**4. Experiment**

In order to verify the effectiveness of the experimental measurement system, we used a truss structure as a reference. It was manufactured by electric discharge machining from a 5 mm thick plate of aluminum. The properties of the glass and air are provided in Table 1. The refractive index of the air at different temperatures was calculated according to the equation of Rüeger [22]. The effects of the translational component and rotational component on the measurement result were verified by the experiment. Then, the CTE along the length of the aluminum sample was measured to verify the validity of this experimental system.

**Table 1.** Properties for the glass [23] and air [22].


#### *4.1. Translational Experiment*

The *x* direction displacement of the laser was adjusted using the adjusting bracket. The *y* and *z* direction displacements of the laser were adjusted by a 2D precision mobile platform as shown in

Figure 7a. The position of the laser was detected by a micrometer (Shahe5313-02, Zhejiang, China) with an accuracy of ±2 μm. According to the measurement steps described in Section 2.3, the temperature was kept at room temperature (i.e., 20 ◦C); the laser was moved respectively along the *x*, *y*, and *z* directions from −50–50 μm; the interval was 10 μm; and each position was measured five times. The effects of the translational component on the length of the air gap are shown in Figure 7, and the measured lengths of the air gap after translating in the *x*, *y*, and *z* directions are provided in Figure 7b–d, respectively. "*Lx*" is the length of the air gap after *x*-axis translation. "*Ly*" is the length of the air gap after *y*-axis translation. "*Lz*" is the length of the air gap after *z*-axis translation. "*LIx*", "*LIy*", and "*LIz*" are the initial lengths of the air gap represented by the dashed lines in Figure 7b–d, respectively. According to the results, the lengths of the air gap fluctuate around the mean length. The amount of fluctuation was less than the accuracy of the measurement (±1 μm). Thus, the lengths of the air gap had no obvious trend with translation. This proves that translation had little e ffect on the length measurement. This is consistent with the previous analysis.

**Figure 7.** Measurement results after artificial additional translation. (**a**) The translation adjustment device. (**b**) Measurement results after *x*-axis translation. "*Lx*" is the length of the air gap after *x*-axis translation. (**c**) Measurement results after *y*-axis translation. "*Ly*" is the length of the air gap after *y*-axis translation. (**d**) Measurement results after *z*-axis translation. "*Lz*" is the length of the air gap after *z*-axis translation. The dashed lines are the initial length of the air gap before translation.

## *4.2. Rotational Experiment*

The rotation of the sample was realized by a rotation platform (KSP-656M, Guangdong, China) with an accuracy of ±52.2" as shown in Figure 8a. The temperature was kept at room temperature (i.e., 20 ◦C); the fine adjustment screw was adjusted to rotate the sample around the *y*-axis from 0 to 0.6◦; the interval was 0.058◦; each position was measured five times. Figure 8b shows the e ffect of the rotation on the sample length measurement at 20 ◦C. "*Lr*" is the measurement length of the air gap after rotation. "*LTh*" is the theoretical length of the air gap after rotation. "*Le*" is length of the air gap after eliminating the measurement error of rotation. The dashed line "*LI*" is the initial length of the air gap before rotation. According to the measurement result, the measurement length of the air gap increased when the rotation angle increased. This proves that the measurement length of the air gap after rotation was larger than the actual length. The rotational component generates extra measurement error. It will reduce the measurement accuracy of the thermal expansion coe fficient. In Figure 8b, the red line provides the length of the air gap after eliminating the measurement error of rotation. This proves that the elimination was e ffective in reducing the measurement errors caused by rotation. Therefore, in order to improve the measurement accuracy, it is necessary to take measures to eliminate the measurement errors caused by rotation.

**Figure 8.** Measurement results after artificial additional rotation. (**a**) The experimental device of the sample rotation. (**b**) Measurement results after rotation. "*Lr*" is the measurement length of the air gap after rotation. "*LTh*" is the theoretical length of the air gap after rotation. "*Le*" is length of the air gap after elimination of the measurement error of rotation. The dashed line is the initial length of the air gap before rotation.

#### *4.3. Measurement of the CTE of the Sample*

According to the measurement steps described in Section 2.3, with interval of 20 ◦C, the sample was heated to 120 ◦C. The length of the air gap was measured five times at each temperature. The sample was heat-treated to reduce the residual stress. This reduces the warping deformation of the sample during measurement.

As shown in Figure 9, The "Air Gap" is the distance between the two lenses fixed on the sample. The length of the air gap between the two lenses increased with the increase in temperature. The blue scale bar represents 0.1 μm in length of the error bar of the air gap. " α" is the equivalent CTE of the sample. The average CTE of the sample was 22.99 ± 0.54 × 10−<sup>6</sup> K−1. Compared with the recommended value (23.1 × 10−<sup>6</sup> <sup>K</sup>−1) in the existing literature [24], this proves that this experimental system is valid and can be used for measuring the thermal expansion of truss structures.

**Figure 9.** CTE measurement of the aluminum truss structure. The air gap is the distance between the two lenses fixed on the sample. It increased with an increase in the temperature. The blue scale bar represents 0.1 μm in length of the measurement error of the air gap. "α" is the CTE of the sample.
