to the mirror edge in the two regions 0„ ± n/Nl ; thus, the effect of the successive moments generated by the forces acting on the pivots is almost cancelled and gives rise to the required increased shear push-pull distribution.

8.6.4 Finite Element Analysis

In addition to their first computational modeling of large astronomical mirrors axially supported by concentric continuous rings or discrete pads in three-fold symmetry, Malvick and Pearson [46] derived the flexure in various cases of lateral support force distributions. Some of these early results from finite element analysis are shown in Fig. 8.13.

Fig. 8.13 Finite element analysis flexures from lateral supports of the 4-m mirror Mayall-KPNO telescope. Left: Special push-pull force distribution. Right: Radial push force distribution [46]

Finite element analysis has now become an indispensable method for the accurate determination of the elastic deflections of a mirror. When finite element analysis is conjugated with Seidel wavefront mode analysis, the residual low spacial frequency modes resulting from the lateral support flexure can also be corrected by closed-loop in situ active optics applied to the axial support system.

8.7 Active Optics and Active Alignment Controls 8.7.1 Introduction and Definitions

The main optics of a large telescope can benefit from active corrections allowing recovery of the image quality for which the telescope was designed. For a ground-based telescope, numerous reasons cause the starlight image to deteriorate: optics manufacturing residuals, theoretical errors of mirror passive support systems - such as described in the latter sections -, focussing and centering errors from the mechanical structure, maintenance errors of the structure and mirror supports, thermal distortions of the mirrors, telescope structure and ambient air, mirror deformation from wind buffeting, atmospheric degradation from turbulence, and tracking errors. All these sources of error were systematically quantified in terms of time-dependent frequency bandpass by Wilson et al. [90]. We hereafter mainly follow and summarize the account by Wilson [93].

• The HST early study: For a space-based telescope, all these image degradations vanish except the thermal gradient effects on the optics and structure, and tracking errors; the effect of the gravity change from 0g to 1 g may also alter the mirror shape. Although a space telescope is the best choice for high-resolution imaging up to the diffraction limit, early investigations (1960s) on the active control of a primary mirror figure were undertaken at Nasa by Creedon and Lindgren [19] for the HST. Howell and Creedon [29] developed a sensing wavefront and mirror reshaping system that was applied on the ground to a 30-inch diameter model mirror with 0.5-inch thickness. Using 58 actuators, the initial error of X/2rms at X = 0.663 nm was reduced to X/50rms. The control system used the natural lateral vibration modes -i.e. minimum energy modes - of the mirror [19].

With 24 actuators at the 2.4-m primary mirror of the HST, the system would have compensated for half a micron of astigmatism sag. This early study must be considered as a pioneering work. However, because of its complexity, its application to the Hst was finally rejected. In can be noted that the correction range would have been far too small to compensate for the spherical aberration due to a calibration error in the optical testing. This problem was then solved by the introduction of an aspherical correcting plate in a post-focal pupil. At that time, such an active optics system would not have been very efficient since CCD detectors appeared later.

• Active optics control and active alignment control: From definitions in Sect. 1.14.3, one may define the active controls of a telescope as follows.

An active optics control system is a low time frequency bandpass process involving corrections of wavefront errors by changing the shape of a telescope mirror, generally the primary mirror; these corrections may include partial wavefront errors caused by misalignments with other mirrors. Concerned actuators are included in the highest level of the mirror axial support system.

An active alignment control system is a low time frequency bandpass process involving corrections of wavefront errors caused by misalignments between the telescope mirrors. Concerned actuators are dedicated to the relative positioning in space of, generally, the secondary mirror. The associated displacements - or "optical despaces" - are the piston (z translation), lateral centering (x and y translations) and tip-tilt (x and y rotations). These five despaces can be obtained by use of an active hexapod truss which links the secondary mirror to its reference support plane.6

6 A hexapod truss is a three-dimensional structure made of three sets of adjacent bar pairs in a "VVV" arrangement in a way that the upper and lower parts of the V's form two distinct triangles located in parallel planes. The system links three hinge points of the mirror to three hinge points

The interdependence conditions between the two above control systems must be strictly set.

For both the above systems, the "open-loop" control can be restrained to passive or quasi-passive corrections whilst "closed-loop" control is relative to active.

A higher time frequency closed-loop level for adaptive optics corrections can be introduced on a telescope mirror if this mirror is extremely thin and possesses its own actuators.

• Aberration evaluation algorithm for the alignment process: Misalignments of the primary and secondary mirrors provide on-axis coma and an astigmatism variation in the field and, in general, the axes of the two mirrors do not intersect. From wavefront analyses of several star beams distributed over the field of view, say at a Cassegrain focus, a tilt, a lateral shift or both of the secondary mirror allow setting the on-axis coma to zero. The result is that now the two axes intersect at the so-called coma neutral point details are given by Wilson et al. [90-93] and Rakich [63], but because of the residual angle formed by the two axes there exist two points in the field where the astigmatism is minimal, usually called field binodal astigmatism. The final alignment consists of dispaces of the secondary mirror about the coma neutral point up to obtaining the superposition of the mirror axes. This is achieved when the binodal astigmatism merges into an axisymmetric distribution all over the field.

8.7.2 Monolithic Mirror Telescopes

• The ESO NTT and VLT: The first ground-based monolithic mirror telescopes equipped with a complete active optics and alignment control systems were the 3.5 m Ntt whose first light was in 1989, and the 8.2 m Vlt of Eso whose first telescope unit started observations in 1998. The principle of the closed-loop control is based on the analysis at Nasmyth or Cassegrain focus of starlight wavefronts by Shack-Hartmann (S-H) systems using a lenslet and CCD. Several guide star beams can either be selected from an on-axis beamsplitter or from offset regions to the field edge (Fig. 8.14).

of its reference support, thus forming a statically rigid 6-hinge truss. Independent system controls of each bar length of the V's provide the above five required displacements.

The conditions for rigid trusses were first discussed by August Ferdinand Möbius [44], in 1837, who was the professor of astronomy at the University of Leipzig and is best known for his work in analytic geometry and in topology. Denoting the number of the hinges by N and that of the bars by B, he found that necessary conditions for two- and three-dimensional trusses are respectively

In addition to these fundamental results, Möbius indicated that even if these conditions are satisfied there are exceptional cases where the truss is not absolutely rigid because the triangulation is improperly done. In studying such exceptional cases, he found that they occur when the determinant of the static equilibrium equation system at the truss joints vanishes (see History of Strength of Materials by Timoshenko [87]).

Fig. 8.14 Principle of active optics and alignment controls for a monolithic mirror telescope [57]

Fig. 8.14 Principle of active optics and alignment controls for a monolithic mirror telescope [57]

Taking into account the azimuth for a non-axisymmetric wavefront, each Seidel mode of a wavefront may be written as where Anm coefficients characterize the sag of the wavefront error and p = 1 is the edge normalized radius of the wavefront pupil.

The principle of the Shack-Hartmann test is to image the telescope mirror input pupil into subpupils by use of a two-dimensional microlens array; thus, the CCD image analysis provides a measure of the local slope errors of the wavefront dw/dx, dw/dy, where w = Zwnm. While the telescope is tracking a star of magnitude V = 12-13, closed-loop cycles - typically 10 minutes and 60 seconds of time for the Ntt and Vlt respectively - deliver resulting correction signals that are transduced either into a new force configuration of the axial support system of the primary mirror or into relative displacement and rotation of the secondary mirror.

A preliminary experiment by Wilson et al. [92] and Noethe et al. [58] on a thin meniscus mirror of one meter diameter was carried out for checking some basic features. With the Ntt, a primary mirror aspect ratio of t/d = 1/15 and Na = 75 axial supports allows corrections of the seven Seidel terms w11, w20, w22, w31, w40, w33 and w44. Because of a calibration error in the optical testing, the primary mirror showed a Sphe3 error of ^3.50¡im in p4. Instead of refiguring the mirror, the problem was solved by the axial support system in introducing a first level of passive correction with springs; the second level for passive diurnal corrections is with astatic levers, and the third active level uses actuators modifying the axial forces. Located on the same site as the conventional - i.e. passive - 3.6-m Eso telescope (La Silla, Chile), the results from the rather thick primary mirror of the Ntt clearly proved the efficiency of active optics and alignment controls. More details are given in Wilson [91].

Much more optimized for active optics control is the thin primary meniscus mirror of the Vlt (Fig. 8.15) with an aspect ratio t/d = 1/47 and Na = 150 axial supports. The flexibility z/F ^ d2 /Et3 to an external force F is ~13-times larger than that of the Ntt. The axial support system consists of a first level with hydraulic

Fig. 8.15 View of a VLT Unit at Cerro Paranal (courtesy ESO)

tripod pads and a second active level with spring-hydraulic actuators [68]. Similarly to the Ntt, all the axial forces can be only generated in the push mode, which then does not allow maintenance operations of the telescope in the horizontal direction. Noethe [57] investigated the modal form of the correction to be generated by the closed-loop control command and determined, from the shallow shell theory, the orthogonal polynomials representing the natural lateral vibration modes of a meniscus; these minimum energy modes, which differ from Zernike's polynomials, provide an optimal process for generating the force distribution corrections. Within the correction range of the Vlt primary mirror, it was shown by Noethe that the active component of the axial forces can be considered as linear with the deformation (Hooke's law). Given a total piston sag Az of a minimum energy mode (n, m), representing a mirror shape error composed of Seidel modes m = constant in the order KO = n + m - 1 (with conditions (1.47) in Sect. 1.8.2), one can derive this linear relation. Denoting K the number of the S-H subpupils, the relation between the column vector of the force change by the Na actuators and the corresponding z-displacement vector of the mirror measured at the sensor writes

where a— is the stiffness matrix. For each minimum energy mode n, m, the associated a-1 coefficients can be determined from experiment but were more accurately determined from theoretical analysis. The largest active optics effect is the configuration conversion from Nasmyth to Cassegrain which, because of an appreciable axial image shift, introduces for m = 0 an error polynomial of about 12¡im in p4;

this leads to generation of maximum active force components of about +413 and -156N within an actuator z-step increment (absolute accuracy) corresponding to a typical force of 0.05 N [56]. A review on active optics control aspects for large optical telescopes is given by Noethe [60].

• Other Active Optics Telescopes: The next ground-based monolithic mirror telescopes with active optics control for visible and infrared observations were designed from preliminary active optics experiments on small mirrors by Itoh et al. [32] and Iye et al. [34], which led to the construction of the 8.2-m Subaru telescope; a combined axial-lateral support system uses 264 cylindrical cavities in the rear side of the meniscus mirror where direct astatic levers act as lateral passive forces [33]. The two 8-m Gemini telescopes are equipped with an active support system [80]; the passive level is by air pressure supporting 75% of the weight whilst the active level is with hydraulic pressure actuators.

8.7.3 Segmented Mirror Telescopes

• Early Approaches: Some preliminary aspects on the recent development of large segmented telescopes may be briefly recalled. For instance, a 4 m, f/1.4/13, segmented infrared collector was built in the 1970s by Chevillard, Connes et al. [15]. The initial goal was to obtain high-resolution infrared spectra of planets and astronomical objects by use of a Michelson-type Fourier transform interferometer installed at coude focus. A detailed review is given by Connes and Michel [16]. The alt-az collector was built with a spherical primary mirror of 36 square segments actively controlled and a highly elliptic secondary provided a stigmatic narrow field; although the active alignment control of the segments showed tracking star images with Fwhm of ^5 arcsec, the project was stopped by lack of support and the interferometer never installed.

Also with a spherical segmented primary mirror concept, Baranne and Lemaitre [7] proposed a five-mirror two-axes multifocus telescope (Temos) where, apart from the Mi mirror and flat 45°M3, one of the other mirrors Mi can be actively in situ aspherized after initial spherical polishing without stress. Let q be an active uniform load applied to the clear aperture of a vase-form mirror Mi of curvature 1/R. Its shape from in situ stressing - at the telescope - can be represented by

where k = 0 and q = 0 when the mirror M; is unstressed (sphere). The total in situ aspherization of a mirror correcting the important Sphe 3 amount of a spherical primary mirror is an interesting challenge. The simplest way for an experimental validation is to choose a two-mirror system and mirror Mi as the Cassegrain mirror, thus M2. This leads to qc2, qc4,... values that provide a quasi-elliptic secondary corresponding to a partial vacuum inside the vase-form secondary. An experimental telescope was built and tested on the sky with 1.4 m Temos 4, f/2/6.6, in z a configuration using four circular spherical segment replicas of 0.5 m diameter, a stainless steel vase-form secondary of 0.35 m diameter and a doublet lens corrector. With a mean aspect ratio t/d = 1/53, mirror M2 was in situ aspherized by a load q = 0.789 x 105Pa corresponding to a center-edge sag of 204.7¡m. Seeing limited images of stars with Fwhm < 0.8 - 1 arcsec were obtained, but the seeing conditions at Haute Provence did not allow full evaluation of the angular resolution [39].

• Alignment and cophasing of the KECK Telescope: The 36 hexagon segments of each primary mirror of the Keck telescope (Fig. 8.16) were mainly aspherized by active optics stress polishing to form a hyperboloid primary mirror of 10 m aperture at f/1.7 (cf. Sect. 7.5.1). A segment is axially supported by three 12-point whif-fletrees and laterally supported by a passive single metal diaphragm mounted in a central cavity. Each of the four subpivots of a whiffletree are equipped with straight springs that can generate moments for partial passive correction of the segment figure residuals. Each of the three whiffletrees is mounted on an axial translation actuator ensuring the piston and tilt control of a segment. Each actuator uses a ball screw pushing a two-stage demultiplying hydraulic bellows that acts on the central pivot of the whiffletree with a translation increment accuracy of 30 nm. Details are given by Mast and Nelson [78]. Pairs of capacitive sensors located at intersegment edges provide accurate measures of surface boundary errors between adjacent segments. When the segments are aligned (see below), the reading of 168 sensors is stored in rms values as a reference set. Then, the readout of the sensors is done every tenth of a second and the piston and tilt corrections by the actuators twice a second.

The alignment of the mirrors and the cophasing of the passive primary segments are determined from a wavefront sensor, the phasing camera system (Pcs) - based

Fig. 8.16 View of the Keck telescopes at Mauna Kea (courtesy W. M. Keck Observatory)

on the Shack-Hartmann principle with starlight -, which can operate in four modes [12]. The "passive tilt mode" collects the light of each segment into one spot per segment and measures the tilt error of the segments. The "fine screen mode," where the light of each segment is sampled in 13 subareas, can measure also the defocus and decentering coma aberrations of the telescope optics, generated by despace (tilt, lateral, and axial shifts) of the secondary mirror. A telescope global defocus and coma introduce, for each segment subaperture, local defocus and astigmatism respectively. After evaluations, these errors are corrected by movements of the secondary mirror. Instead of conventional lenslets, the first mode uses 36 prisms and a lens and the second uses additional multi-aperture masks [14]. The "ultra fine screen mode" is a classical Shack-Hartmann test with ^200 lenslets for precise checking of a segment shape. The "segment phase mode", developed by Chanan, Troy et al. [13], is a physical optics generalization of the Shack-Hartmann technique. The phasing procedure uses the starlight reflected by 78 circular subapertures, 120 mm in diameter, which are located at the middle of the intersegment edges. For an atmospheric coherence diameter of r0(0.5¡um) = 20cm and rms piston errors varying in the range ^Zrms e [0, 200 jum], theoretical results show that the Fwhm of the images is of 0.5 arcsec at X = 0.5 jum whatever the 4zrms-value and that the phasing is effectively irrelevant. However, since r0 scales as X6/5 , the results also show that the phase errors significantly reduce the image central intensity for X > 1 jm; for X = 5 jum and ^Zrms = 500nm, the central intensity is reduced by ^60%. For this reason, and also for efficient use of adaptive optics, the phasing tolerance goal of the Keck telescopes is ^Zrms < 30nm, i.e. corresponding to an actuator incremental step. The Strehl ratio due to piston errors ^Zrms for a telescope made of ns segments is [11]

ns which, for ^Zrms = 30nm and X = 0.7 jum, leads to S = 0.76 for ns = 36 segments.

For the first three modes, the integration time of the Pcs is of the order of 30 seconds. The passive mode tilt uses stars of magnitude V = 9 whilst the other three modes require V = 45. The segment phase mode can take 20 or 90 minutes with the narrow- or broad-band algorithms respectively. In a routine situation, a full alignment takes approximately one hour.

• LAMOST and in-situ aspherized segments: The large sky area multi-object spectroscopic telescope (Lamost) (Fig. 8.17) is a 4-m all-reflective Schmidt telescope of the siderostat mount type with a 5° field of view (cf. Sects. 1.6.3,4.3.4, and 5.3.6). It was designed by Su, Cui et al. [84]. Both plane in situ aspherized primary mirror M1 and concave spherical secondary mirror M2 are made of hexagon segments, 1.1 m in diameter. Input pupil mirror M1 is mounted at a nodal plane of an alt-az hemispherical truss. Mirror M2 is in a fixed position. For typical integration times of 1.5 hour, the effect on the images of the atmospheric differential refraction over the large field of view was analyzed by Su and Wang [85].

Three purposes are required for the active optics control: (i) in situ aspherization of the plane mirror M1 into a shape determined by homothetic ellipse level lines,

Fig. 8.17 View of LAMOST primary mirror M1 in its alt-az hemispheric mount. This mirror is the telescope input pupil and reflects 4-m circular beams whatever the declination angle S e [-10°, 90°] of the observed region in the sky (courtesy Niaot/Cas)

whose sag is a function of the declination angle S e [-10°, 90°], (ii) alignment and co-focusing of 24 M1 segments, and (iii) alignment and co-focusing of 37 M2 segments.

Preliminary active optics experiments for in situ aspherization were carried out by Su, Jiang et al. [82] with an Mi-type outer-most segment of 500 mm diameter and 6 mm thickness (aspect-ratio t/d = 1/83); a set of 58 actuators and three fixed points distributed over concentric circles allowed correcting the spherical aberration of a spherical M2. Shack-Hartmann analysis of up to ten Seidel modes in the form (8.53) showed resulting wavefront errors < 30nmrms. Using an on-scale Lam-ost segment pair M1 - M2 in the outer-most configuration, active optics tests on the sky by Cui et al. [21, 22] with M1 aspherized by 35 actuators and three fixed points provided S-H wavefront errors < 80nmrms. Experiments by Su, Zou et al. [83] concerned the alignment and co-focusing of three subsegments by six actuators and capacitive sensors; Shack-Hartmann tests showed tilt errors < 0.035 arcsec and diffraction limited images. The final design for the active optics aspherization of the M1 segments, 1.1m long in the diagonal, uses 35 actuators including three fixed points including 18 actuators near the hexagonal edge.

• Other segmented telescopes: Other built large telescopes are the 11 m Het [36] at McDonald Observatory and 10 m Salt in South Africa [81]. With a segmented spherical primary mirror, both designs are a tilted-Arecibo concept where the telescope platform can rotate 360° in azimuth. A four-mirror spherical aberration corrector (cf. for instance O'Donoghue [61]) mounted a the prime focus enables a celestial object to be tracked across 12° sky. These telescopes are mainly used for spectroscopy with fibre feed options as by D. Buckley for Salt.

8.7.4 Cophasing of Future Extremely Large Telescopes

The next generation of astronomical telescopes with aperture diameters reaching 30 or 40 m, referred to as extremely large telescopes (ELTs), will have highly segmented primary mirrors, probably in a range ns e [750,1500] in number.

Other sensing systems than the Shack-Hartmann are applicable to segmented mirrors. For instance, the wavefront curvature method, originated by Roddier [66] for the testing of ground-based telescopes, is developed by Cuevas et al. [20] and Montoya-Martinez et al. [51]. Esposito et al. [25] have shown that piston and tip-tilt errors can also be derived from a pyramidal beam separator. Wavefront sensing based on the principle of Mach-Zehnder interferometry have been investigated and experimented by Montoya-Martinez et al. [50] and Yaitskova et al. [99]; the wave-front in one of the interferometer arms is spatially filtered to provide the reference wavefront so the two recombined beams produce an interferogram of the wavefront.

Theoretical investigations on the effects of segment cophasing errors are of fundamental importance to obtain high resolution imaging. For an ns segment mirror, the effect of piston errors (z translation) on the Strehl ratio is given by (8.56). The effect of tip-tilt errors (x, y rotations) on the point-spread-function have been analyzed by Yaitskova and Dohlen [94, 98] showing that, for ns hexagonal segments, the Strehl ratio can be approximated by where 4zrms is the sag of the tilt from flat to flat edges, and y a shape contour factor whose value is y = V55/6 for a hexagon. To the second order, this expression coincides with Marechal's approximation. Unlike the case of piston errors [cf. (8.56)], this equation has no strong dependence on the segment number ns. For a tip-tilt error of Orms = 2n/30, corresponding to 4zrms = 30nm at X = 0.7¡m, we obtain

In these two latter papers, other effects such as the diffraction caused by the intersegment gaps and segment edge misfigure have also been evaluated.

Active optics corrections of the figure residual errors of each segment is under investigation by Noethe [59]; such systems may use an appropriate number of controlled three-directional stressing strips distributed on the rear side of the segments.

8.8 Special Cases of Highly Variable Thickness Mirrors 8.8.1 Introduction - Mirror Flexure in Fast Tip-Tilt Mode

Secondary or tertiary telescope mirrors may be required to generate either tip-tilt functions for the adaptive optics correction of first-order atmospheric degradation modes - field stabilization mirror - or a single axis rotation function for the thermal noise substraction of infrared observations - wobbling mirror.

Because of the necessarily high frequency of the tip-tilt mode (at least 100 Hz), a special mirror design with a variable thickness distribution (VTD) allows minimizing the elastic deformation. The problem of the determination of the mirror geometry can be enounced as follows: Given a material and a finite volume - or mass -, what are the mirror VTD and associated support geometry that provide a minimal flexure during tip-tilt motions?

A preliminary approach for solving this problem consists of determining a VTD that minimizes the flexure due to gravity g. For a mirror with rotational symmetry, VTD solutions for several supporting cases are investigated. In the g case, one will see that the resulting thickness geometries lead to "linear prismatic edge mirrors." Such thickness shapes are useful distributions for minimizing the kg flexure from fast tip-tilt motions.

8.8.2 Minimum Flexure in Gravity of a Plate Supported at its Center

Within the Love-Kirchhoff hypotheses of elasticity theory, one preliminarily investigates the basic theoretical case of minimizing the flexure of an axisymmetric plate in the field of gravity g when in a horizontal plane and supported at its center.

The static equilibrium of an elementary segment of size dr, rdQ, t, is derived about its local tangential axis. This involves the components of the radial and tangential bending moments Mr, Mt and the shearing force Qr arising at the element. The equilibrium is achieved if (cf. for instance Sect. 2.1.2)

dr where the bending moments are defined from the rigidity D(r) = Et3(r)/[12(1 -v2)] and the flexure z(r) by w r,fd2z v dz\ / d2z 1 dz\

dr2 r dr dr2 r dr

For a plate supported at its center, one can easily define the radial shearing force Qr at radius r e [0, a] as the partial weight AW of the plate corresponding to the region r e [r, a] taken per length unit, i.e. Qr = AW/2nr. At the outer edge r = a of the plate, this entails Qr = 0. Hence, the radial shearing force is expressed by

where ¡1 and g are the density of the plate and the acceleration of gravity. From the latter equations, the substitutions into (8.58) give d3z fl dD 1\d2z (v dD 1\ dz ¡igfa J n drz + DdD+-rjdrz + rDdD-72)dZ+nglrtdr=0 (8.61)

Only using the flexure z(r) and thickness t(r) as unknowns, we obtain dh ( 1 dt! I ^ d!z (v dtL 1 ^ dz dr3 \t3 dr r J dr2 \rt3 dr r2) dr

E rt3 Jr which is the general equation of the flexure of an axisymmetric VTD plate under its own weight and supported at its center.

For comparisons of various VTD mirrors, it is useful to define a mean thickness t for which the mass of the mirrors M = Kjia2t is the same, so the mean thickness is

a2 Jo

This leads to definition of the associated dimensionless thickness T(r) such as

where p = r/a is the normalized radius.

Two flexure cases are investigated and compared hereafter: a constant thickness plate and a plate which shows a parabolic flexure.

• Constant thickness plate: The basic case of a constant thickness plate provides a reference center-edge flexure for further comparisons with VTDs. In this case [37], t (r) =t = constant, T(r) = 1, and since dV2 • /dr = d/dr[d2 • /dr2 + (1/r)d • /dr], (8.62) reduces, after a first integration, to

V2z = -^ (2a2lnr - r2 + 2d), ¡3 = 12(1 - v2)^, (8.65)

where d is a constant. Since the Laplacian also writes V2- = (1/r)d/dr(rd • /dr), the second integration leads to dz = _ 3 dr = 16

The thickness of the plate is finite and also is the flexure. Since dz/dr\r=0 ^ one must set C2 = 0. The bending moment Mr must be zero for the edge, thus, from (8.59), d2z + v dz = o dr2 r dr '

This boundary condition entails, after calculation, that

which leads to the slope dZ

0 0

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