3.1

Mirrors light-weighted by foam structures

The mirror is filled inside with a stochastic Voronoi foam [14]. The density distribution of the foam is based on static load cases, resulting in a smaller cell size next to the interfaces for mounting at the housing. In addition to the front side, on which the functional areas are located, the backside has a closed surface as well, and thus, a high bending stiffness is generated (see Figure 5).

Figure 5:

Mirror substrate M1M3 light-weighted by Voronoi foam

The mass could be reduced to 44 % compared to the massive component and is thus slightly below the conventional light-weighted design, which was produced by cross-drilling. In addition, the stiffness can be increased, resulting in a significant increase for the specific stiffness.

Table 1:

Mass and eigenfrequencies of the optimized mirror substrate

Due to the high eigenfrequencies, which are above the typical incitation frequencies, the substrate is insensitive to random vibration.

3.2

Topology optimization of the housing

For the housing, a topology optimization is used. A defined envelope curve is required for the optimization, in which the material can be manipulated by the optimization algorithm. The envelope has to provide the interfaces to the other components and the straylight baffle, as shown in Figure 6.

Figure 6:

Envelope for the topology optimization of the housing

The most critical load case for this type of mechanical structure is typically random vibration. Since topology optimizations based on random analyses are currently not possible and at the same time the combination of two time-intensive simulations is not reasonable, substitute loads have to be generated. The generation of these substitute loads is similar to a primary notching process, because only the force reaction at the boundary conditions and not the interface forces can be determined in a random analysis.

Figure 7:

FE-model with envelope and the definition of forces

Because the stresses caused by random vibration depend not only on the excitation, damping, and eigenfrequencies, but also on mass and stiffness, the Young’s modulus and density of the envelope are reduced to produce resilient loads. Therefore, the Young’s modulus and the density are multiplied by the expected lightweight factor ϕ.

This model is now used to perform random analyses for the x-, y-, and z-directions and the forces reactions at the boundary conditions can be determined. In the next step, the model is accelerated with a quasi-static load (QSL) in x, y, and z and the results are scaled until the same forces occur at the boundary conditions.

For the scaled QSL, the interface forces can be simulated and used for topology optimization.

Figure 8:

Substitute forces for topology optimization

A fine and uniform mesh is used for the topology optimization. By a linear element order, the number of nodes can be reduced. The increased stiffness of the elements obtained is accepted for the optimization. In the recalculation of the result, a quadratic element order is then used. With an element size of 1.5 mm, the model consists of approx. 1.5 million elements.

The interfaces and the stray light baffle must remain after the optimization and are therefore excluded from the design region.

Figure 9

left: uniform mesh; right: design & exclusion region

The following objectives are used for the optimization:

The used quasi-static-loads (QSL) were determined in the previous section. With an increase of the 7^th mode under freefree conditions, the result has only six rigid-body modes and thus, a coherent body is obtained. Increasing the 1^st mode with iso-static mounting reduces the sensitivity to random vibration. The minimum structure size should be at least twice the element size to avoid hourglass effects.

3.3

Smoothing and regeneration

After optimization, the result must be prepared for additive manufacturing, otherwise the non-continuous removal of elements would result in a very rough surface and there might not be enough material at the interfaces for a subsequent CNC milling process. Three steps are necessary:

Figure 10:

Steps for smoothing and regeneration of the optimized result

3.4

Results of topology optimized housing

After smoothing the model, simulations can be performed with the optimized housing to validate the results. The results of the topology optimization are compared again to the massive envelope model and the conventional ripped housing. Compared to the massive envelope, the mass was reduced to 32 % and thus, the specified lightweight factor of 30 % was well met and is below the mass of the ribbed housing. The difference is more significant for the eigenfrequencies. Thus, the topology-optimized housing is 20 % more stiffly than the ribbed one. In summary, the specific stiffness could also be increased significantly.

Table 2:

Mass and eigenfrequencies of the optimized housing

The eigenfrequencies were determined under free-free conditions to evaluate the stiffness independent of boundary conditions. The 1st mode shows a global deformation of the housing in each case (see Figure 11).

Figure 11:

1^st Mode shape of the massive, ripped, and topology optimized housing

5.1

Low-level sine search

For the low-level sine search, the TMA was accelerated in the frequency range 5-2000 Hz with a sine-incitation of 0.5 g. The acceleration sensors show a significant increase of the response acceleration, when reaching the eigenfrequencies due to resonances. The comparison with the simulated harmonic analyses shows that only small deviations below 8 % occur between test and simulation of the eigenfrequency. For most modes, the deviation is much smaller.

Figure 14:

Results of the low-level sine search; left: test; right: simulation

5.2

Random vibration

To verify the loading capacity of the TMA, a random vibration test is performed with a broadband excitation in the x-, y-, and z-directions. A low-level sine test is performed before and after each random load case in order to determine the eigenfrequencies of the TMA. A change in the eigenfrequencies would indicate damage to the assembly.

Figure 15

left: test-flow for random vibration testing; right: random spectrum (x-,y-, and z-direction)

The change in the dominant eigenfrequencies (high effective masses in the translatory directions) are less than 1 %. It can therefore be assumed that the TMA is not damaged by the random vibration test.

Table 3:

Changes in the dominant eigenfrequencies due to random vibration testing