Piezoelectric Energy Harvesting Under Uncertainty S Adhikari 1 M I - PowerPoint PPT Presentation

Piezoelectric Energy Harvesting Under Uncertainty S Adhikari 1 M I Friswell 1 D J Inman 2 1 School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP , UK 2 CIMSS, Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA Bristol Energy Harvesting Workshop Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 1 / 26

Outline Introduction 1 Brief review of piezoelectric energy harvesting The role of uncertainty Brief Overview of Stationary Random Vibration 2 Single Degree of Freedom Electromechanical Model 3 Circuit without an inductor Circuit with an inductor Summary & Future Directions 4 Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 2 / 26

Introduction Brief review of piezoelectric energy harvesting Brief review of piezoelectric energy harvesting The harvesting of ambient vibration energy for use in powering low energy electronic devices has formed the focus of much recent research [1–6]. Of the published results that focus on the piezoelectric effect as the transduction method, almost all have focused on harvesting using cantilever beams and on single frequency ambient energy, i.e., resonance based energy harvesting. Soliman et al. [7] considered energy harvesting under wide band excitation. Liu et al. [8] proposed acoustic energy harvesting using an electro-mechanical resonator. Shu et al. [9–11] conducted detailed analysis of the power output for piezoelectric energy harvesting systems. Several authors [12–15, 15] have proposed methods to optimize the parameters of the system to maximize the harvested energy. Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 3 / 26

Introduction The role of uncertainty Why uncertainty is important for energy harvesting? In the context of energy harvesting of ambient vibration, the input excitation may not be always known exactly. There may be uncertainties associated with the numerical values considered for various parameters of the harvester. This might arise, for example, due to the difference between the true values and the assumed values. If there are several nominally identical energy harvesters to be manufactured, there may be genuine parametric variability within the ensemble. Any deviations from the assumed excitation may result an optimally designed harvester to become sub-optimal. Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 4 / 26

Introduction The role of uncertainty Types of uncertainty Suppose the set of coupled equations for energy harvesting: L { u ( t ) } = f ( t ) (1) Uncertainty in the input excitations For this case in general f ( t ) is a random function of time. Such functions are called random processes. In this work we consider stationary Gaussian random processes, characterised by the standard deviation σ and two-point autocorrelation function R ( t 1 , t 2 ) . Uncertainty in the system The operator L {•} is in general a function of parameters θ 1 , θ 2 , · · · , θ n ∈ R . The uncertainty in the system can be characterised by the joint probability density function p Θ 1 , Θ 2 , ··· , Θ n ( θ 1 , θ 2 , · · · , θ n ) . Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 5 / 26

Brief Overview of Stationary Random Vibration Stationary random vibration Mechanical systems driven by this type of excitation have been discussed by Lin [16], Nigam [17], Bolotin [18], Roberts and Spanos [19] and Newland [20] within the scope of random vibration theory. When x b ( t ) is a weakly stationary random process, its autocorrelation function depends only on the difference in the time instants: E [ x b ( τ 1 ) x b ( τ 2 )] = R x b x b ( τ 1 − τ 2 ) . (2) This autocorrelation function can be expressed as the inverse Fourier transform of the spectral density Φ x b x b ( ω ) as � ∞ R x b x b ( τ 1 − τ 2 ) = Φ x b x b ( ω ) exp [ i ω ( τ 1 − τ 2 )] d ω. (3) −∞ Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 6 / 26

Brief Overview of Stationary Random Vibration Stationary random vibration We are interested in the average harvested power given by � � � v 2 ( t ) � v 2 ( t ) = E E [ P ( t )] = E . (4) R l R l For a damped linear system of the form V ( ω ) = H ( ω ) X b ( ω ) , it can be shown that [16, 17] the spectral density of V is related to the spectral density of X b by Φ VV ( ω ) = | H ( ω ) | 2 Φ x b x b ( ω ) . (5) Thus, for large t , we obtain � ∞ � � v 2 ( t ) = R vv ( 0 ) = | H ( ω ) | 2 Φ x b x b ( ω ) d ω. E (6) −∞ This expression will be used to obtain the average power for the two x b ( t ) is cases considered. We assume that the base acceleration ¨ Gaussian white noise so that its spectral density is constant with respect to frequency. Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 7 / 26

Brief Overview of Stationary Random Vibration Stationary random vibration The calculation of the preceding expressions requires the calculation of integrals of the following form � ∞ Ξ n ( ω ) d ω I n = (7) Λ n ( ω )Λ ∗ n ( ω ) −∞ where the polynomials have the form Ξ n ( ω ) = b n − 1 ω 2 n − 2 + b n − 2 ω 2 n − 4 + · · · + b 0 (8) Λ n ( ω ) = a n ( i ω ) n + a n − 1 ( i ω ) n − 1 + · · · + a 0 (9) Following Roberts and Spanos [19] this integral can be evaluated as det [ D n ] I n = π det [ N n ] . (10) a n Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 8 / 26

Brief Overview of Stationary Random Vibration Stationary random vibration In the preceding expression the m × m matrices are   b n − 1 b n − 2 b 0 ··· − a n a n − 2 − a n − 4 a n − 6 ··· 0 ···   − a n − 1 a n − 3 − a n − 5 ··· D n =   0 0 ··· (11)   a n − a n − 2 a n − 4 0 ··· 0 ··· 0 ··· ··· 0 ··· ··· − a 2 a 0 0 0 and   a n − 1 − a n − 3 a n − 5 − a n − 7 − a n a n − 2 − a n − 4 a n − 6 ··· 0 ···   − a n − 1 a n − 3 − a n − 5 ··· N n =  0 0 ···   . (12)  a n − a n − 2 a n − 4 0 ··· 0 ··· 0 ··· ··· 0 ··· ··· − a 2 a 0 0 0 Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 9 / 26

Single Degree of Freedom Electromechanical Model SDOF electromechanical models Proof�Mass Proof�Mass � x � x Piezo- Piezo- v v R l ceramic L R l ceramic + + Base Base x b x b Schematic diagrams of piezoelectric energy harvesters with two different harvesting circuits. (a) Harvesting circuit without an inductor, (b) Harvesting circuit with an inductor. Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 10 / 26

Single Degree of Freedom Electromechanical Model Circuit without an inductor Circuit without an inductor DuToit and Wardle [21] expressed the coupled electromechanical behavior by the linear ordinary differential equations m ¨ x ( t ) + c ˙ x ( t ) + kx ( t ) − θ v ( t ) = − m ¨ x b ( t ) (13) v ( t ) + 1 x ( t ) + C p ˙ v ( t ) = 0 θ ˙ (14) R l Transforming both the equations into the frequency domain and dividing the first equation by m and the second equation by C p we obtain � � X ( ω ) − θ − ω 2 + 2 i ωζω n + ω 2 mV ( ω ) = ω 2 X b ( ω ) (15) n � � i ω θ 1 X ( ω ) + V ( ω ) = 0 i ω + (16) C p C p R l Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 11 / 26

Single Degree of Freedom Electromechanical Model Circuit without an inductor Circuit without an inductor The natural frequency of the harvester, ω n , and the damping factor, ζ , are defined as � k c ω n = and ζ = . (17) m 2 m ω n Dividing the preceding equations by ω n and writing in matrix form one has �� � � X 1 − Ω 2 � � � Ω 2 X b � − θ + 2 i Ω ζ k = , (18) V i Ω αθ ( i Ω α + 1 ) 0 C p where the dimensionless frequency and dimensionless time constant are defined as Ω = ω α = ω n C p R l . and (19) ω n α is the time constant of the first order electrical system, non-dimensionalized using the natural frequency of the mechanical system. Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 12 / 26

Single Degree of Freedom Electromechanical Model Circuit without an inductor Circuit without an inductor Inverting the coefficient matrix, the displacement and voltage in the frequency domain can be obtained as � � � � � � � X � = 1 θ ( i Ω α + 1 )Ω 2 X b / ∆ 1 ( i Ω α + 1 ) Ω 2 X b k = , (20) V Cp X b / ∆ 1 − i Ω αθ Cp ( 1 − Ω 2 ) + 2 i Ω ζ − i Ω 3 αθ ∆ 1 0 where the determinant of the coefficient matrix is � � ∆ 1 ( i Ω) = ( i Ω) 3 α + ( 2 ζ α + 1 ) ( i Ω) 2 + α + κ 2 α + 2 ζ ( i Ω) + 1 (21) and the non-dimensional electromechanical coupling coefficient is κ 2 = θ 2 . (22) kC p Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 13 / 26