Probe guidance and focus drive for the Nanomefos optical sensor

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Probe guidance and focus drive for the Nanomefos optical

sensor

Citation for published version (APA):

Ravensbergen, S. K. (2006). Probe guidance and focus drive for the Nanomefos optical sensor. (DCT rapporten; Vol. 2006.127). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/2006

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Probe guidance and focus drive

for the Nanomefos optical sensor

S.K. Ravensbergen BSc (Student no: 0509840) DCT report #2006.127

Report Master Traineeship (4W809) Technische Universiteit Eindhoven Department of Mechanical Engineering

Section Dynamics and Control (prof.dr. Henk Nijmeijer) subsection Constructies en Mechanismen

Coach: dr.ir. Nick Rosielle November 21, 2006

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Abstract

For the measurement machine NANOMEFOS (Nanometer Accuracy NOn-contact MEa-surement of Freeform Optical Surfaces), currently developed by Rens Henselmans, an elas-tic guidance for the opelas-tical probe is needed. This guidance has to full following demands: no/minimal friction and hysteresis, an optimal inherent straightness, high bandwidth and minimal heat dissipation. Two dierent guidance types with matching actuator have been designed that can meet these requirements. The guidance based on six folded plate springs, three on each side under 120o_{, turned out to be a better solution than the one based on}

the double parallellogram principle. Although they have, by coincidence, the same heat dissipation during a normal measurement, the guidance using a double parallellogram will be more dicult to manufacture. In this setup a proper synchronization is designed but the correct mechanical connection with the parallellogram will be hard to realize. The rst disturbing eigenfrequency of this guidance is 1030 Hz, which is much lower than the 1920 Hz of the guidance using the folded plate springs.

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Introduction

In this traineeship the goal is to design an elastic guidance with minimal friction and hysteresis, optimal straightness, high bandwidth and minimal heat dissipation for the measurement machine Nanomefos (Nanometer Accuracy NOn-contact MEasurement of Freeform Optical Surfaces). This machine is designed at the moment by Ir. R. Henselmans as basis of his doctorate. The objective is to measure optical surfaces that have non-axial-symmetrical shapes, called freeform (optics), in a measurement time of approximately 15 minutes. To do so, the freeform is placed on a turntable below a measurement machine with an axis in radial (r) and an axis in height direction (z), see gure 1.1.

Below the z-axis the -axis is mounted on the machine to rotate the measurement head above the freeform. This -axis can rotate between an angle of 45o _{to 135}o _{to measure}

convex and concave freeform optics. To measure the position of the -axis, two laser beams are used as shown in gure 1.1 and a rotational encoder on one end. The elastic guidance that has to be designed is mounted on that -axis and preferably has to t within the dimensions of 55 mm in diameter and 50 mm in height. To be able to measure a large variety of freeform optics the stroke of the guidance has to be 5 mm. Other main requirements of the guidance are listed in table 1.1. The uncertainty goal of the measurement of a freeform is 10 nm at an perpendicular measurement, and 35 nm with a 5o _{tilt of the measurement head w.r.t the optical surface.}

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Table 1.1: Current probe characteristics and guide requirements of the guidance. focal length (stand-o) 1.5 [mm]

focal depth 2 [m]

spot size 2 [m]

objective and mirror mass 3 [g]

maximum speed 100 [mm/s]

maximum acceleration 20 [m/s2_]

orientation -45o _{to 135}o _{with respect to g#.}

dimensions 100 [mm] between spot and centerline -axis. -axis diameter is 70 [mm]

range 5 [mm]

straightness 25 [nm], (after calibration)

alignment 10 [rad] (to optical axis)

actuator voice coil

lowest disturbing eigenfrequency 200 [Hz]

Z R c z _x y + 1 3 5 - 4 5 o o - a x i s m e a s u r e m e n t b e a m c - a x i s m e a s u r e m e n t b e a m c T u r n t a b l e f r e e f o r m c - a x i s

Figure 1.1: Machine layout, showing the radial (r) and height (z) directions, also the local axis-system is dened.

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1.1 Optical measurement

The measurement itself is done by a beam with a diameter of 2 mm. This beam comes from a corner cube in the middle of the -axis. The design of the non-contact probe (see [Cac06]) is shown in gure 1.2. A cilinder of 5 mm round that beam is kept free to be able to make some adjustments and alignments in the optical layout. The beam is focused on the freeform using an a-spherical lens. On both sides of the beam two beams of 3 mm are placed to measure the position of the guidance. At the bottom a mirror is placed with in the middle the aspherical lens. The two outer beams re ect on the mirror and enable a optical interference measurement of the position of the lens, the beam in middle is focused on the surface of the freeform. Using the focus error signal during a whole rotation of the freeform, the height of the lens and mirror are known and thus the shape of the freeform. The measured data of the outer two beams are used to actuate the guidance and thus to move the mirror with lens.

Figure 1.2: Optical probe design, the guidance will be placed between the cube corners and the mirror.

The design space is shown in gure 1.3, where the outer distance of the beams is 13.5 mm and the overall diameter of the guidance is 55 mm. It is also possible to place a square shaped guidance in the design space of 55 55 mm. To minimize measuring errors due to diused light, it is preferable to leave the space between the beams open. So a tube with some inner spacing around the three beams is allowed but something around only the middle beam is rejected.

The guidance has to translates along a (nearly) straight line i.e. it should be a parallel guidance for the mirror and focal lens. Two potential best solutions that meet this property are one based on the double-parallellogram principle and one based on six folded plate-springs. Both types of the guidance will be explained further on in this report dealing with

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Figure 1.3: Design space overview, the space between the beams is preferably left open.

the pros and cons of the specic solution and what type of actuator will be used to drive that kind of guidance.

1.2 Material choice

To be able to make a relatively long stroke, with an elastic guidance that ts in the design space, a proper selection has to be made to nd a suitable material. For both of the above mentioned elastic guidances a material is needed that has a low elastic modulus (E) to minimize the actuation forces and a high allowable stress (), combining both gives a parameter that denes the allowable strain: =E. Another material parameter (using also the density ) that helps to choose the material is =pE, representing the maximal stroke that can be obtained when making a plate spring out of this material. In [Bon95, page 64] a group of materials is compared resulting in the titanium alloy TiAl6V4, also called

tialv, as the most suitable material to use in plate springs (E = 114 GPa, = 880 MPa, = 4430 kg/m3_{). Using this material in the design of the probe guidance a lifetime of 10}

years has to be taken into account, resulting in 108 _{cycles. As shown in the fatigue gure}

1.41 _{the maximal stress at 10}7 _{cycles is approximately 500 MPa. A more specic value}

can be found on matweb (see appendix A for the data), where the unnotched (meaning no grooves, holes etc. in the utilized region) value is 510 MPa. To meet the requirements the maximal stresses are restricted to 450 MPa.

To increase the buckling resistance thickened plate springs are used. Normally the length of the thin parts is 1=6 of the length of a normal plate spring, and the thick part (6 to

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Figure 1.4:Fatigue properties of tialv: maximum stress as function of number of cycles.1

10 times thicker) is 5=6 l long. Doing so the buckling force is 9 times higher, the stresses are the same and the stiness in direction perpendicular to the plate spring is only 20 % increased. An overview of dierent plate springs and corresponding formulas is given in appendix B.

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Chapter 2

Double parallellogram guidance

2.1 Overview

The basic concept of a so called double parallellogram is shown in gure 2.1, where body B is called the intermediate body and A is the main body. When B translates over a distance z=2 the plate springs c and d will de ect and shorten a little (for formulas see appendix B), resulting in a parabolic motion of B. When at the same time A translates over a distance z the plate springs e and f will shorten the same as c; d and therefore A moves along a straight line. This operation is only correct when the of motion of bodies A and B satisfy the proper ratio of 2:1. So only if body B makes exactly half of the stroke of body A a straight movement of A is obtained. This requires a synchronization between A and B and will be treated in section 2.2, but the use of a 1 over 2 'lever' is obvious.

B

A

e

c

f

d

z / 2

z

Figure 2.1: Basic double parallellogram setup.

Within the design space a few dierent orientations of the double parallellogram are possi-ble, they are shown in gure 2.2. In 2.2a the guidance is placed with a radial distance with respect to the three beams. An advantage of this setup is that there is enough room for the actuator placed radial opposite to the guidance. Disadvantage of this setup is the fact

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that there is little room for the synchronization within the double parallellogram. This can be solved by placing the 1 over 2 lever behind the guidance. However this will result in a longer stroke of the actuator (1.5 times the translation of the main body) when the actuator is connected to the lever to preform a correct actuation of the guidance. This is necessary because actuating the plate springs in the wrong way results in moments in the bodies and springs, and thus in a tilt of the guided body A. The correct location of the actuation force is in the middle between the bodies A and B, see [vE85, chapter 5] and [Ein03, chapter 6]. Elongating the lever leads to the location of the pivot of the lever to lay outside the design region which is not preferable. Placing the pivot at the height of body B and choosing the length of the lever the same as the plate spring length (approximately 17 mm), then the lever will be to small to manufacture.

In gure 2.2b the guidance is placed in the middle of the design space with a hole to allow the three optical measurement beams to pass. In this setup it is possible to use longer plate springs, a practical length could be 28 mm. Now it is also possible to place the synchronization (partially) inside the guidance. Two actuators can be placed, as shown, at each side of the double parallellogram. A disadvantage is now that the lever passes through the beams and the guidance. This problem will be solved in section 2.2. Figure 2.2c could be an improvement of 2.2b, because the stiness could be nearly the same in the translation direction but higher for the direction perpendicular to that. A disadvantage of this setup is the fact that it is more dicult to manufacture due to the slopes of the plate springs. Further comparison between the two setups can be based on nite element analysis, which is done in section 2.4.

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a

b c

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2.2 Synchronization of the double parallellogram

To synchronize the double parallellogram dierent solutions are possible, but they all have to meet the requirement of a straightness (after calibration) of 25 nm over a stroke of 5 mm, or they must be adjustable to do so, or the movement has to be calibrated and the error must stay beneath it.

2.2.1 'Simple' lever B A z 2 z l b l_a B A a _b

Figure 2.3: Dierent setups for synchronization of the double parallellogram, using in (a) a lever and in (b) a rocker to prescribe the motion of A and B.

As shown in section 2.1, particularly gure 2.1, a proper synchronization between inter-mediate and main body is required. Taking a look at the simple lever rst, see gure 2.3a, the connection rods also shorten during the movement. So the main body moves along a nearly straight line (over a distance 2z), the intermediate body along a parabolic curve (over a distance z) and the lever makes a circular motion round the pivot. This behavior given in formulas will result for the shortening of the lever in:

lever = Llever(1 cos()) (2.1)

where Llever represents the lever length, and the rotation angle of the lever is given by

= 2z=Llever. The vertical displacement of the intermediate body is given by (see also

appendix B):

B= ₁₂7 z 2

L : (2.2)

where L is the length of the plate springs. The shortening of the rod connecting the main body to the lever is the same as the motion of the lever, a= lever. The shortening of the

rod between the intermediate body and the lever is given by:

b = B+1₂a (2.3)

build up by the rst term, representing the parabolic motion of the intermediate body, and a term describing half the shortening of the lever itself. The actual displacement of the intermediate body and the main body can now be determined (see also gure 2.4):

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z _bz

2 z z _a

Figure 2.4: Shortening at the left half of the double parallellogram, the dotted circles are the desired displacement and the dotted lines the actual one.

The synchronization will be perfect if these two displacements where the same, thus using formula 2.2 for the shortening of the connection rods and choosing the length of the rod connecting the main body as la, the length of the other one (lb) is determined by:

lb = 2 2 b la

2

a (2.5)

Thus depending on the length of the plate springs in the double parallellogram and the length of the lever the dierent lengths of the rods can be chosen in such a way that there is a minimal deviation of the main body to the perfect straight translation. In gure 2.5 the position of the main body is shown as function of the lever angle. In this case the full scale represents a motion of 2.5 mm in one direction. For a correct length of laand lbthere

is a small (below one nanometer) displacement in radial direction due to the mismatch between the circular motion of the lever and the parabolic motion of the intermediate body even if there is a small deviation of 10 m in length. On the other hand, choosing the same length for la and lb, say 7 mm, then the straightness is 12 nm which is already

half of the required value for the guidance, still neglecting the manufacturing tolerances. Actually this deviation is repeatable so there is the possibility to calibrate it and removing the error in software but this is not advisable because there is a way to do it mechanical. In the derivation above, the assumption is made that the pivot of the lever rotates around one pole. But with a normal elastic cross spring pivot this is not the case. During the rotation the pole moves along a straight line, see [Bon95, page 14] and B.1e, so the correct 1 over 2 prescribed motion is disrupted with the linear motion of the pivot. To avoid this a pivot is used where the crossing of the springs is at 1=8 of the length. It is also possible to made this pivot using thickened plate springs, see appendix B for more details. In this application wire electric discharge machining (wire EDM) and normal EDM is used to create the lever. A benet of the material used, Tialv, is that it works easily using EDM. The process enables the manufacturing of free standing lms of 100 m thick, when the length of the wire is kept as short as possible (in this case 30 mm) and when the wire is surrounded with material during the erosion. But with a wire length of < 10 mm and very sensitive work free standing lms of 25 m are already made.

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 φ [rad] Straightness [nm] l_b l_b − 10 µ m l_b + 10 µ m

Figure 2.5: The straightness of the main body as a function of the angle of the synchronizing lever. Also the straightness is shown if lb is 10 m above or below the perfect value.

2.2.2 Rocker

In gure 2.3b synchronization using a rocker is shown where the concept is based on the motion of a rolling wheel. At the axle of the wheel half the displacement is made with respect to the top of the wheel. The connection to the main body A can be realized using strings under tension as shown in gure 2.6a. Connecting the rocker in this way will introduce no moments around the vertical axis. To link body B with the rocker two folded plate springs can be used at each side of the rocker, as shown in gure 2.6b. The torsion in the springs is not a normal use but the dimensions (length and width) can be optimized to ensure a constraint even under torsion. Using a cilinder at the bottom of the rocker and loading it in vertical direction, the rocker is fully constrained. Another option is by using two rolling cilinders in a channel, see gure 2.6c. These cilinders are pressed to the walls by a tensioned strip. Now the cilinders are able to roll within the slit with minimal slip and hysteresis. Connecting the top of the rocker with body A and one cilinder with body B a 1 over 2 ratio is obtained when the radius of the rocker is 3 times higher than the radii of the cilinders.

To choose between the two proposed options of synchronization, the question rises: is it manufacturable? The rocker needs some adjustments at the connection with the main body and the tension of them is also not easy. Notice that the rocker still have some shortening in those connections, depending on the type of the connection with the intermediate body, gure 2.6b or c. This can be minimized using another shape of the rocker instead of a circular one. Then the best way of making the rocker would be by using wire EDM and make it out of one piece of material. But now the lever and the rocker are both equal in diculty and on top of that the lever has the advantage of a hole in the middle where the measurement beams are able to pass through together whereas the rocker will be larger to enable the same. This is due to the dierence in distance between pivot point and

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the middle of the measurement beams, the rocker makes a larger rotation. The choice for the 'simple' lever is based on the manufacturing diculty, the feasibility of the dierent connections including the connection to the xed world: the -axis. Final geometry is shown in gure 2.9b and the dimensions in appendix C.

B A L o a d a b A B c

Figure 2.6:Connections between rocker and double parallellogram, in (a) the link using tensioned strings to connect the rocker with the main body. In (b) the rocker is held using two folded plate springs that are connected to the intermediate body B and a loaded cilinder at the bottom to x the vertical position. (c) Represents the guidance of the rocker using two rolling cilinders within a slit. The cilinders are pushed against the side walls with a strip under tension and the rocker is connected to the lower one. It is also possible that the rocker itself is one of the cilinders.

2.3 Linear actuator

To actuate the double parallellogram guidance, rst of all the required static and dynamic force has to be calculated. The rst part is implemented in a Matlab script using the static formulas of appendix B. In gure 2.7 that static force is plotted for a varying thickness of the plate spring and also the minimal length of the springs is shown where the stress reaches the maximum of 450 MPa. A normal and machinable thickness (using wire EDM) of the thin parts of the plate springs is 100 m, so the minimal length is 11 mm but then at the same time the force needed to make a translation of 2.5 mm is approximately 6 N. An actuator that can produce that force however would be too large. So iteratively

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0.024 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 6

8 10 12

plate spring thickness [mm]

plate spring length [mm]

0.020 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

0.5 1 1.5

plate spring thickness [mm]

Static force for 2.5mm displacement [N]

Figure 2.7: The minimal length L of a plate spring as function of the thickness and underneath it the static force needed to de ect the double parallellogram (L = 28 mm) over a distance of 2.5 mm as function of the thickness.

the dimensions are coupled together to design a solution for a guidance that has good straightness and minimal heat dissipation.

Taking an estimated value for the moving mass (containing the coil, holder and the equiv-alent masses of the main and intermediate body) the dynamic force is given by the second law of Newton, using the maximal acceleration given in table 1.1. To design an optimal actuator the free software of Ansoft: Maxwell 2D1 _{is used. Drawing an actuator in 2D,}

the force on the current carrying coil can be estimated as well as the distribution of the magnetic eld. To compare dierent actuators, and be able to choose between them, a relation between the force and the dissipated power is derived below. The power is given by:

P = I2R (2.6)

where the coil resistance R is given by:

R = _Acl (2.7)

using the electrical resistivity of copper c = 16.78 nm, the length of the coil l and the

cross sectional area A. Within the coil the current density is limited to J = 10 A/mm2

resulting in a maximum current through the coil (neglecting the winding factor). Increasing this value will cause a break down of the isolation of the wires used in the coil due to the high temperature. This value is also used to calculate the maximal power in table 1.1, using equation 2.6 and 2.7. The lorentz force can be calculated using:

F = ~I ~B l (2.8)

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h e i g h t w i d t h c o i l d i s t a n c e m a g n e t t h i c k n e s s c o i l h o l d e r t h i c k n e s s m a g n e t s c o i l a i r g a p

Figure 2.8:Variable dimensions of a lineair actuator.

Combining above equations results in:

P = _BF₂2_l₂ _Acl = F_B2₂_Vc (2.9) where V is the coil's copper volume. The (motor) constant to compare the dierent actu-ators therefore is:

= _FP₂ (2.10)

The smaller this value is, the better the actuator is, meaning: supplying more force with less power. Notice, using formula 2.9, that increasing the magnetic eld in the gap is more ecient than increasing the volume of the coil.

A basic setup for a linear actuator is shown in gure 2.8, where the design variables are shown. The outer measurements of the actuator width and height are limited due to the remaining build-in volume. Changing the cross sectional area of the coil A and the thickness of the magnets an optimal actuator is designed taking into account that normal soft magnetic material saturates at 1.2 tesla. Final dimensions and orientation within the assembly with the double parallellogram and synchronization lever are given in section 2.4 and in appendix C. The specications of the actuator are shown in table 2.1, where the rst set is made by an actuator using normal neodymium iron boron (NdFeB) magnets of grade N352 _{and normal iron as soft magnetic core. Using the stronger, but more expensive,}

grade N453 _{and a special soft core material hiperco50A from cartech}4 _{(see appendix A.2}

for material properties) the actuator is substantially improved.

2.4 The double parallellogram assembly

The above discussed elements: double parallellogram, synchronization and actuator are assembled into one nal design. An overview is shown in gure 2.9, the synchronization lever (b) is placed partially in the double parallellogram (c). The beams are surrounded by a rectangular tube that can hold the optics (mirror and focal lens) to the right and, at the top, connects the lever to the main body of the double parallellogram (d). Around the tube the synchronization lever is placed with one connection rod at the top and two for the intermediate body, this is done to minimize the rotation of the lever around the

2_{see: http://www.matweb.com/search/SpecicMaterial.asp?bassnum=NALLIAN01}

for material properties

3_{matweb: bassnum=NALLIAN05}

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a b

c d

e f

Figure 2.9:Assembly of the double parallellogram guidance, (a) the double parallellogram. In (b) the synchronization lever showing the hole in the middle for the three measurement beams, the connections to the main and intermediate body and the 1=8 cross spring pivot. Figure (c) shows the connection between the lever and the intermediate body and (d) the connection between the lever and the tube that holds the optics. The connection between the tube and the main body still has to be designed. Figure (f) shows the nal lay-out including the two linear actuators (e).

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Table 2.1: Specications of the lineair actuator Normal NdFe35 magnets and iron soft magnetic material:

maximum force 1.3 [N]

mean coil length 71 [mm]

coil cross sectional area 9 [mm2_]

maximum power 1.1 [W]

motor constant 0.64 [W/N2_]

Stronger NdFe45 magnets and hiperco50A soft mag. material:

maximum force 1.6 [N]

vertical axis. Also the connectors between the lever and the intermediate body are shown, the connector between the tube and the main body could be a plate at the left end and a rod between tube and body to totally constrain it to the main body. This connection is not fully designed yet, and needs some further attention if this setup turned out to be better than the six folded plate-spring guidance of chapter 3. The actuators (e) t exact on both sides inside the double parallellogram and are connected to the world on both sides. This is also the reason why this rectangular setup is preferred above the one using A-shaped plate springs of gure 2.2d. The maximal width of the plate springs is reached already and using plate springs with an A-shape of the same width will result in a decrease of the stiness in y-direction (use equation 2.11). The coils are connected to the main body using a coil-holder that is xed at the side of the body and two rods at the bottom to minimize the motion of the coils in y-direction; this mode is now at 1500 Hz (see gure C.5c). Other specic dimensions can be found in appendix C.

The mode of the guidance (the moving mass is 56 g) round the z-axis is approximately 1030 Hz. This value is obtained using a static force in y-direction to obtain the stiness of the double parallellogram and the following equation:

f = ₂1

r

c

m : (2.11)

The maximal stresses that occur in the thin parts of the thickened plate springs at maximal stroke is approximately 100 MPa, mainly based on the choice to get a low overall stiness in the guidance and have long plate springs to have sucient room for the synchronization and the actuators.

A disadvantage of this setup is the need of a connection to the -axis of the measurement machine at the top, bottom and sides of the assembly to connect the double parallellogram, the synchronization lever and the actuators.

To calculate the total dissipated power during a normal measurement, rst this 'normal' measurement has to be dened. Propose an averaged freeform the motion of the measuring head, and thus the guidance, would come close to a sinusoidal motion of 10 Hz with 1 mm amplitude. The maximal acceleration during this motion is !2 _{= (2f)}2 _{= 3:95}

m/s2_{. The dynamic force needed to accelerate 56 g is 0.22 N, the static stiness of double}

parallellogram and synchronization lever is 470 N/m, so the maximal static force is 0.47 N. The absolute mean force needed for this motion is:

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and equals 0.6 N. Using the best obtained lineair actuator constant = 0:41 W/N2 _the

dissipated power reads 0.15 W. If the guidance is for example in the lower position, the direction of the static force is opposite to the direction of the dynamic force. This is neglected in the analysis above.

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Chapter 3

Guidance with six folded plate

springs

Figure 3.1:Example of a guidance made of a monolith with six folded plate springs. (see [Ein03, chapter 1])

Another way to elastically guide a body along a (nearly) straight line is by using six folded plate springs: three on either end of a moving tube oriented around in 120 degrees. This setup is shown in gure 3.1. Using six springs the -direction is once overconstrained, but using ve springs the guidance isn't symmetrical anymore. However there is still the possibility after manufacturing to cut one plate spring if the 'click clack' eect is disrupting the measurement or the straightness of the guidance. A challenge is now to place the plate springs within the design space. Figure 3.2 shows some dierent possible solutions that enables the three measurement beams passing through one tube. An improvement can be made by changing the rectangular plate springs into A-shaped ones (gure 3.2b). The stiness in radial direction will nearly be the same but the stiness in z-direction will decrease from 2500 to 1700 N/m.

As shown above it is more dicult to place the three folded plate springs in the design space. To simplify the design also for this setup a Matlab-script is written to predict the stresses and required actuator forces at dierent thicknesses and lengths of the springs

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a b

Figure 3.2: Dierent setups of six folded plate springs within the design space. Within (a) also three linear actuators can be used, but they are much less ecient than a ring shaped-actuator that ts within all designs.

(used formulas are explained in appendix B). A good starting point are folded plate springs with a length of 17 mm for a spring that makes the whole stroke (horizontally placed) and 8 mm for the spring that has to compensate for the shortening of the longer ones. This combination has the potential to t within the design space and requires an actuator that also ts. The minimal plate length as function of the thickness of the thin plate parts and the total static force are shown in gure 3.3. Using a thickness of 0.09 mm, the maximal stresses in this setup are 390 MPa and the stiness is 2460 N/m. The static force needed to make the 2.5 mm translation is 6.2 N (as also shown in gure 3.3). To optimize the design A-form shaped plate springs are used where the stiness linearly decreases with the width of the thin parts of the springs. A nal stiness of 1700 N/m is obtained. Normally the guidances of this type are made monolithically out of a simple piece of material (see gure 3.1), to minimize errors in the production proces and assembly. In this design however, the choice is made to assemble the guidance. Three cilinders are separately turned on a lathe to make a circular tting and are provided with pins to x the rotation compared to each other. Then that assembly is manufactured further. Out of the two outer cilinders the folded plate springs are made, the inner cilinder is the basis of a ring-shaped actuator. The folded plate springs are made in the following steps: rst some larger pieces are cut out using normal EDM, to make a raw A-shape. Then the folded plate springs will be made using wire EDM. The 'legs' of the A-shaped springs can be milled before the wire EDM or eroded afterwards with normal EDM.

3.1 Ring-shaped actuator

To dene what type of ring-shaped actuator is the best, some dierent magnet locations are investigated and shown in gure 3.4. The comparison is made using the analysis of section 2.3. The setup of (a) is used by the company BEI1 _{and gives the best result at a}

low actuator diameter. The best thermal option is shown in 3.4b, where the soft magnetic material do not introduce stresses in the magnet in uctuating thermal conditions. Within this setup however the height of the actuator is a more critical design parameter, and the

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0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 10 12 14 16 18

Main leafspring thickness [mm]

Main leafspring length [mm]

0.050 0.06 0.07 0.08 0.09 0.1 0.11 0.12

5 10 15

Leafspring thickness [mm]

Static force for 2.5mm displacement [N]

Figure 3.3: The minimal length L of a plate spring as function of the thickness and below it the total static force needed to de ect the six folded plate springs (L = 17 mm) over a distance of 2.5 mm as function of the thickness.

Table 3.1: Specications of the ring-shaped actuator. Normal NdFe35 magnets and iron soft magnetic material:

force 5.4 [N]

mean coil length 134 [mm]

coil cross sectional area 9 [mm2_]

power 2.0 [W]

Stronger NdFe45 magnets and hiperco50A soft mag. material:

force 6.0 [N]

best actuator with the lowest height is shown in 3.4d. As displayed in the picture the actuator has a chamfer in the middle to allow a cone to connect the coil with a central tube. The specications of this actuator are given in table 3.1. It is possible to make the magnet in for example four segments to reduce stresses due to dierences in thermal expansion coecients.

3.2 Assembly of the six folded plate spring guidance

The design is shown in gure 3.5 in a cross-section, the length of the thickened plate springs is 17 mm, the thin parts are 90 m thick. The base of the A-shape is 18 mm and the thickness of the thick parts of the springs is 6 mm. Overall outside diameter of the guidance is 58 mm, 3 mm larger than the target value but this will cause no further problems. Other nal dimensions are shown in appendix D. At the top in gure

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c o i l m a g n e t s c o i l m a g n e t a _b c o i l m a g n e t c o i l m a g n e t c d

Figure 3.4: Cross sections with centerline displaying dierent orientations of the magnet within the ring-shaped actuator. The magnetization direction is shown and the coil as well. Figure (d) shows the nal design.

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Figure 3.5: Cross-section of the assembly of the six folded plate spring guidance with the ring shaped actuator.

3.5 is a part that connects the guidance to the -axis. There is on two sides some room for the measurement beams that measure the position of the axis (also shown in gure 3.7a). Underneath it, the upper monolith with three folded plate springs is placed using a cylindrical t. Below them a cone with a tube connects the upper and lower monolith with each other and makes a connection with the coil. The outer upper ring is also the limit for the maximal translation of 2.5 mm in the upward direction (so: away from the freeform). The cone contacts the monolith directly under the outer connection of the plate springs, best seen in the upper left part of the picture. Below that, the limit of the downward direction is made between the upper part of the actuator and the cone. Below the actuator, the monolith with the lower three plate springs is placed, also using cylindrical ts to position it. At the bottom, a cover is applied with a taper shape with an top angle of 125o _{to protect the plate springs. The individual parts are shown in gure}

3.6 to 3.8.

Analyzing the eigenfrequencies of the guidance (see gure D.2 for the mode-shapes) results in a rst disturbing frequency of 1924 Hz, this is the rotation of the tube round the z-axis. The displacement of the mean tube in radial direction is at 2225 Hz and the dierent modes where the thickened parts of the plate springs are moving in z-direction occur at 2311 Hz. Total mass of the assembly is approximately 400 gram, including the translating mass of 50 g. The mode in radial direction is also calculated using a force in that direction and formula 2.11. The obtained stiness is 7700 kN/m, so the frequency is 2400 Hz (using the moving mass of 50 g). The advantage of this calculation is the possibility to use the non-lineair solver in Unigraphics NX3 instead of the linear modal solver, but the results are comparable as shown above. Using the stiness of 1700 N/m and the actuator eciency of = 0:056 W/N2_{the total dissipated power during a normal measurement can}

be calculated in the same manner as in section 2.4. For this design the static force is 1.7 N, the dynamic force 0.2 N, then the mean force is 1.65 N and thus the dissipated power

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a b

c d

Figure 3.6:Monoliths with the three folded plate springs, showing in (a) the top view of the upper set springs that are close to the -axis and in (b) a bottom view. In (c) and (d) the same for the lower set springs that are close to the freeform.

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a (up side down) b

Figure 3.7: The connector with the -axis in (a) where the three holes for the springs can be distinguished and on the opposite side the two places that are left open for the -axis measurement beams. In (b) the lower housing with the taper shape of 125o_.

a b

Figure 3.8: The left gure shows the actuator with the circular soft magnetic core, the magnet with the magnetization in diameter direction, the coil with a cross-section of 1.56 mm and the coil holder. At the right the cone shaped spacer that connects the coil holder to the upper and lower set of plate springs.

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is 0.15 W. By coincidence this value is the same as the power dissipated by the double parallellogram guidance so the selection has to be made based on machinability and the simplicity of the assembly.

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Chapter 4

Conclusion and recommendations

To summarize and be able to give some recommendations on which type of guidance to choose, the main dierent properties and results are shown in table 4.1. As already men-tioned the dissipated heat during a normal measurement is the same for the two guidances. The rst disturbing eigenfrequency of the double parallellogram is approximately at 1030 Hz. In this mode the whole moving mass of the guidance (= main body, coil with carrier, tube with optics and partially the lever and the intermediate body) is rotating around the z-axis and will in uence the optical measurement. This frequency is a lot lower than the lowest disturbing frequency of the six folded plate springs, 1924 Hz. This rotation mode has less aect on the optical measurement itself. The rst considerable disturbing motion is the movement of the guidance in x or y-direction at 2220 Hz.

Table 4.1: Specications of both the guidance types.

Double parallellogram guidance Six folded plate springs

Stiness: 470 [N/m] 470 [N/m] 1700 [N/m]

Moving mass: 56 [g] 56 [g] 50 [g]

Motor const. 0.41 [W/N2] 0.41 [W/N2_] _{0.056 [W/N}2_]

Dissipated power during

a normal measurement 0.15 [W] 0.15 [W]

1st _{disturbing eig. freq:} _{1030 [Hz]} _{1924 [Hz]}

Looking at manufacturability of the components used in the both designs the lever of the double parallellogram and the three folded plate springs out of one piece are comparable in diculty. The advantage of the design with the six folded springs is the feasibility to easily connect the dierent parts using a circular tting made on a lathe. Also the connection with the -axis is more simple, one connector at one end of the guidance is sucient for a proper mounting. The mounting of the double parallellogram is more dicult to realize because a xed world is needed at four sides of the guidance.

So a guidance using six folded plate springs is recommended as a good solution for design problem stated the in the introduction. In the present layout there are still some improve-ments to be made in gap width of the ring shaped actuator. Now the air distance between coil and magnet is 1 mm, but optimizing the connection with the coil holder the actuator can be made even more ecient. Also the orientation of the folded plate springs must be investigated again. In the current design the folding line of the springs are inside, but placing those to the outside (for example in the upper set), the rotation stiness can be

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considerably improved (>3000 Hz). The disadvantage is a higher stress in the connector to the inner tube, but this perhaps can be solved using a good solid basis for the A-shape.

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Appendix A

Overview material properties

A.1 TiAl

6

V

4

All property values are taken from mathweb1_.

Component Weight percent

Al 6 Fe < 0.25 O < 0.2 Ti 90 V 4 Density 4430 [kg/m3_]

Yield stress 880 [MPa]

Young's modulus 113.8 [GPa]

Poisson's Ratio 0.342

Fatigue Strength 510 [MPa] (Unnotched 107 _Cycles)

Shear Modulus 44 [GPa]

CTE, linear 20o_C _{8.6 [m/(m}o_{C)] (20 ! 100}o_C)

Specic Heat Capacity 526.3 [J/(kgo_C)]

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A.2 Hiperco50A

Component Weight percent

C 0.01 Co 48.75 Fe 49 Mn 0.05 Nb 0.05 Si 0.05 V 1.9 Density 8120 [kg/m3_]

Yield stress 1275 [MPa]

Young's modulus 207 [GPa]

CTE, linear 20o_{C 9.5 [m/(m}o_{C)] (20 ! 200}o_C)

A.3 Neodymium Iron Boron magnetic material

Material properties of the rare earth magnets Nd2Fe14B:

Density 7500 [kg/m3_]

Ultimate tensile strength 80 [MPa]

Young's modulus 150 ! 160 [GPa]

CTE, linear 20o_C _{5 [m/(m}o_{C)] (20 ! 200}o_C)

N35 grade:

Magnetic coercive force Hc 12000 [Oe] = 955 [kA/m]

Magnetic remanence Br 11900 [Gauss] = 1.19 [T]

N45 grade:

Magnetic coercive force Hc 12000 [Oe] = 955 [kA/m]

Magnetic remanence Br 13500 [Gauss] = 1.075 [T]

Using 1 Oe = 103

4 A/m and 10000 Gauss = 1 T.

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Appendix B

Overview plate spring formulas

In gure B.1a the deformation of the normal plate spring (cantilever beam) is shown. De ection and slope can be calculated using:

x = F L3

3EI =

F L2

2EI (B.1)

where F is the load, L the length of the spring, E the modulus of elasticity and where the moment of inertia I is given in case of a rectangular cross section by: I = 1

12ht3. Figure

B.1b shows the displacement of a thickened plate spring. The stiness of the spring and the stress in the thin parts are given by:

czz = 1:2Eht 3

l3 z = 3Etz_l2 (B.2)

where t is the thickness and h the width. In gure B.1c a pure displacement of a plate spring is shown. The displacement, stiness and stress can be calculated using:

z = F l3 12EI czz = 12EI l3 z= 3Etz l2 : (B.3)

Figure B.1d shows the dierences between normal and thickened plate springs with the same bending stress. For a rst order approximation a rod with two hinges can be used, resulting in the same de ection. The simple cross spring pivot in gure B.1e can be made of two perpendicular plate springs of width h=2 or symmetrically using two (horizontal) outer springs of h=4 and one (vertical) inner spring of h=2 width. In both cases the stresses due to bending and the stiness of the pivot are given by:

= Et_2l c = 2EI_l = Eht 3

6l : (B.4)

In the image also the motion of the pivot point is shown as a dotted line. In gure B.1f the 1=8 cross spring pivot is shown, where the bending stress and stiness is:

= 3:3Et_2l c = 2:672EI_l = 2:67Eht 3

6l (B.5)

Figure B.1g shows the use of thickened plate springs in a 1=8 cross spring pivot where the distance between the middle of the spring and the pivot point is 3₈l = 3₈ 6₇L = ₂₈9L, thus

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F L z f _l L = 7 / 6 l z F a b F z l l 1 / 7 L = 1 / 6 l 5 / 7 L = 5 / 6 l L ~~ 5 / 6 l = 5 / 7 L c d l f l 1 / 8 l f e f L = 6 / 7 l 5 / 2 8 L j z L = 7 / 6 l g _h j ₁ 2 1 z i

Figure B.1:Dierent deformation gures showing plates (a) to (d), cross spring pivots (e) to (g) the vertical displacement of a parallel guidance and the deformation of a folded plate spring.

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the height of the pivot point is 5

28L. In gure B.1h the shortening of a parallel guidance

is shown, this is as function of the displacement z:

= 1₂ z_l2 = ₁₂7 z_L2 (B.6)

The deformation of a folded plate spring is shown in gure B.1i, where the de ection of the second spring is a function of z, like equation B.6: 1= 7=12 _Lz2₁. The stiness (czz; tot)

can be calculated using equation B.2 for spring 1 and: 1 czz; tot = 1 czz1 + _c1 zz2 with czz2 = 1:2 Eh2t32 4l3 2 (B.7)

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Appendix C

Double parallellogram design

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a b

c _d

Figure C.5: Finite element results of the double parallellogram guidance, (a) stiness test to calculate rst disturbing eigenfrequency. Used force is 1 N , and quad elements of 1 mm with 8 element nodes for the solid bodies and quad elements with 4 nodes for the sheets. The face of the basis is xed in all directions. In (b) the maximal stresses are shown during a translation of the main body over 2.5 mm: 83 MPa. In gure (c) the rst mode of the actuator coils (1490 Hz) is shown and in (d) the second mode: 1971 Hz, using tri-elements of 1 mm and as boundary the faces of the coil holder (on the right side) are xed in all directions.

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Appendix D

Six folded plate springs design

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a b

c d

Figure D.2:Finite element results of the guidance using six folded plate springs. Mode 1 is shown: 43 Hz, (b) mode 2, rotation around the z-axis at 1924 Hz, (c) mode 3, the displacement in x-direction at 2219 Hz. Figure (d) shows the ap motion of the thickened parts of the folded plate springs at 2392 Hz. The solid bodies are meshed using tri-elements of 1 mm with 4 nodes, the plate springs are meshed sheets using tri-elements with three nodes.

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Bibliography

[Bon95] M.N. Boneschanscher. Elastische geleidingen, een literatuurstudie. TNO Tech-nische physische dienst, 1995.

[Cac06] L Cacace. Development of a nanometer accuracy non-contact probe for nanome-fos. Internal report, dct 2006.032, TUE, 2006.

[Ein03] Techinsche Universiteit Eindhoven. Constructieprincipes 1, 2003. Lecture note 4007 (in dutch).

[vE85] Jan v Eijk. On the design of plate-spring mechanisms. PhD thesis, TU Delft, 1985.