Flexure Mounts for High Resolution Optical Elements



Paper Review: Flexure Mounts for High Resolution Optical Elements

Vukobratovich D, Richard R M, Proc of SPIE Vol. 0959, Jan 1988

Mir Salek

1- Introduction

Flexures are passive mechanical structures which isolate an optical element from the mechanical and thermal effects of the structural support. By definition, flexure is an elastic element which provides controlled motion. Thermal effects are both the transient and steady-state temperature variations. Generally, an optical mount should have the following specifications:

1- It must exert low stress on the optics

2- It should have high stiffness to maintain the alignment of optics

3- It must maintain the specified tolerance in the operational temperature range

4- It must maintain the position of the optical element throughout its assigned life time.

5- The mount size and weight should be minimized

6- It must have the minimum cost

2- Flexures as Comparison to Kinematics and Semi-Kinematics Design

A flexure mount has the following advantages over kinematic or semi-kinematic design

1- Free of slick-slip and friction effects of semi-kinematic design

2- Less hysteresis than rolling or sliding contacts

3- More robust to adverse environment effects such as extreme temperatures, vacuum, and abrasive dust

4- Needs very little maintenance if any

These advantages make flexures ideal for space applications. Flexure mounting is similar to semi-kinematic mounting; the flexure location is determined by kinematics principles: the forces on the optics should act through its center of gravity.

3- Flexure Material

Material selection is usually the final step in the flexure design. Generally, flexure material should provide the required compliance within the length limitation as well as dimensional stability for repeated use throughout time. Fracture toughness and thermal properties are other important parameters of flexure material.

Compliance of a flexure is dependent both on its shape and material. For a given length of material, the maximum compliance is achieved by a material which has the highest reduced tensile modulus. Reduced tensile modulus is the ratio of yield strength to modulus of elasticity.

Stability is important because of continuous stress in flexure material. Micro-yield strength (MYS) is a common figure of merit of dimensional stability of material. Although MYS is usually considered a safe limit, dimensional instability or room temperature creep, can happen in stresses less than MYS as well.

Room temperature creep in most cases can be predicted with Andrade’s Beta Law which states that creep is proportional to the cube root of time of applied stress:

ε = βtm

where ε is the creep strain as a function of time, β is constant dependent on material, temperature and stress. “m” is also dependent on material and is usually about 0.33.

Flexure toughness determines its resistance to fracture or cracking. Most high-strength materials have low fracture toughness. Especial alloys alleviate this problem. To reduce thermal effects, a good match of coefficient of thermal expansion (CTE) of optics and flexure material is helpful. However, thermal conductivity in most cases is not an important parameter.

4- Flexure Designs

Mounting flexures are a combination of simple or primary flexure geometries. In many cases compliance is added to a flexure mounting point by another flexure to compensate assembly-error-induced moments. This situation is shown in Figure 1. An alternative for this in some cases is higher precision fabrication. However, in many cases such precision is not practical.

[pic]

Figure 1: Additional flexure helps to compensate assembly error

4-1- Single Strip Flexure

A single strip flexure as shown in figure 1 is simply a cantilever. It can be used to guide both translation and rotation. The strain is a function of axial preload. The relationship between moment and tilt and also force and displacement of simple strip flexure under different axial stress is summarized in Table 1.

In the table L is the flexure length; E is the elastic modulus; I is the moment of inertia; P is the applied axial load; θ is the end slope of the flexure; M is the applied torque; δ is the end displacement of the flexure; F is the applied force; and

[pic]

[pic]

Figure 2: Single-strip cantilever[1]

Table 1: Relations between moment and tilt and force and displacement of simple strip flexure under compressive/tensile or no axial stress

|Relation |Axial Preload |The Relation |

|Moment and Flexure End |0 |[pic] |

|Slope | | |

| |Compressive |[pic] |

| |Tensile |[pic] |

|Force and edge translation|0 |[pic] |

| |Compressive |[pic] |

| |Tensile |[pic] |

4-2- Cross-Strip Rotational Hinge

Two single stripped flexures at right angles provide a rotational hinge. The motion is not pure rotation but the center of rotation shifts as a function of angle of rotation:

[pic]

The rotation angle as a function of applied moment to the flexure and the axial load on it is summarized in Table 2. The relations are valid as long as θ < 0.1 radian.

[pic]

Figure 3: Cross-strip rotational flexure[2]

Table 2: Rotation angle as a function of applied torque for cross-strip rotational flexure

|Relation |Axial Preload |The Relation |

|Moment and Flexure |0 |[pic] |

|Rotation | | |

| |Compressive |[pic] |

| |Tensile |[pic] |

4-3- Parallel Spring Guide Flexure

A pair of parallel single strip guides provides linear translation. The range of motion is limited to 1-2mm. The structure is shown in Figure 4. Here also the motion is not purely linear and there is a height shift as well:

[pic]

And “h” is the slab widths

Also if the force is not applied exactly at the midpoint of the blades, the flexure will tilt. Fabrication errors can introduce tilt in the blades as well.

The displacement-force relations as a function of axial preload are summarized in Table 3. In the table

[pic]

[pic]

Figure 4: Parallel-spring guide

Table 3: Force-displacement relation for parallel-spring guide flexure

|Relation |Axial Preload |The Relation |

|Force-displacement |0 |[pic] |

|relation | | |

| |Compressive |[pic] |

| |Tensile |[pic] |

4-4- Cruciform and Tapered Uniform-Stress Cantilever Flexure

Cruciform flexures provide limited rotation in very confined spaces. Figure 5 illustrates this flexure. Rotation-torque relation for this type of flexure is given by

[pic]

[pic]

Figure 5: Cruciform Flexure

The tapered uniform-stress cantilever flexure is used to provide a small range of translation motion in very confined space. The structure is shown in Figure 6. The displacement as a function of the force is given by:

[pic]

[pic]

Figure 6: Uniform-stress cantilever flexure

5- Flexure Mounts

A three-point flexure mount is shown in figure 7. This flexure mount was first introduced by Chin to be used in a space telescope. The flexures can be either tangent bars or cantilever. Tangent bars are connected to the optics from their center.

[pic][pic]

Figure 7: tangent bar and Cantilever mounting of lens

Figure 8 shows a bipod flexure mirror mount. Three bipod flexures attached to the mirror back restrain it in all 6 degrees of freedom. Each bipod acts like a two-strip rotational flexure. By changing the angles of the two flexures, the center location of the flexure assembly could be tuned to the mid-plane of the mirror. This is not possible with kinematic mounts at the back of the mirror.

[pic]

Figure 8: Bipod flexure mount

6- Conclusion

Basics of flexure design were introduced. Flexures can isolate the optics from stresses caused by temperature/pressure change as well as assembly error. Flexures relax tight tolerances which are often required by other mount types. In some cases, such as mounting in cryogenic temperatures, flexures might be the only choice.

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[1] Figure 6 of the summarized paper

[2] Figure 7 of the summarized paper

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