# Low Thermal Expansion Machine Frame Designs Using Lattice Structures
## 1 Introduction
### 2. Methodology
#### Motivation and Lattice Design Concept
Polyamide 12 (Nylon 12) was chosen as a relatively low-CTE material for the lattice outer part because it processes well by LPBF to enable complex parts to be manufactured with a good combination of geometrical accuracy and mechanical performance [32]. Ultra-high-molecular-weight polyethylene (UHMWPE) was chosen as a relatively high-CTE material for the cylindrical inner part because it provides a high value of CTE, good impact resistance, very low coefficient of friction, and self-lubricating performance [33], which is advantageous in the assembly and performance of the proposed two-material structure. The two parts were manufactured separately then hand assembled by inserting the cylindrical inner part into the centre of the lattice outer part. The design of the lattice outer part is illustrated in Figure 3. The design comprised four CTE-minimising layers, each connected to a load-bearing pillar along one of its edges. The deformation mechanism section was designed as a hollow octagonal prism with eight connecting struts, intended to bend when the inner part expanded in response to increased temperature. This bending provides a complementary deformation that minimises the structure's planar CTE. The length of the hollow octagonal prism (in the \(Y\)-axis) was 0.94 \(L\), where \(L\) is the lattice side length. The prism is offset (in the \(Y\)-axis) by 0.03 \(L\) from both the top and bottom surfaces. In addition, the lattice was designed to expand along the diagonal directions, as shown in Figure 3b, providing more space for the load-bearing pillars at the cell's corners where they can be thicker and stiffer than they would be if they were placed at the cell's faces. The four CTE-minimising layers had different orientations around the \(Y\)-axis: 0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), and 270\({}^{\circ}\), as illustrated in Figure 3b. The layers formed two pairs, according  
Figure 2: A concept of a single-unit cell (**a**) and a 4 \(\times\) 4 cell lattice structure (**b**). to their connectivity with the load-bearing pillars along the cell's edges. The two layers connected to diagonally-opposite pillars were connected by the struts of the octagonal prism discussed above. Tessellation of the unit cell in order to construct a lattice structure was then investigated. This posed a challenge because the unit cell was not rotationally symmetric in the XZ plane, resulting in different CTEs in each direction; however, it is desirable to engineer a structure with isotropic planar CTE. In Figure 4, the layers in the unit cell are numbered according to the position of the load-bearing pillar (top view of Figure 4a). Layers 1, 2, 3, and 4 (front view of Figure 4a) are the lowest layer, the second-lowest layer, etc., respectively. The symbol 'P' was used to represent the orientation of the unit cell, as shown in Figure 4b. Five potential patterns were selected and created by changing the orientation around the Y-axis and combining them into 2 \(\times\) 1 \(\times\) 2 lattices, as shown in Figure 4b. We investigated the 1 st pattern to demonstrate the CTEs of a tessellated lattice and the difference between the mean CTEs in the X- and Z-axes of each composed lattice. A 1 \(\times\) 1 \(\times\) 1 lattice up to a 4 \(\times\) 1 \(\times\) 4 lattice (Figure 5) were chosen for this investigation. Figure 4: Annotation and patterns of the lattices. (**a**) Top view and front view of the unit cell; (**b**) Patterns of 2 \(\times\) 1 \(\times\) 2 lattices. Figure 3: The concept of low-CTE design. (**a**) Top view of a drafted layer, where \(L\) is the original length of the lattice and D is the original diameter of the cylindrical inner part; (**b**) Isometric view of a unit cell (the two-way arrows indicate the deformation movement of the lattice).