Dynamic response of structures with uncertain properties S. Adhikari 1 1 Chair of Aerospace Engineering, College of Engineering, Swansea University, Bay Campus, Fabian Way, Swansea, SA1 8EN, UK International Probabilistic Workshop 2015, Liverpool, UK Adhikari (Swansea) Dynamic response of structures with uncertainies November 5, 2015 1
Swansea University New Bay Campus Adhikari (Swansea) Dynamic response of structures with uncertainies November 5, 2015 2
Stochastic dynamic systems Adhikari (Swansea) Dynamic response of structures with uncertainies November 5, 2015 3
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 5 The Galerkin approach Model Reduction Computational method Numerical illustrations 6 Conclusions 7 Adhikari (Swansea) Dynamic response of structures with uncertainies November 5, 2015 4
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) Dynamic response of structures with uncertainies November 5, 2015 5
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) Dynamic response of structures with uncertainies November 5, 2015 6
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) Dynamic response of structures with uncertainies November 5, 2015 7
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) Dynamic response of structures with uncertainies November 5, 2015 8
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) Dynamic response of structures with uncertainies November 5, 2015 9
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) Dynamic response of structures with uncertainies November 5, 2015 10
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) Dynamic response of structures with uncertainies November 5, 2015 11
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) Dynamic response of structures with uncertainies November 5, 2015 12
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) Dynamic response of structures with uncertainies November 5, 2015 13
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) Dynamic response of structures with uncertainies November 5, 2015 14
Single degree of freedom damped stochastic systems Equivalent damping factor Equivalent damping For small damping, the maximum determinestic amplitude at ω = ω n 0 is 1 / 4 ξ 2 e where ξ e is the equivalent damping for the mean response Therefore, the equivalent damping for the mean response is given by √ 2 3 ǫξ ( 2 ξ e ) 2 = √ (9) tan − 1 ( 3 ǫ/ 2 ξ ) For small damping, taking the limit we can obtain 1 ξ e ≈ 3 1 / 4 √ ǫ � √ π ξ (10) The equivalent damping factor of the mean system is proportional to the square root of the damping factor of the underlying baseline system 1 Adhikari, S. and Pascual, B., ”The ’damping effect’ in the dynamic response of stochastic oscillators”, Probabilistic Engineering Mechanics, in press. Adhikari (Swansea) Dynamic response of structures with uncertainies November 5, 2015 15
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