# Low Thermal Expansion Machine Frame Designs Using Lattice Structures
## 1 Introduction  
Temperature fluctuations, even in controlled laboratory environments, can cause alignment drift in high-precision measuring machines, as their support frames undergo thermal expansion and contraction [1]. This can lead to significant measurement uncertainty and, therefore, represent a significant issue in precision engineering [2]. It is desirable to use structures with low, or ideally zero, coefficient of thermal expansion (CTE) in such systems, but low-CTE structures are difficult to design and fabricate [1]. The geometrical complexity enabled by additive manufacturing (AM) provides new opportunities [3, 4, 5, 6, 7] for low-CTE structures for precision measuring machines but, to the authors' knowledge, there is no established design route to achieve this. The design of structures with low or tailorable CTE is a growing research topic, with applications identified in aerospace, optical measurement systems, and precision instruments [8, 9, 10]. For example, [PERSON] et al. [11] proposed three-phase composites (Invar, nickel and voids) using topology optimisation to design structures with positive, zero, and negative CTE. [PERSON] et al. [12] and [PERSON] et al. [13] applied the same approach to the internal geometry of porous composites, resulting in planar negative CTE structures. [PERSON] et al. [13] concluded that topology optimisation is the most effective structural design method for minimising thermally induced stress and maximising structural stiffness. Their structures had CTEs in the range of 300% to 1000% of those of the constituent materials, thus demonstrating that advanced computer design techniques can be used to create tailorable CTE structures [13]. [PERSON], following another novel design route, based their tailorable-CTE structure on a repeating hexagonal lattice with bi-material ribs [14]. The ribs were made of bonded layers, resulting in bending under increasing temperature. From this concept, [PERSON] proposed a tetrakaidecahedral foam cell, resulting in large positive, zero, and large negative CTE structures. [PERSON] et al. [15] similarly proposed two-dimensional (2D) and three-dimensional (3D) lattices with bi-material elements. Their 2D lattices had aluminium and copper as the constituent materials (with CTE = 2.3 \(\times\)\(10^{-5}\) K\({}^{-1}\) and 1.85 \(\times\)\(10^{-5}\) K\({}^{-1}\), respectively), resulting in an effective CTE of \(-\)68.1 \(\times\)\(10^{-6}\) K\({}^{-1}\) for the lattice structure. An innovation in the design of low-CTE structures came from [PERSON] et al. [16], who proposed a lattice with two dissimilar materials and an assembly step, in which the voids of a continuous honeycomb structure were filled with 'inserts' of a higher-CTE material; this formed a hybrid structure with a predicted CTE of close to zero. Bi-material solutions for low and tailorable CTE structures were examined extensively, with the results summarised in Table 1. Based on Table 1, previous bi-material lattice designs have been fabricated in a wide range of metals and polymers, and have provided effective CTE in the range of \(-\)4 \(\times\)\(10^{-4}\) K\({}^{-1}\) to 1 \(\times\)\(10^{-3}\) K\({}^{-1}\). The production of low-CTE bi-material lattice structures requires advanced manufacturing techniques or several sequential manufacturing steps. Such structures have been fabricated previously using multi-material AM processes [12; 13; 15; 17; 18; 19], conventional manufacturing techniques followed by joining [15; 17; 20; 21; 22], and combinations of both techniques [8; 23]. From these publications, it is clear that the designer of a low-CTE bi-material lattice needs to consider both the complexity and manufacturability of the lattice, as well as the challenge of assembly. Our proposed approach is to simplify as far as possible the design requirements for a low-CTE lattice by utilising high-CTE inserts of a simple cylindrical geometry, which are made from a commonly available polymer.