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ADDITIVELY MANUFACTURED LATTICE STRUCTURES FOR

PRECISION ENGINEERING APPLICATIONS

Waiel A. Elmadih1, Wahyudin P. Syam1, Ian Maskery2 and Richard Leach1

1Manufacturing Metrology Team

2Centre for Additive Manufacturing

University of Nottingham

Nottingham, NG7 2RD, UK

INTRODUCTION

Additive manufacturing (AM) is limited by fewer

design constraints than conventional

manufacturing [1]. Therefore, AM provides higher

design flexibility and allows fabrication of highly

complex geometries, for example, lattice

structures. Compared to solid structures, lattice

structures allow designs that incorporate highly-

tailored mechanical properties so that the

requirements of applications are met with efficient

use of resources [2].

Representative stress-strain curves for solid and

lattice structures are shown in FIGURE 1. In

FIGURE 1, the area under the curve up to the

densification strain is the energy absorbed per

unit volume of material. The energy absorption is

small for a solid, compared to lattice structures

that benefit from long plastic plateau regions in

their stress-strain curves.

FIGURE 1. Representative stress-strain curves

of a full density solid and lattice structures of

different density fraction, adapted from [1].

The length of this plateau region, and the stress

at which it occurs, depends on the lattice material,

volume fraction, unit cell configuration, and the

number of lattice cells in a structure [1]. As such,

with the same total volume, different lattice

configurations can have different mechanical

properties. There has been considerable

research focus on the manufacturability and

strength-to-weight ratio of some lattice structures

[3]–[5]. However, there is little published work

investigating lattice structures for vibration

isolation and thermal expansion control; both of

which are significant for precision engineering

applications. Many applications need structures

that can be isolated from external vibration, for

example, the tool in a CNC machine and the

probe of an optical metrology system. High

stiffness structures are often used, which

increase the natural frequency, achieving

vibration damping [6]. Taking an alternative and

complementary approach, this paper discusses

applications of lattice structures for vibration

isolation.

FIGURE 2 shows the vibration response of a

structure, with natural frequency to an external

vibration source with frequency . When =,

a resonance occurs, leading to a large

displacement amplitude. [6]. To have higher

vibration isolation efficiency, a structure should

have a low , which can be achieved by lowering

the stiffness of the structure. But, there is a

practical limit: if the stiffness is too low, the

structure will be unable to sustain the mass load

of the body of interest (work tool/probe). Lattice

structures possess high strength-to-weight ratio

for their mass, providing low stiffness at relatively

high strength [1].

In this study, various configurations of strut-based

lattice structure are investigated with a

combination of finite element analysis (FEA) and

experimental assessment. Information gained in

this study shows that lattice structures can be

used to tune the design of a structure to have a

desired for a specific vibration isolation

application, effectively setting up a mechanical

band-gap. In practice, external vibration sources

commonly have multiple dominant frequencies,

so should be optimised for a specific range of

frequencies. A case study in which a metrology

frame for an optical system is to be designed, will

be used to demonstrate how for the lattice

structures can be optimised for a specific

frequency range.

FIGURE 2. Vibration damping and isolation.

LATTICE DESIGN AND FABRICATION

Strut-based lattices are used in this study as they

are periodic open-cell cellular structures that

possess bending characteristics that give longer

plateau regions than closed cell cellular

structures, and thus can absorb more energy

than periodic closed-cell cellular structures

[1],[7].

CAD design

The design of lattice structures for AM poses a

significant challenge. We have developed a

simple design tool, in the form of equation (1).

This allows us to construct a range of lattice unit

cells with predefined volume fraction, V. Lattice

strut diameters were calculated using equation

(1), thus

16 +

16 +2

8+3

4+

+

16 +

16 +

16 +2

8+3

4

= 0

(1)

where the length of a unit cell is fixed at 30 mm;

is the strut diameter; is the cell length; is

the number of horizontal struts of length as

they appear in top and bottom views of a single

lattice cell; is the total number of vertical struts

of length as they appear in top and bottom

views of a single lattice cell; is the total number

of vertical struts of length as they appear in

front, back and side views; is the total number

of diagonal struts on all faces; is the number of

diagonal members across the centre of the lattice

cell; is the filled volume of the lattice cell; and

is the constant of intersection volume equal to 0

when = 0, 0.181021 when = 2, and 0.636638

when = 4 [8].

CAD was used for the design of single unit cells

which were then tessellated to create lattice

structures (as illustrated in FIGURE 3).

FIGURE 3. (a) Example of single lattice cell as

designed with Creo Parametric CAD software,

and (b) example of 2×2×2 periodic lattice cells as

tessellated with Autodesk Inventor CAD software.

Samples with lattice behaviour

There is a minimum required number of lattice

unit cells that a structure should have in order for

it to exhibit lattice behaviour, hence, the designed

lattices should be tested to determine whether

they exhibit lattice behaviour [9]. Experimental

compression tests were carried out on samples of

similar overall size, volume fraction and, cell

configuration, but of variable number of

tessellations: 2×2×2, 3x3×3 and 4×4×4. The

samples are shown in FIGURE 4. The purpose of

these experimental compression tests is to

determine the minimum number of cells that

represent a lattice structure, which will later be

used in the study.

Three replicas of each sample were compressed

at a constant compression speed of 1 mmmin-1.

The results of the compression tests with the

three samples are shown in the form of

load-displacement curves in FIGURE 5. All

samples took the curves up to the

densification area, which essentially means all

tested samples exhibited lattice behaviour. This

means that 2×2×2 lattices represent the minimum

number of lattices that can be used in this study.

FIGURE 4. (a) Design of compression test

samples, and (b) compression test samples as

manufactured from Nylon 12 powder on an EOS

P100 selective laser sintering machine with a

building powder layer height of 100 µm.

FIGURE 5. Compression test curves of lattice

samples of different unit cells.

Lattice structure identification

The lattice structures used in this study are of

different volume fraction, different number of

nodes and different cell configurations. For the

purpose of individual characterisation, every

lattice cell was assigned a unique code in the

form of (-), where is the number of

nodes in a 2D view of the single cell,

denotes the volume fraction taking, for example,

a value of ‘010’ for a 10 % lattice cell, and is

an identification number of the struts

configuration. A total of twenty-three lattice cell

configurations were developed based on a fixed

number of nodes. Each configuration formed the

basis for three different volume fraction cells:

10 %, 20 % and 30 %. FIGURE 6 shows the

lattice cells of 10 % volume fraction in a 2×2×2

tessellation.

FIGURE 6. CAD models of the lattices used in

this study, all in 2×2×2 tessellation and of 10 %

volume fraction.

MODELLING ISOLATION PROPERTIES

The CAD models shown in FIGURE 6, along with

their corresponding versions of 20 % and 30 %

volume fraction, were analysed using ANSYS

Workbench FEA. To ensure the calculations had

converged with respect to the element size, we

determined the first of the structures using a

range of element sizes from 25 mm to 0.25 mm.

At an element size of 1 mm, the difference

between subsequent results had fallen below 1 %

for Nylon-12 samples of (60×60×60) mm size.

Therefore, we used elements of 1 mm for all

subsequent calculations. The base surfaces of

the models were constrained in all degrees of

freedom. The harmonic response to a vertical

load of 0.1 N, vibrating at a frequency in the range

0.1 Hz to 1000 Hz (with 10 Hz frequency

intervals) acting on the upper surface was

simulated. The displacements due to the

generated harmonic vibration in the vertical

direction were calculated. The simulated spectra

of the 4010-x and 5010-x lattice models are

shown in FIGURE 7, where the peaks represent

of each model. The 5010-x lattice types have

peaks at higher frequencies than the

corresponding 4010-x lattices with the same

number of unit cells, except for the 5010-3 lattice,

which has the lowest natural frequencies among

all the tested samples. This makes the 5010-3

cell a good candidate for vibration isolation.

However, the 5010-3 cell has more resonance

frequencies within the testing range than the

4010-8 cell. Consequently, the 5010-3 cell can

complicate the control of vibration isolation

0

10,000

20,000

30,000

40,000

0 5 10 15 20

Compressive force/N

Extension/mm

2×2×2 Sample 3×3×3 sample

4×4×4 sample

because the 5010-3 cell has high number of

resonance peaks that need to be controlled for

providing vibration isolation, and thus, the 4010-8

cell is more appropriate for vibration isolation

control.

FIGURE 7. Results from harmonic response

simulations of the 4010-x and 5010-x lattice types

in 2×2×2 tessellations.

The models with lower volume fraction showed

lower natural frequencies than models of higher

volume fraction. This is most likely due to the

lower stiffness of the lower volume fraction

models.

For experimental verification of these results,

three samples each of the 4010-8 and 4030-1

lattices were manufactured and the harmonic

responses in the vertical direction were tested.

Each sample was fixed to a shaker (shown in

FIGURE 8) which vibrated in the range of 0.1 Hz

to 1000 Hz for the 4010-8 samples, and a range

of 0.1 Hz to 3000 Hz for the 4030-1 samples (all

with < 1 Hz frequency intervals) and the reaction

of the bottom surface of the sample was

measured using a force sensor attached to the

shaker. The frequency range for the experimental

testing of each sample was chosen based on the

range at which was obtained using FEA for

each sample. The positions of the natural

frequency peaks are in 90.6 % average

agreement with the frequencies at which they

appeared in the simulation results. Comparison

between average experimental results and

simulation results are shown in FIGURE 9.

FIGURE 8. (a) Selective laser sintering

manufactured samples for experimental

verification, (b) vibration experiment setup.

FIGURE 9. Experimental verification of the

natural frequencies for vertical excitation of the

4030-1 (top) and 4010-8 (bottom) lattice

structures.

The effect of the size of the lattice cells on the

natural frequencies was simulated. The following

sizes were chosen: 30 mm, 22.5 mm and

16.875 mm, while keeping the number of

tessellations constant at 2×2×2. From the FEA

results, it can be concluded that decreasing the

cell size results in higher natural frequencies.

This is likely due to the excitation of shorter

wavelength vibration modes than are permitted in

larger structures of the same volume fraction, as

shown in FIGURE 10.

FIGURE 10. The effect of lattice cell size on .

FIGURE 11. The effect of the number of

tessellations of lattice cells on .

In addition, increasing the number of

tessellations, for a constant cell size and volume

fraction, reduces of the structure, which is

thought to be related to increased strut bending

in structures with a higher number of

tessellations. This was confirmed by testing

samples of tessellations of 2, 4, 5 and 10, in

three-dimensions, as shown in FIGURE 11.

CASE STUDY

The structural frame of a new all-optical

coordinate measuring system was manufactured

with selective laser sintering from Nylon-12 using

lattice structures to hold the mass of a 1 kg probe

and isolate mechanical vibrations in the -

direction arising from the movement of the stage

which drives the probe, which vibrates in the

range from 40 Hz to 80 Hz.

Determining the lattice cell size

There are two options for the selection of the cell

size and number of cell tessellations that can fill

a volume in space with low : (a) large size cells

with low number of tessellations, or (b) smaller

cells with a higher number of tessellations. The

results, so far, do not suggest which route is more

appropriate. As a result, a study was conducted

with the aim of suggesting the most appropriate

route to obtain lower . The study comprised the

design of (30×30×30) mm lattice structure

samples using 4010-8 cells of different sizes:

15 mm, 7.5 mm, and 3.75 mm. By default, cells of

less size need a higher number of tessellations to

fill the (30×30×30) mm space. FEA (results are

summarised in TABLE 1) showed that larger size

cells had lower first mode than smaller ones

with a higher number of tessellations.

TABLE 1.The coupled effect of cell size and

number of tessellations at constant volume

fraction on the first of the lattice structures.

Cell size/

mm Sample

size/mm Number

of cells 1st mode

/ Hz

15

30

8

409.97

7.5

30

64

421.51

3.75

30

512

467.88

As a result, the most suitable cell size for the

structural frame must have two tessellations in

the direction of the shortest dimension (two is the

lowest number of tessellations that can represent

a lattice as discussed previously). Subsequently,

30 mm cells were used to construct the body of

the structural frame because the shortest

dimension was the thickness of the structural

frame (60 mm). The structure used the 4010-8

cell, for the reasons discussed above, and then

varied the volume fraction, and the number of

tessellations to get the isolation to the desired

range.

Verification with in-situ harmonic test

The design of the structural frame comprises

three identical sub-frames each kinematically

coupled to the stage of the optics through balls in

v-grooves located on top of each sub-frame. The

sub-frame, shown in FIGURE 12, is stiffened at

the back through the use of a solid swept

stiffener, and in both sides through the use of

perforated swept surfaces to further damp the

vibration in - and - directions. The harmonic

response to the compressive working load of

10 N, with excitations ranging from

0.1 Hz to 1500 Hz, was modelled with frequency

intervals of 1 Hz. A safety factor of two was used

for the maximum working load which was applied

to the top of the frame.

0

500

1000

1500

123456

Natural frequency/Hz

Vibration mode

4010-8 2x2x2 30 mm

4010-8 2x2x2 22.5 mm

4010-8 2x2x2 16.875 mm

0

200

400

600

800

1000

123456

Natural frequency/Hz

Vibration mode

4010-8 2x2x2 4010-8 3x3x3

4010-8 4x4x4 4010-8 10x10x10

FIGURE 12: Design of the structural sub-frame

for isolating vertical excitations arising from the

movement of the stage of the optics using lattice

structure, dimensions in millimetres.

FIGURE 13: Harmonic response of one

sub-frame.

The structure succeeds in isolating excitations in

the range of 38 Hz and 83 Hz, with maximum

displacement of 2 % of the length of the frame,

see FIGURE 13. Additional metal support was

used to mount the sub-frame on the shaker which

affected the harmonic results obtained from

experiment. The positions of peaks from FEA

of similar conditions to the conditions of the

experiment are in 89.8 % average agreement

with the frequencies at which they appeared in

the experiment results.

CONCLUSION

Before lattice structures can be used for vibration

isolation of a system, the dynamic behaviour of

the system has to be understood. Then, the

lattice characteristics can be tailored to isolate the

resonance peaks of the system. The control of

the natural frequency is the major step towards

tailoring the lattice structure. The study reveals

that the natural frequency of a lattice structure

can be reduced by increasing cell size, reducing

volume fraction, and/or increasing the number of

tessellations of a singular lattice cell, and vise

versa. The work was funded by the EPRSC

project EP/M008983/1.

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