Nonlinear Dynamics of Energy Harvester Based on ... - Science Direct

Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 144 (2016) 936 – 944

12th International Conference on Vibration Problems, ICOVP 2015

Nonlinear Dynamics of Energy Harvester Based on Flow Induced Vibration A. Raj*, A. Garg, S. K. Dwivedy Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India.

Abstract In this present work, a vertical cantilever piezoelectric energy harvester is proposed that exploits wind energy for vibration. The vertical cantilever is attached with a torsional spring at the base and a tip mass. A square bluff body is attached at the end of the cantilever so that the wind energy could be exploited effectively. The governing nonlinear electromechanical equations of motion for the system are developed using Lagrange principle and were discretized to the temporal form by using generalized Galerkin’s method. The equations are solved using method of multiple scales and also using numerical methods. The responses of the system are determined for different wind speeds and load resistances. Time response and phase portraits are plotted to study the system response and influence of different system parameters on voltage generation. © 2016 2016Published The Authors. Published by Elsevier Ltd. © by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICOVP 2015. Peer-review under responsibility of the organizing committee of ICOVP 2015 Keywords: Piezoelectric energy harvesting; Bluff body; Method of multiple scales; Wind energy

1. Introduction Energy Harvesting is the process of capturing minute amounts of energy from one or more naturally-occurring energy sources, accumulating them and storing them for future purposes. Vibration energy harvesters based on linear vibrations can be designed for a particular excitation frequency only. Introduction of stiffness-type nonlinearities to the harvesters design could help in harvesting more power as the effective bandwidth of the harvester will be extended. The mechanical energy conversion to electrical energy is done through electromagnetic electrostatic or piezoelectric induction. Researches on piezoelectric energy harvesting are very promising and have been evolving over the past several years, because of the increasing energy needs and its capability to be installed in

*

Corresponding author. Tel:+919085965564 E-mail address: [email protected]

1877-7058 © 2016 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICOVP 2015

doi:10.1016/j.proeng.2016.05.120

937

A. Raj et al. / Procedia Engineering 144 (2016) 936 – 944

remote locations. According to Priya[1] the energy density of piezoelectric energy harvesting devices is 3-5 times higher than electrostatic and electromagnetic devices. Due to the drop of the power output at vibrations apart from resonant conditions, a large part of the research was concentrated on increasing the frequency band for the energy harvesting. Soliman et al. [2] used an amplitude limiter to gain broader bandwidth. Saadon and Sidek [3] constructed a series triple layer bimorph out of a metallic layer, sandwiched between two piezo electrics and the piezoelectric patches are electrically connected in series. Hua et al. [4] modeled a novel compact 2 DOF energy harvester using cut out beam. It was having larger bandwidth than usual single and two degrees of freedom systems. Kluger et al. [5] modeled a nonlinear spring mechanism composed of a cantilever wrapping as it deflects. The 2DOF nonlinear system gave best average performance over various excitation signals. Scientists have attempted to harvest the energy from the vibration of bodies due to the passage of wind. Yan and Abdelkefi [6] studied the harvester’s response under combined loading, that is base excitation and galloping excitations. Paxson et al. [7] studied the design considerations for small-scale wind energy harvesters driven by broadband vortex-induced vibrations making minute changes to the trailing edge geometry of a D shaped body and fluid flow characteristics. Miao Shi [8] studied the vibration energy harvesting from wind induced vibration of suspension bridges. The most famous energy harvester based on Karman Vortex Street is the ‘Energy Harvesting Eel [9]. Abdelkefi et al. [10] studied the vibration of a horizontal cantilever beam with a bluff body at the end subjected to wind induced vibrations. The present study deals with a piezoelectric cantilever energy harvester as a forced vibrating system; with the excitation force being provided by the wind force. A torsional spring is attached to the fixed part of the beam. The harvester is considered as an Euler Bernoulli beam and the equation of motion of the system has been derived using Lagrange and Galerkin’s principle. Nomenclature v u L Mt ρ Kt s Uw

horizontal displacement of the beam vertical displacement of the beam length of the beam tip mass mass per unit length of the beam torsional spring stiffness distance along the neutral axis of the beam wind velocity

2. Mathematical modeling A cantilever beam of length L, tip mass Mt, a torsional spring of stiffness K t , along with piezoelectric patch and a square bluff body is shown in Fig. 1 to study the possibility of harvesting energy. The bluff body is half of the length of the beam and is made of very light material such that the weight is considered to be insignificant. 2.1. Strain Displacement relationship The strain displacement relation is considered as nonlinear due to large transverse displacement. Due to wind excitation, any arbitrary point undergoes a transverse displacement of v and an axial displacement of u. The curvature k(s, t) of the beam is given by k(s, t) = wI / ws I c . Using extensibility condition, axial displacement can be given as [11],

uc

1 vc2 | 1/ 2 vc2

s

u ( s, t )

1 c v ([ , t ) 2 d [ 2 ³0

(1)

938


1 6

I s, t sin 1 v vc vc3 Here I denotes rotation of the beam at a distance s from the base.

(b)

(a)

Fig. 1. Schematic diagram of the system under consideration.(a) Front view (b) side view

2.2. Equation of motion The beam has a cross sectional area A, mass density ρ, Young’s modulus E and second moment of inertia I. The expression for the total kinetic energy of the system (T), total potential energy of the system (U) and dissipation energy (D) in terms of curvature k(s,t) and displacements u and v are given below. L

T

1 1 2 2 U A³ [v s, t u s, t ]ds M t [v 2 u 2 ] I tI 2 2 2 0

U

1 1 § wv · EI ³ k ( s, t ) 2 ds U Ag ³ u s, t ds M t gu ³ G (s d ) K t ¨ ¸ ds 2 0 20 © ws ¹ 0

D

1 § wv · Ca ¨ ¸ 2 © wt ¹

L

L

L

2

(2)

2

Here Ca is the damping coefficient. The moment M(s,t) about the beam neutral axis produced by a voltage V across the piezoelectric layers can be given by the following relation

M s, t J cV t ,

(3)

Here J c is a constant which depends on the geometry, configuration and piezoelectric material. Piezoelectric layers with thickness hp and width bp are attached to the beam in such a way that if strains are applied in first direction, then electric charge is generated in the third direction.

Jc

Ed31bp c h hp

(4)

939


where d31 is a piezoelectric constant. The work done by moment about the beam neutral axis produced by the piezoelectric patches in moving or extracting the electric charge is W

³

L

0

(5)

M (s, t ) k (s)ds

The aerodynamic force acting on the bluff body per unit length can be represented as [10]

Fy

3 ª v § v · º 1 a3 ¨ UairU w2 b « a1 ¸ » 2 «¬ U w © U w ¹ »¼

(6)

Where, U air is the density of air, b the width of the bluff body, a1 and a3 are empirical coefficients obtained by polynomial fitting of total aerodynamic force coefficient in the direction normal to the incoming flow versus the tangent of angle of attack. The Lagrangian L is given by L=T-U. To determine the temporal equation of motion, one may discretize the governing equation of motion by using following assumed mode expression, v s, t q t \ s . The displacement may be approximated by the following function which satisfies the boundary conditions

§Ss · ¸. © 2L ¹

\ ( s) 1 cos ¨

Using similar nondimensional paratamets as in [12], the resulting governing equations of motion can be written as § q · (7) q D 1q HD 2q HD 3q 3 HD 4(q 2 q q 2 q) HD5V H 2D 6q 2V H f 01U w q H f 02 ¨ ¸ © Uw ¹ 2 2 V rV 1 v H r2 vv

(8)

0

where,

D1 D4 f 01

EIN 6 N9 U Ag N 4 M 4 g N10 K t ; ( N52 I t M t U AN1 )Z 2

D2

Ca ; ( N5 I t M t U AN1 )HZ

T2T1

U AN3 M t N 4 2 N54 I t 2 Lg ; D 6 N52 I t M t U AN1

C p ( N52 I t M t U AN1 )H 2Z 2

.157 U airbL

.157 U airbL

HD 0Z

u .146a1 ;

f 02

HD 0

D3

2

u .0463a3Z Lg 2 ; 2

L2g ;

r1

D5

L2g 2 EIN7 u 2 N5 I t M t U AN1 HZ 2

T12 Lg 2 C p ( N52 I t M t U AN1 )HZ 2

1 T 2 Lg 2 ; r2 CpRlZ H 2T 1

L L ° s ½° 2 2 N1 ³\ ( s) ds ; N 2 ³\ ( s)ds ; N3 ³ ® ³ ª¬\ c(] ) º¼d ] ¾ ds ; N 4 ³\ c( s)2 ds ; N 6 ³ >\ cc( s)@ ds ° ° 0 ¯0 ¿ 0 0 0 0 L s L L L ½ ° ° 2 4 2 2 N7 ³ >\ cc( s)\ c( s)@ ds ; N8 ³ >\ c( s)@ >\ cc( s)@ ds ; N9 ³ ® ³ ª¬\ c(] ) º¼d] ¾ ds ; N10 ³ G ( s d )\ c( s)2 ds ° ° 0 ¯0 ¿ 0 0 0 L L 1 2 T1 J c ³\ cc( s)ds ; T 2 3J c ³ \ cc( s) >\ c( s)@ ds 2 0 0 L

L

2

L

940


2.3. Solution by perturbation method A uniform first order approximate solution of the governing equation is obtained by using the standard method of multiple scales. The time dependence can be expressed in multiple time scales as Tn=εnt where ε is a book keeping parameter. The solution for the system can be assumed to be as

q0 (T 0, T 1) H q1 (T0 , T1 ),

q(t )

V (t ) V0 (T 0, T 1) HV1 (T 0, T 1)

(9)

Substituting Eq. (9) in Eq. (7) and Eq. (8) and equating the coefficients of H 0 and H , A(T 1)eiZ0T0 A(T 1)eiZ0T0 ,

q0

V0

Z02 ir1Z0 A(T 1)eiZ T A(T 1)eiZ T r12 Z02 0 0

0 0

(10)

Collecting the secular terms and equating to zero, and substituting A a eiE leads to

ac

F a, U w

§ r1Z0 · a a 3f U w f 01 02 a 3Z0 2 ¨ D12 D15 2 2 ¸ 2 8U w r1 Z0 ¹ 2 ©

The equilibrium point is obtained by substituting F a, U w

(11)

0

So, one can obtain two solutions, the trivial solution that is a 0 and the nontrivial solution

a

4U w § r1Z0 · 4U w2 f 01 D D ¨ ¸ 15 12 3 f 02Z0 2 © r12 Z0 2 ¹ 3 f 02Z0 2

(12)

The trivial solution is stable for all values of Uw such that

Uw

r1Z0 · 1 § ¨ D12 D15 2 ¸ f 01 © r1 Z0 2 ¹

(13)

and is unstable for all higher values of Uw. For the nontrivial branch, the solution is stable for all values of Uw such that

Uw !

r1Z0 · 1 § ¨ D12 D15 2 ¸ f 01 © r1 Z0 2 ¹

(14)

3. Results and discussion The geometric and material parameters of the harvester are given in Table 1 which is same as those considered by Friswell et al. [2]. An excitation wind velocity ranging from 1m/s to 15m/s was used to study the behavior of the system.

941


Table 1: Geometric and material parameters of harvester Properties of beam 2

Properties of piezoelectric

E

Steel Young’s modulus(GN/m )

210

Lc

Length of Piezoelectric (m)

ρ

Steel density(kg/m3)

7850

γc

Constant(Nm/V)

-4×10-5

L

Length of beam(m)

.3

bp

Width of piezoelectric layer(m)

.014

Mt

Tip mass(g)

7

hp

Piezoelectric layer thickness(mm)

0.3

b1

Width(m)

.016

Cp

Capacitance(nF)

51.4

h1

Thickness(mm)

.254

Rl

Resistance(ohm)

105-108

b

Width of bluff body(m)

.016

Ca

Damping coefficient(Ns/m)

.001

.028

The damping coefficient (Ca) is considered to be as 0.001Ns/m due to the air resistance of the system and the torsional spring stiffness (Kt) is 15Nm/rad. The time domain displacement, voltage response and the phase portrait are shown in Fig. 2 and these are obtained from numerically solving the Eq. (6) and (7). 3.1 Numerical solution The behaviour of the system for a load resistance of 105 : is shown in Fig. 2. The displacement and voltage amplitude and phase portrait for wind velocities of 3m/s, 5m/s and 7m/s are plotted in the figure. At a wind velocity of 3m/s, the vibration amplitude is 0.0058m, whereas at a wind velocity of 5m/s amplitude is 0.0753m and nearly 0.1182m at a wind velocity of 7m/s. A large variation of more than 10 times has been observed as the wind speed varies from 3m/s to 5m/s. A similar trend is being followed by the voltage as well. The voltages obtained for wind velocities of 3m/s, 5m/s and 7m/s are 0.0486V, 6.54V and 11.75V respectively

(a)

(b)

942


(c) Fig. 2. Displacement time response, Phase portrait and Voltage time response for a load resistance of 105Ώ and wind speed of (a) 3m/s (b) 5m/s (c) 7m/s

Similar plots were made for a range of wind speeds ranging from 1m/s to 15 m/s. The variation of voltage with wind speeds are plotted in Fig. 3(a). The voltage value is very low at small wind velocities. The voltage value increases suddenly after a particular value and a linear variation is followed afterwards. The higher load resistance is leading to higher voltage and power values as shown in Fig. 3. At a wind velocity of 10m/s and a load resistance of 105 : , a voltage of nearly 20V is obtained, while at the same speed and load resistance of 2×10 5 : , nearly 38V is obtained. The analysis predicts a power of 4.6mW at load resistance of 105 : while a power of 10mW is predicted for a load resistance of 2×105 : s shown in Fig. 3(b)

(a)

(b)

Fig. 3. (a) Voltage variation with wind speed for varying load resistances (b) Power variation with wind speed for varying load resistances

3.2. Solution by perturbation methods The stability of the system is studied by plotting the variation of amplitude with wind velocity. A pitchfork type of bifurcation is obtained for the present system. The trivial solution is stable upto a wind velocity of 3.58 m/s for a load resistance of 105 : as shown in Fig. 4. The wind speed at which the trivial solution cease to be stable and some measurable displacement is obtained can be called as the onset of galloping wind speed. Onset of galloping wind speed is varying for different load resistances as shown in Fig. 5.


943

Fig. 4. Variation of wind speed with amplitude for a load resistance of 10 5 Ώ

The variation of this onset of galloping wind speed, was plotted against various load resistances.The onset of galloping wind speed versus load resistance was found to have a dome like shape where the galloping wind speed was found to vary between 3.3m/s at 103 : to a maximum of nearly 4.35 m/s at 7 u105 : .

Fig. 5. Variations of the onset of galloping speeds with the electrical load resistance

4. Conclusion In this present work, a vertical cantilever piezoelectric energy harvester is proposed that exploits wind energy for vibration. The study of voltage and displacement response characteristics for the system shows that a reasonable amount of voltage and power could be developed from this harvester. The voltage and power output is influenced by the attached load resistance. Also, the load resistance has a significant effect on the onset of galloping wind force. The voltage and power produced are sufficient enough to power wireless sensor nodes and LEDs.

944


References [1] P. Shashank, Advances in energy harvesting using low profile piezoelectric transducers, Journal of electroceramics 19.1 (2007) 167-184. [2] M.S.M. Soliman, E.M. Abdel-Rahman, E. El Saadanay, A wideband vibration-based energy harvester, Journal of micromechanics and microengineering 18.11 (2008) 115021. [3] S. Saadon, O.B. Sidek, O.S. Hamad, Vibration-based MEMS piezoelectric energy harvesters using cantilever beams. [4] H. Wu, L. Tang, Y. Yang, C.K. Soh, A compact 2 degree-of-freedom energy harvester with cut-out cantilever beam. Japanese Journal of Applied Physics 51.4R (2012) 040211. [5] J.M. Kluger, T.P. Sapsis, A.H. Slocum, Robust energy harvesting from walking vibrations by means of nonlinear cantilever beams, Journal of Sound and Vibration 341 (2015) 174-194. [6] Z. Yan, A. Abdelkefi, Nonlinear characterization of concurrent energy harvesting from galloping and base excitations, Nonlinear Dynamics 77.4 (2014): 1171-1189. [7] B. Paxson, A.M. Wickenheiser. Design considerations for small-scale wind energy harvesters driven by broadband vortex-induced vibrations. SPIE Smart Structures and Materials+ Nondestructive Evaluation and Health Monitoring, International Society for Optics and Photonics, 2014. [8] M. Shi, Energy harvesting from wind-induced vibration of suspension bridges. Diss. Massachusetts Institute of Technology, 2013. [9] G.W. Taylor, J.R. Burns, S.A. Kamman, W.B. Powers, T.R. Welsh, The energy harvesting eel: a small subsurface ocean/river power generator. Oceanic Engineering, IEEE Journal of 26.4 (2001) 539-547. [10] A. Abdelkefi, Z. Yan, M.R. Hajj, Performance analysis of galloping-based piezoaeroelastic energy harvesters with different cross-section geometries. Journal of Intelligent Material Systems and Structures (2013) 1045389X 13491019. [11] M. Friswell, S.F. Ali, S. Adhikari, G. Litak, Non-linear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass, Journal of Intelligent Material Systems and Structures 23.13 (2012) 1505-1521. [12] R. Masana, M.F, Daqaq. Electromechanical modeling and nonlinear analysis of axially loaded energy harvesters, Journal of Vibration and Acoustics 133.1 (2011) 011007. .