Dynamics of structures with uncertainties: Applications to piezoelectric vibration energy harvesting S. Adhikari 1 1 Chair of Aerospace Engineering, College of Engineering, Swansea University, Bay Campus, Fabian Way, Swansea, SA1 8EN, UK Structural Mechanics and Coupled Systems Laboratory (CNAM), Paris Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 1
Swansea University Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 2
Swansea University Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 3
My research interests Development of fundamental computational methods for structural dynamics and uncertainty quantification A. Dynamics of complex systems B. Inverse problems for linear and non-linear dynamics C. Uncertainty quantification in computational mechanics Applications of computational mechanics to emerging multidisciplinary research areas D. Vibration energy harvesting / dynamics of wind turbines E. Computational nanomechanics Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 4
Stochastic dynamic systems - ensemble behaviour Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 5
Outline of the talk Introduction 1 Single degree of freedom damped stochastic systems 2 Equivalent damping factor Multiple degree of freedom damped stochastic systems 3 Spectral function approach 4 Projection in the modal space Properties of the spectral functions Error minimization The Galerkin approach Model Reduction Computational method Numerical illustrations 5 Piezoelectric vibration energy harvesting 6 The role of uncertainty Single Degree of Freedom Electromechanical Models 7 Linear systems Nonlinear systems Optimal Energy Harvester Under Gaussian Excitation 8 Circuit without an inductor Circuit with an inductor Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 6 Nonlinear Energy Harvesting Under Random Excitations 9
Introduction Few general questions How does system stochasticity impact the dynamic response? Does it matter? What is the underlying physics? How can we efficiently quantify uncertainty in the dynamic response for large dynamic systems? What about using ‘black box’ type response surface methods? Can we use modal analysis for stochastic systems? Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 7
Single degree of freedom damped stochastic systems Stochastic SDOF systems u(�t� )� k� f� (� t� )� m� f� d� (� t� )� �� Consider a normalised single degrees of freedom system (SDOF): u ( t ) + ω 2 u ( t ) + 2 ζω n ˙ ¨ n u ( t ) = f ( t ) / m (1) � √ Here ω n = k / m is the natural frequency and ξ = c / 2 km is the damping ratio. We are interested in understanding the motion when the natural frequency of the system is perturbed in a stochastic manner. Stochastic perturbation can represent statistical scatter of measured values or a lack of knowledge regarding the natural frequency. Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 8
Single degree of freedom damped stochastic systems Frequency variability 4 4 uniform uniform normal normal 3.5 3.5 log−normal log−normal 3 3 2.5 2.5 p x (x) p x (x) 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x x (a) Pdf: σ a = 0 . 1 (b) Pdf: σ a = 0 . 2 Figure: We assume that the mean of r is 1 and the standard deviation is σ a . Suppose the natural frequency is expressed as ω 2 n = ω 2 n 0 r , where ω n 0 is deterministic frequency and r is a random variable with a given probability distribution function. Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 9
Single degree of freedom damped stochastic systems Frequency samples 1000 1000 uniform uniform normal normal 900 900 log−normal log−normal 800 800 700 700 600 600 Samples Samples 500 500 400 400 300 300 200 200 100 100 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency: ω n Frequency: ω n (a) Frequencies: σ a = 0 . 1 (b) Frequencies: σ a = 0 . 2 Figure: 1000 sample realisations of the frequencies for the three distributions Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 10
Single degree of freedom damped stochastic systems Response in the time domain 1 1 deterministic deterministic random samples random samples 0.8 0.8 mean: uniform mean: uniform mean: normal mean: normal 0.6 0.6 mean: log−normal mean: log−normal Normalised amplitude: u/v 0 0.4 Normalised amplitude: u/v 0 0.4 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 0 5 10 15 0 5 10 15 Normalised time: t/T n0 Normalised time: t/T n0 (a) Response: σ a = 0 . 1 (b) Response: σ a = 0 . 2 Figure: Response due to initial velocity v 0 with 5% damping Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 11
Single degree of freedom damped stochastic systems Frequency response function 120 120 deterministic deterministic mean: uniform mean: uniform mean: normal mean: normal 100 100 mean: log−normal mean: log−normal Normalised amplitude: |u/u st | 2 Normalised amplitude: |u/u st | 2 80 80 60 60 40 40 20 20 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Normalised frequency: ω / ω n0 Normalised frequency: ω / ω n0 (a) Response: σ a = 0 . 1 (b) Response: σ a = 0 . 2 Figure: Normalised frequency response function | u / u st | 2 , where u st = f / k Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 12
Single degree of freedom damped stochastic systems Key observations The mean response response is more damped compared to deterministic response. The higher the randomness, the higher the “effective damping”. The qualitative features are almost independent of the distribution the random natural frequency. We often use averaging to obtain more reliable experimental results - is it always true? Assuming uniform random variable, we aim to explain some of these observations. Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 13
Single degree of freedom damped stochastic systems Equivalent damping factor Equivalent damping Assume that the random natural frequencies are ω 2 n = ω 2 n 0 ( 1 + ǫ x ) , where x has zero mean and unit standard deviation. The normalised harmonic response in the frequency domain u ( i ω ) k / m = √ (2) [ − ω 2 + ω 2 f / k n 0 ( 1 + ǫ x )] + 2 i ξωω n 0 1 + ǫ x � Considering ω n 0 = k / m and frequency ratio r = ω/ω n 0 we have u 1 f / k = √ (3) [( 1 + ǫ x ) − r 2 ] + 2 i ξ r 1 + ǫ x Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 14
Single degree of freedom damped stochastic systems Equivalent damping factor Equivalent damping The squared-amplitude of the normalised dynamic response at ω = ω n 0 (that is r = 1) can be obtained as � | u | � 2 1 ˆ U = = (4) ǫ 2 x 2 + 4 ξ 2 ( 1 + ǫ x ) f / k Since x is zero mean unit standard deviation uniform random variable, its √ √ √ pdf is given by p x ( x ) = 1 / 2 3 , − 3 ≤ x ≤ 3 The mean is therefore � � � 1 ˆ E U = ǫ 2 x 2 + 4 ξ 2 ( 1 + ǫ x ) p x ( x ) d x � � √ 1 3 ǫ ξ 1 − ξ 2 tan − 1 � � � = √ 1 − ξ 2 − 4 3 ǫξ 2 ξ 1 − ξ 2 � � √ 1 3 ǫ ξ 1 − ξ 2 tan − 1 � � � + √ 1 − ξ 2 + (5) 4 3 ǫξ 2 ξ 1 − ξ 2 Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 15
Single degree of freedom damped stochastic systems Equivalent damping factor Equivalent damping Note that � � 1 tan − 1 ( a + δ ) + tan − 1 ( a − δ ) = tan − 1 ( a ) + O ( δ 2 ) (6) 2 Provided there is a small δ , the mean response � � √ � � 3 ǫ 1 ˆ tan − 1 + O ( ζ 2 U ≈ � � n ) . (7) E √ 1 − ζ 2 1 − ζ 2 2 3 ǫζ n 2 ζ n n n Considering light damping (that is, ζ 2 ≪ 1), the validity of this approximation relies on the following inequality √ 3 ǫ 2 ≫ ζ 2 ζ 3 ǫ ≫ √ n . (8) or n 2 ζ n 3 Since damping is usually quite small ( ζ n < 0 . 2), the above inequality will normally hold even for systems with very small uncertainty. To give an example, for ζ n = 0 . 2, we get ǫ min = 0 . 0092, which is less than 0 . 1 % randomness. In practice we will be interested in randomness of more than 0 . 1 % and consequently the criteria in Eq. (8) is likely to be met. Adhikari (Swansea) Stochastic dynamics / vibration energy harvesting January 15, 2016 16
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