Vibration Transmission in Complex Vehicle Structures

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1 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 and Acoustics Laboratory Department of Mechanical Engineering and Applied Mechanics The University of Michigan Ann Arbor, MI

2 Outline Power Flow in Complex Structures Component Mode Synthesis (CMS) Characteristic Constraint Modes (CC Modes) Illustration: Two-Span Beam Reduced CMS Models using CC Modes Example: Cantilever Plate Power Flow Analysis using Reduced CMS Models Application: Composite Armor Vehicle Conclusions

3 Scalar Measure of Vibration Quantitative rating of vibration severity A-to-B comparison of competing designs Natural implementation in design optimization Power Flow Analysis Path of Vibration Transmission Critical sections or component structures Influence of design parameters Ensemble Average of Mid- to High-Frequency Vibration Statistical Energy Analysis Monte Carlo Simulation Vibration Levels in Upper Hull? Power Input Hull of Composite Armor Vehicle

4 Power Flow Analysis Methods Finite Element Analysis Low frequency Computationally intensive FEA Statistical Energy Analysis High frequency SEA Computationally efficient Mid-Frequency Analysis? Efficient Low-Frequency Analysis? Power 1 2 = c(e 1 - E 2 )

5 FEA of Component Structures

6 Component Mode Synthesis (CMS) Modes of component structures Example: Flexural Beam Vibration Synthesis at interface Component Structure Craig-Bampton Method (NASTRAN superelement) Normal modes: modes of component structure with fixed interface Normal Modes of Component Structure Constraint modes: motion due to unit deflection at one interface DOF while all other interface DOF held fixed Rotational DOF at interface Constraint Mode

7 Accurate Advantages of CMS Based on component structure Finite Element Models (FEMs) In the limit (all component modes taken), no approximation is made Efficient Modal analysis Few DOF relative to FEM Good modal convergence Component structure approach Analogous to SEA Convenient formulation for power flow calculations

8 CMS Matrices m 1C m C m CN m CN k 1C k C M = m 1 CN T m 1 N 0 K = 0 k 1 N 0 m 2 CN T 0 m 2 N 0 0 k 2 N Normal (N) mode DOF = # of selected modes Constraint (C) mode DOF =# of FEM DOF in the interface Matrix size may be dominated by constraint modes!

9 Characteristic Constraint Modes Perform eigen-analysis on the constraint mode partitions k 1C k 2 C v λ m m C + = 1C 2 + v Eigenvectors are Characteristic Constraint (CC) modes Capture characteristic interface motion Can be truncated Transform CMS matrices using selected set of CC modes Reduction of CMS DOF Correlation of power flow with interface motion K = V T k 1 C C + k 2 V k 1 N k 2 N

10 Illustration: Two-Span Beam Two-span beam on simple supports Each span is considered a component structure Y X Beam 1 Beam 2

11 Constraint Modes 1 Y Axis X Axis Single constraint mode for beam-element FEM (beam 1) Y X Beam 1 Beam 2 One of 60 constraint modes for solid-element FEM (beam 1)

12 Characteristic Motion 1 Y Axis X Axis Single constraint mode for beam-element FEM (beam 1) Y First characteristic constraint mode for solid-element FEM (beam 1) X Beam 1 Beam 2

13 CC Modes (Beam 1 Shown) Second CC Mode Y Fifth CC Mode X Beam 1 Beam 2

14 Example: Cantilever Plate Example model from Craig and Bampton (1968) 2024-T3 Aluminum thickness = in FEM constructed using ABAQUS shell elements (element S4R, 4 nodes/element, 6 DOF/node) mesh size: 48 x 24 x 1 Full CMS model 10 normal modes for each component structure 144 constraint modes 164 DOF Y X 12 in 16 in 8 in Plate 1 Plate 2

15 Natural Frequency Error Relative error compared to full (164-DOF) CMS model Natural Frequency Error (%) st Nat. Freq. 2nd Nat. Freq. 3rd Nat. Freq. 4th Nat. Freq. 5th Nat. Freq Number of Characteristic Constraint Modes 16 in 8 in 20 DOF 30 DOF Y X 12 in Plate 1 Plate 2

16 Natural Frequency Error (6-15) Natural Frequency Error (%) th Nat. Freq. 7th Nat. Freq. 8th Nat. Freq. 9th Nat. Freq. 10th Nat. Freq. Natural Frequency Error (%) th Nat. Freq. 12th Nat. Freq. 13th Nat. Freq. 14th Nat. Freq. 15th Nat. Freq Number of Characteristic Constraint Modes Number of Characteristic Constraint Modes 16 in 8 in Y X 12 in Plate 1 Plate 2

17 CC Modes (Plate 2 Shown) 16 in 8 in Y X 12 in Plate 1 Plate 2 CC Mode 1 CC Mode 2 CC Mode 3 CC Mode 4 CC Mode 5 CC Mode 6

18 Global Modes Mode 1 Mode 2 Mode 3 Mode 4

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

20 Composite Armor Vehicle (CAV) Prototype hull for a military vehicle Concept vehicle Composite structure Composite material properties determined from design optimization (ARC / Kikuchi) Homogenization method Multi-objective optimization: static and dynamic stiffnesses

21 CAV Substructures Investigate power flow from lower substructure to upper substructure Wide-band, random excitation at each road arm attachment

22 CC Modes CC Mode 2 CC Mode 17

23 Global Natural Frequencies 20 modes for each substructure + 6 rigid body modes Reduced CMS model = (46 + # of CC Modes) DOF Full CMS model = 484 DOF Global Mode Frequency (Hz) DOF (Full CMS Model) 76 DOF (30 CC modes) 66 DOF (20 CC modes) 56 DOF (10 CC modes) Global Mode Number Relative Frequency Error (%) DOF (30 CC modes) 66 DOF (20 CC modes) 56 DOF (10 CC modes) Global Mode Number

24 Frequency Error vs. Model Size Natural frequency error relative to full (484-DOF) CMS st Nat. Freq. 2nd Nat. Freq. 3rd Nat. Freq. 4th Nat. Freq. 5th Nat. Freq. 6th Nat. Freq. 7th Nat. Freq. 8th Nat. Freq. 9th Nat. Freq. 10th Nat. Freq. Relative Frequency Error (%) 10 0 Number of Frequency Error (%) Number of Characteristic Constraint Modes Number of Characteristic Modes 46 DOF 56 DOF 66 DOF 76 DOF 46 DOF 56 DOF 66 DOF 76 DOF

25 Comparison: Global Mode Shapes 76-DOF model 484-DOF model

26 Power Flow Transmitted Power DOF (10 CC modes) 76 DOF (30 CC modes) 96 DOF (50 CC modes) 484 DOF (Full CMS) Frequency (Hz)

27 Computational Cost 96 DOF 30 s 484 DOF 1200 s 1000 CPU Time (s) Number of Degrees of Freedom

28 Full vs. Reduced CMS Model Size 96 DOF CC 1 2 Finding and using CC modes is a secondary modal analysis determine characteristic motion select participating modes Constraint Mode Partition of the Stiffness Matrix for the Full CMS Model Note that refining the finite element mesh: increases # of constraint modes and thus the size of the full CMS model does not change the number of CC modes nor the size of the reduced CMS model 1 2

29 Conclusions Component Mode Synthesis (CMS) provides an excellent framework for power flow analysis New CMS technique improves efficiency and analysis capability of CMS Secondary modal analysis Characteristic Constraint (CC) Modes CC Modes capture characteristic interface motion CMS with CC Modes allows efficient vibration modeling and power flow analysis for complex structures

30 Ongoing and Future Work Determine proper selection of Characteristic Constraint modes Minimize DOF Achieve desired accuracy in target frequency range Track power flowing through individual CC modes Investigate mid-frequency issues Approximations of power flow statistics Assessment of accuracy Possible implementation of quasi-static mode compensation (ARC / Hulbert, Ma) Develop general code Apply technique to other vehicle structure models

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