17th AIAA Non-Deterministic Approaches Conference on January 5-9, 2015 at Kissimmee , FL, USA Free vibration analysis of angle-ply composite plates with uncertain properties S. Dey, T. Mukhopadhyay, S. Adhikari Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom 1
Outline • Introduction • Laminated Composites - Plate Model / Shell Model • Governing Equations • Uncertainty Propagation using surrogate models Ø Random Sampling - High Dimensional Model Representation (RS-HDMR) model Ø Polynomial Regression Model using D-Optimal design Ø Kriging Model • Results and Discussion • Conclusions 2
Introduction q Suppose f(x) is a computationaly intensive multidimensional nonlinear (smooth) function of a vector of parameters x. q We are interested in the statistical properties of y=f(x) , given the statistical properties of x. q The statistical properties include, mean, standard deviation, probability density functions and bounds. q This work considers computational methods for dynamics of composite structures with uncertain parameters. q General categories of uncertainties are: Ø Aleatoric: Due to variability in the system parameters Ø Epistemic: Due to lack of knowledge of the system Ø Prejudicial: Due to absence of variability characterization q To start with Bottom up approach to quantify the uncertainty from its sources of origin for the composite laminate. 3
Introduction q Prime sources of uncertainties are: Ø Material and Geometric uncertainties Ø Manufacturing uncertainties Ø Environmental uncertainties q Monte Carlo Simulation is an accurate but expensive computational method for Uncertainty Quantification. q If the model is computationally expensive, this cost has a cascading effect on the increasing cost of computation. q To save sampling time and hence the computational cost, the following three surrogate modelling methods are employed: Ø Random Sampling-High Dimensional Model Representation Ø Polynomial Regression Model using D-Optimal Ø Kriging Model q These metamodels can be employed to general stochastic problems. 4
Factors affecting uncertainty in composites 5
Laminated Composites – Plate / Shell Model Figure : Laminated composite cantilever plate. 6
Governing Equations Ø If mid-plane forms x-y plane of the reference plane, the displacements can be computed as Ø The strain-displacement relationships for small deformations can be expressed as where Ø The strains in the k -th lamina: Ø In-plane stress resultant {N}, the moment resultant {M}, and the transverse shear resultants {Q} can be expressed 7
Bottom Up Approach All cases consider an eight noded isoparametric quadratic element with five degrees of freedom for graphite-epoxy composite plate / shells Material properties (Graphite-Epoxy)**: E 1 =138.0 GPa, E 2 =8.96GPa, G 12 =7.1GPa, G 13 =7.1 GPa, G 23 =2.84 GPa, ν =0.3 0 N [ ]{ A } [ ]{ } B k { } 4 4 2 2 2 2 m n 2 m n 4 m n = ε + ⎡ ⎤ ⎢ ⎥ 4 4 2 2 2 2 n m 2 m n 4 m n 0 { } M [ ]{ } B [ ]{ } D k ⎢ ⎥ = ε + ⎢ ⎥ 2 2 2 2 4 4 2 2 m n m n ( m n ) 4 m n + − [ ] [ Q ( )] ⎢ ⎥ Q ω = ij ij 2 2 2 2 2 2 2 2 m n m n 2 m n ( m n ) ⎢ ⎥ − − n = Cos ( ω ) m = Sin ( ω ) θ θ ⎢ ⎥ 3 3 3 3 3 3 m n mn ( mn m n ) 2 ( mn m n ) − − ⎢ ⎥ ( ω ) θ = Random ply orientation angle ⎢ ⎥ 3 3 3 3 3 3 mn m n ( m n mn ) 2 ( m n mn ) − − ⎣ ⎦ A ( ) B ( ) 0 ⎡ ⎤ ω ω ij ij ⎢ ⎥ [ D ' ( )] B ( ) D ( ) 0 ω = ω ω ⎢ ⎥ ij ij ⎢ ⎥ 0 0 S ( ) ω ⎣ ⎦ ij z n k 2 [ A ( ), B ( ), D ( )] [ Q ( )] [ 1 , z , z ] dz i , j 1 , 2 , 6 ∑ ∫ ω ω ω = ω = ij ij ij ij k k 1 = z k 1 − z n k [ S ( )] [ Q ( )] dz i , j 4 , 5 ∑ ∫ ω = α ω = ij s ij k 8 k 1 = z k 1 −
Eigenvalue Problem t f H [ T U W ] dt 0 Ø From Hamilton’s principle: δ = ∫ δ − δ − δ = t i Ø Potential strain energy: Ø Kinetic energy: Ø Mass matrix: z n k [ M ( )] [ N ] [ P ( )] [ N ] d ( vol ) ω = ∫ ω where P ( ) ( ) dz ∑ ∫ ω = ρ ω Vol k 1 = z k 1 − 1 1 T [ K ( )] [ B ( )] [ D ( )] [ B ( )] d d Ø Stiffness matrix: ω = ω ω ω ξ η ∫ ∫ 1 1 − − � � � [ M ( )] ( t ) [ C ] ( t ) [ K ( )] ( t ) f ( t ) Ø Dynamic Equation: ω δ + δ + ω δ = For free vibration, the random natural frequencies are determined from the standard eigenvalue problem, solved by the QR iteration algorithm 9
Modal Analysis Ø The eigenvalues and eigenvectors satisfy the orthogonality relationship Ø Using modal transformation, pre-multiplying by X T and using orthogonality relationships, equation of motion of a damped system in the modal ~ coordinates is obtained as T 2 � � � y ( t ) X C X y ( t ) y ( t ) f ( t ) + + Ω = Ø The damping matrix in the modal coordinate: Ø The generalized proportional damping model expresses the damping matrix as a linear combination of the mass and stiffness matrices Ø Transfer function matrix T X X n j j 2 2 1 T H ( i ) ( ) X [ I 2 i ] X − ∑ ω ω = − ω + ω ζ Ω + Ω = 2 2 2 i − ω + ω ζ ω + ω j 1 j j j = Ø The dynamic response in the frequency domain with zero initial conditions: 10
Random Sampling – High Dimensional Model Representation (RS-HDMR) 11
Random Sampling – High Dimensional Model Representation (RS-HDMR) Model Ø Use of orthonormal polynomial for the computation of RS-HDMR component k functions: i f x ( ) ( ) x ∑ α ≈ ϕ i i r r i r 1 = l l ' ij f ( , x x ) ( ) x ( x ) ∑∑ ≈ β ϕ ϕ ij i j pq p i q j p 1 q 1 = = Ø Check for Coefficient of determination ( R 2 ) and Relative Error ( RE): where, 12
Uncertainty Quantification - Flow chart Random Sampling – High Dimensional Monte Carlo Simulation Model Representation (RS-HDMR) Model 13
Global Sensitivity Analysis based on RS-HDMR Ø The orthogonal relationship between the component functions of Random Sampling – High Dimensional Model Representation (RS-HDMR) expression implies that the component functions are independent and contribute their effects independently to the overall output response. Ø Sensitivity Index partial varianceof theinput parameter Sensitivity index an input parameter ( S ) = i total variance n n Such that, S S .. . S 1 ∑ ∑ + + + = i ij 1,2,... n i 1 1 i j n = ≤ < ≤ Input uncertainty model • Ply angles are varied +/- 5 degrees in each layer of the laminate • All material properties are varied +/- 10% 14
Validation – Random Sampling – High Dimensional Model Representation (RS-HDMR) Model Figure : Probability distribution function (PDF) with respect to model response of first three natural frequencies for variation of ply-orientation angle of graphite-epoxy angle-ply (45°/-45°/45°) composite cantilever plate, considering E 1 =138 GPa, E 2 =8.9 GPa, G 12 = G 13 =7.1 GPa, G 23 =2.84 GPa, ρ =3202 Kg/m 3 , t =0.004 m, ν =0.3 Table: Convergence study for coefficient of determination R 2 (second Figure: Scatter plot for fundamental frequencies for order) of the RS-HDMR expansions with different sample sizes for variation of ply-orientation angle of angle-ply variation of only ply-orientation angle of graphite-epoxy angle-ply (45°/-45°/45°) composite cantilever plate (45°/-45°/45°) composite cantilever plate, considering E 1 =138 GPa, E 2 =8.9 GPa, G 12 = G 13 =7.1 GPa, G 23 =2.84 GPa, t =0.004 m, ν =0.3 Sample Size ¡ Frequency ¡ 32 64 128 256 512 1st ¡ 65.68 ¡ 93.48 ¡ 99.60 ¡ 99.95 ¡ 99.96 ¡ 2nd ¡ 69.67 ¡ 93.74 ¡ 99.47 ¡ 96.38 ¡ 97.81 ¡ 3rd ¡ 66.44 ¡ 97.85 ¡ 99.40 ¡ 98.86 ¡ 99.61 ¡ 15
Sensitivity Analysis Figure: Sensitivity index for combined variation (10,000 samples) of ply-orientation angle, elastic modulus and mass density for graphite-epoxy angle-ply (45°/-45°/45°) composite cantilever plate, considering E 1 =138 GPa, E 2 =8.9 GPa, G 12 = G 13 =7.1 GPa, G 23 =2.84 GPa, ρ =3202 Kg/m 3 , t =0.004 m, ν =0.3 16
Modes Shapes : Random Sampling – High Dimensional Model Representation (RS-HDMR) Model Figure: Mode shapes by RS-HDMR of first three modes due to combined stochasticity in ply-orientation angle, elastic modulus and mass density for three layered graphite-epoxy angle-ply (45°/-45°/45°) composite cantilever plate considering E 1 =138 GPa, E 2 =8.9 GPa, G 12 = G 13 =7.1 GPa, G 23 =2.84 GPa, ρ =3202 Kg/m 3 , t =0.004 m, ν =0.3 Mode 1 Mode 2 17 Mode 3
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