable reinforced composite laminate. This device achieved a broad bandwidth by the elastic buckling of the bistable composite plate. A bistable composite laminate has two stable shapes, giving rise to potentially useful vibration behavior, with nonlinear large amplitude oscillations occurring over a wide frequency range. This is due to ‘snap through’ between the two stable states, which is strongly nonlinear. The stresses allow the reinforced composite laminates to exhibit multistability results from mismatched thermal residual stress. When cured in a press or autoclave, the initially flat composite laminate develops curvature when cooled to room temperature [25]. Bistability can also be accomplished at the material level by prestretching two isotropic plates and then bonding them to opposite sides of an unstressed middle third plate. Inspired by bistable natural systems such as the Venus flytraps plant [26] which can switch between different functional shapes upon actuation, Chen et al. [27] developed a prestretched bistable structure. Here the terminologies of prestretch and prestrain are used interchangeably to create prestress. Two thin latex rubber sheets were prestretched by equal amount and bonded to an elastic strip to create a multi-layered structure. Releasing the prestresses will then result in a stress distribution within the structure and cause out-of-plane displacement into bistable configurations (cylindrical shapes) or monostable configuration (saddle shape). Comparing to the reinforced composite laminates, this prestretching technique is much easier to implement because it only involves processes at room temperature and simple plain structural material without any filling materials. In this paper, square three-layer soft PDMS structures have been fabricated using the prestretching method. An analytical model of this multi-layered structure based on classical laminate theory (CLT) has been developed to predict the configurations and consequently to aid the design of energy harvester. The dependence of configurations on various geometrical and mechanical parameters were studied and characterized. A multi-layered PDMS cylindrical structure was used to demonstrate its potential as a low frequency

Top and bottom thin PDMS films are subjected to the same amount of strain and will sandwich a thicker PDMS film. The bonded structure will deform after the stresses were released. The following details are the fabrication process of a multi-layered PDMS structure. 2.1. Prestretched PDMS In our previous work, it has been shown using an orthogonal experiment design that the Young’s modulus of PDMS can be predicted by three parameters: curing time (t), curing temperature (T) and mixing ratio (R) [3]. In order to achieve a soft PDMS of Young’s modulus 8 × 105 Pa, Sylgard 184 A:B = 20:1 in weight was used to prepare the PDMS films. PDMS base A and curing agent B were mixed thoroughly, degassed under vacuum, and poured into three molds. These molds were 3-D printed from acrylonitrile butadiene styrene (ABS) plastic with different depth. The PDMS was then cured in an oven for 30 min at 90 ◦ C. In order to ensure that the accuracy of the thickness of the thin PDMS films, the weight of PDMS mixture poured into the mold has been measured by an electronic scale. Fig. 2 shows the PDMS bonding process. After prestretching the top (A) and bottom (C) layers by an equal amount of stress f, three PDMS films were bonded together to achieve the final PDMS structure using oxygen plasma. Oxygen plasma treatment is a simple and effective method to activate the surface of PDMS layers, so that a strong PDMS to PDMS bond can be produced [28]. Here, two prestretched PDMS thin films (A, C) and one unstretched PDMS square film (B) were surface treated for 30 seconds with a RF power of 50 W and pressure of ∼450 mTorr (Fig. 2(a)). After oxygen plasma treatment, three films were quickly brought into contact (Fig. 2(b)). A strong bond is immediately produced. The fixtures used to prestretch layers A and C before bonding are shown as in Fig. 3.

Fig. 2. Bonding process. (a) Oxygen plasma treatment on one side of prestreched PDMS A, C and both sides of unstretched PDMS B. (b) Three layers are bonded together.

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curvatures kx , ky and kxy are defined as in Eq. (3), in which w is the out-of-plane displacement.

⎧ 2 ⎪ ∂ w ⎪ ⎪ kx = − ⎪ ⎪ ∂x2 ⎪ ⎪ ⎨ 2 k =−

∂ w

(3)

y ⎪ ∂ y2 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ∂ w ⎪ ⎩ kxy = −2

∂x∂y

In the large displacement case, considering Föppl–Von Karman equations, the mid-plane strains εx0 , εy0 ,  xy0 are

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

Fig. 3. The fixtures used to prestretch PDMS films A and C.

3. Analytical model Through the prestretching technique, bonded plate can deform into either cylindrical shape or saddle shape. However, only cylindrical configuration is desirable for the purpose of designing broadband nonlinear energy harvester. In this paper, a mathematical model is to be developed to qualitatively predict the configuration of the bonded prestretched PDMS multi-layered structure under different, arbitrarily chosen prestrain levels, geometric parameters and elastic properties. Generally, the deformation behavior of bistable laminate can be modeled with classical laminate theory [25,29,30] or complex continuum elasticity theory [27]. Considering the particular purpose of predicting cylindrical shape rather than study the detailed deformation behavior of the bonded structure, classical laminate theory has been naturally selected in this work. Therefore, the analysis shown in this section is based on the classical laminate model developed by Hyer et al. [25,29,30] where the strain energy of the PDMS three-layer structure is minimized by using a Rayleigh–Ritz technique to analyze the energetic content of the structure.

3.1. Strain–stress relations For thin linear elastic isotropic materials, the stress to strain relations is shown in Eq. (1). Here  is Poisson’s ratio, E is Young’s modulus and G is shear modulus.  x ,  y ,  xy are the stresses and εx , εy ,  xy are the strains.

⎡

⎤

⎡

E

⎢ (1 − 2 ) ⎢ ⎢ ⎥ ⎢ E ⎣ y ⎦ = ⎢ (1 − 2 ) ⎢ ⎣ xy x

⎤

E (1 − 2 ) E (1 − 2 )

0 0 E (1 + )

0

0

⎡ ⎤ ⎥ εx ⎥ ⎥⎢ ⎥ ⎥ ⎣ εy ⎦ ⎥ ⎦ xy

(1)

In a general situation, the above stress-to-strain relations can be rewritten as

⎡

x

⎤

⎡

Q11

Q12

Q16

⎤⎡

εx

⎤

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ y ⎦ = ⎣ Q12 Q22 Q26 ⎦ ⎣ εy ⎦ xy

Q16

Q26

Q66

εx0 =

∂u0 1 + 2 ∂x

εy0 =

∂v0 1 + 2 ∂y





∂w ∂x ∂w ∂y

2

2

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂u0 ∂v0 ∂w ∂w ⎪ ⎩ xy0 = + + ∂y

∂x

∂x

(4)

∂y

where u and v are the in-plane displacements in x and y directions. The total strains εx εy ,  xy are given by Eq. (5), in which z is the thickness of the plate.

⎧ ε = εx0 + zkx ⎪ ⎨ x ⎪ ⎩

εy = εy0 + zky

(5)

xy = xy0 + zkxy

It is more convenient to define force and moment by integrating the thickness of the plate. For plate with thickness h, the in-plane loads, Nx , Ny , Nz and Nxy and bending moments, Mx , My , Mz and Mxy are defined in [31] as Eqs. (6) and (7) respectively.

⎧  h/2 ⎪ ⎪ ⎪ x dz Nx = ⎪ ⎪ ⎪ −h/2 ⎪ ⎪ ⎪  h/2 ⎪ ⎪ ⎪ ⎪ N = y dz ⎪ ⎨ y −h/2  h/2 ⎪ ⎪ ⎪ N = z dz ⎪ z ⎪ ⎪ ⎪ −h/2 ⎪ ⎪  h/2 ⎪ ⎪ ⎪ ⎪ ⎪ Nxy = xy dz ⎩

(6)

−h/2

⎧  h/2 ⎪ ⎪ ⎪ M = x zdz x ⎪ ⎪ ⎪ −h/2 ⎪ ⎪ ⎪  h/2 ⎪ ⎪ ⎪ ⎪ My = y zdz ⎪ ⎨ −h/2  h/2 ⎪ ⎪ ⎪ Mz = z zdz ⎪ ⎪ ⎪ ⎪ −h/2 ⎪ ⎪  h/2 ⎪ ⎪ ⎪ ⎪ ⎪ = xy zdz M ⎩ xy

(7)

−h/2

(2)

xy

The configurations of the multi-layered PDMS structure can be expressed by the principal curvatures. Here, the principal curvatures in the x and y directions are denoted as kx and ky respectively. Applying the approximations of Kirchhoff–Love plate theory, the

Therefore, the force–strain relation and moment–strain relation can be obtained by using Eq. (2) and integrating through the plate thickness, as shown in Eqs. (8) and (9).

⎡

Nx

⎤

⎡

A11

A12

A16

⎤⎡

εx

⎤

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ Ny ⎦ = ⎣ A12 A22 A26 ⎦ ⎣ εy ⎦ Nxy

A16

A26

A66

xy

(8)

M. Xie et al. / Sensors and Actuators A 222 (2015) 140–148

⎡

⎤

Mx

⎡

D11

D12

D16

⎤⎡



⎤

kx

U1 =

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ My ⎦ = ⎣ D12 D22 D26 ⎦ ⎣ ky ⎦ Mxy

D16

D26

n 

A

+ My ıky + Mxy ıkxy )dxdy

where Aij and Dij are Aij =

Qij (zk − zk−1 )

U= n  1



Qij zk 3 − zk−1 3

3



⎤

⎡

A11

A12

⎤⎡

A16

εx

⎤

⎢ N ⎥ ⎢A ⎥⎢ ε ⎥ 0 ⎢ y ⎥ ⎢ 12 A22 A26 ⎥⎢ y ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ Nxy ⎥ ⎢ A16 A26 A66 ⎥ ⎢ xy ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ Mx ⎥ ⎢ D11 D12 D16 ⎥ ⎢ kx ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ 0 D12 D22 D26 ⎦ ⎣ ky ⎦ ⎣ My ⎦ ⎣ Mxy

D16

D26

D66

(10)

kxy

3.2. Minimum potential energy For many practical situations, the governing equations are sufficiently complex to make obtaining a direct algebraic solution very difficult. The application of numerical methods to study a related problem considering the energy in the structure can be a powerful and convenient alternative. The deformation and stress analysis of a structural system can usually be accomplished using the principle of minimum potential energy. This type of energy method can yield useful and reasonably accurate solutions for the prestretched PDMS structure. The internal potential energy U1 without prestresses can be obtained by integrating all contributions of products of stresses and incremental strains as in Eq. (11)

 U1 =





A11 εx0 2 + 2A12 εx0 εy0 + 2A16 εx0 xy0 + 2A26 εy0 xy0



+ A22 εy0 2 + A66 xy0 2 dxdy

Aij and Dij are the membrane stiffnesses and bending stiffnesses of coordinate i acting in the direction j respectively, which depends on the Young’s modulus and Poisson’s ratio of the PDMS (0.5 [32]) in addition to the layer thickness. zk and zk−1 are the upper and lower coordinates of the kth layer. Therefore, for a thin symmetrical structure of isotropic material with linear elastic behavior, the relationship between the bending moments, in-plane loads, strains and curvatures are shown in Eq. (10). Nx

1 2

A

k=1

⎡

(12)

Let f1 and f2 as the prestresses of the top and bottom PDMS films in x and y directions, respectively, the total potential energy with prestresses f1 and f2 can be written as in Eq. (13) by substituting Eq. (10) into (12), where H is the overall thickness.

k=1

Dij =

(Nx ıεx0 + Ny ıεy0 + Nxy ıxy0 + Mx ıkx

(9)

kxy

D66

143

1 2



V

(11)

Oxygen plasma bonding causes the surface of each PDMS to be intimately bonded to the surface of the neighboring layer with no intermediate material, hence, it is considered justifiable to assume that the strain in each bonded layer is equal at the interface between this pair of layers. In addition, the three PDMS layers have the same Young’s modulus, which significantly reduces the complexity of the calculation. (Thus various PDMS stiffness situation will not be considered here.) For thin PDMS structures with thickness of h, it — can be assumed that  z — —  yz =  zx — 0. After substituting Eqs. (5), (8) and (9) into Eq. (11), the potential energy without prestresses reduces to:



2

D11 kx + 2D12 kx ky + 2D16 kx kxy + 2D26 kx kxy

A 2

+ D22 ky + D66 kxy

  

+

A

f1

εx0 −

2

H kx 2



(13)

dxdy



 + f2

εy0 −

H ky 2

 dxdy

In this paper, kxy is assumed to be negligible if comparing with x and y direction curvatures because the applied prestresses are along x and y directions only. If ky = 0 or kx = 0, in other words, if the multi-layered PDMS structure turns into a cylindrical configuration, two equilibrium equations from minimizing potential energy U with respect to kx and ky can be obtained as in Eqs. (14) and (15). kx = 6(1 − 2 )f1 /EH 2 2

ky = 6(1 −  )f2 /EH

(14)

2

(15)

This result is similar to that reported in [27] for thin prestretched latex rubber structures. They obtained this result using reduced parameter model by defining dimensionless geometric parameter based on continuum elasticity theory. From these two equations, it is obvious that curvatures of the multilayer prestretched structures that exhibit cylindrical configuration can be determined by the prestresses (f1 and f2 ), thickness (h) and mechanical properties (Young’s modulus E and Poisson’s ratio ) of the materials. 3.3. Multi-layered prestretched PDMS configuration The total potential energy function chosen to approximate the deformed shape of the structure must satisfy the boundary conditions at the edges or center of the PDMS. The analysis here is based on the assumption proposed by Hyer et al. [30]. In order to use the Rayleigh–Ritz method [29], the out of plane displacement w is modeled with Eq. (16), where a and b are the negative curvatures in x and y directions: w=

(x ıεx + y ıεy + z ıεz + yz ıyz + xz ıxz + xy ıxy )dxdydz

+

1 (ax2 + by2 ) 2

(16)

The approximated in-plane elongation strains εx0 , εy0 using set of polynomials are



εx0 = a1 + a2 x2 + a3 y2 + a4 xy

(17)

εy0 = b1 + b2 y2 + b3 x2 + b4 xy

where a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 are coefficients to be determined. According to Eq. (4), integrating Eqs. (16) and (17) with respect to x and y, respectively, result in the mid-plane displacements u0 and

v0

⎧ a a ax3 ⎪ ⎨ u0 (x, y) = a1 + 2 x2 + a3 xy2 + 4 x2 y − 3

2

6

⎪ ⎩ v0 (x, y) = b + b2 y2 + b x2 y + b4 xy2 − by3 1 3 3

2

6

(18)

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M. Xie et al. / Sensors and Actuators A 222 (2015) 140–148

Thus, the in-plane shear strain, ␥xy0 can be computed from the third strain–displacement relation in Eq. (4), resulting into Eq. (17): xy0 =

a4 2 b4 2 x + y + 2a3 xy + 2b3 xy + abxy 2 2

(19)

The total potential energy can be found by substituting all the strains and curvatures into Eq. (13). Thus, the potential energy can be determined by 10 unknown coefficients, as described in Eq. (18). U = U(a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 , a, b)

(20)

According to the minimum potential energy theory, the first variation of the potential energy can be expressed as: ıU = ıU(a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 , a, b)

(21)

Solving these 10 equations in Eq. (19) gives the equilibrium configurations of the fabricated multi-layered structure. Moreover, by changing one of geometrical and mechanical parameters (Lx , Ly , H, ε, E) in the potential energy Eq. (19), we obtain the dependence of equilibrium configuration on various parameters. This dependence allows the prediction of a stable configuration of PDMS structure. 4. Analysis of the PDMS shapes The material properties and geometry of PDMS films studied here were chosen to match the structures used in the experimental work. The computation of minimum potential energy has been carried out using Matlab® with its default optimization function ‘fmincon’. This is a nonlinear multivariable optimization problem, and convergence at meaningful minima (where the residual energy

(a)

in the structure is minimal) depends on the initial conditions. In this study, 4000 random starting points have been tested to better cover the search space, and the overall best solutions selected that have the smallest remaining potential energy. It is noted that the curvature of multi-layered PDMS structure is a function of thickness, prestrain levels, side-length and Young’s modulus. After Matlab® convergence, the shape configurations in terms of ratio of side-length to thickness (L/H), prestrains (ε), and Young’s modulus (E) are shown in Fig. 4. Here, the multi-layered PDMS structures are square (Lx = Ly = L) and the applied prestrains in x and y direction are equal, which means x direction and y direction curvatures are symmetrical. Hence, in Fig. 4, only the x direction curvature is shown. Fig. 4(a) presents the relationship between curvature and ratio of side-length to thickness of the multi-layered PDMS structure. In Fig. 4(a), prestrain is 0.3 and Young’s modulus is 800 kPa. The shape of the structure is saddle when the ratio is less than a critical value, which is ∼7 in this study. As the ratio increases further than this critical value, the multi-layered structure switches from a saddle into a cylindrical configuration. Fig. 4(b) demonstrates the linear relationship between the amount of prestrain and curvature. Ratio of side-length to thickness is 8, and Young’s modulus is 800 kPa. In addition, in Fig. 4(c), ratio of sidelength to thickness is 8 and prestrain is 0.3. The stiffness of the material can control the curvatures of the structure as well and their relationship is presented. A stiffer the material will cause smaller bending. These two numerical results for cylindrical configuration in Fig. 4(b) and (c) demonstrate two functions as shown in Eqs. (14) and (15). Therefore, it can be concluded that the shape of the PDMS structure can be controlled by geometrical and mechanical parameters (Lx , Ly , H, ε, E).

Curvature in x-direction (m-1)

Curvature in x-direction (m-1)

(b)

120 100 80 60 40 20 0 0.00

0.05

0.10

0.15

(c)

0.20

0.25

0.30

0.35

0.40

0.45

Prestrain

Ratio of sidelength to thickness 220

Curvature in x-direction (m-1)

200 180 160 140 120 100 80 60 40 20 0 500

1000

1500

2000

2500

3000

3500

4000

4500

Young’s modulus (KPa) Fig. 4. Shapes of square three-layer PDMS structure with curvature in x direction versus (a) ratio of side-length to thickness, (b) the prestrains applied and (c) the Young’s modulus.

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145

Fig. 5. Analytical configurations of thin PDMS structure ((a) and (b) are two cylindrical shapes).

Table 1 Thin and thick multi-layered PDMS structure properties. PDMS properties

Thin

Thick

Young’s modulus Poisson ratio Prestrain PDMS plate length/width PDMS top/bottom layer thickness PDMS middle layer thickness

8 × 10 Pa 0.5 0.3 25 mm 0.3 mm 0.4 mm 5

8 × 105 Pa 0.5 0.3 25 mm 1 mm 2 mm

With the aid of this analytical model, the configuration of the prestressed multi-layered PDMS structure can be predicted. It is obvious that ratio of side-length to thickness plays important role in configuration prediction. In order to verify the model, numerical computation and experiments of three-layer PDMS structure with the properties presented in Table 1 were performed. Fig. 5 shows two predicted cylindrical configurations obtained after Matlab® iteration based on the thin PDMS multi-layered structure while Fig. 6 displays the predicted saddle configuration with thick PDMS multi-layered structure. Two multi-layered PDMS structures have been fabricated based on the properties in Table 1. By using the same PDMS curing, bonding and prestretching procedure mentioned in section 2, the thin PDMS multi-layered structure deformed into one of two stable configurations, which are cylindrical in shapes as shown in Fig. 7 (marked as × in Fig. 4(a)). The thick PDMS structure exhibited one stable configuration, which has a saddle shape as shown in Fig. 8 (marked as + in Fig. 4(a)) and is undesirable for energy harvester. The good agreement between predicted and experimental configuration indicates that this analytical model is able to predict the configuration of multi-layered PDMS structure.

Fig. 6. Analytical configuration of the thick PDMS structure (saddle shape).

5. Energy harvester As mentioned in the introduction section, a nonlinear energy harvester can be implemented using laminate structure as a bistable oscillator. Between two configurations a prestressed multilayered structure can exhibit, only cylindrical shape has geometric nonlinearity and is able to contribute to nonlinear behavior [27]. Even without ‘snap through’ at small excitation, a bistable oscillator will exhibit a nonlinear resonant behavior similar to that of a monostable Duffing oscillator [33]. Large input excitation can induce strong nonlinearity due to ‘snap through’. Here, the focus is on the dynamic response before the occurrence of ‘snap through’, as it is this region that is most relevant to the performance of an energy harvester subjected to low amplitude excitation. Fig. 9 shows the clamped bistable PDMS structure. Here one end of the multi-layered PDMS structure was clamped to a 3-D printed ABS plastic stage. Two magnets (0.2 g × 2) were attached to the free end with the assumption it will be used as an electromagnetic

Fig. 7. The bonded three-layer thin PDMS structure ((a) and (b) are two cylindrical shapes).

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Fig. 8. The bonded prestreched three-layer thick PDMS structure (one saddle shape).

Fig. 9. The clamped nonlinear geometric PDMS. (x, y and ± z direction in this figure are not the same as the directions as in analytical model).

induction vibrational energy harvester in the future. A schemetric diagram of the experimental setup is provided as Fig. 10. A signal generator and amplifier were used to provide a sinusoidal sweep frequency signal input to a Derritron VP5 shaker. During each frequency sweep, the accerleration of excitation is kept constant at all times. In this work, three different base excitations are considered, namely 2.68 m/s2 , 4.41 m/s2 and 9.29 m/s2 and are labeled as 0.2 g, 0.5 g and 1 g. Two laser sensors were used to determine the displacement of the multi-layered PDMS relative to the supporting structure. One laser measures the free end’s tip displacement of the multi-layered PDMS structure while the other records the vibration of the shaker. The subtrction of these two laser data gives the relative cantilever’s displacement. As PDMS is a low stiffness material, this bistable multi-layered PDMS structure operates well at low vibration frequency. In this study, the input sweep frequency applied to the system was from

Labview Two lasers 1

2

Magnet

ABS clamp Waveform

Nonlinear PDMS

Signal Generator Oscilloscope Shaker Amplifier Fig. 10. The experimental setup.

11 Hz to 18 Hz and then 18 Hz back to 11 Hz. This multi-layered PDMS structure allows good performance at low frequencies in a device significantly smaller than those studied in other references [8,34]. When the amplitude of the applied excitation is small, the system oscillates periodically at the excitation frequency within one potential well. Fig. 11 presents the results for frequency sweep forward and reverse in the range of 11–18 Hz, and for excitations ranging from 2.68 m/s2 up to 9.29 m/s2 . Fig. 11(a)–(c) shows the +z direction displacement of the tip magnet during a frequency sweep forward from 11 Hz to 18 Hz and sweep back from 18 Hz to 11 Hz with three different base excitations. Fig. 11(d)–(f) demonstrates the −z direction displacement of the tip magnet. Directions x, y, + -z are labeled in Fig. 9 and they are not the same with the directions in analytical model. Comparing the tip displacements along +z direction and −z direction under the same excitation, it can be concluded that the relative tip displacement is not symmetrical and +z direction displacement is always higher than −z direction displacement. This is due to the asymmetrical restoring force in the cylindrical structure when it travels up and down because one end of the multilayered PDMS structure was fixed rather than center fixed. In this work, when the excitation is traveling up along −z direction, the excitation force is not sufficient to cause ‘snap through’ between two stable states. However, only force along −z direction can cause ‘snap through’. When the excitation is in +z direction, excitation force can only maintain the structure at one stable state. In addition, it is obvious that this bistable energy harvester under small input excitation is a nonlinear system and exhibits hysteresis depending on the direction of the frequency sweep. This hysteresis response becomes more pronounced as the excitation level is increased. Frequency response in +z direction under 2.68 m/s2 , 4.41 m/s2 , and 9.29 m/s2 excitations during frequency reverse sweep from 18 Hz to 11 Hz can be re-illustrated in Fig. 12. It can be noted that at low excitation (2.68 m/s2 ), the PDMS structure has already started to exhibit nonlinear behavior. In Fig. 12(a), the resonant frequency has been shifted from 14.4 Hz to 11.3 Hz when the base excitation increases up to 9.29 m/s2 and the bandwidth of the energy harvester increases as the excitation level increases. During the reverse frequency sweep, the frequency response can be hypothesized due to the effect of the nonlinear restoring force, which is similar to that of a Duffing oscillator and this PDMS structure behaves as a softening spring [16]. A numerical study proved that softening spring is better than linear and hardening spring for energy harvester because it can provide both wider bandwidth and larger harvested power [35]. Fig. 12(b) shows the theoretical/ideal average power can be transferred to the tip mass, which was calculated by P = 82 mA2 f3 (A is displacement, m is mass, and f is frequency). From this figure, it is clear that the power is available over a wider frequency range than a simple cantilever beam would. However the power generated from actual electromagnetic harvester could not be determined as it is dependent on the design/construction of the coil, coupling loss and power management circuit and is not the main intention of this paper. In this work, the softening behavior can be tuned by changing PDMS dimensions and properties, while for most hardening or softening spring based energy harvesters the spring forces are controlled by adjusting the magnetic strength. Changing PDMS dimensions and properties is able to control the curvature of structure and therefore determines the depth of potential wells. The effects of potential well depth on the nonlinearity of monostable and bistable energy harvesters have been investigated in [23]. The softening behavior shows the prestretched PDMS structure has the potential to broaden the bandwidth of a vibrational energy harvester. In addition, when the amplitude of excitation is increased further, the cross-well motion ‘snap through’ is expected to occur and increases the nonlinearity of the system. Fig. 13 demonstrates the dynamic response of the frequency forward

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Fig. 11. Frequency sweep of the bistable energy harvester under three different excitations. Red lines are forward frequency sweeps, while blue lines are reverse frequency sweeps. (a)–(c) demonstrate the tip displacement in +z direction during frequency sweep at constant acceleration 2.68 m/s2 , 4.41 m/s2 and 9.29 m/s2 respectively. (d)–(f) show the −z direction displacements at 2.68 m/s2 , 4.41 m/s2 and 9.29 m/s2 . (Labels 0.2 g, 0.5 g and 1 g represent accelerations of 2.68 m/s2 , 4.41 m/s2 and 9.29 m/s2 .)

0.012

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0.008 0.006 0.004 0.002

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15

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17

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0 11

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13

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15

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( b)

Fig. 12. The frequency response of the bistable PDMS energy harvester with reverse frequency sweep. (a) Shows the tip displacement and (b) is the theoretical average power can be transferred to the tip mass.

sweep from 11 Hz to 18 Hz at 9.31 m/s2 excitation, which is slightly higher than the excitation in Figs. 11(c), (f) and 12 (9.29 m/s2 ). It can be noted in Fig. 13 that between 12.3 Hz and 12.8 Hz, the chaotic intrawell dynamic responses is about to be initiated. However, further response after ‘snap though’ is not pursued because the aim is to extract energy from human motion at low excitation (<1 g). 6. Conclusions

Fig. 13. Frequency sweep of the bistable energy harvester under excitation of 9.31 m/s2 .

A square multi-layered PDMS structure created by bonding two prestretched thin PDMS films on both sides of a thick middle PDMS film has been proposed, modeled and characterized. An analytical model capable of predicting the configuration of the structure has

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been developed based on the classical laminate theory, Föppl–Von Karman theory and Rayleigh–Ritz approximation method. The deformed shapes of the prestretched PDMS predicted by the model shows good agreement with the shapes obtained experimentally. Three parameters (ratio of side-length to thickness, prestrain levels, and Young’s modulus) determine the configuration of the multilayered PDMS structure and a critical sidelength to thickness ratio of ∼7 separates the saddle and cylindrical configurations. A low frequency wide band energy harvester has been developed using cylindrical multi-layered PDMS structure. The dynamic response of the bistable PDMS structure under small input excitation indicates a monostable Duffing oscillation before ‘snap through’ occurs and successfully broadened the bandwidth of vibration energy harvester. Acknowledgement The main author (Mengying Xie) would like to express her gratitude to China Scholarship Council for funding her doctoral study at the University of Auckland, New Zealand. References [1] S.P. Beeby, M.J. Tudor, N.M. White, Energy harvesting vibration sources for microsystems applications, Meas. Sci. Technol. 17 (2006), R175-R95. [2] N. Wang, D.P. Arnold, Fully batch-fabricated MEMS magnetic vibrational energy harvesters, in: Proc Power-MEMS, Washington, DC, 2009, pp. 348–351. [3] M. Xie, K. Aw, B. Edwards, W. Gao, P. Hu, Orthogonal experimental design of polydimethylsiloxane curing for the design of low-frequency vibrational energy harvester, J. Intell. Mater. Syst. Struct. (2014). [4] C.B. Williams, R.B. Yates, Analysis of a micro-electric generator for microsystems, in: Solid-State Sensors and Actuators, 1995 and Eurosensors IX Transducers ‘95 The 8th International Conference on 1995, 1995, pp. 369–372. [5] J.-Q. Liu, H.-B. Fang, Z.-Y. Xu, X.-H. Mao, X.-C. Shen, D. Chen, et al., A MEMSbased piezoelectric power generator array for vibration energy harvesting, Microelectron. J. 39 (2008) 802–806. [6] M.S.M. Soliman, E.M. Abdel-Rahman, E.F. El-Saadany, R.R. Mansour, A wideband vibration-based energy harvester, J. Micromech. Microeng. 18 (2008) 115021. [7] F. Cottone, H. Vocca, L. Gammaitoni, Nonlinear energy harvesting, Phys. Rev. Lett. 102 (2009) 080601. [8] A.F. Arrieta, P. Hagedorn, A. Erturk, D.J. Inman, A piezoelectric bistable plate for nonlinear broadband energy harvesting, Appl. Phys. Lett. 97 (2010) 104102. [9] A.M. Wickenheiser, E. Garcia, Broadband vibration-based energy harvesting improvement through frequency up-conversion by magnetic excitation, Smart Mater. Struct. 19 (2010) 065020. [10] K. Ashraf, M.H. Md Khir, J.O. Dennis, Z. Baharudin, A wideband, frequency up-converting bounded vibration energy harvester for a low-frequency environment, Smart Mater. Struct. 22 (2013) 025018. [11] B. Andò, S. Baglio, C. Trigona, N. Dumas, L. Latorre, P. Nouet, Nonlinear mechanism in MEMS devices for energy harvesting applications, J. Micromech. Microeng. 20 (2010) 125020. [12] B. Marinkovic, H. Koser, Smart Sand—a wide bandwidth vibration energy harvesting platform, Appl. Phys. Lett. 94 (2009) 103505. [13] B.P. Mann, N.D. Sims, Energy harvesting from the nonlinear oscillations of magnetic levitation, J. Sound Vib. 319 (2009) 515–530. [14] H. Kulah, K. Najafi, Energy scavenging from low-frequency vibrations by using frequency up-conversion for wireless sensor applications, Sensors J. IEEE 8 (2008) 261–268. [15] H. Vocca, I. Neri, F. Travasso, L. Gammaitoni, Kinetic energy harvesting with bistable oscillators, Appl. Energy 97 (2012) 771–776. [16] R. Ramlan, M.J. Brennan, B.R. MacE., I. Kovacic, Potential benefits of a non-linear stiffness in an energy harvesting device, Nonlinear Dyn. 59 (2010) 545–558. [17] S.G. Burrow, L.R. Clare, A resonant generator with non-linear compliance for energy harvesting in high vibrational environments, in: IEEE International Electric Machines & Drives Conference, 2007 (IEMDC’07), 2007, pp. 715–720. [18] M. Renaud, P. Fiorini, R.v. Schaijk, C.v. Hoof, Harvesting energy from the motion of human limbs: the design and analysis of an impact-based piezoelectric generator, Smart Mater. Struct. 18 (2009) 035001. [19] Q. Tang, Y. Yang, X. Li, Bi-stable frequency up-conversion piezoelectric energy harvester driven by non-contact magnetic repulsion, Smart Mater. Struct. 20 (2011) 125011. [20] L. Tang, Y. Yang, C.-K. Soh, Improving functionality of vibration energy harvesters using magnets, J. Intell. Mater. Syst. Struct. 23 (2012) 1433–1449.

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Biographies

Mengying Xie received her BS degree in Applied Physics from Tianjin University, China, in 2010. She is currently pursuing her PhD in the Department of Mechanical Engineering at the University of Auckland and is expected to complete in December 2014. Her research focuses on the various applications of soft organic materials such as PDMS.

Kean C. Aw is an Associate Professor at the Department of Mechanical Engineering, University of Auckland, New Zealand since 2004. Prior to his academic position, he worked at Intel, Altera and Navman for a total of 11 years. His main interests are in micro-systems and deployment of smart/functional materials and structures such as conducting polymers, metallic oxides, etc. as sensors and actuators in various applications such as bio-sensors, medical/rehabilitation robots, micro-pumps, micro-manipulators, MEMS, energy harvester, etc. He has over 130 refereed publications.s

Wei Gao is a Professor of Materials Science and Engineering at the University of Auckland New Zealand. Received DPhil degree from Oxford University UK, he worked at MIT, USA for 4 years as a Research Associate. His research covers areas of Nanomaterials, Thin Films, Coatings, Light Alloys, Electronic Materials, Corrosion and Oxidation. He leads a group of 25 people, has published >600 refereed publications. He is the Fellow of Royal Society NZ, Advisory Professor for Universities in China, Taiwan, Korea and Saudi Arabia, and received prestigious awards, including “Scott Medal”, “James Cook Award” and “Distinguished Materials Scientist of China”.