Uncertainty propagation in structural dynamics
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The document discusses uncertainty propagation in structural dynamics, focusing on methods for uncertainty quantification and management in aircraft design. It outlines computational techniques for analyzing the effects of parametric and nonparametric uncertainties on structural performance, using examples like a cantilever beam and a vibrating plate. The conclusion emphasizes the importance of a unified mathematical representation for propagating uncertainty and its implications for model calibration and experimental validation.
![Projection in the Modal Basis
Adhikari, S., "A reduced spectral function approach for the stochastic finite element analysis", Computer Methods in Applied Mechanics and
Engineering, 200[21-22] (2011), pp. 1804-1821.](https://image.slidesharecdn.com/dynamicuncertainty-130116084430-phpapp01/85/Uncertainty-propagation-in-structural-dynamics-15-320.jpg)
![Spectral functions
2 2
10 (4) 10 (4)
E[ ( , ( )] E[ 1
( , ( )]
1
(4) (4)
E[ ( , ( )] E[ 2
( , ( )]
3 2 3
10 (4) 10 (4)
E[ ( , ( )] E[ 3
( , ( )]
3
Mean spectral function
Mean spectral function
(4) (4)
E[ ( , ( )] E[ 4
( , ( )]
4 4 4
10 (4) 10 (4)
E[ ( , ( )] E[ 5
( , ( )]
5
(4) (4)
E[ 6
( , ( )] E[ 6
( , ( )]
5 5
10 E[ (4)
( , ( )] 10 E[
(4)
( , ( )]
7 7
6 6
10 10
7 7
10 10
0 100 200 300 400 500 600 0 100 200 300 400 500 600
Frequency (Hz) Frequency (Hz)
σa = 0.05 σa = 0.2
Mean of the spectral functions (4th order)](https://image.slidesharecdn.com/dynamicuncertainty-130116084430-phpapp01/85/Uncertainty-propagation-in-structural-dynamics-26-320.jpg)
![Wishart random matrix model
Distribution of the systems matrices should be such that
they are
• Symmetric, and
• Positive definite
Using these as constraints, it can be shown that the mass,
stiffness and damping matrices can be represented by
Wishart random matrices such that
[1] Adhikari, S., Pastur, L., Lytova, A. and Du Bois, J. L., "Eigenvalue-density of linear stochastic dynamical systems: A random matrix approach",
Journal of Sound and Vibration, 331[5] (2012), pp. 1042-1058.
[2] Adhikari, S. and Chowdhury, R., "A reduced-order random matrix approach for stochastic structural dynamics", Computers and Structures,
88[21-22] (2010), pp. 1230-1238.
[3] Adhikari, S., "Generalized Wishart distribution for probabilistic structural dynamics", Computational Mechanics, 45[5] (2010), pp. 495-511.
[4 Adhikari, S., and Sarkar, A., "Uncertainty in structural dynamics: experimental validation of a Wishart random matrix model", Journal of Sound
and Vibration, 323[3-5] (2009), pp. 802-825.
[5] Adhikari, S., "Matrix variate distributions for probabilistic structural mechanics", AIAA Journal, 45[7] (2007), pp. 1748-1762.
[6] Adhikari, S., "Wishart random matrices in probabilistic structural mechanics", ASCE Journal of Engineering Mechanics, 134[12] (2008), pp.
1029-1044.](https://image.slidesharecdn.com/dynamicuncertainty-130116084430-phpapp01/85/Uncertainty-propagation-in-structural-dynamics-37-320.jpg)
Uncertainty propagation in structural dynamics
- 1. Uncertainty propagation in structural dynamics Professor Sondipon Adhikari College of Engineering, Swansea University Uncertainty Quantification and Management in Aircraft Design 8th-9th November 2012 Advanced Simulation Research Centre, Bristol
- 2. Outline of the Talk • Background & Motivation • Uncertainty Quantification • Uncertainty propagation in complex dynamical systems – Parametric uncertainty propagation – Nonparametric uncertainty propagation – Unified representation • Computational method and validation – Representative experimental results – Software integration • Conclusions
- 3. Actual Performance of Engineering Designs On-target Off-target High variability Low variability Off-target On-target High variability Low variability
- 4. Overview of Computational Modeling Challenge 3: Model calibration under Challenge 1: uncertainty Uncertainty Modeling Challenge 2: Fast Uncertainty Propagation Methods
- 5. Why Uncertainty: The Sources Experimental error Parametric Uncertainty uncertain and unknown uncertainty in the geometric error percolate into the parameters, boundary model when they are conditions, forces, strength calibrated against of the materials involved experimental results Computational uncertainty Model Uncertainty machine precession, error arising from the lack of tolerance and the so called scientific knowledge about ‘h’ and ‘p’ refinements in the model which is a-priori finite element analysis unknown (damping, nonlinearity, joints) A low-fidelity answer with known uncertainty bounds is more valuable than a high-fidelity answer with unknown uncertainty bounds (NASA White Paper, 2002).
- 6. Uncertainty Modeling • Random variables Parametric • Random fields Uncertainty • Probabilistic Approach Ø Random matrix theory Non-parametric Uncertainty • Possibilistic Approaches Ø Fuzzy variable Ø Interval algebra Ø Convex modeling
- 7. Equation of Motion of Dynamical Systems
- 8. Uncertainty modeling in structural dynamics Uncertainty modeling Nonparametric uncertainty: Parametric uncertainty: mean matrices + a single mean matrices + random dispersion parameter for each field/variable information matrices Random variables Random matrix model
- 9. Parametric uncertainty propagation Title of presentation Click to edit subtitle style
- 13. What should be the form of the response?
- 15. Projection in the Modal Basis Adhikari, S., "A reduced spectral function approach for the stochastic finite element analysis", Computer Methods in Applied Mechanics and Engineering, 200[21-22] (2011), pp. 1804-1821.
- 16. Outline of the derivation
- 17. Projection in the Modal Basis
- 18. Projection in the Modal Basis
- 19. Projection in the Modal Basis
- 20. Projection in the Modal Basis
- 23. Model reduction by a reduced basis
- 24. Summary of the spectral functions • Not polynomials in random variables, but ratio of polynomials • Independent of the nature of the random variables (i.e. applicable to Gaussian, non-Gaussian or even mixed random variables • Not general, but specific to a problem as it utilizes the eigenvalues and eigenvectors of the system matrices. • The truncation error depends on the off-diagonal terms of the random part of the modal system matrix • Show ‘peaks’ when the frequency is close to the system natural frequencies
- 25. Numerical illustration • An Euler-Bernoulli cantilever beam with stochastic bending modulus (nominal properties L=1m, A=39 x 5.93mm2, E=2 x 1011 Pa) • We use n=200, M=2 • We study the deflection of the beam under the action of a point load on the free end. • The bending modulus of the cantilever beam is taken to be a homogeneous stationary Gaussian random field with exponential autocorrelation function (correlation length L/2) • Constant modal damping is taken with 1% damping factor for all modes. • The standard deviation of the random field σa is varied up to 0.2 times the mean.
- 26. Spectral functions 2 2 10 (4) 10 (4) E[ ( , ( )] E[ 1 ( , ( )] 1 (4) (4) E[ ( , ( )] E[ 2 ( , ( )] 3 2 3 10 (4) 10 (4) E[ ( , ( )] E[ 3 ( , ( )] 3 Mean spectral function Mean spectral function (4) (4) E[ ( , ( )] E[ 4 ( , ( )] 4 4 4 10 (4) 10 (4) E[ ( , ( )] E[ 5 ( , ( )] 5 (4) (4) E[ 6 ( , ( )] E[ 6 ( , ( )] 5 5 10 E[ (4) ( , ( )] 10 E[ (4) ( , ( )] 7 7 6 6 10 10 7 7 10 10 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Frequency (Hz) Frequency (Hz) σa = 0.05 σa = 0.2 Mean of the spectral functions (4th order)
- 30. Summary of the Proposed Method
- 31. Frequency domain response of the beam 2 1 10 10 MCS MCS 2nd order Galerkin 2nd order Galerkin 3rd order Galerkin 2 3rd order Galerkin 3 4th order Galerkin 10 4th order Galerkin 10 deterministic deterministic 4th order PC 4th order PC : 0.1 : 0.2 3 10 f f 4 10 damped deflection, damped deflection, 4 10 5 10 5 10 6 10 6 10 7 7 10 10 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Frequency (Hz) Frequency (Hz) σa = 0.1 σa = 0.2 Mean of the dynamic response (m)
- 32. Frequency domain response of the beam 2 1 10 10 MCS MCS 2nd order Galerkin 2nd order Galerkin 3rd order Galerkin 3rd order Galerkin 3 4th order Galerkin 2 4th order Galerkin 10 10 : 0.1 : 0.2 4th order PC 4th order PC f f Standard Deviation (damped), Standard Deviation (damped), 4 10 10 3 5 10 4 10 6 10 5 10 7 10 6 0 100 200 300 400 500 600 10 Frequency (Hz) 0 100 200 300 400 500 600 Frequency (Hz) σa = 0.2 σa = 0.1 Standard deviation of the dynamic response (m)
- 33. PDF of the Response Amplitude 5 5 x 10 x 10 2 2 direct MCS direct MCS 1st order spectral 1st order spectral 2nd order spectral 2nd order spectral 3rd order spectral 3rd order spectral Probability density function Probability density function 1.5 1.5 4th order spectral 4th order spectral 4th order PC 4th order PC 1 1 0.5 0.5 0 0 0.5 1 1.5 2 0 Deflection (m) 5 0 0.5 1 1.5 2 x 10 Deflection (m) 5 x 10 σa = 0.1 σa = 0.2 Standard deviation of the dynamic response (m)
- 34. Plate with Stochastic Properties • An Euler-Bernoulli cantilever beam with stochastic bending modulus (nominal properties 1m x 0.6m, t=03mm, E=2 x 1011 Pa) • We use n=1881, M=16 • We study the deflection of the beam under the action of a point load on the free end. • The bending modulus is taken to be a homogeneous stationary Gaussian random field with exponential autocorrelation function (correlation lengths L/5) • Constant modal damping is taken with 1% damping factor for all modes.
- 35. Response Statistics 3 10 3 10 deterministic direct MCS direct MCS 2nd order spectral Standard deviation of deflection (m) 4 2nd order spectral 4 3rd order spectral 10 10 4th order spectral 3rd order spectral Mean of deflection (m) 4th order spectral 5 5 10 10 6 6 10 10 7 7 10 10 8 10 10 8 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (Hz) Frequency (Hz) Mean with σa = 0.1 Standard deviation with σa = 0.1 Proposed approach: 150 x 150 equations 4th order Polynomial Chaos: 9113445 x 9113445 equations
- 36. Non-parametric uncertainty propagation Title of presentation Click to edit subtitle style
- 37. Wishart random matrix model Distribution of the systems matrices should be such that they are • Symmetric, and • Positive definite Using these as constraints, it can be shown that the mass, stiffness and damping matrices can be represented by Wishart random matrices such that [1] Adhikari, S., Pastur, L., Lytova, A. and Du Bois, J. L., "Eigenvalue-density of linear stochastic dynamical systems: A random matrix approach", Journal of Sound and Vibration, 331[5] (2012), pp. 1042-1058. [2] Adhikari, S. and Chowdhury, R., "A reduced-order random matrix approach for stochastic structural dynamics", Computers and Structures, 88[21-22] (2010), pp. 1230-1238. [3] Adhikari, S., "Generalized Wishart distribution for probabilistic structural dynamics", Computational Mechanics, 45[5] (2010), pp. 495-511. [4 Adhikari, S., and Sarkar, A., "Uncertainty in structural dynamics: experimental validation of a Wishart random matrix model", Journal of Sound and Vibration, 323[3-5] (2009), pp. 802-825. [5] Adhikari, S., "Matrix variate distributions for probabilistic structural mechanics", AIAA Journal, 45[7] (2007), pp. 1748-1762. [6] Adhikari, S., "Wishart random matrices in probabilistic structural mechanics", ASCE Journal of Engineering Mechanics, 134[12] (2008), pp. 1029-1044.
- 38. How to obtain the dispersion parameters? Suppose a random system matrix is expressed as It can be shown that the dispersion parameter is given by Therefore, it can be calculated using sensitivity matrices within a finite element formulation
- 39. Dynamic Response • Taking the Fourier transform of the equation of motion • Transforming into a reduced modal coordinate we have • Solving a random eigenvalue problem for the random matrix Ω2, the uncertainty propagation can be expressed • The matrix Ω2 is a Wishart matrix (called as a reduced diagonal Wishart matrix) who's parameters can be obtained explicitly from the dispersions parameters of the mass and stiffness matrices.
- 40. An example: A vibrating plate 1 Input 0.5 5 0 3 6 1 4 2 0.5 Fix ed 0.8 edg e Outputs 0.6 1 0.8 0.4 0.6 0.2 0.4 0.2 Y direction (width) 0 0 X direction (length) A thin cantilever plate with random properties and 0.7% fixed modal damping.
- 41. Physical properties The data presented here are available from: http://engweb.swan.ac.uk/~adhikaris/uq
- 42. Uncertainty type 1 The Young's modulus, Poissons ratio, mass density and thickness are random fields of the form
- 43. Uncertainty type 2 • Here we consider that the baseline plate is `perturbed' by attaching 10 oscillators with random spring stiffnesses at random locations • This is aimed at modeling non-parametric uncertainty only. • This case will be investigated experimentally also.
- 44. Mean of a cross-FRF: Utype 1 80 M and K are fully correlated Wishart 90 Generalized Wishart Reduced diagonal Wishart Direct simulation 100 Mean of amplitude (dB) 110 120 130 140 150 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)
- 45. Mean of the driving-point-FRF: Utype 1 80 90 M and K are fully correlated Wishart Generalized Wishart Reduced diagonal Wishart 100 Direct simulation Mean of amplitude (dB) 110 120 130 140 150 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)
- 46. Standard deviation of a cross-FRF: Utype 1 80 90 M and K are fully correlated Wishart Generalized Wishart Reduced diagonal Wishart 100 Direct simulation Standard deviation (dB) 110 120 130 140 150 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)
- 47. Standard deviation of the driving-point-FRF: Utype 1 80 90 M and K are fully correlated Wishart Generalized Wishart Reduced diagonal Wishart 100 Direct simulation Standard deviation (dB) 110 120 130 140 150 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)
- 48. Computational method and validation Title of presentation experimental results • Representative • Plate with randomly placed oscillator Click to edit subtitle style • Software integration • Integration with ANSYS
- 49. Plate with randomly placed oscillators 10 oscillators with random stiffness values are attached at random locations in the plate by magnet
- 50. Mean of a cross-FRF 60 50 40 30 Mean of amplitude (dB) 20 10 0 10 20 30 Reduced diagonal Wishart Experiment 40 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz)
- 51. Mean of the driving-point-FRF 60 50 40 30 Mean of amplitude (dB) 20 10 0 10 20 30 Reduced diagonal Wishart Experiment 40 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz)
- 52. Standard deviation of a cross-FRF 1 10 Reduced diagonal Wishart Experiment Relative standard deviation 0 10 1 10 2 10 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz)
- 53. Standard deviation of the driving-point-FRF 1 10 Reduced diagonal Wishart Experiment Relative standard deviation 0 10 1 10 2 10 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz)
- 54. Integration with ANSYS Input The Finite Element (FE) model of an aircraft wing (5907 degrees-of-freedom). The width is 1.5m, length is 20.0m and the height of the aerofoil section is 0.3m. The Output material properties are: Young's modulus 262Mpa, Poisson's ratio 0.3 and mass density 888.10kg/m3. Input node number: 407 and the output node number 96. A 2% modal damping factor is assumed for all modes.
- 55. Vibration modes
- 56. Mean of a cross-FRF 60 70 Deterministic =0.10, M=0.10 Amplitude (dB) of FRF at point 2 80 k =0.20, M=0.20 k =0.30, M=0.30 k 90 =0.40, M=0.40 k 100 110 120 130 140 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)
- 57. Standard deviation of a cross-FRF 60 =0.10, M=0.10 k =0.20, M=0.20 k 70 =0.30, M=0.30 k =0.40, M=0.40 k Amplitude (dB) of FRF at point 2 80 90 100 110 120 130 140 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)
- 58. Summary and Conclusions Title of presentation Click to edit subtitle style
- 59. Dynamic Response • For parametric uncertainty propagation: • For nonparametric uncertainty propagation • Unified mathematical representation • Can be useful for hybrid experimental-simulation approach for uncertainty quantification
- 60. Summary • Response of stochastic dynamical systems is projected in to the basis of undamped modes • The coefficient functions, called as the spectral functions, are expressed in terms of the spectral properties of the system matrices in the frequency domain. • The proposed method takes advantage of the fact that for a given maximum frequency only a small number of modes are necessary to represent the dynamic response. This modal reduction leads to a significantly smaller basis. • Wishart random matrix model can used to represent non- parametric uncertainty directly at the system matrix level. • Reduced computational approach can be implemented within the conventional finite element environment
- 61. Summary • Dispersion parameters necessary for the Wishart model can be obtained, for example, using sensitivity matrices • Both parametric and nonparametric uncertainty can be propagated via an unified mathematical framework. • Future work will exploit this novel representation for model validation and updating in conjunction with measured data.