**2. Mirror modeling**

2 Adaptive Optics

sensors. The use of voice coil actuators allows for large stroke and exact positioning of the mirror while it is floating in a magnetic field. By changing the spatial properties of the magnetic field, the mirror shell can be deformed into a desired shape to compensate for optical aberrations measured by the AO system. Thereby, a low-contact bearing combined with little intrinsic damping of the mirror shell draws the need for adequate damping in closed loop operation. For the MMT deformable mirror, this problem is tackled by a 40 *µ*m air-gap between the mirror shell and a reference plate behind the shell. The induced viscous damping is sufficient to operate the mirror within the designed specifications [11]. However, recent deformable mirrors for the LBT and VLT shall operate within a larger air-gap to provide more stroke. As a side effect, the requirement for a larger air-gap reduces the natural viscous damping and electronic damping is needed to achieve the same control bandwidth [14]. Therefore, the LBT and VLT type deformable secondary mirrors are controlled by local PD-control for each actuator/sensor pair in combination with feedforward force compensation [15]. Each actuator is driven by a dedicated local position controller running at 40-70 kHz. The shape command for the mirror unit is generated by a higher level wavefront control loop running at about 1 kHz. For small set-point changes of the mirror shell, this control concept is well-suited and has proven to be applicable in practice [16].

For high speed deformations over large amplitudes (e.g. chopping of deformable secondary mirrors), disadvantages of local PD-control must be considered. First of all, there is a shape-dependent stiffness and damping variation of the mirror shell. In particular, each deformation of the shell requires a specific amount of external force by the distributed actuators [11]. If local control instead of global control is used for position control of the shell, then robust and subsequently conservative controller design is necessary for all local control loops. Secondly, interaction of neighboring actuators has to be studied carefully. The control loop gains have to be chosen such that only little interaction with neighboring control loops is caused. Otherwise, the local control concept can lead to instability of the shell.

In order to further investigate practical concepts for DM shape control, there have been studies on MIMO optimal feedback control [17–19]. Certainly, the closed loop performance of a global optimal feedback control concept is superior to local PD-control. But still, the computational load of a global MIMO controller may not be suitable for large deformable mirrors with more than thousand actuators. Only in [17], the circular symmetry of the mirror shell is used to reduce the controllers complexity. Thereby, it is shown that symmetry can effectively reduce computational loads without loosing control performance. Although, even when symmetry is fully exploited, the computational effort for a global feedback control

The most difficult task for control of deformable mirrors clearly is not the stabilization of the shell in a static shape, but changing the mirrors deflection in a predefined time and maintaining system stability. Instead of using a pure feedback controller for this task, model-based feedforward control concepts are proposed as in [20, 21]. In the past, the concept of model-based feedforward control has successfully been applied to various classes of dynamical systems. This technique is widely used in control practice as an extension of a feedback control loop to separately design tracking performance by the feedforward part and closed-loop stability and robustness by the feedback part. By using model-based feedforward control for shape control a deformable mirror, the feedback control loop is only needed to stabilize the shell in steady state and along precomputed trajectories. Thereby, the feedback loop can be designed to achieve high disturbance rejection. The computational load for feedforward control is comparably low because the feedforward signals may not be

concept of future deformable mirrors is considerably high.

Deformable membrane mirrors with non-contacting actuators are often represented by static models approximating the nonreactive deformation of the mirror surface. Thereby, both Kirchoff and van Kármán theory is used to describe plate deformations smaller than the plate thickness [23–26] and deformations close to the thickness of the mirror plate [27–29], respectively. Additionally, finite element methods are used to model deformable mirrors, in particular for the development large deformable secondary mirrors in astronomy [30–36]. In order to describe actuator influence functions, radially symmetric Gauss functions or splines are commonly used [2, 37], also. A more detailed static analysis is performed in [38] and [39] for a circularly clamped deformable mirror using a Kirchoff plate model. Additionally, in

<sup>1</sup> In this context, feedback control is mainly needed to account for model uncertainties and to reject external disturbances (e.g. mechanical vibrations, wind loads).

10.5772/52726

103

http://dx.doi.org/10.5772/52726

Modeling and Control of Deformable Membrane Mirrors

shearing stresses *τxy*, *τyx*. The shear forces per unit length *Qx*, *Qy* arise from shearing stresses

**Figure 2.** Illustration of shear forces, bending and twisting moments, and external loads affecting a differential plate element

 *Qy* + *∂Qy ∂y* d*y* 

*∂Mxy ∂x* d*x* 

*∂Myx ∂y* d*y* 

+ *q* = *ρh*

*∂*2*w*

*<sup>∂</sup>y*<sup>2</sup> <sup>+</sup> *<sup>q</sup>* <sup>=</sup> *<sup>ρ</sup><sup>h</sup>*

d*x* + *q*d*x*d*y* = *ρh*d*x*d*y*

d*y* − *Qy*d*x*d*y* = 0, (2b)

d*x* − *Qx*d*x*d*y* = 0, (2c)

*<sup>∂</sup>t*<sup>2</sup> , (3a)

*<sup>∂</sup>t*<sup>2</sup> . (4)

*<sup>∂</sup><sup>x</sup>* <sup>−</sup> *Qy* <sup>=</sup> 0, (3b)

*<sup>∂</sup><sup>x</sup>* <sup>−</sup> *Qx* <sup>=</sup> 0. (3c)

*∂*2*w*

*∂*2*w <sup>∂</sup>t*<sup>2</sup> , (2a)

d*y* − *Qy*d*x* +

*∂Qx ∂x* + *∂Qy ∂y*

*<sup>∂</sup><sup>y</sup>* <sup>−</sup> *<sup>∂</sup>Mxy*

*∂My*

*∂Mx ∂y* + *∂Myx*

single equation in terms of various moments can be achieved reading

*∂*2*Mxy*

*<sup>∂</sup>x∂<sup>y</sup>* <sup>−</sup> *<sup>∂</sup>*2*Mxy*

 *Mxy* +

 *Myx* +

with *x*, *y*, *t* ∈ **R**. Rotary-inertia effects of plate elements as well as higher-order contributions to the moments from loading *q* have been neglected in the moment equations (2b) and (2c).

By solving the last two equations for *Qx* and *Qy* and substituting in the first equation, a

+ *∂*2*My*

*∂y∂x*

Of particular interest are the resulting three equations of motion

d*x* − *My*d*x* + *Mxy*d*y* −

d*y* − *Mx*d*y* − *Myx*d*x* +

By canceling terms, the equations of motion reduce to

*∂*2*Mx <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

*∂Qx ∂x* d*x* 

*τxz*, *τyz* [41].

*h*d*x*d*y* of the deformable mirror.

−*Qx*d*y* +

*∂My ∂y* d*x* 

*∂Mx ∂x* d*x* 

 *My* +

 *Mx* +  *Qx* +

**Figure 1.** Schematics of a circular deformable mirror with deflection *w*(*r*, *θ*, *t*), inner radius *r*1, outer radius *r*2, plate thickness *h*, and external forces *q*(*r*, *θ*, *t*) acting on marked actuator positions over the mirror area *S*

[40] a deformable mirror with a free outer edge is modeled based on a Kirchoff plate model and resulting actuator influence functions for point forces are given.

Unfortunately, many of these modeling approaches only concentrate on the static characteristics of the deformable mirror and neglect dynamic properties like temporal eigenfrequencies and spatial characteristics of inherent eigenmodes. For model-based controller design addressed here, particularly these eigenfrequencies and eigenmodes are of substantial interest and can be found with the following modeling approach.

At first, circular deformable membrane mirrors with considerable out-of-plane stiffness and possible finite inner radius *r*<sup>1</sup> and outer radius *r*<sup>2</sup> centered at the origin of the *r* − *θ* plane are considered. Hereby, *r*, *θ* are polar coordinates. The plate is modelled as an isotropic Kirchoff plate with constant thickness *h* as shown in Figure 1.

For dynamic analysis, large deflections of the plate are not considered and nonlinear effects such as tensile stresses of the plate are neglected [28, 41, 42]. Secondly, it is assumed that a native curvature of the shell can be neglected due to only small deflections perpendicular to the surface.<sup>2</sup> The time varying deflection of the plate is measured by *w*(*r*, *θ*, *t*) relative to the undeflected reference and *t* ∈ **R**<sup>+</sup> is the time. External forces caused by non-contacting voice coil actuators are represented by *q*(*r*, *θ*, *t*) and are defined by

$$q(r,\theta,t) = \sum\_{m=1}^{M} \frac{1}{r\_m} \mu\_m(t) \delta(r - r\_m) \delta(\theta - \theta\_m) \tag{1}$$

where *um*(*t*) corresponds to the actuator force at position (*rm*, *θm*) and *δ*(*r* − *rm*)*δ*(*θ* − *θm*) describe the point-shaped force transmission.

The governing forces for a differential element of the plate are derived in Carthesian coordinates for simplicity. The differential element *h*d*x*d*y* is effected by various shear forces, bending and twisting moments, and external loads as illustrated in Figure 2. The bending moments per unit length *Mx*, *My* arise from distributions of normal stresses *σx*, *σy*, while the twisting moments per unit length *Mxy*, *Myx* (shown as double-arrow vectors) arise from

<sup>2</sup> As discussed in [43], a possibly large radius of curvature and a large diameter of the deformable mirror spherical face-sheet (e.g. 2*m*) in comparison to the considered deflections (usually around 100*µm*) allows to approximate the spherical shell by a Kirchhoff plate. Similar assumptions are drawn in [44] for a comparable system.

shearing stresses *τxy*, *τyx*. The shear forces per unit length *Qx*, *Qy* arise from shearing stresses *τxz*, *τyz* [41].

**Figure 2.** Illustration of shear forces, bending and twisting moments, and external loads affecting a differential plate element *h*d*x*d*y* of the deformable mirror.

Of particular interest are the resulting three equations of motion

4 Adaptive Optics

*S*

*q*

*h*, and external forces *q*(*r*, *θ*, *t*) acting on marked actuator positions over the mirror area *S*

and resulting actuator influence functions for point forces are given.

plate with constant thickness *h* as shown in Figure 1.

coil actuators are represented by *q*(*r*, *θ*, *t*) and are defined by

*q*(*r*, *θ*, *t*) =

describe the point-shaped force transmission.

*M* ∑ *m*=1

1 *rm* 1

**Figure 1.** Schematics of a circular deformable mirror with deflection *w*(*r*, *θ*, *t*), inner radius *r*1, outer radius *r*2, plate thickness

[40] a deformable mirror with a free outer edge is modeled based on a Kirchoff plate model

Unfortunately, many of these modeling approaches only concentrate on the static characteristics of the deformable mirror and neglect dynamic properties like temporal eigenfrequencies and spatial characteristics of inherent eigenmodes. For model-based controller design addressed here, particularly these eigenfrequencies and eigenmodes are

At first, circular deformable membrane mirrors with considerable out-of-plane stiffness and possible finite inner radius *r*<sup>1</sup> and outer radius *r*<sup>2</sup> centered at the origin of the *r* − *θ* plane are considered. Hereby, *r*, *θ* are polar coordinates. The plate is modelled as an isotropic Kirchoff

For dynamic analysis, large deflections of the plate are not considered and nonlinear effects such as tensile stresses of the plate are neglected [28, 41, 42]. Secondly, it is assumed that a native curvature of the shell can be neglected due to only small deflections perpendicular to the surface.<sup>2</sup> The time varying deflection of the plate is measured by *w*(*r*, *θ*, *t*) relative to the undeflected reference and *t* ∈ **R**<sup>+</sup> is the time. External forces caused by non-contacting voice

where *um*(*t*) corresponds to the actuator force at position (*rm*, *θm*) and *δ*(*r* − *rm*)*δ*(*θ* − *θm*)

The governing forces for a differential element of the plate are derived in Carthesian coordinates for simplicity. The differential element *h*d*x*d*y* is effected by various shear forces, bending and twisting moments, and external loads as illustrated in Figure 2. The bending moments per unit length *Mx*, *My* arise from distributions of normal stresses *σx*, *σy*, while the twisting moments per unit length *Mxy*, *Myx* (shown as double-arrow vectors) arise from

<sup>2</sup> As discussed in [43], a possibly large radius of curvature and a large diameter of the deformable mirror spherical face-sheet (e.g. 2*m*) in comparison to the considered deflections (usually around 100*µm*) allows to approximate the

spherical shell by a Kirchhoff plate. Similar assumptions are drawn in [44] for a comparable system.

*um*(*t*)*δ*(*r* − *rm*)*δ*(*θ* − *θm*), (1)

of substantial interest and can be found with the following modeling approach.

$$-Q\_{\mathbf{x}}\mathbf{d}y + \left(Q\_{\mathbf{x}} + \frac{\partial Q\_{\mathbf{x}}}{\partial \mathbf{x}}\mathbf{dx}\right)\mathbf{d}y - Q\_{\mathbf{y}}\mathbf{dx} + \left(Q\_{\mathbf{y}} + \frac{\partial Q\_{\mathbf{y}}}{\partial \mathbf{y}}\mathbf{dy}\right)\mathbf{dx} + q\mathbf{dx}\mathbf{dy} = \rho\hbar \mathbf{dx}\mathbf{dy}\frac{\partial^2 w}{\partial t^2},\tag{2a}$$

$$\left(M\_y + \frac{\partial M\_y}{\partial y} \text{d}x\right) \text{d}x - M\_y \text{d}x + M\_{xy} \text{d}y - \left(M\_{xy} + \frac{\partial M\_{xy}}{\partial x} \text{d}x\right) \text{d}y - Q\_y \text{d}x \text{d}y = 0,\tag{2b}$$

$$\left(M\_{\mathbf{x}} + \frac{\partial M\_{\mathbf{x}}}{\partial \mathbf{x}} \mathbf{dx}\right) \mathbf{d}y - M\_{\mathbf{x}} \mathbf{d}y - M\_{\mathbf{y}\mathbf{x}} \mathbf{dx} + \left(M\_{\mathbf{y}\mathbf{x}} + \frac{\partial M\_{\mathbf{y}\mathbf{x}}}{\partial \mathbf{y}} \mathbf{dy}\right) \mathbf{dx} - Q\_{\mathbf{x}} \mathbf{dx} \mathbf{d}y = \mathbf{0},\tag{2c}$$

with *x*, *y*, *t* ∈ **R**. Rotary-inertia effects of plate elements as well as higher-order contributions to the moments from loading *q* have been neglected in the moment equations (2b) and (2c). By canceling terms, the equations of motion reduce to

$$\frac{\partial Q\_x}{\partial x} + \frac{\partial Q\_y}{\partial y} + q = \rho h \frac{\partial^2 w}{\partial t^2} \, \text{\AA} \tag{3a}$$

$$\frac{\partial M\_y}{\partial y} - \frac{\partial M\_{xy}}{\partial x} - Q\_y = 0,\tag{3b}$$

$$\frac{\partial M\_{\mathbf{x}}}{\partial \mathbf{y}} + \frac{\partial M\_{\mathbf{y}\mathbf{x}}}{\partial \mathbf{x}} - Q\_{\mathbf{x}} = \mathbf{0}.\tag{3c}$$

By solving the last two equations for *Qx* and *Qy* and substituting in the first equation, a single equation in terms of various moments can be achieved reading

$$
\frac{
\partial^2 M\_x
}{
\partial x^2
} + \frac{
\partial^2 M\_{xy}
}{
\partial x \partial y
} - \frac{
\partial^2 M\_{xy}
}{
\partial y \partial x
} + \frac{
\partial^2 M\_y
}{
\partial y^2
} + q = \rho h \frac{
\partial^2 w
}{
\partial t^2
}.
\tag{4}
$$

In [41], the relationship between the moments and the deflection is described and it is shown that the bending moments per unit length read

$$M\_{\mathbf{x}} = -D\left(\frac{\partial^2 w}{\partial \mathbf{x}^2} + \nu \frac{\partial^2 w}{\partial y^2}\right),\tag{5a}$$

10.5772/52726

105

*w*(*r*1, *θ*, *t*) = 0 (10a)

Modeling and Control of Deformable Membrane Mirrors

= 0 , (10b)

http://dx.doi.org/10.5772/52726

= 0 (11a)

= 0. (11b)

= 0, (12a)

= 0. (12b)

or a simply supported edge at *r* = *r*<sup>1</sup> reading

*ν r*2

or a free edge at *r* = *r*<sup>1</sup> reading

(*ν* − 2) *r*2

−*c*

*∂*2*w*(*r*, *θ*, *t*) *∂θ*<sup>2</sup> <sup>+</sup>

> *ν r*2

*∂*3*w*(*r*, *θ*, *t*)

can be extended by a righting moment *c* reading

+ *ν r*2

*∂*3*w*(*r*, *θ*, *t*)

simply supported edge, (c) a free edge, and (c) a spring supported edge type.

the PDE will be analyzed further in the following section.

*∂w*(*r*, *θ*, *t*) *∂r*

(*ν* − 2) *r*2

*∂*2*w*(*r*, *θ*, *t*) *∂θ*<sup>2</sup> <sup>+</sup>

*<sup>∂</sup>r∂θ*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*3*w*(*r*, *<sup>θ</sup>*, *<sup>t</sup>*)

*∂*2*w*(*r*, *θ*, *t*) *∂θ*<sup>2</sup> <sup>+</sup>

*<sup>∂</sup>r∂θ*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*3*w*(*r*, *<sup>θ</sup>*, *<sup>t</sup>*)

−1 *r*

(a) (b) (c) (d)

**Figure 3.** Schematics of four typical boundary conditions for deformable membrane mirrors showing (a) a clamped edge, (b) a

The PDE (7) and the boundary conditions (9)-(11) fully describe both the temporal and the spatial behavior of the modelled deformable mirror. In order to quantify the resulting eigenfrequencies and eigenfunctions for a given mirror geometry and material properties,

−1 *r*

*∂*2*w*(*r*, *θ*, *t*) *<sup>∂</sup>r*<sup>2</sup> <sup>+</sup>

*ν r*

*∂*2*w*(*r*, *θ*, *t*) *<sup>∂</sup>r*<sup>2</sup> <sup>+</sup>

*<sup>∂</sup>r*<sup>3</sup> <sup>+</sup> (<sup>3</sup> <sup>−</sup> *<sup>ν</sup>*)

*∂*2*w*(*r*, *θ*, *t*) *<sup>∂</sup>r*<sup>2</sup> <sup>+</sup>

In order to include a flexible support at *r* = *r*<sup>1</sup> (see Figure 3 (d)), boundary conditions (11)

*∂*2*w*(*r*, *θ*, *t*) *<sup>∂</sup>r*<sup>2</sup> <sup>+</sup>

*<sup>∂</sup>r*<sup>3</sup> <sup>+</sup> (<sup>3</sup> <sup>−</sup> *<sup>ν</sup>*)

*∂*2*w*(*r*, *θ*, *t*) *<sup>∂</sup>r*<sup>2</sup> <sup>+</sup>

*r*3

*r*3

*∂w*(*r*, *θ*, *t*) *∂r*

> *ν r*

1 *r*2

> *ν r*

1 *r*2 *∂w*(*r*, *θ*, *t*) *∂r*

*∂*2*w*(*r*, *θ*, *t*) *∂θ*<sup>2</sup>

> *∂w*(*r*, *θ*, *t*) *∂r*

 *r*=*r*<sup>1</sup>

 *r*=*r*<sup>1</sup>

 *r*=*r*<sup>1</sup>

*∂w*(*r*, *θ*, *t*) *∂r*

*∂*2*w*(*r*, *θ*, *t*) *∂θ*<sup>2</sup>

> *∂w*(*r*, *θ*, *t*) *∂r*

 *r*=*r*<sup>1</sup>

 *r*=*r*<sup>1</sup>

$$M\_{\mathcal{Y}} = -D\left(\frac{\partial^2 w}{\partial y^2} + \nu \frac{\partial^2 w}{\partial x^2}\right),\tag{5b}$$

$$M\_{\rm xy} = -M\_{\rm yx} = D(1 - \nu) \frac{\partial^2 w}{\partial x \partial y'} \tag{5c}$$

$$D = \frac{Eh^3}{12(1 - \nu^2)},\tag{5d}$$

with Young's modulus *E* and Poisson's ratio *ν*. By using the relationship *Mxy* = −*Myx* the biharmonic partial differentail equation (PDE)

$$D\left(\frac{\partial^4 w}{\partial x^4} + 2\frac{\partial^4 w}{\partial x^2 \partial y^2} + \frac{\partial^4 w}{\partial y^4}\right) - q = -\rho h \frac{\partial^2 w}{\partial t^2} \tag{6}$$

is derived.

Using (1) for a point actuated circular plate, Equation (6) can be written in polar coordinates as

$$D\nabla^4 w(r,\theta,t) + (\lambda\_d + \kappa\_d \nabla^4) \frac{\partial w(r,\theta,t)}{\partial t} + \rho h \frac{\partial^2 w(r,\theta,t)}{\partial t^2} = \sum\_{m=1}^M \frac{\mu\_{\text{fl}}(t)}{r\_m} \delta(r - r\_m) \delta(\theta - \theta\_m),\tag{7}$$

with the biharmonic operator ∇<sup>4</sup> given by

$$
\nabla^4 = \nabla^2 \nabla^2 = \left(\frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}\right)^2. \tag{8}
$$

The parameters *λ<sup>d</sup>* and *κ<sup>d</sup>* are used to characterize additionally included viscous and Rayleigh damping and need to be identified in practice.

In order to fully describe the spatio-temporal behavior of deformable mirrors, the biharmonic equation (7) must be completed by physically motivated boundary conditions of the mirror plate. Typical boundary conditions for deformable mirrors are illustrated in Figure 3 (a)-(c) and can either be a clamped edge at *r* = *r*<sup>1</sup> reading

$$w(r\_1, \theta, t) = 0\tag{9a}$$

$$\frac{\partial w(r,\theta,t)}{\partial r}\Big|\_{r=r\_1} = 0 \,,\tag{9b}$$

or a simply supported edge at *r* = *r*<sup>1</sup> reading

$$w(r\_1, \theta, t) = 0 \tag{10a}$$

$$\frac{\nu}{r^2}\frac{\partial^2 w(r,\theta,t)}{\partial \theta^2} + \frac{\partial^2 w(r,\theta,t)}{\partial r^2} + \frac{\nu}{r}\frac{\partial w(r,\theta,t)}{\partial r}\Big|\_{r=r\_1} = 0,\tag{10b}$$

or a free edge at *r* = *r*<sup>1</sup> reading

6 Adaptive Optics

is derived.

as

that the bending moments per unit length read

biharmonic partial differentail equation (PDE)

*D*∇4*w*(*r*, *θ*, *t*)+(*λ<sup>d</sup>* + *κd*∇4)

with the biharmonic operator ∇<sup>4</sup> given by

*D ∂*4*w*

In [41], the relationship between the moments and the deflection is described and it is shown

 *∂*2*w <sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>ν</sup>*

 *∂*2*w <sup>∂</sup>y*<sup>2</sup> <sup>+</sup> *<sup>ν</sup>*

*Mxy* <sup>=</sup> <sup>−</sup> *Myx* <sup>=</sup> *<sup>D</sup>*(<sup>1</sup> <sup>−</sup> *<sup>ν</sup>*) *<sup>∂</sup>*2*<sup>w</sup>*

with Young's modulus *E* and Poisson's ratio *ν*. By using the relationship *Mxy* = −*Myx* the

Using (1) for a point actuated circular plate, Equation (6) can be written in polar coordinates

*∂*4*w ∂y*<sup>4</sup>

*∂*2*w*(*r*, *θ*, *t*) *<sup>∂</sup>t*<sup>2</sup> <sup>=</sup>

> 1 *r ∂ ∂r* + 1 *r*2 *∂*2 *∂θ*<sup>2</sup>

*∂*2*w ∂y*<sup>2</sup> 

*∂*2*w ∂x*<sup>2</sup> 

*∂x∂y*

− *q* = −*ρh*

*M* ∑ *m*=1

2

*w*(*r*1, *θ*, *t*) = 0 (9a)

= 0 , (9b)

*um*(*t*) *rm*

, (5a)

, (5b)

, (5d)

*∂*2*w*

, (5c)

*<sup>∂</sup>t*<sup>2</sup> (6)

*δ*(*r* − *rm*)*δ*(*θ* − *θm*),

. (8)

(7)

*Mx* = − *D*

*My* = − *D*

*<sup>D</sup>* <sup>=</sup> *Eh*<sup>3</sup>

*<sup>∂</sup>x*<sup>4</sup> <sup>+</sup> <sup>2</sup> *<sup>∂</sup>*4*<sup>w</sup>*

*∂w*(*r*, *θ*, *t*)

∇<sup>4</sup> = ∇2∇<sup>2</sup> =

Rayleigh damping and need to be identified in practice.

and can either be a clamped edge at *r* = *r*<sup>1</sup> reading

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>ρ</sup><sup>h</sup>*

 *∂*<sup>2</sup> *<sup>∂</sup>r*<sup>2</sup> <sup>+</sup>

The parameters *λ<sup>d</sup>* and *κ<sup>d</sup>* are used to characterize additionally included viscous and

In order to fully describe the spatio-temporal behavior of deformable mirrors, the biharmonic equation (7) must be completed by physically motivated boundary conditions of the mirror plate. Typical boundary conditions for deformable mirrors are illustrated in Figure 3 (a)-(c)

> *r*=*r*<sup>1</sup>

*∂w*(*r*, *θ*, *t*) *∂r*

12(1 − *ν*2)

*<sup>∂</sup>x*2*∂y*<sup>2</sup> <sup>+</sup>

$$\frac{\nu}{r^2} \frac{\partial^2 w(r, \theta, t)}{\partial \theta^2} + \frac{\partial^2 w(r, \theta, t)}{\partial r^2} + \frac{\nu}{r} \frac{\partial w(r, \theta, t)}{\partial r} \Big|\_{r = r\_1} = 0 \tag{11a}$$

$$\begin{split} \frac{(\nu - 2)}{r^2} \frac{\partial^3 w(r, \theta, t)}{\partial r \partial \theta^2} - \frac{\partial^3 w(r, \theta, t)}{\partial r^3} + \frac{(3 - \nu)}{r^3} \frac{\partial^2 w(r, \theta, t)}{\partial \theta^2} \\ - \frac{1}{r} \frac{\partial^2 w(r, \theta, t)}{\partial r^2} + \frac{1}{r^2} \frac{\partial w(r, \theta, t)}{\partial r} \Big|\_{r = r\_1} = 0. \end{split} \tag{11b}$$

In order to include a flexible support at *r* = *r*<sup>1</sup> (see Figure 3 (d)), boundary conditions (11) can be extended by a righting moment *c* reading

$$-c\frac{\partial w(r,\theta,t)}{\partial r} + \frac{\nu}{r^2} \frac{\partial^2 w(r,\theta,t)}{\partial \theta^2} + \frac{\partial^2 w(r,\theta,t)}{\partial r^2} + \frac{\nu}{r} \frac{\partial w(r,\theta,t)}{\partial r} \Big|\_{r=r\_1} = 0,\tag{12a}$$

$$\frac{(\nu-2)}{r^2} \frac{\partial^3 w(r,\theta,t)}{\partial r \partial \theta^2} - \frac{\partial^3 w(r,\theta,t)}{\partial r^3} + \frac{(3-\nu)}{r^3} \frac{\partial^2 w(r,\theta,t)}{\partial \theta^2}$$

$$-\frac{1}{r} \frac{\partial^2 w(r,\theta,t)}{\partial r^2} + \frac{1}{r^2} \frac{\partial w(r,\theta,t)}{\partial r} \Big|\_{r=r\_1} = 0.\tag{12b}$$

**Figure 3.** Schematics of four typical boundary conditions for deformable membrane mirrors showing (a) a clamped edge, (b) a simply supported edge, (c) a free edge, and (c) a spring supported edge type.

The PDE (7) and the boundary conditions (9)-(11) fully describe both the temporal and the spatial behavior of the modelled deformable mirror. In order to quantify the resulting eigenfrequencies and eigenfunctions for a given mirror geometry and material properties, the PDE will be analyzed further in the following section.
