PSI - Issue 12

514 8 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 The FEM simulations are performed at different folding angles , obtaining the deformations in the panels shown in Fig. 3b. As can be seen from Fig. 3c, where the x- axis represents the spatial coordinate along the panel’s contour and the y -axis refers to the deformations, it is possible to approximate the dependence between the angle and the resulting deformation by a linear relation. This linearity was also verified from a numerical point of view. The optical errors resulting from deformations, in terms of rotation , must multiplied by the factor 2, as shown by Eq. 5. Therefore, the error in the sunlight reflection can be estimated as: = 2 ∙ atan( ′) (24) where ′ represents the slope of the function of the deformed shape of the panel. For what concern the parabolic jig, both errors on the production and the assembly operations might cause a loss in the efficiency of the system. The errors related to the jig production involve an incorrect surface waviness, that can be identified by two parameters: is the amplitude of the deformed jig surface, whereas is the half of the wavelength of the profile, as schematized in Fig.4. As discussed later, these two parameters are strictly connected, thus only was selected for the sensitivity analysis. It is sufficient to derive the deformed profile to obtain the local rotation, and then multiply it by factor 2 to get the corresponding sunlight reflection error . Fig.4. Schematization of the jig’s deformed shape caused from the waviness identified by and . The errors related with the jig positioning involve the horizontal and the vertical offsets from the ideal position, and the wrong rotation, respectively identified by , , , and . Considering the horizontal offset, referring to the scheme in Fig.5a, it is possible to state that the error, represented by the angle , corresponds to the sunlight reflection error due to the horizontal displacement of the parabola (it does not require to be multiply by the factor 2). Relying on a simply trigonometric approach, the error can be directly computed as: = − (25) where the angles and can be obtained by geometrical considerations: = atan ( − ) (26) = atan ( +Δ , − ) (27) where , represents the horizontal offset and and represent the correct horizontal and vertical parabolic coordinates. The same procedure can be applied to compute the errors in the cases of the vertical offset and the rotation. The only differences are in the needed parameters ( , and ) and in the geometric references (Fig.5b and Fig.5c). It worth to mention that the position and rotations of the jig in the third direction were not considered since they had a much lower influence on the intercept factor. F. Cadini et al. / Procedia Structural Integrity 12 (2018) 507–520

(a)

(b)

(c)

Fig.5. Schematization of: (a) Horizontal offset of the jig , ; (b) Vertical offset of the jig , ; (c) Rotation of the jig .