Additively manufactured lattice structures for precision engineering applications

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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 mmmin-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.
REFERENCES
[1] L. J. Gibson and M. F. Ashby. Cellular
Solids: Structure and Properties.
Cambridge University Press, Cambridge:
1999.
[2] O. Rehme and C. Emmelmann. Rapid
Manufacturing of Lattice Structures with
Selective Laser Melting, Proc. SPIE 2006;
6107:192-203.
[3] C. Yan, L. Hao, A. Hussein, P. Young, and
D. Raymont. Advanced Lightweight 316L
Stainless Steel Cellular Lattice Structures
Fabricated via Selective Laser Melting,
Mater. Des. 2014; 55: 533-541.
[4] C. Yan, L. Hao, A. Hussein, and D.
Raymont. Evaluations of Cellular Lattice
Structures Manufactured Using Selective
Laser Melting, Int. J. Mach. Tools Manuf.
2012; 62: 32-38.
[5] M. Santorinaios, W. Brooks, C. J.
Sutcliffe, and R. A. W. Mines. Crush
Behaviour of Open Cellular Lattice
Structures Manufactured Using Selective
Laser Melting, WIT Trans. Built Environ.
2006; 85: 481-490.
[6] T. L. Schmitz and K. S. Smith. Mechanical
Vibrations: Modeling and Measurement.
Springer, New York: 2011.
[7] O. Rehme. Cellular Design for Laser
Freeform Fabrication. Gottingen: 2010.
[8] M. Moore. Symmetrical Intersections of
Right Circular Cylinders, Math. Gaz. 1974;
58: 181-185.
[9] W. P. Syam, J. Wu, B. Zhao, I. Maskery,
W. Elmadih, and R. K. Leach. Design and
Analysis of Strut-based Lattice Structures
for Vibration Isolation. Precision
Engineering. 2017; In press.