Finite element modeling and testing of a deformable carbon fiber reinforced polymer mirror Christopher C. Wilcox,1,* Michael S. Baker,2 David V. Wick,2 Robert C. Romeo,3 Robert N. Martin,3 Brian F. Clark,2 Nicole L. Breivik,2 and Brad L. Boyce2 1
U.S. Naval Research Laboratory, 4555 Overlook Avenue SW, Washington, D.C., USA
2
Sandia National Laboratories, P.O. Box 5800, MS 1188, Albuquerque, New Mexico, USA
3
Composite Mirror Applications, Inc., 1638 S Research Loop, Suite 100, Tucson, Arizona, USA *Corresponding author:
[email protected] Received 29 September 2011; revised 31 January 2011; accepted 8 February 2012; posted 13 February 2012 (Doc. ID 155161); published 18 April 2012
Thin-shelled composite mirrors have been recently proposed for use as deformable mirrors in optical systems. Large-diameter deformable composite mirrors can be used in the development of active optical zoom systems. We present the fabrication, testing, and modeling of a prototype 0.2 m diameter carbon fiber reinforced polymer mirror for use as a deformable mirror. In addition, three actuation techniques have been modeled and will be presented. © 2012 Optical Society of America OCIS codes: 080.4035, 110.1080, 120.0120, 120.4820, 220.0220.
1. Introduction
The Naval Research Laboratory (NRL) and Sandia National Laboratories (SNL) have been researching the use of carbon fiber reinforced polymer (CFRP) materials for optical components in imaging applications for several years. The use of CFRP offers advantages over glass in many applications due to its strength-to-weight ratio and controllable coefficient of thermal expansion. Replacing glass components in large-aperture optical systems with CFRP optics can dramatically reduce the mass of the overall system [1–3]. Since minimizing mass is a priority for air–and space-based applications, the development of lightweight, adaptive mirrors is an important component for these systems. Working closely with Composite Mirror Applications, Inc. (CMA), a prototype 2 m radius of curvature, 0.2 m diameter CFRP mirror has been designed and fabricated for testing and evaluation. This prototype CFRP mirror weighs 91.5 g, while
1559-128X/12/122081-07$15.00/0 © 2012 Optical Society of America
typical glass mirrors of this diameter can weigh in the range of 2–3 kg. Extensive modeling of the CFRP mirror has led to the development of a finite element model (FEM) at SNL [4]. This FEM is to be validated with optical measurements of the mirror’s surface and, additionally, three sample actuation techniques will be examined. Currently, an optical system for a reflective zoom system utilizing a 0.2 m CFRP mirror and a smaller, high-actuator-density microelectromechanical deformable mirror has been designed. An all-reflective zoom system requires two active elements to achieve optical zoom [5]. The current designs will require the CFRP mirror to be able to change its radius of curvature from either 2 m to 1.25 m or from 2 m to 4 m. Starting with the fabricated 2 m radius of curvature, the CFRP mirror will be deformed to either of these radii of curvature. 2. Finite Element Modeling Method
A FEM of a material or structure extends from the finite element method of solving partial differential equations. It allows the detailed visualization of a structure or material as it bends or endures stress. 20 April 2012 / Vol. 51, No. 12 / APPLIED OPTICS
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There is great need to maintain high optical surface quality of the CFRP mirror before and after actuation for use in an imaging application. The CFRP mirror FEM development began by using the published parameters of commercial CFRP and is being refined with testing of sample CFRP. The structural analysis of this material is also being adjusted via the testing of a fabricated prototype 0.2 m CFRP mirror. Through testing sample material, an accurate FEM can be developed and thus actuation schemes can be tested and evaluated in modeling rather than by costly prototyping. Parameters needed for the FEM are the nine orthotropic stiffness matrix values for a single layer of CFRP material. These include Young’s modulus, Ei , along each axis i; the shear modulus, Gij , in the direction j on the plane whose normal is in the direction i; and Poisson’s ratio, νij , which corresponds to a contradiction in direction j when an extension is applied in direction i [6]. A four-point bend test was performed to evaluate the parameters of the CFRP for use with the FEM. A.
Four-Point Bend Test
A four-point bend test is conducted by placing a specimen, or coupon, of material on a simply supported beam of constant cross-sectional area. A flat rectangular coupon is supported close to its ends and pressed by two loads placed symmetrically between the supports, giving a four-point test [6]. Figure 1 shows an illustration of a typical bend test of this type. The test coupon of thickness, t, and length, C, is point loaded at two points separated by a distance, A, with a force, F. The test coupon is supported at two points separated by a distance, B, creating the four points for the test. The force is then increased linearly to the point the sample fails and the stiffness can then be extrapolated. The test coupons used in the four-point bend test performed at SNL were manufactured by CMA. Several coupons of CFRP were cut from two large specimens with five specimens of each. The two thicknesses used were 16 and 24 plies of CFRP. Each piece was 6 in. × 1 78 in. (152.4 mm × 47.6 mm) and set up in a four-point bend test to calculate stiffness defined as the slope of the force versus displacement. Figures 2 and 3 show the resulting force-versusdisplacement plots from the four-point bend test for CFRP coupons of 16 and 24 plies of CFRP, respectively. The four-point bend tests showed an average stiffness of 134.7 lbf ∕in. (23.6 N∕mm) and
Fig. 1. Four-point material bend test configuration. 2082
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Fig. 2. (Color online) Results from four-point bend test for coupons made from 16 plies of CFRP.
384.6 lbf ∕in. (67.4 N∕mm), respectively. The first 0.1 in. (2.54 mm) of displacement was used as the “linear region” to calculate this slope, although all of the test coupons exhibit linear behavior past 0.2 in. of displacement. There is slight variation in the slopes of the different coupons because the test piece of CFRP from which the coupons were cut had a slight wedge to its profile. Each specimen was tested to the point of catastrophic failure to complete each test. Results from the four-point bend test yielded more accurate material parameters that were then utilized by the commercial finite element analysis software package ANSYS to simulate the orthotropic material behavior. Each material layer in the shell element definition had a specified layer orientation, thickness, and material property. When comparing to the measured data, the properties were adjusted to match the test sample. Each layer was given the same parameters for the orthotropic properties and the x direction of the first layer was aligned to the primary fiber direction of the coupons in the model. 3. Prototype CFRP Deformable Mirror
While the deformation of a mirror made of 16 plies of CFRP requires less force for actuation than one of 24 plies, increasing the thickness to 24 plies of material has yielded mirrors with much higher optical quality.
Fig. 3. (Color online) Results from four-point bend test for coupons made from 24 plies of CFRP.
Fig. 4. (Color online) Ronchi image comparison between CFRP mirrors of (a) 16 and (b) 24 plies.
Figure 4(a) shows a Ronchigram of a mirror from CMA made with 16 plies of CFRP. Figure 4(b) shows a Ronchigram of a mirror made with 24 plies. A Ronchigram is the result of a Ronchi test, which is an interferometric method of determining surface quality [7]. The use of 24 plies increases the optical quality significantly. It can be seen in Fig. 4 that slope of the wavefront is significantly improved when the mirror is made with 24 plies rather than 16 plies. The prototype CFRP deformable mirror developed for this project will be made with 24 plies of CFRP and will be discussed in the next sections. The prototype CFRP deformable mirror currently being tested is a 0.2 m diameter, 2 m radius of curvature thin mirror. The mirror’s thickness is 1.79 mm consisting of 24 plies of CFRP layers. This mirror will be one of the active elements in a forthcoming active optical zoom system. By actuating this mirror and changing its radius of curvature in conjunction with a smaller microelectromechanical deformable mirror, optical zoom can be achieved without the need for physically moving the position of the mirrors in the system [5]. Testing was performed with a phase-shifting interferometer from Zygo, using an f ∕7.2 reference sphere [5]. The resulting interference pattern, or fringe pattern, is a representation of the phase difference of the test and reference beams [8]. This phase difference from a perfectly spherical surface can be represented with Zernike coefficients. By using a force gauge at
Fig. 6. (a) Interference pattern and (b) surface measurement.
the center of the rear of the mirror, force-versusdeflection measurements were gathered to compare with the FEM. Figure 5 shows a picture of the optical setup to measure the CFRP mirror’s surface. Figure 6(a) shows the interference pattern resulting from the measurement of the 0.2 m CFRP prototype mirror. Figure 6(b) shows the surface variation of the 0.2 m prototype CFRP mirror from a best-fit sphere. The grayscale bar to the right is in units of waves of a He-Ne laser (λ 632.8 nm). The peak-tovalley surface roughness is measured to be 6.4λ. These measurements are of the CFRP mirror in its unsupported state with a radius of curvature of 2 m. Performing singular value decomposition into Zernike modes, or aberrations, tabulated in Table 1, it can be seen (neglecting piston and tilts) that there are astigmatism and spherical aberrations present in the mirror. Because this mirror was so thin, it could not be mounted regularly in a standard optical mount, so this mirror was taped in place and, even without adequate mounting, the mirror’s figure was still reasonable. Much of the astigmatism is likely due to the gravitational effects and employed lack of proper mounting. Because an adaptive optical zoom system requires a second active element, other uncorrected aberrations may be compensated using the smaller diameter, high-actuator-density microelectromechanical deformable mirror. A. Testing of the Prototype CFRP Mirror
To validate the FEM, surface-versus-force measurements were taken and compared to the theoretical predictions. In the first attempt to take Table 1.
Fig. 5. (Color online) The prototype 0.2 m CFRP mirror measured with a Zygo interferometer.
Aberrations in Relaxed 0.2 Meter CFRP Mirror
Zernike mode, i
Aberration
1 2 3 4 5 6 7 8 9 10 11
Piston Tip Tilt Focus Astigmatism X Astigmatism Y Coma X Coma Y Trefoil X Trefoil Y Spherical
Weights 0.0000 0.2948 1.7008 0.1658 1.0718 0.1497 −0.1656 −0.2779 0.3768 0.1914 1.4365
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Fig. 7. Predicted (left) and measured (right) surfaces supported with three points under single-point force load from the rear of (a) 0.86 N, (b) 1.26 N, (c) 2.00 N, (d) 2.06 N, (e) 2.60 N, (f) 3.10 N, and (g) 3.52 N.
measurements with the Zygo interferometer, the CFRP mirror was placed in a standard optical mount. Unfortunately, applying any force to the rear of the mirror created pinch points around the edge of the mirror, causing great difficulty in acquiring data. This was largely due to the fact that the mount is not optically flat and the edge of the CFRP mirror is not perfectly smooth around its circumference as a result of the CFRP cutting process. To resolve this issue, Table 2.
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Resulting RMS Error between Modeling and Measurement
Force
Peak-to-valley
RMS % error
0.86 1.26 2.00 2.06 2.60 3.10 3.52
14.8λ 21.1λ 33.2λ 34.0λ 41.4λ 48.3λ 54.3λ
6.48% 8.65% 8.85% 10.42% 13.02% 14.45% 14.97%
N N N N N N N
three ball bearings were glued at equal distances around the optical mount, thus defining a plane for the mirror’s surface. A force gauge was used with a ball bearing epoxied to ensure a single-point contact. Each measurement obtained with the Zygo
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Fig. 8. (Color online) Radius of curvature versus pressure for uniformly distributed pressure.
Fig. 9. (Color online) (a) Surface plot and (b) cross-sectional profile of deviation from a best-fit sphere with radius of curvature of 1.25 m for uniformly distributed pressure.
predicted the deformation of the CFRP mirror very well. All surfaces are represented as the difference from a 2 m radius of curvature sphere. Table 2 lists the peak-to-valley deformation values in waves of the mirror’s surface with the associated forces applied to the back of the mirror. Table 2 also lists the root mean square (RMS) percentage error between the predicted and measured surface deformations. The RMS percentage error was calculated using the formula
Fig. 10. (Color online) Radius of curvature versus force for radial inward force.
interferometer was recorded with the associated force applied to the back of the mirror. 4. Comparison to Modeling
With the aforementioned FEM and laboratory setup, direct comparisons between theoretical and experimental results at different force levels were made. Figure 7 shows the comparison of the predicted and measured surfaces under single-point force load for each force applied to the back of the CFRP mirror. It can be seen from these surfaces that the model has
RMS 100 ×
s PN PN jP − M j i;j i;j i1 j1 N2
(1)
where P is the predicted surface from the FEM, M is the measured surface, and both P and M are N × N matrices with the surface data. We were pleased with the relative accuracy of the FEM, as shown by the relatively small RMS errors. These measurements validate the predictions from the FEM. 5. Modeled Actuation Techniques
With validation of the CFRP mirror model, further simulations of the prototype CFRP mirror can be performed with a fair amount of confidence. Different actuation techniques were simulated to visualize
Fig. 11. (Color online) (a) Surface plot and (b) cross-sectional profile of deviation from a best-fit sphere with radius of curvature of 1.25 m for radial inward force. 20 April 2012 / Vol. 51, No. 12 / APPLIED OPTICS
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Fig. 13. (Color online) (a) Surface plot and (b) cross-sectional profile of deviation from a best-fit sphere with radius of curvature of 1.25 m for tangential force.
how the mirror’s surface is affected as the mirror’s radius of curvature is changed from 2 m to 1.25 m and 2 m to 4 m. Three different actuation methods have been simulated for comparison: uniformly distributed pressure applied to the back, radial force applied to the edge, and rotation angle applied tangentially to the edge. A uniform pressure applied to the back of the CFRP mirror can be achieved by sealing the back of the mirror and applying a vacuum or positive pressure. In this simulation, the mirror is held in place and fixed at the edges. Figure 8 shows a radius of curvature versus pressure plot of the predicted forces necessary to push/pull the CFRP mirror in a range of 1.25–4 m radius of curvature with a resting radius of curvature of 2 m. Predicted pressures of −31.2 kPa and 9.95 kPa are needed in order to achieve radii of curvature of 1.25 m and 4 m, respectively. Figure 9 shows the predicted surface and cross-sectional profile of the CFRP mirror with a uniformly distributed vacuum of −31.2 kPa applied. This surface is the deviation from a best-fit sphere with a radius of curvature of 1.25 m resulting in a peak-to-valley surface error of about 105 μm. Another simulated actuation method is a distributed radial force applied at the edge of the CFRP mirror. This can be achieved by using a ring-clamp-type configuration [9] or a variable iris mechanism to squeeze or stretch the mirror. In this simulation, the
Fig. 12. (Color online) Radius of curvature versus rotation angle for tangential force. 2086
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mirror is held in place and clamped/stretched at the edge to cause deformation. Figure 10 shows a plot of the radius of curvature versus distributed force required to deform the mirror from a range of 1.25–4 m. Predicted distributed radial forces of 20.2 kN∕m and −21.5 kN∕m would be needed to achieve radii of curvature of 1.25 m and 4 m, respectively. Figure 11 shows the predicted surface and cross-sectional profile of the CFRP mirror with a radial inward distributed force of 20.2 kN∕m applied to the edge. This surface is the deviation from a best-fit sphere with radius of curvature of 1.25 m resulting in a peakto-valley surface error of about 108 μm. Actuation of this mirror can also be performed by rotation of the mirror’s edge. Figure 12 shows the rotation angles needed at the edge of the CFRP mirror to vary the mirror throughout the radius of curvature range of 1.25–4 m. Predicted angles of −3.38° and 1.77° are needed to rotate the edge and achieve radii of curvature of 1.25 m and 4 m, respectively. Figure 13 shows the predicted surface and crosssectional profile of the CFRP mirror with a rotation angle of −3.38° uniformly applied at the edge of the mirror. This surface is the deviation from a best-fit sphere with radius of curvature of 1.25 m resulting in a peak-to-valley surface error of about 380 μm. 6. Conclusion
The NRL and SNL are currently working with CMA to develop thin-shelled replicated mirrors to be used in an active optical zoom system. In this paper, we have introduced a FEM developed from measurements of sample CFRP to model the prototype CFRP mirror. Currently, a 0.2 m diameter prototype CFRP mirror has been fabricated and initial measurements have been performed, and it has been used to validate the FEM. The measurements performed have shown that the FEM is an accurate and powerful tool to model actuation techniques to explore their viability prior to costly prototyping. The FEM has been used to model three different actuation techniques and their results have been presented. These particular techniques introduce aberrations that may be corrected with a high-density-actuator, small
microelectromechanical deformable mirror. These actuation techniques are being explored further and system development is under way. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. This work was supported, in part, by Sandia National Laboratories—Laboratory Directed Research and Development program, the Office of Naval Research, and the Naval Research Laboratories. The authors would especially like to thank Dr. Mike Duncan, Dr. Sergio Restaino, and Dr. Ty Martinez for their support. References 1. S. R. Restaino, C. C. Wilcox, J. R. Andrews, T. Martinez, F. Santiago, S. W. Teare, R. Romeo, R. Martin, and D. M. Payne, “16ʺ OTA prototype telescope project at the Naval Research
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