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Design Innovation Paper

Natural Frequency Optimization of Variable-Density Additive Manufactured Lattice Structure: Theory and Experimental Validation

[+] Author and Article Information
Lin Cheng, Xuan Liang, Xue Wang

Department of Mechanical Engineering
and Materials Science,
University of Pittsburgh,
Pittsburgh, PA 15261

Eric Belski

Aerotech, 101 Zeta Drive,
Pittsburgh, PA 15238

Jennifer M. Sietins

Materials Manufacturing Technology Branch,
Army Research Laboratory,
Aberdeen Proving Ground,
Aberdeen, MD 21005

Steve Ludwick

Aerotech,
101 Zeta Drive,
Pittsburgh, PA 15238

Albert To

Department of Mechanical Engineering
and Materials Science,
University of Pittsburgh,
Pittsburgh, PA 15261
e-mail: albertto@pitt.edu

Manuscript received October 10, 2017; final manuscript received June 15, 2018; published online July 27, 2018. Assoc. Editor: Qiang Huang. This work is in part a work of the U.S. Government. ASME disclaims all interest in the U.S. Government's contributions.

J. Manuf. Sci. Eng 140(10), 105002 (Jul 27, 2018) (16 pages) Paper No: MANU-17-1626; doi: 10.1115/1.4040622 History: Received October 10, 2017; Revised June 15, 2018

Additive manufacturing (AM) is now capable of fabricating geometrically complex geometries such as a variable-density lattice structure. This ability to handle geometric complexity provides the designer an opportunity to rethink the design method. In this work, a novel topology optimization algorithm is proposed to design variable-density lattice infill to maximize the first eigenfrequency of the structure. To make the method efficient, the lattice infill is treated as a continuum material with equivalent elastic properties obtained from asymptotic homogenization (AH), and the topology optimization is employed to find the optimum density distribution of the lattice structure. Specifically, the AH method is employed to calculate the effective mechanical properties of a predefined lattice structure as a function of its relative densities. Once the optimal density distribution is obtained, a continuous mapping technique is used to convert the optimal density distribution into variable-density lattice structured design. Two three-dimensional (3D) examples are used to validate the proposed method, where the designs are printed by the EOS direct metal laser sintering (DMLS) process in Ti6Al4V. Experimental results obtained from dynamical testing of the printed samples and detailed simulation results are in good agreement with the homogenized model results, which demonstrates the accuracy and efficiency of the proposed method.

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Figures

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Fig. 1

Two categories of artificial lattice materials: (a) metal foam and (b) 3D printed regular lattice material

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Fig. 2

Microstructure of the bamboo [16]

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Fig. 3

Lattice structure topology optimization of an aerospace component for stiffness maximization

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Fig. 4

Homogenization of RVE for lattice materials

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Fig. 5

Lattice unit of cubic lattice structure

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Fig. 6

Effective elastic constants as a function of relative density of the lattice structure

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Fig. 7

Benchmark study for the overhang and ligament size: (a) CAD model of the benchmark for Ti6Al4V and (b) photo of the printed-out benchmark in Ti6Al4V by using EOS DMLS

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Fig. 8

Specimens printed by the EOS DMLS system with Ti6Al4V for tensile testing: (a) geometry and dimension of the samples and (b) printed samples and tensile test

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Fig. 9

Comparison of the effective elastic constants by the homogenized model and experiments

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Fig. 10

Model of a cantilever beam: (a) CAD model and boundary condition and (b) mesh model

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Fig. 11

Optimization results: (a) optimized density distribution of the beam, (b) longitudinal sectional view of the density distribution, (c) optimized design using variable-density lattice structure, and (d) convergence history of the first eigenfrequency

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Fig. 12

Full-scale simulation for the cantilever beam with uniform lattice structure (V*=0.6): (a) CAD model of the uniform beam and (b) first mode of the cantilever beam ω1=676.4Hz

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Fig. 13

Full-scale simulation for the cantilever beam with optimized lattice structure (V*=0.6): (a) CAD model of the nonuniform beam and (b) first mode of the optimized beam ω1=1245.3Hz

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Fig. 14

Computer-aided design model of the cantilever beams with the fixture for experiments: (a) uniform lattice structured beam and (b) optimized lattice structured beam

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Fig. 15

Beams printed by EOS M290 with Ti6Al4V: (a) uniform lattice structured beam and (b) optimized lattice structured beam

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Fig. 16

Full-scale simulation for the beams with fixture: (a) uniform beam, ω1=631.8Hz and (b) optimized beam,ω1=1094.3Hz

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Fig. 17

Vibration tests for two beams: (a) uniform beam and (b) optimized beam

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Fig. 18

Vibration experiments for the two beams: (a) uniform beam and (b) optimized beam

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Fig. 19

Model of a dual-fixed beam

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Fig. 20

Optimization results of the dual-fixed beam: (a) optimized density distribution, (b) optimized design using variable-density lattice structure, and (c) convergence history of first eigenfrequency

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Fig. 21

Full-scale simulations for the dual-fixed beam: (a) CAD model of uniform beam and vibration mode of first eigenfrequency ω1=3730.9Hz and (b) CAD model of optimized beam and vibration mode of first eigenfrequency ω1=5208.3Hz

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Fig. 22

Computer-aided design model of dual-fixed beams after assembling with the fixtures used for experiments: (a) uniform beam and (b) optimized beam

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Fig. 23

Dual-fixed beams printed by EOS M290 with Ti6Al4V: (a) uniform beam and (b) optimized beam

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Fig. 24

Full-scale simulations for the dual-fixed beams with the fixtures: (a) uniform beam, ω1=3501.4Hz and (b) optimized beam, ω1=4610.1Hz

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Fig. 25

Experimental setup for the dual-fixed beam: (a) uniform beam and (b) optimized beam

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Fig. 26

Experimental results of the dual-fixed beams: (a) uniform beam, ω1=3504.6Hz and (b) optimized beam, ω1=4480.9Hz

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Fig. 27

Model of test component: (a) CAD model and boundary conditions and (b) mesh model

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Fig. 28

Optimization results for the test component: (a) optimal density distribution and (b) convergent history of first eigenfrequency

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Fig. 29

Reconstruction of lattice infills and full-scale simulation for uniform design and optimal designs: (a) CAD model of reconstructed uniform design (V*=0.5), (b) CAD model of reconstructed optimal design (V*=0.5), (c) full-scale simulation of uniform part,ω1=2960Hz, and (d) full-scale simulation of optimized part, ω1=3713.1Hz

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