Chapter 3 Motion system

Figure 3.23: Base block

3.2.2   Vibration isolation

The base block is suspended onto four Newport I-2000S passive vibration isolators (1 in Figure 3.24). These isolators consist of a vertical air-spring with damping via a porous restriction, and horizontal isolation via an oil-damped pendulum. Mechanical levelling valves increase or decrease the pressure inside the air volume to adapt to shifting of the load due to moving of the stages. Each isolator is rated up to 900 kg, so with a total suspended mass of around 3000 kg, four are required. This also gives more tilting stability compared to three. The block is hanging in two triangular frames (2), with two plates (3) attaching the block to the top beams of these frames to constrain the sideways swinging degree of freedom. A rubber layer (4) is applied in this connection to provide damping between the steel beams and the block. This prevents acoustic vibrations in the low-damped steel triangles to propagate into the base block. 

Figure 3.24: Base vibration isolation

The two struts cross below the centre of gravity of the machine (CG), resulting in almost pure tension and compression in the beams. This setup results in a centre of gravity that is about 450 mm below the mounting points (5), which gives inherent tilting stability and lowers the tilting frequency of the base. It also allows for an almost square footprint that is no wider than the base block itself. The final footprint size is 2.1 m x 1.85 m.

Isolation quality

When the machine is simplified to a dual mass-spring system (Figure 3.25), the isolation quality is given by (3.4), similar to (Vermeulen, M., 1999), and is equal to the ratio between the amplitude δ at the probe tip and the floor amplitude z_g. The Newport I-2000S isolators have a mass independent resonance frequency f_isol of around 1 Hz in vertical and 1.5 Hz in horizontal direction. The first (measured) eigenfrequency f_eq at the probe tip is about 75 Hz in vertical and 50 Hz in horizontal direction (see section 3.8.2). 

Figure 3.25: Base dual mass model and tilt model

Metrology laboratories typically have independent foundations providing floor frequencies f_g of around 5 Hz (Figure 3.30), which gives an isolation quality of 7?10^-6 in vertical and 8?10^-5 in horizontal direction. Translated to floor amplitude this gives a typical z_g of 3 m, resulting in a displacement δ at the probe tip of 0.02 and 0.25 nm in vertical and horizontal direction, respectively. These values are only an indication of the actual displacements. In practice, these values will be higher due to parasitic stiffness of cables and non-ideal behaviour of the isolators. To calculate the remaining vibration amplitude spectrum at the probe tip more accurately, one may also use the Power Spectral Density of the actual laboratory in combination with the isolator and machine frequency response.

Levelling

When the stages move from left to right, the base will tilt in ψ-direction (Figure 3.26). 

Figure 3.26: Machine levelling model

In case of a tilt, the equation of moments consists of a moment of the static mass (stat), the moved mass (mov) and the isolator springs (isol):

M stat + Misol ? Mmov = 0

M stat = mstat ? g ?sin(ψbase )?hCG

Misol = kψ?ψbase                                                                                               (3.5)

Mmov = mmov ? g ?ΔR

Using the small angle approximation, the base tilt can herewith be expressed as:

The total moving mass m_mov (stages, probe and interferometry system) is approximately 275 kg and the stationary mass m_stat (base, spindle and metrology frame) is approximately 2600 kg. This brings the total mass m_tot to 2875 kg that is suspended at 1 Hz. Using (3.7), this gives a vertical isolator stiffness of 2.8?10⁴ N/m.

The isolators are spaced 1860 mm apart, resulting in a tilting stiffness k_ψ of the base of 1?10⁵ Nm/rad (3.8). 

When the R-stage moves from the centre position to the outer right position, it displaces 200 mm. The centre of gravity of the stationary mass is approximately 450 mm below the mounting points on the isolators. When no active levelling is applied, this results in a base tilt of 4.8 mrad. This is equivalent to 4.5 mm displacement at the isolators. Since there is no friction and a direct drive motor is applied, the R-stage motor would have to counter this tilt with about 13 N. 

The afore-mentioned only happens when the R-stage moves swiftly (tens of mm/s) from left to right. The valves of the levelling system respond to approximately 0.1 mm displacement at the valve. The tuning of the levelling system restriction is a compromise between settling time and levelling speed. Since the primary mode of operation will be circular scanning, low settling time (high damping) is preferred above levelling speed. In practice, base tilt is negligible when scanning circular tracks, as is thus also the case for the required R-stage motor force.

3.2.3   Transportation

A forklift frame is present under the base block (Figure 3.27, gray). When the isolators are pressurized, the base is lifted 5 mm above this frame. Four bolts in inserts in the bottom of the base, extend into oversized holes in the forklift frame. The isolator columns are also attached to this frame. The bolts limit the range of motion of the base relative to the isolators to prevent the whole assembly from falling over.

Figure 3.27: Transportation frame

When the isolators are depressurized, the block lands on the frame. After removing the triangular frames and the isolator columns, a footprint of 2.1 x 1.25 m remains. This is narrow enough to fit through a double door. In the top view (Figure 3.27, right), the transportation frame (gray) extends from under the machine in all directions, which protects the sensitive parts of the machine for instance from hitting a doorpost during transport.

During development of the machine, the TU/e GTD workshop moved to a new building which forced the transportability of the machine to be tested in practice (Figure 3.28). To prevent air bearing damage due to vibrations, air supply was maintained using a pack of compressed air cylinders. Small jacks were used to lift the machine and mount rollers underneath. It was then pushed towards to loading dock and into a truck by hand. At the new location, a forklift then lifted the machine out of the truck and onto the loading dock, after which it was rolled towards its new position.

Figure 3.28: Machine transportation during GTD move

3.2.4 Base assembly

Figure 3.29 shows the base assembly. Due to late delivery of the isolators, the spindle is already installed in this photograph. Warnings are placed on the bearing surfaces, to prevent damaging the surfaces. The total mass of the assembly is 2850 kg (spindle not included), of which 2350 kg is suspended on the vibration isolators. The footprint is 2.1 x 1.85 m, and the height is about 1.7 m.

Figure 3.29: Base assembly

3.2.5 Vibration measurement

To measure the quality of the vibration isolation system, measurements were taken with an acceleration sensor (B&K 8318C). The vertical vibration levels were measured on the base and on the floor. Since only one sensor was available, no frequency response could be derived. It should further be noted that the measured vibration levels on the base are around the minimum specified sensitivity of 10^-5 m/s² of this sensor. The achieved reduction can still clearly be seen in Figure 3.30. Around 2-3 Hertz, the vibration level on the base is slightly higher than on the floor. This is due to the amplification around the isolator resonance frequency. A sharp 50 Hz peak can also be seen from EMC interference of the mains. The source of the other peaks in the base measurement has not been investigated.

Figure 3.30: Measured vertical vibration levels on floor and base

The above measurements were taken at the former TU/e GTD workshop. The final machine validation measurements are performed at the new location at the TNO Eindhoven facility. No accelerometer measurements were performed here. The actual influence of the remaining vibration between probe tip and spindle is measured in section 3.8.2. The measured vibrations originate from floor vibrations, acoustic excitation and internal sources. The measured vibration levels are sufficiently low, so it can be concluded that the vibration isolation performs as required. 

(6). The base and flange faces were manufactured to high flatness tolerances, allowing them to be bolted together face to face with minimal distortion of the spindle. To measure the rotation angle of the spindle, a Heidenhain ERP880 optical encoder (7) was installed. At the time, this was the best encoder available with a central hole. This central hole was to remain open for calibration purposes, as will be further discussed in sections 4.3.5 and 7.3. The encoder has 90.000 lines, resulting in 180.000 signal periods of 1 V_pp per revolution. According to Heidenhain, 200-fold interpolation may be reliably applied, resulting in 0.17 rad resolution, or 44 nm in tangential direction at the outer edge of the measurement volume. The encoder has one reference mark, and system accuracy is specified at 1 arcsec, or 4.8 rad.

To drive the spindle, an MCS B20-13 brushless DC motor (8) was installed by the manufacturer. This motor has a high motor constant of 0.65 Nm/√W, to limit the heat produced during operation. At startup, Hall sensors provide the rotor orientation for commutation. To assure smooth commutation, the amplifier uses the encoder signal to interpolate between the Hall sensors. Regular amplifiers can not cope with the high line count of the ERP880 encoder. A second encoder, a Renishaw RESR ring encoder with 20.000 lines (9), was therefore installed to aid commutation. The total inertia of the rotor assembly is 0.3 kgm². The assembly was obtained from the spindle manufacturer to prevent distortions from affecting the bearings air gaps.

3.3.2   Product mounting table and intermediate body

Stress-free mounting of optics is a science in itself and not included in the scope of this thesis. The stiffness required during machining is generally much higher than the stiffness required during operation. Statically determined mounting, for instance using three hinged leaf-springs generally is not sufficient (Van Venrooy and Ploeg, 2005). Another complication is that machining and measurement is usually done in vertical and/or horizontal setup, while the product may have any orientation relative to gravity in the final application. In space applications, gravity is not even present.

In polishing, it is common to attach the optic to a counter formed intermediate body using pitch or wax. By pushing the product into the molten wax, it is attached stressfree and with even support stiffness over its entire surface. In diamond turning, surfaces with a flat backside are generally clamped onto a vacuum chuck. Provided that the backside and the chuck are (diamond turned) flat, this provides relatively low mounting stresses and high stiffness. In some cases, for instance in open back structures, this is not possible and bolting the product to the chuck is necessary. Much care must be taken not to deform the product in this case. 

Ideally, a universal intermediate body is applied in single-piece production, to which the product is waxed or bolted. This body serves as a quick and repeatable interface between manufacturing and measuring machine. Some first ideas are explained in Appendix D.

Mounting table

The mounting table is a solid steel disc, 600 mm in diameter and 40 mm thick. A solid disc provides highest stiffness within the available dimensions, and mass is no issue since the higher inertia only aids in keeping a constant speed. Steel was chosen to match the thermal expansion coefficient of the spindle, and to provide a hard, damage resistant mounting face. The total mass is 75 kg and the inertia is 2.9 kgm². The table has twelve rows of M5 holes for bolting or clamping a product or intermediate body to the table (Figure 3.33). 

Figure 3.33: Mounting table and edge profile

As will be explained in section 4.4, the error motion of the spindle is measured by capacitive probes measuring to the edge of the table. This reference edge is machined onto the side and bottom of this table using a conventional 1960s carousel lathe. The steel disc was taken from a rod of bulk material instead of plate to have an axial symmetric stress distribution that allows for better flatness. After warming up the machine overnight, the edge was machined to a PV roundness of 2.3 m and a PV flatness of 5.1 m (Figure 3.33, right). The table was Nickel plated to protect it from corrosion. The profiles shown are actual capacitive probe signals with their first order (eccentricity and tilt) removed. Axially, the profile is mainly a 2^nd order profile, indicating warp. Radially, the profile is less clear, but some elliptical form can be distinguished.

3.3.3 Spindle brake

When performing a radial scan or when installing a product, the spindle is clamped. A ring (1 in Figure 3.34) has hereto been lasercut from a stainless steel plate. This ring is bolted to the bottom of the table via 12 spacers (2). The brake clamp (3) is similar to the clamp mechanism described in section 3.1.2, but has a 90° bend in it. This way it can be bolted to inserts on the front of the bottom base block (4) via a spacer (5). 

Figure 3.34: Spindle brake

As explained in section 3.1.2, the brake clamp has a stiffness of 5?10⁷ N/m between the contact points and the base. The contact is at a radius of 230 mm, giving a rotation stiffness of 2.6 ?10⁶ Nm/rad. The total inertia of the rotor, motor, encoders and table is 3.3 kgm², giving a resonance around the centre line of 141 Hz. 

3.3.4   Spindle assembly

The spindle assembly with mounting table and brake is shown in Figure 3.35. The lower metrology system is already installed around the spindle in this photograph. 

Figure 3.35: Realized spindle assembly

The total mass of the spindle assembly is about 215 kg. The rotor, motor, encoder and mounting table mass is 120 kg. Table 3.1 summarizes the theoretical spindle properties. Experimental validation will be done in section 3.8.2.

Table 3.1: Spindle assembly theoretical rigid body properties

3.3.5 Error motion measurement

Several experiments were conducted to check the spindle after delivery. First of all, the air gap, stiffness and load capacity were verified. Using three LVDT probes, one central and two on the edge of the rotor, an air gap of only 3.5 m was measured. This is relatively small for an air bearing and probably explains the high stiffness of this particular bearing. 

After installing the spindle into the base, and bolting on the product mounting table, the error motion was measured to verify that no distortion was introduced during mounting, and to provide comparable results for testing the multi-probe method (section 4.4.2). The roundness of a steel reference sphere was first calibrated at NMi VSL. The minimum zone roundness profile is shown in Figure 3.36 (left). A contact stylus with a tip radius of 1.8 mm was used for this measurement. Capacitive probes with a spot diameter of 2.5 mm will be used in the experiment, which have different filtering properties. The roundness profile is hereto filtered by a flat averaging filter with 2.5 mm width, to approximate this effect.

Figure 3.36: Reference sphere roundness (by NMi) and spindle error motion setup

The reference sphere was centred on the spindle, and an axial and radial capacitive probe were attached to the Ψ-axis rotor (Figure 3.36, right). The probes were aligned with the centre of the sphere by moving the stages towards the maximum readout of the capacitive probes. The Z-stage was landed for optimal stability. Tests were performed at 0.1, 0.5 and 1 rev/s with a sample rate of 1 kHz. At each speed, 20 revolutions were recorded.

Figure 3.37 (left) shows the axial measurement, which is by definition equal to the synchronous axial error motion (ASME, 1985). It can be seen that the synchronous axial error motion is 9 nm PV with a clear first order component. 

Figure 3.37: Axial synchronous error motion and radial capacitive probe signal

In Figure 3.37 (right) the radial measurement is shown. The reference sphere profile is also shown, from which a large radial error motion could be concluded. A multistep measurement (ASME, 1985; Whitehouse, 1976; Grejda, 2002), however, indicates the contrary, as will be explained in the next paragraph.

The deviation from the average at 1 rev/s is shown for 20 revolutions in Figure 3.38, with a mean rms of 2.9 nm and 3.2 nm respectively. The noise level of the probes used is about 0.2 nm rms. At standstill, the noise level originating in air fluctuations and external vibrations is similar to Figure 3.38. The part of the asynchronous error motion coming from the rotation of the spindle is therefore only a small part of this signal. 

3.4 Z-stage

The Z-stages provides 6 degrees of freedom to the Ψ-axis. It should constrain these such that sub-m uncertainty is achieved at the probe tip. This especially concerns its y, ?, ψand-direction; the r and z-direction are measured by the interferometry system on the Ψ-axis rotor. The Z-stage is aligned with three bearings to a vertical plane, constraining it in y, ? and -direction. The other three degrees of freedom are constrained by the R-stage. Two bearings along the side of the main tube constrain the r and ψ-direction, and the z-direction is constrained by the Z-motor or Z-brake.  The Z-stage front and back view are shown in Figure 3.40. The Z-stage consists of a position frame (1), a preload frame (2) and a motor (3), brake (4) and linear scale (5). Further, a weight compensation (6), connected to the R-stage (7), and an emergency brake (not shown) are applied. The main components of the Z-stage will be explained in the coming sections.

Figure 3.40: Z-stage assembly

3.4.1   Position frame

The basis of the Z-stage position frame is the vertical tube that is straight guided on its sides by two air bearing pairs on the R-stage. Main characteristic for the material selection is a high specific stiffness, to obtain high eigenfrequencies but also because every kilogram has to be weight compensated. Further, a high hardness is desired to enable grinding and lapping of the air bearing faces. As can be seen in Table 4.1, Silicon Carbide is the best material here again, but Alumina is a good second. This is more common and therefore is available in more cross-sections, has a shorter lead time and has a lower cost.

The Al₂O₃ tube (1) has outer dimensions of 150.5 x 155 mm, a wall thickness of 15 mm and a mass of 29 kg. The guidance face (2) and preload face (3) are lapped to a straightness of less than 2 m. Holes are provided in the front and backside for connecting the preload frame and the top air bearing (Figure 3.41).

Figure 3.41: Z-stage Alumina tube with bonded interfaces

Three metal interfaces (4-6) are bonded to the ceramic tube. Due to the large difference in thermal expansion coefficient, special alloys such as Vacon^TM, Kovar^TM or Nilo^TM (www11) that also have an α of almost 6 m/m/K are sometimes used. These metals are hard to obtain and have high cost, and are only required when large temperature differences are expected. 

The difference in expansion between steel and Al₂O₃ is 6 m/m/K, or 0.45 m/K at the corners of the 150 mm tube. Araldite 2020 adhesive has an E-modulus of 2?10⁹ N/m², a shear modulus G of ~6?10⁸ N/m² and a shear strength of approximately 25 N/mm². When the Al₂O₃ and steel parts are supposed to be rigidly expanding (Figure 3.42, left), a temperature change of 2.2 K would break a 0.025 mm thick layer of adhesive. Larger temperature shifts must however be expected, for instance during transport.

Figure 3.42: Deformation of the adhesive layer under thermal expansion

The steel and Al₂O₃, however, are not rigid (Figure 3.42, right). For the bottom interface, the shear stress of 25 N/mm² causes deformations in the steel and Al₂O₃ of about 7 and 5 m, respectively. Together with the allowable deformation of the adhesive of 1 m, the total expansion difference can be 13 m. This occurs at a (homogeneous) temperature shift of 29 K, which is a safe margin.

To mount the ?150 mm top bearing (1 in Figure 3.43), a steel cup (2) is bonded into the ceramic tube. A ring (3) is bonded on the inside to create a symmetrical connection. The cardanic hinge (4) of the bearing is bolted into this cup. The cup has a conical socket at the backside to connect the force frame (5) via a sphere (6), and the topside of the cup is flattened to mount the Z-stage motor and brake strip (7) as will be further explained in section 3.4.4. This way, the bearing has a stiff connection to the ceramic tube, while the preload force can be applied directly behind the bearing without distorting the position frame.

Figure 3.43: Connection of the top bearing

At the steel interface on the bottom of the ceramic tube (1 in Figure 3.44), the two lower ?100 mm bearings are to be attached, together with the mount for the Ψ-axis and the Z-stage optics assembly. The two bearings and their cardanic hinges are mounted to an aluminium hollow bridge. This bridge consists of 4 parts: a centre part (2), two arms (3, 4) and a bottom cover plate (5). The two arms are bolted to the centre part and form a closed box together with the cover plate. The preload frame (6) will be mounted inside (but separated from) this box, as is further explained in section 3.4.2. 

Figure 3.44: Bridge carrying the lower bearings

The y-position of the Ψ-axis and thus of the probe tip is determined by the y-position of the bottom of the ceramic tube, which gets its y-position from the two bearings. To avoid hysteresis between the bridge and the interface, the aluminium centre part of the bridge is attached to the steel interface using two leaf springs (7) and two folded leaf springs (8), manufactured into the centre part by wire EDM. A folded leaf spring (9) constrains the remaining sideways degree of freedom. This way thermal snapping due to the difference in thermal expansion coefficient is avoided, and the route from the ceramic tube to the two bearings in y-direction is shortest.

The 2 ?100 mm bearings are spaced 350 mm apart, and the distance between these bearings and the top ?150 mm bearing is 595 mm. The ?100 mm bearings with the cardanic hinges each have a stiffness of 1?10⁸ N/m, and the ?150 mm bearing has a stiffness of 1.6?10⁸ N/m. Supposing the Z-stage as rigid, this gives a stiffness c_Z,y of 3.6?10⁸ N/m, k_Z,? of 3.2?10⁷ Nm/rad, and k_Z, of 6.1?10⁶ Nm/rad. The 2 ?125 mm  

bearings on the side of the ceramic tube have a stiffness of 1.3?10⁸ N/m and are spaced 595 mm apart. This gives a stiffness between the R-stage and Z-stage in r and ψ-direction of c_Z,r = 2.6?10⁸ N/m and k_Z,ψ = 2.3?10⁷ Nm/rad.

The total mass of the Z-stage position frame is 35 kg, including the position determining bearings. These bearings also have to support the preload frame, the Ψaxis and the probe. The total mass, inertia and resulting rigid body modes of the Zstage assembly will be explained in section 3.4.6.

The Z-stage preload frame will be mounted directly onto the bearings, as will be explained in the next section. This mainly lowers the rigid-body mode frequencies of the Z-stage, but does not influence the flexural body modes of the position frame. The 1^st and 2^nd flexural eigenfrequencies of the Z-stage are bending modes of the bridge in phase and counter phase at 367 Hz and 381 Hz (Figure 3.45). The 3^rd eigenfrequency is in-plane bending of the bridge at 600 Hz, where the 15 kg of Ψ-axis mass is taken into account.

Figure 3.45: Z-stage position frame eigenmodes

Figure 3.46 shows the partially realized Z-stage position frame with the cup and the motor mount on the left. The assembled bridge, with the two ?100 mm bearings, is shown on the right.

Figure 3.46: Alumina Z-stage tube and lower bearings bridge

3.4.2 Preload frame

The ?150 mm bearing requires 2000 N and the ?100 mm bearings require 1000 N. The Z-stage preload frame applies the preload forces while preventing distortion of the Z-stage position frame. The frame consists of a steel tube (1) inside the ceramic tube (2), as shown in Figure 3.47. The central pull tube (3) is bolted to this vertical tube with four spacers (4) extending through holes in the ceramic tube. A box structure (5) at the end of the pull tube carries the preload bearings. The vertical tube is a welded fabrication. The top part (6) contains a conical socket that is aligned with the conical socket in the cup (8), shown in Figure 3.43. A 25 mm hardened steel sphere (9) in between constrains the 3 translations of the preload frame, relative to the position frame. As will be shown later, the motor and brake are also connected to the cup (8), constraining the vertical position of the position and preload frame in a short and direct route.

Figure 3.47: Z-stage preload frame

At the bottom, a plate (10) that preloads the bottom bearings via 2 contact cylinders (11) is bolted to the vertical tube. This plate is mounted inside the hollow bridge of Figure 3.44 and constrains the ? and -direction of the preload frame. A folded leaf spring (12), attached to the lower interface on the ceramic tube, constrains the remaining ψ-direction.

To mount the Z-stage into the R-stage, the ceramic tube with the vertical preload frame tube are lowered in from the top. Next, the bridge with the preload plate inside, together with the two bearings, is installed. Then, the backside part of the preload frame is assembled from the backside of the machine, and bolted to the vertical preload tube through holes in the front and the backside of the ceramic tube. Besides the sphere, two cylinders and the folded leaf spring that constrain the six degrees of freedom, the preload frame does not contact the position frame.

The wall thickness of the vertical and horizontal tubes is 4 mm and the box thickness is 1 mm. As can also be seen in Figure 3.47, all plates extend into the opposite plate at sharp corners, to increase the torsional stiffness of the box. The box is welded from lasercut sheets using plug welding. The maximum stress is 60 N/mm² and the maximum deflection is 0.38 mm, as shown in Figure 3.48. 

Figure 3.48: Z-stage force frame deformation due to preload forces

The combined mass of the preload bearings and pistons is 4 kg, which has to be constrained by torsion and bending of the vertical and horizontal tube. The relatively large thickness of 4 mm is required to constrain this mass that is far away from the mounting points. The first mode (Figure 3.49) is torsion in the vertical tube, causing the backside to swing from left to right at 84 Hz. The second mode is bending in the pull tube, causing the backside to swing up and down at 105 Hz. The total preload frame mass is 22.6 kg, including the preload bearings and pistons. 

Figure 3.49: Z-stage preload frame eigenmodes

Figure 3.50 shows the preload frame and the lower plate with the contact cylinders inside the hollow aluminium bridge (bottom cover plate removed).

Figure 3.50: Z-stage preload frame

As will be explained in Chapter 5, the probe electronics should be close to the probe to minimize capacity of the cables. They are therefore mounted into the backside box of the Z-stage. From there BNC cables transport the ±10 V signals to the data acquisition system. Inside the horizontal tube of the preload frame, air valves and the solenoid valve for the Ψ-axis brake are mounted. This limits the number of air hoses that have to move along with the Z-stage. 

With the two valves, the air of the Ψ-axis and the air to the fixed front bearings of the Z-stage can be closed. The air pressure on the preload bearings and pistons remains, clamping the Z-stage to the vertical base plate. This is useful for testing of the machine since it eliminates the R- and Z-bearings and brakes from the structural loop.

3.4.3 Weight compensation

The total vertically moving mass is about 85 kg. To avoid the motor having to continuously lift this weight and thereby forming a disturbing heat load, the weight is compensated. Also, the brake can be much lighter if no weight is to be held up. Friction must be minimal to avoid virtual play that limits the positioning resolution. Several methods can be applied for generating a constant upward force over a large stroke (Frank et al., 2005). 

Most common is balancing with a mass via a pulley, using a wire or strip (Figure 3.51A). This doubles the moving mass and introduces an extra body of which five degrees of freedom have to be constrained relative to the R-stage. The friction introduced by the pulley may be acceptable, but introducing another 85 kg of moving mass to the machine is not preferred. 

A Tensator? spring (www18) generates a constant force over a considerable stroke

(Figure 3.51B). It is, however, not capable of providing 850 N over 150 mm. The rolling of the windings further also is not frictionless. It is also difficult to adjust to the increasing weight during assembly and to finetune when assembly of the machine is completed.

An air piston (Figure 3.51C) has a much higher force density. A ?52 mm piston at 4 bar delivers the required force. Sealing the gap between cylinder wall and piston usually introduces unacceptable amounts of friction. Using air bearings for guidance and a narrow gap as a seal eliminates the friction, but has very tight manufacturing tolerances. 

Figure 3.51: Weight compensation concepts

To avoid the air bearing, one may also use a pressurized volume with a rolling membrane and a lever (Figure 3.51D). If the pressurized volume is large enough, the force is almost constant over the range. The stroke of the membrane is generally limited to a few tens of millimeters, requiring a lever ratio in the order of 1:10. The piston area or air pressure thus have to be increased by one order of magnitude to compensate 85 kg over 150 mm this way, which is not practical.

A more mechatronic solution would be to drive a lead screw with a motor (Figure 3.51E). With a spring between the nut and the Z-stage, it can still move frictionless. A control system should maintain a constant force in the spring. Lifting 85 kg over 150 mm requires 125 J to be inserted in for instance 1 second. Quite a powerful motor is thus required, and a new heat source would be introduced into the machine.

An air bearing piston is chosen to be applied. Since a single round bore could not provide enough surface area within the available volume, a rectangular piston (70 x 30 mm) was chosen (Figure 3.52). The piston (1) has 12 porous air bearing plugs (2) similar to those used in the Ψ-axis (see section 3.6.1). The individual air bearing pads

(3) are separated by atmospheric grooves (4). The air bearings are fed from the pressurized volume (5) through holes (6) in the bottom of the piston. The normalized air consumption of the air bearing is approximately 0.8 l/min. 

Figure 3.52: Z-stage weight compensation

The entire available volume is used as an air chamber to minimize the relative volume reduction when the piston is moving. This is done by machining the bore (7) and chambers (8), interconnected by holes (9) from a solid block of aluminium and covering it with a lid (10). The piston was then made to fit the bore. A passive precision pressure regulator is used to finetune the pressure to the actual mass of the Z-stage and to (passively) keep this pressure constant. Some hysteresis will be present in this valve. In combination with the tuning error this will give a remaining force that is to be countered by the motor.

The pressure chamber is sealed with a parallel gap resistance (11). The resistance R of a parallel gap with width w, gap height h and length l, and dynamic viscosity η is

 R

The air bearing gap h is 7 m. The 10 mm long parallel gap seals the pressure chamber with a circumference of 200 mm. The dynamic viscosity η = 17?10^-6 Pa.s. This results in a resistance R of 3.0?10¹⁰ Pa.s/m³. With a pressure inside the cylinder of 4 bar, the normalized leakage is equal to 3.2 l/min. The leakage is a function of the 3^rd power of the gap height, so small manufacturing tolerances or for instance scratches cause a large increase in the leakage. The total normalized air consumption of the weight compensation is 4 l/min.

The combined centre of gravity of the Z-stage, Ψ-axis, probe and Z-stage optics is located at the backside of the ceramic tube, as shown in Figure 3.53. The weight compensation system is nearly aligned with this centre of gravity. The piston rod is connected to the Z-stage preload frame and the pressure chamber is connected to the R-stage preload frame as shown later in section 3.5.2. Connecting the weight compensation to the preload frames minimizes varying deflection in the position frames due to varying compensation forces.

Figure 3.53: Z-stage centre of gravity and weight compensation location

Figure 3.54 shows five measured hysteresis loops after careful alignment, obtained by applying a triangular force as a disturbance to a pure gain controller. The controller stiffness has been subtracted from this graph. The discontinuity in the force in combination with the absence of damping causes the oscillations. Around the centre of the stroke (z = 40 mm), the residual friction is about 1 N and at the ends of the stroke (z = -20 mm and z = 120 mm) this is 4 N and 5.5 N, respectively.

Figure 3.54: Weight compensation of Figure 3.52 and its hysteresis loop

3.4.4 Motor, brake and linear scale

For optimal dynamic behaviour, the motor should be aligned with the centre of gravity. The centre of gravity of the combined Z-stage, Ψ-axis and probe is situated approximately on the back face of the ceramic tube as shown in Figure 3.53. An aluminium plate (1) is hereto bolted to the cup and the interface block on top of the ceramic tube (Figure 3.55). Hinges (2, 3) are machined into the plate to cope with the difference in expansion coefficient. The magnet yokes (4) of the motor are bolted to this plate, and the coils (5) are mounted to the R-stage position frame (6). Placing the motor this high, keeps the heat source away from the metrology loop. Next to the motor magnets, the brake strip (7) is mounted. The motor plate is bolted directly to the interface cup, which provides a stiff coupling between the motor and brake and the Z-stage position and preload frame. The brake clamp (8), as discussed in section 3.1.2, is mounted to the R-stage position frame, next to the motor coils. 

Figure 3.55: Z-stage motor and brake

The remaining friction and tuning error of the weight compensation was measured to vary between 1 and 5.5 N along the stroke. The selected Tecnotion UL3 ironless linear motor has a motor constant of 97 N²/W, a peak force of 240 N and a continuous force of 35 N. This particular motor has the highest motor constant and thus the least power dissipation to fit the available volume.

When measuring circular tracks, the amount of Z-stage work depends on the height variation of the product. Making a 1 mm step in 1 s with a 3^rd order profile requires a peak acceleration of 6.3 mm/s², which is equal to a peak force of 0.5 N on 85 kg. When opening the brake, the residual weight compensation error thus determines the required force. Generating 5 N during 1 s requires 0.25 J, and for 250 steps during 15 minutes 62 J. When assuming ideal operation of the amplifier, this is a negligible average heat dissipation of 69 mW.

An optical linear scale is used for position feedback (Figure 3.56). The Heidenhain LIP481R Zerodur scale is bonded to the front of the Al₂O₃ tube via small flexure blocks to allow differences in thermal expansion. The bottom block, which is closest to the probe, has no flexure and determines the vertical position of the scale. The mounting and alignment mechanism of the read head on the R-stage position frame will be discussed in section 3.5.4.

The linear scale is positioned relatively far from the motor, which may cause control instabilities. The loop between motor and linear scale is stiff, consisting of only the motor plate and the Alumina tube. The motor drives the Z-stage through its centre of gravity, so parasitic rotations will be small. This non-collocated setup is preferred to have the Z-position measurement close to the probe.

The brake, motor and scale are shown in Figure 3.57. Two pairs of limit switches are also shown, for detecting the ends of the Z-stage stroke. These will be further discussed in Chapter 6.

Figure 3.57: Z-stage brake (left), motor (centre) and linear scale (right)

3.4.5 Emergency brake

When air pressure fails, the weight compensation pressure will also drop. Since the brake is not able to hold the 85 kg mass, the Z-stage will plummet down. Shock absorbers are present to dampen this, but when the probe is oriented vertically, it will crash into the table or into a product that is rotating underneath. To prevent this, an emergency brake system is included. A pressure vessel is present, giving a few minutes of spare pressure supply. 

Figure 3.58: Z-stage emergency brake

If an electronic or passive sensor detects an air pressure failure, a series of valves diverts a slightly higher pressure to the weight compensation, which causes the Zstage to be swiftly lifted. In the mean time, the pressure also drops from the spring preloaded piston (1) shown in Figure 3.58. This causes the lever (2) to lock into the slot (3) in the Z-stage motor mount, locking it at the top position. Tests have shown that this takes less than a second, which is probably fast enough to save the probe. This procedure is executed when the air pressure fails, when the control software stops responding, when the emergency stop is pressed or when electric power fails.

3.4.6   Z-stage assembly

The Z-stage is shown in Figure 3.59 as it is mounted in the R-stage. The ceramic tube with the preload tube inside was lowered into the R-stage from the top, after which the lower bearing bridge was bolted on. Next, the central pull tube and the backside box were bolted on. The alignment of the Z-stage will be discussed in section 3.7, along with the assembly of the rest of the motion system.

Figure 3.59: Z-stage assembly

Table 3.2 summarizes the (theoretical) properties of the Z-stage. The Ψ-axis, probe and Z-stage optics assembly have been taken into account in the mass and inertia calculations. The total mass is 80 kg, and about 5 kg is added for cables and the probe electronics. The vertical stiffness c_Z,z is the brake stiffness only, so the stiffness of the Z-stage relative to the R-stage. This is also the case for c_Z,y and k_Z,ψ. The complete structural loop stiffness from probe tip to base will also be explained in section 3.7.

Experimental validation will be done in section 3.8.2.

Table 3.2: Z-stage assembly theoretical rigid body property summary

3.5 R-stage

The R-stage constrains the r, ψ and z degrees of freedom of the Z-stage, so it must provide high stiffness in these directions. Since the r and z-position are measured directly at the Ψ-axis rotor, especially the ψ-direction must be provided with high repeatability. Since the Z-stage is directly aligned to the base in y, ? and θ-direction, errors of the R-stage in these directions do not influence the position of the probe tip. The R-stage, however, also carries the scale read heads and R-stage optics assembly, of which the alignment is critical. The other three degrees of freedom of the R-stage should thus be constrained with high stiffness and repeatability as well. 

Figure 3.60: R-stage assembly

Figure 3.60 shows the front and back view of the R-stage assembly. In the coming sections the main components will be explained, starting with the position frame (1), the preload frame (2) and the Z-stage r and ψ constraining bearings (3). Next, the Rstage motor and brake (4) and linear scale (5) will be explained, followed by the cable guidance (6).

3.5.1    Position frame

The R-stage position frame (Figure 3.61) has two horizontal bearings (1 and 2) at the front constraining the z and ψ-direction, in the same plane as the ceramic tube axis of the Z-stage. Two plates extend backwards to two horizontal bearings (3 and 4) at the backside that constrain the ?-direction. The position frame has a torsional degree of freedom to prevent overconstraining. It is hereto built from two box structures (5 and 6) on the left and right, which are connected by a U-shaped bridge (7). The plate (8) and strut (9) were added to increase the bending stiffness in the horizontal plane, and also to provide a mounting point for the Z-stage motor and brake. Two horizontal bearings (10 and 11) against the vertical base plane constrain the y and-direction to the same plane as to which the Z-stage is aligned.

Figure 3.61: R-stage position frame

The ?150 mm bearings and cardanic hinges have an in series stiffness of 1.6?10⁸ N/m. The distance l_R between bearing pairs (1) and (2), and (10) and (11) is 1.4 m. The distance l_Z between these bearing pairs and bearings (3) and (4) is 470 mm. This gives a stiffness in y-direction of c_R,y = 3.2?10⁸ N/m, in z-direction of c_R,z = 6.4?10⁸ N/m, a tilting stiffness in ψ and θ-direction of k_R,ψ = k_R,θ = 1.6?10⁸ Nm/rad and a tilting stiffness in ?-direction of k_R,? = 3.5?10⁷ Nm/rad.

The R-stage position frame is constructed from 10 mm thick aluminium plate, mainly for its high thermal conductivity and good machinability. These plates can be connected by bolting, bonding or welding. Bolting requires tight tolerances and many threaded holes in the (inclined) side faces of the plates. The limited contact area between the plates forms a high heat flow resistance, which increases thermal gradients over the structure. Main advantage is, however, that the stage can be disassembled in case of damage or a design change. Bonding requires narrow bond gaps to create high stiffness. This requires high dimensional tolerances, adhesive forms a resistance to heat flux and disassembly is not possible. Welding requires less tight dimensional tolerances since small gaps can be closed with filler material when welding. Welds have high stiffness and good conductivity, but disassembly is not possible. Due to the high conductivity, the structure has to be pre-heated to create good welds. During welding, much heat is further brought in very locally. This may lead to distortions and internal stresses during cooling. Annealing is applied to reduce these, but this may further increase distortion. Machining the interfaces to other parts after welding and annealing solves this problem.

The left and right boxes (Figure 3.62) consist of a vertical back plate (1) to which the vertical plate (2) containing the support to the backside bearing is joined. A horizontal plate (3) is mounted in between these boxes. The inclined top plate (4) and front plate (5) form a triangular box. Another plate (6) is added at 45° on the bottom to form a horizontal triangular tube. A stiffening plate (7) is added inside the top box to stiffen the horizontal plate where the 45° plate is attached. After the box structure has been welded, the interface faces were machined and the end plate (8) was bolted to it to connect the bearings.

Figure 3.62: R-stage position frame box structure

The outside corners (L-connections) of the box are made by adding a chamfer, such that the weld is at the intersection of the plate centre lines (A in Figure 3.63). At the end plate of the triangular tube, plate 3 and 4 are welded and the end plate is bolted such that the plate mid-planes intersect (B). All T-joints are made by plug welding (C), where the plug is slightly recessed to have the weld at the mid-plane and reduce the weld bulging out of the slot. At the intersection of plate 3, 6 and 7 an alternate combination of joint A and C has been made (D).

Figure 3.63: R-stage position frame weld connections

The total mass of the R-stage position frame is about 68 kg. Figure 3.64 shows the first and second eigenfrequency, at 139 Hz and 227 Hz respectively. In the first mode, the stage is mainly rotating in ?-direction, which does not influence the Z-stage position since this makes the bearings that guide the Z-stage, slide over the sides of the ceramic tube. The second mode is bending of the stage, which causes a ψ-rotation of the Z-stage since the two bearings that guide it move in r-direction in counter phase. The calculated stiffness in z-direction at the Z-stage brake clamp mounting position is 7.1?10⁷ N/m. The tilting stiffness in ψ-direction at the Z-stage bearing points is 1.15?10⁷ Nm/rad.

Figure 3.64: R-stage position frame eigenmodes

The structure is designed such that it can be completely assembled jig less before being welded, to minimize distortions. Figure 3.65 shows the assembly and welding of an R-stage position frame box structure. The assembled position frame is shown before painting.

Figure 3.65: R-stage position frame welding

3.5.2 Preload frame

Each ?150 mm bearing must be preloaded with 2 kN, with minimal distortion of the R-stage position frame. The combined mass of all stages, which is about 275 kg, partially preloads the horizontal bearings (1-4) as shown in Figure 3.66. In this figure, the end plates (5 and 6) of the R-stage position frame are also shown. The centre of mass is located in the front half of the stage, resulting in a higher preload of the two front bearings (1 and 2). The four horizontal bearings (1 – 4) combined require 8 kN, of which 2750 N is provided by the stage mass, and the remaining 5250 N should be provided by the horizontal preload bearings. This is done by two ?150 mm bearings (7), each with a piston that generates 2625 N, at the cost of a slightly smaller air gap.

The two bearings with the highest load capacity as measured in Figure 3.2 are used here. These bearings are positioned such that the resulting force vector of the mass and the preload bearings results in 2 kN at each position determining horizontal bearing.

Figure 3.66: R-stage preload frame assembly

The preload frame has to connect these bearings to the R-stage position frame and deliver the forces directly onto the bearings to minimize distortion. It hereto consists of a horizontal front beam (8), which is connected to the endplates of the position frame with two shoulder bolts. A flexure at these two points constrains the y, z, ψ and θ-degrees of freedom. From this beam, a tube (9) extends backwards through the gap in the base, which has its side faces aligned with the two fixed bearings (3 and 4) at the backside. This constrains the ?-degree of freedom. The r-direction will be constrained by the motor and brake, as shown in section 3.5.4. Several cut-outs have been made to allow space for the Z-stage and its bearings, and to introduce a torsional degree of freedom just as with the position frame. 

From the central tube (9), two plates (10 and 11) extend downwards to a box structure (12) that carries the two preload bearings (7) and pistons. The front plate runs through the slit between the bottom and middle base block. To prevent parallelogram motion, two ?50 mm bearings (13 and 14) are added, the front one fixed, the backside one with a piston.

From the central tube, two arms (15 and 16) extend to the left and right, carrying the horizontal preload bearings and pistons. Two support struts (17) provide stiffness in vertical direction to the arms.

The preload frame is mainly constructed of 1 mm thick steel sheets. Only the vertical sides of the central tube are 4 mm thick. Steel is chosen here because of the high load the frame has to withstand, and for the better weldability of thin sheets compared to Aluminium. The total mass of this frame is 55 kg. Figure 3.67 shows the 1^st and 2^nd eigenfrequency at 68 Hz and 73 Hz respectively. In practice, the pistons will provide some stiffness and damping, increasing the eigenfrequencies. The preload forces cause a maximum deflection of 0.56 mm at the ends of the arms on the backside, and a stress of 240 N/mm². 

Figure 3.67: R-stage preload frame eigenmodes

The preload frame is constructed of 1 and 4 mm thick lasercut steel sheets. T-joints are made using plug welding, and L-joints are simply welded together. Figure 3.68 shows the bottom box assembled just before welding and the completed preload frame before paint. 

Figure 3.68: R-stage preload frame realization

3.5.3   Z-stage tube bearings

The fixed ?125 mm bearings that guide the Z-stage tube are attached to the right half of the R-stage position frame (Figure 3.69). A stiff arm (1) hereto extends from the Rstage position frame towards the lower position determining bearing (2). A bracket (3) is used to bolt the upper position determining bearing (4) to. 

These bearings require a preload of 1500 N. The opposing preload bearing and piston are hereto mounted to a plate (5) that is bolted to the R-stage position frame, constraining only the in-plane degrees of freedom. At the upper bearings, the force loop is closed with a C-yoke (6). The lower bearings are bonded to two cups (7 and 8) which are connected with two pull rods (9) due to space constraints. In both cases the force loop is separated from the position frame this way.

Figure 3.69: Z-stage tube bearings

The lower Z-stage tube bearing assembly is hidden behind the R-stage preload frame. The visible preload bearing cup is shown in Figure 3.70 (left). The upper Z-stage tube bearing assembly with the C-yoke is shown in Figure 3.70 (right).

Figure 3.70: Z stage lower (left) and upper (right) tube bearings

3.5.4 Motor, brake and linear scale

The location of the centre of gravity of the stages, with the Z-stage in its middle position, is shown in Figure 3.71. The R-stage motor and brake are aligned with this point. Just as with the Z-stage, an aluminium plate (1) is bolted to the backside of the position frame. The magnet yoke (2) and the brake strip (3) are mounted to this plate. A folded leaf spring (4) constrains the r degree of freedom of the R-stage preload frame (5) to create a short route from motor and brake to this frame. The motor coils (6) and brake clamp (7) are mounted to a plate (8) that is bolted onto inserts in the top base block. 

Figure 3.71: Stage assembly centre of gravity and R-stage motor and brake assembly

As explained in section 3.2.2, a maximum base tilt of 4.8 mrad is expected when the R-stage moves from left to right, requiring a counterforce from the R-stage motor of 13 N. Similar to the Z-stage, the R-stage makes 1 mm steps in 1 s between measurements. This requires a peak acceleration of 6 mm/s², or 1.6 N. During these small steps, base tilt is negligible so only acceleration and deceleration forces are required from the motor. The mean force during this 3^rd order profile is 1.4 N, requiring 0.02 J for a single step. For 250 steps in 15 minutes, this is equal to 5 J. Assuming ideal operation of the amplifier, the mean power dissipation is only 6 mW.

As with the Z-stage, a Heidenhain LIP481R linear scale is applied for feedback (Figure 3.72). This scale (1) is mounted onto the metrology frame (not shown) using flexure blocks similar to those used for the Z-stage. The R- and Z-stage read heads (2 and 3) are mounted to an aluminium arm (4) that extends from the R-stage position frame (5) via two alignment mechanisms (6 and 7). These monolithic mechanisms consist of a plate spring (8) that constrains three degrees of freedom, and three spring preloaded adjustment screws (9) to adjust tip, tilt and focus. Rotating the read head in its mounting hole provides the fourth degree of freedom required for proper alignment.

Figure 3.72: R-stage linear scale and read head alignment mechanisms (upside down)

Figure 3.73 shows the motor and brake as assembled on the machine. It also shows the R and Z scales with their read heads and alignment mechanisms. The R-stage optics assembly can also be seen in this figure, which is attached to the read-head arm on one side.

Figure 3.73: R-stage motor and brake, and linear scale read head assembly

3.5.5 Cable guidance

Measuring many parameters inside the machine has many cables, fibres and connectors as a consequence. The cables and air supply hoses mostly run inside the machine. 3 cables come from the probe and 5 from the Ψ-axis. They run through the Z-stage preload frame, together with the air supply hoses for the Z-stage and Ψ-axis bearings. The 2 probe signal cables are connected to the probe electronics in the backside of the Z-stage, and changed into 7 BNC cables and 2 power supply cables for the electronics. All 17 cables then come out at the bottom of the Z-stage preload frame and are guided through a cable chain onto the R-stage. From here, they are joined by about 10 more cables and air hoses from the R-stage and go into the R-stage cable chain to be transferred to a bar on the base. From here, the cables go to the electronics cabinet, together with the about 20 cables from the spindle, the lower metrology frame and the R-stage motor. This makes a total of 47 cables coming off the machine.

Figure 3.74: Cable chain assembly

The applied cable chains have rubber flexures instead of links, which reduces friction. In practice, the friction was found to be small enough to not influence the positioning resolution. The chains are connected to the preload frames. The R-stage cable chain has a minimum in its gravitational energy content when most of it is resting on the horizontal bar. This causes a small disturbing force on the R-stage. Mounting the cable chain horizontally would have kept the gravitational energy constant, but this was difficult due to space constraints.

Air hoses are thick and quite rigid. To minimise their disturbing forces, a single air line passes from the base to the R-stage and is split in a valve block (Figure 3.74) inside a backside arm of the preload frame. With this valve block, the R-stage bearings can be turned off to land the stage, the Z-brake can be operated manually and the air pressure of the weight compensation can be adjusted. One air hose passes from the R-stage to the Z-

3.6.1   Air bearing

The main dynamic force acting on the rotor is the reaction force of the acceleration of the probe objective. A single sided bearing (Figure 3.77, left) is preferred for the large stroke, but dynamic simulations (section 3.1.4) show that this would result in large tilts of the rotor, with large out-of-plane errors at the probe tip as a consequence. A symmetrical radial bearing layout converts these forces into radial error motion, which is measured by the metrology loop. The large range of ψ-motion of -45° – +120° requires a substantial part of the connecting housing between the two radial bearings to be open. The housing outer diameter is limited for measuring deeply concave optics (dotted line in Figure 3.77), which thus also limits the thrust bearing dimension and location.

Figure 3.77: Ψ-axis bearing configurations

Porous plug air bearings

For a radial bearing to have stiffness, it must be composed of at least three individual bearing segments. This is generally done by supplying pressure to restrictions around the bearing. These restrictions can be small holes, in the order of tenths of millimeters (Figure 3.78A), in orifice or inherent compensated form (Holster, 1967; Van Beek, 2006). A chamber is sometimes added for extra load capacity and stiffness, but with the risk of pneumatic hammering. Small hole restrictions tend to get easily clogged with dirt particles. A porous graphite ring (Figure 3.78B) essentially has an infinite amount of restrictions. Machining this ring to the roundness tolerances required is very difficult, since graphite is a difficult material to work with. An intermediate solution is to use porous plugs as restrictions (Figure 3.78C). These may have the same flow resistance as the small holes, but have a much larger surface area which makes them much less vulnerable to dirt. The carrier ring in which they are mounted can be machined to the required roundness.

Figure 3.78: Air bearings with hole (A), porous (B) and porous plug (C) restrictions

A porous plug bearing unit can be characterized as shown in Figure 3.79, with the supply pressure P_S, the pressure after the restriction P_R, and the atmospheric pressure P_a. The porous plug has radius r₁ and thickness s, and the bearing surface has radius r₀. The bearing gap has height h.

Figure 3.79: Porous plug bearing and pressure distribution

The mass flow ^M_plug through the porous plug can be calculated with the permeability k_p, the surface area A_p, the viscosity η, the gas constant R and temperature T (Van Beek, 2006):

Equalling the two mass flows results in:

Where the geometry factor g is:

            g = π s                                                                                    (3.14)

r0 k Ap p 6 ln( )

r1

This results in the pressure after the restriction P_R and the exact solution for the pressure distribution P(r):

         PR =  PS2 + gh P33 a2 and  P r( )a 2 =  PRa 2 ? ln((r r0 / 11)) PPRa 2 ?1        (3.15)

                        1+ gh                     P           P         ln r /r

The pressure distribution may also be approximated by a cone:

        P r( ) = P_R     for   r ≤ r₁

         and                                                       

(3.16)

          P( )r P  (r    r₁)   for   r₁        r      r₀

Integrating the pressure over the area gives the load capacity F_unit of a single bearing unit:

Funit = π(PR ? Pa )(r02 ? r12 + r r0 1 )

         and                                                                                                    (3.17)

        Funit = π(PR ? Pa )r02     when    r1  r0

           unit              dh

The permeability of the batch of carbon to be used was experimentally determined to be 5.5?10^-15 m² (Aerts, 2007). For a bearing unit radius of 10 mm, a plug diameter of 5 mm and a thickness 3 mm, the load capacity, stiffness and airflow are shown in Figure 3.80. Here, r₁ is not supposed to be much smaller than r₀.

Figure 3.80: Bearing unit load capacity, stiffness and flow rate 

(r₀ = 10 mm; r₁ = 2.5 mm; s = 3 mm)

An air gap of 7 m is considered a good compromise between manufacturability, robustness to contamination and bearing properties. Figure 3.81 and Figure 3.82 show the influence of the porous plug dimensions on load capacity, stiffness and airflow for this gap height. A plug diameter of 5 mm and a thickness of 3 mm was selected for the Ψ-axis bearings, giving a load capacity of 29.6 N, a stiffness of 6.5?10⁶ N/m and a normalized air consumption of 0.18 l/min per bearing unit.

Figure 3.81: Load capacity and stiffness as a function of the plug dimensions  at 7 m gap height

Figure 3.82: Flow rate as function of plug dimensions at 7 m gap height

Axial bearing properties

The designed radial and thrust bearings consist of 18 bearing units. For the axial bearings this has 18 times the properties of an individual unit, or c_ax = 1.2?10⁸ N/m, F_ax = 533 N and Q_ax = 3.2 l/min (normalized). Since the thrust bearing will be form closed, stiffness is doubled to c_Ψ,ax = 2.4?10⁸ N/m.

To verify the bearing properties, an axial bearing was made as shown in Figure 3.83, left. The flow rate of each plug was verified first after gluing in, as shown in the photo. The bearing stiffness was tested using the same setup as used for the porous air bearing (Figure 3.1). The measured and theoretical results are shown in Figure 3.83, right. The parameters were somewhat different from the final design, so this graph is not equal to the final Ψ-axis thrust bearing stiffness. The theoretical curve agrees quite well with the experimental values.

Figure 3.83: Test thrust bearing and test results

Radial bearing properties

For a radial bearing with relative eccentricity e, the bearing gap around the circumference is calculated with:

h( )ψ = h₀ (1? e cos( )ψ)                                                                  (3.19)

With this gap height, the properties of each bearing unit along the circumference can be calculated. The resulting force as a function of the eccentricity can herewith be calculated along with the stiffness and air flow. For a radial bearing with 18 bearing units, a diameter of 70 mm and an initial gap height h₀ of 7 m, the force, stiffness and airflow as a function of the eccentricity are shown in Figure 3.84.

Figure 3.84: Radial bearing force, stiffness and flow rate

The rotating mass (rotor, motor, encoder, probe and counterbalance) is about 5.6 kg. At this load, the eccentricity for the two radial bearings is 0.5 m. The stiffness per bearing is 5.5?10⁷ N/m, giving a total radial stiffness of c_Ψ,rad = 1.1?10⁸ N/m. The bearings are spaced 123 mm apart, giving a tilting stiffness of k_Ψ,? = k_Ψ,θ = 4.2?10⁵ Nm/rad. The normalized air consumption per radial bearing is 2.5 l/min.

Bearing design

The bearing housing (1 in Figure 3.85) is machined from a solid block of aluminium, as is the rotor (2). The front radial bearing (3) is a separate ring that is bolted onto a seat in the housing. The second ring (4) is a combined radial and thrust bearing. On a spacer (5), the second thrust bearing (6) is bolted. Three holes (7) in the housing, together with three further holes in the rotor (8), function as an exhaust for the radial and thrust bearing.