Efficient Reduced Order Modeling of Low- to Mid-Frequency Vibration and Power Flow in Complex Structures

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1 Efficient Reduced Order Modeling of Low- to Mid-Frequency Vibration and Power Flow in Complex Structures Yung-Chang Tan Graduate Student Research Assistant Matthew P. Castanier Assistant Research Scientist Christophe Pierre Professor Vibrations and Acoustics Laboratory Department of Mechanical Engineering and Applied Mechanics

2 Outline Vibration and Power Flow Analysis of Complex Structures Component Mode Synthesis (CMS) Efficient Models of Vibration and Power Flow via Characteristic Constraint Modes (CC Modes) Examples: Two-Span Beam, Plate Application: Military Vehicle (coarse mesh) Multi-Level Substructuring Method Application: Military Vehicle (fine mesh) New Method for Power Flow Statistics

3 Power Flow Scalar Measure of Vibration Paths of Vibration Transmission Ensemble Average of Mid-Frequency Vibration Vibration Levels in Upper Hull? Power Input Hull of Composite Armor Vehicle

Power Flow Analysis Methods Finite Element Analysis Low-frequency range Accurate Computationally intensive Statistical Energy Analysis FEA

4 Power Flow Analysis Methods Finite Element Analysis Low-frequency range Accurate Computationally intensive Statistical Energy Analysis FEA High-frequency range Approximate Computationally efficient SEA Mid-Frequency Analysis? Efficient Low-Frequency Analysis? Power 2 = c(e - E 2 )

5 Component Structure Approach Reduced FEA cost Power flow analysis Distributed modeling and design FEMs of Component Structures Finite Element Model (FEM)

6 Component Mode Synthesis (CMS) Example Structure: Two-Span Beam Component Structure with Fixed Interface Craig-Bampton Method (NASTRAN superelement) Normal Modes of Component Structure 2 Constraint Modes Unit displacement at at one interface DOF Constraint Mode

7 Some Pros and Cons of CMS Accurate Based on FEMs In the limit (all component modes), no approximation is made Efficient Modal analysis of smaller-size FEMs (components) Few DOF relative to FEM Component Structure Approach Analogous to SEA Convenient formulation for power flow Cost Associated with Constraint Modes Add one CMS-model DOF for each FEM-interface DOF Generally have limited physical meaning

8 CMS Matrices m C m CN M = CMS m NC m 2 NC m N m 2 CN 0 0 m 2 N k C 0 0 K = CMS 0 k N k N 2 Normal (N) mode DOF = # of selected modes Constraint (C) mode DOF = # of FEM DOF in the interface Size may be dominated by constraint-mode DOF Idea: perform secondary modal analysis on the constraint-mode partitions of the matrices

9 Secondary Modal Analysis m C m CN M = CMS m NC m 2 NC m N m 2 CN 0 0 m 2 N k C 0 0 K = CMS 0 k N k N 2 K CMS v = λ M CMS v Full Matrices Eigenvectors give global modes Eigenvalues give global natural frequencies k C v = λ m C v Constraint-Mode Partitions Eigenvectors give characteristic constraint modes (CC Modes) Meaning of eigenvalues is still under investigation...

10 Illustration Y Beam-Element FEM: -DOF Interface X Beam Beam 2 Y Axis X Axis Single constraint mode (Beam shown) Solid-Element FEM: 60-DOF Interface. of 60 constraint modes (Beam shown) First characteristic constraint mode for solid-element FEM

11 Example: Cantilever Plate Shell elements Mesh size: 48 x 24 x CC Modes (Plate 2 Shown): Y X 2 in 6 in 8 in Plate Plate 2 CC Mode CC Mode 2 CC Mode 3 CC Mode 4 CC Mode 5 CC Mode 6

12 Natural Frequency Error 6 in 8 in Error relative to full (64-DOF) CMS model Y X 2 in Plate Plate 2 Natural Frequency Error (%) 0 0. Global Modes -5 st Nat. Freq. 2nd Nat. Freq. 3rd Nat. Freq. 4th Nat. Freq. 5th Nat. Freq Number of Characteristic Constraint Modes 2 DOF 30 DOF Natural Frequency Error (%) Natural Frequency Error (%) Global Modes th Nat. Freq. 7th Nat. Freq. 8th Nat. Freq. 9th Nat. Freq. 0th Nat. Freq Number of Characteristic Constraint Modes Global Modes -5 th Nat. Freq. 2th Nat. Freq. 3th Nat. Freq. 4th Nat. Freq. 5th Nat. Freq Number of Characteristic Constraint Modes

13 CC Modes: Characteristic Motion Natural Frequency Error (%) 0 0. st Nat. Freq. 2nd Nat. Freq. 3rd Nat. Freq. 4th Nat. Freq. 5th Nat. Freq Number of Characteristic Constraint Modes Mode 3 CC Mode 2 Mode 2

14 Derivation of Power Flow Equations Traction forces t = t 2 = λ on interface Γ Applied forces f Ω 2 t Ω t 2 2 f Component FEM matrices Γ Component reduced CMS matrices M i = m iγγ 0 0 m iωω Transform M i m icc m icn = m icnt m in K i = k iγγ k iγω k iγω k iωω for i = and 2 by CC modes & normal modes K i k icc 0 = 0 k in

15 Derivation of Power Flow (cont d) Substructure jω mcc m CN m CNT m N v CC v N + jη k CC 0 v CC λ = jω 0 k N v N 0 + f CC f N Substructure 2 jω m2cc m 2CN m 2CNT m 2N v CC v 2N + jη k 2CC 0 v CC λ = jω 0 k 2N v 2N 0 + f 2CC f 2N Coupled system jω m CC m CN m 2CN m CNT m N 0 m 2CNT 0 m 2N v CC v N v 2N + jη jω k CC k N k 2N v CC v N v 2N = f CC f N f 2N

16 Derivation of Power Flow (cont d) Solve the coupled equations of motion for v CC and v N Calculate constraint forces for CC modes λ = z CC v CC + jωm CN v N f CC where z CC = η ---k CC j ωm CC + ---k CC ω ω Power flow (spectral density) is given by the product of the constraint forces and the corresponding velocities Πω ( ) = π lim T E [ Re{ λ ( ω; T )v CC ( ω; T )}] 2T

17 Derivation of Power Flow (cont d) Power flow (out of substructure i) where Π i ( ω) = tr{ S i Re[ Y ]} 2π power input tr{ SY C i Y } 2π power dissipation All matrices are square matrices corresponding to CC-mode DOF has collected terms of power spectral density (PSD) of the applied S i N ss forces of the subsystem i projected onto the interface and S = Si Y represents projection of the mobility of the system on the interface C i represents projection of the damping of the subsystem i on the interface i =

18 Ensemble Average Approximation Consider that only substructure is disordered, while the excitation is only applied to substructure 2 E[ Πωε ( ; )] = Π( ω; ε)p( ε) dε = Approximate the matrix multiplication by including only the diagonal terms of the matrices The equation reduces to E[ Πωε ( ; )] = The integral with respect to ε may be carried out by calculating its residues 2π Re[ tr( SỸ C Ỹ )]p( ε) dε M C rr S ( ε) 2π rr D rr ( ε) 2 r = α π dε 2 + ε 2 α

19 Example: Two-Span Beam Ordered 3% Monte Carlo 3% Approx. 0% Monte Carlo 0% Approx. 0 3 Power Y X w (rad/s) Beam Beam 2

20 Application: Composite Armor Vehicle Investigate power flow from lower substructure to upper substructure Wide-band, random excitation at each road arm attachment

21 Full vs. Reduced CMS Model Size Stiffness Matrix for the full CMS model (484 DOF) Stiffness Matrix for a reduced CMS model (96 DOF) CC 2 Constraint Mode Partition Refining the FEM mesh: Increases # of constraint modes and size of CMS model Does not change # of CC modes nor size of reduced CMS model 2

22 CC Mode 2 and Natural Frequencies Global Mode Frequency (Hz) DOF (Full CMS Model) 76 DOF (30 CC modes) 66 DOF (20 CC modes) 56 DOF (0 CC modes) Global Mode Number Relative Frequency Error (%) DOF (30 CC modes) 66 DOF (20 CC modes) 56 DOF (0 CC modes) Global Mode Number

23 Power Flow Transmitted Power DOF (0 CC modes) 76 DOF (30 CC modes) 96 DOF (50 CC modes) 484 DOF (Full CMS) Frequency (Hz)

24 Military Vehicle Body Structure (Fine Mesh) Finite Element Model max element size = 2.0 in 20,000 plate elements 5,000 DOF Modal Analysis # modes under 00 Hz = 27 # modes under 000 Hz = 043 Design Cases Baseline Variation Track Stiffness = 0.5*Baseline Torsion Bar Stiffness = 0.5*Baseline Variation 2 Variation and... Torsion Bar Damping = 2*Baseline

25 Split structure into component structures Multi-Level Substructuring Split components into sub-components Perform FEA on sub-components Calculate global modes, vibration, and power flow Use sub-component modes to find component modes and CC modes

26 Natural Frequency Results (Baseline) 500 Comparison of Lowest 385 Natural Frequencies frequency (Hz) cc mode model (56 DOF) 94 cc mode model (649 DOF) FEM (5,344 DOF) mode # 3 2 % error 0 6 cc mode model 94 cc mode model mode #

27 Forced Response Results (Baseline) 0 Comparison of displacement at node 994 FEM (5,344 DOF) 6 cc mode model (56 DOF) 0 2 Node 994 Displacement (in/lb) Comparison of displacement at node FEM (5,344 DOF) 6 cc mode model (56 DOF) 0 5 Displacement (in/lb) frequency (Hz) frequency (Hz) Displacement at Node 994: FEM (5,344 DOF) vs. ROM (56 DOF)

28 Power Flow Results Power flow from lower body to upper body (Baseline) 0 Baseline 0 0 Power Frequency (Hz) Power flow from lower body to upper body (Variation ) Power flow from lower body to upper body (Variation 2) 0 Variation 0 Variation Power 0 Power Frequency (Hz) Frequency (Hz)

29 New Method for Power Flow Statistics Equations of motion with random parameters jωm sys K + jη sys + jη jω 0 jω sys K s θ s v sys = f sys N θ s = θ s, s=,2,, N θ The modal response quantities can be approximated by the truncated series expansion N CC CC v r ( ωθ, ) w rl l 2 l N ( ω) θ = P θ ls ( θ s ) r = 2,,, N cc v irn P ls ( ) : orthogonal polynomials θ s 0 L N p N ( ωθ, ) w irl l 2 l N θ N θ s = ( ω) P ls ( θ s ) r 2,,, N i ; i 2,,, N ss = = = 0 L N p s =

30 Power Flow Statistics (cont d) Choice of polynomial basis P ls ( θ s ) depends on pdf of the random variables Uniform distribution: Legendre polynomials Normal distribution: Hermite polynomials Example: Legendre polynomials orthogonality condition recurrence condition W ( θ )P ( θ )P ( θ )dθ s = δ i s j s s ij where W ( θ s ) = for θ θ s θ s P n ( θ s ) = C n P ( θ n s ) + C n + P n + ( θ s ) Definition of ensemble-averaged power flow out of ith substructure [ ( )] = Π i ( ω; θ) f ( θ s ) dθ dθ 2 dθ N θ s = E Π θ i ω; θ θ θ 2 θ N θ N θ

31 Power Flow Statistics (cont d) Solving equations of motion Apply orthogonality and recurrence relations b N θ + jη jωm ab k + jη jω 0ab wbl l 2 l k N θ jω sab C ls w bl l 2 l s l C N θ ls + w + ( + bl l 2 l s + l ) N θ s = N θ N θ =22 f a δ ls 0 s = Rearrange the coefficients CC into an N cc N d dimensional vector w rl l 2 l N θ w CC and the coefficients N into an N i N d dimensional vector w irl l 2 l N θ w i N Same form as equations of motion of ordered system, but larger size Ensemble-averaged power flow out of ith substructure E Π θ i ω; θ [ ( )] = Re tr ŜiŶ ŜY ˆ C ˆ iŷ 2π

32 Example: Cantilever Plate Ensemble-averaged power flow for two random parameters (plate stiffnesses) 2 in Plate Plate 2 Y X 6 in 8 in Power (N*m/s) Nominal Monte Carlo 3rd order approx Frequency (Hz) Power (N*m/s) Nominal Monte Carlo 5th order approx Frequency (Hz)

33 Conclusions Component Mode Synthesis (CMS) provides an excellent framework for power flow analysis New technique improves efficiency and analysis capability Secondary Modal Analysis Characteristic Constraint (CC) Modes CC Modes capture physical mechanisms of power flow CMS with CC Modes allows efficient vibration modeling and power flow analysis Multi-level substructuring allows analysis of large structural models

34 Ongoing and Future Work Determine proper selection of CC modes Minimize DOF Achieve desired accuracy in target frequency range Track power flow through individual CC modes Investigate mid-frequency issue Approximations of power flow statistics Assessment of accuracy Possible implementation of quasi-static mode compensation Investigate use of multi-level substructuring Develop general code

Vibration Transmission in Complex Vehicle Structures

Vibration Transmission in Complex Vehicle Structures Christophe Pierre Professor Matthew P. Castanier Assistant Research Scientist Yung-Chang Tan Dongying Jiang Graduate Student Research Assistants Vibrations