A THREE Axis CNC ROUTER DESIGN

by

Alexander D. Sprunt

SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

BACHELOR OF SCIENCE AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY

JANUARY 2000 __~~X (j'T-)(D

@ 2000 Alexander D. Sprunt All rights reserved

The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part.

S ignature of A uthor ...... Dep rtment of Mechanical Engineering January 14, 2000

Ce rtified by ...... { ...... Alexander H. Slocum Margaret MacVicar Faculty Fellow Professor of Mechanical Engineering Thesis Supervisor

Accepted by ...... Ernest George Cravalho MASSACHUSETTS INSTITUTE Professor of Mechanical Engineering OF TECHNOLOGY Chairman, Undergraduate Thesis Committee JUN 2 8 2000 ARCHIVESACIE LIBRARIES1

A THREE Axis CNC ROUTER DESIGN

by

Alexander D. Sprunt

Submitted to the Department of Mechanical Engineering on January 14, 2000 in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in Mechanical Engineering

ABSTRACT

A preliminary design of a low-cost, three-axis, computer numerically controlled (CNC) router was completed with the goal of bringing the advantages of numerically controlled machine tools to the woodshop. To reduce cost, a novel single rail design was employed. The number of custom parts was kept to a minimum and, where necessary, every effort was made to minimize manufacturing cost. The novel features of the design include vacuum clamping, the ability to cut joinery at miter angles from 0* to 90*, and force controlled cutting. Many of the components are used quite aggressively (particularly with regard to stiffness), but the accuracy goal of ±0.005" in X and Y and +0.010" in Z was met.

Thesis Supervisor: Alexander H. Slocum Title: Margaret MacVicar Faculty Fellow Professor of Mechanical Engineering

3 List of Symbols ball screw lead angular velocity Mioad mass of the load being moved Jtrans inertia of the transmission Fload external load Not optimal transmission ratio x a length (context sensitive) r constant for ball screw selection - accounts for external load c constant for ball screw selection - accounts for trajectory shape Cbuckle buckling constant kwhi, shaft whip constant E Young's modulus P mass density

I 2 nd Moment of inertia (for beam bending, etc.) AX Cross-sectional area te Cycle time (ball screw selection) a Acceleration time (ball screw selection) b Deceleration time (ball screw selection) s, Shaft whip safety factor Sb Shaft buckling safety factor d Ball screw diameter

Wd Heat dissipation in motor armature coil Ra Motor coil resistance K, Motor torque constant T, Torque i F, Force i J Inertia i L Life constant for calculating linear guide life {1 05 or 5-1 04} C Load capacity (bearing and ball screw life) dalong Distance between two trucks along a linear guide daco,, Distance between two trucks perpendicular to a linear guide n Bearing life equation coefficient a Angular acceleration PO Outside pressure

4 P Inside pressure

Pam Atmospheric pressure Q Volumetric flow rate Cd Coefficient of discharge 91 Reynold's Number t Thickness v Velocity P Viscosity T Temperature R Ideal gas constant Ai Area i q7 Efficiency #8 An angle for ball screws 0 An angle G Shear modulus 3, Displacement i 3 Friction force Fn Normal force P, Static coefficient of friction Nholes Number of holes in the optical table

All units are in Systeme Internationale unless otherwise noted.

5 Introduction This thesis describes the preliminary design of a low-cost router table that brings the benefits of computer numerical control (CNC) tools to the woodshop. The router introduces entirely new capabilities such as full three-dimensional shaping to the shop in addition to obviating the need for several conventional woodworking implements such as joinery jigs and conventional table routers. Tiresome and repetitive tasks can also be automated for superior output and increased throughput. Cost reduction was accomplished using a novel, single rail, design that minimizes part count and eases assembly. There are relatively few custom parts, and those required were designed with attention to manufacturability. During the course of the project, numerous safety, performance, and ease of use advantages were developed for a force controlled cutting system that could easily be implemented using the hardware that was the principal focus of this thesis. Naturally, there is a balance between cost and performance. With that in mind, a reasonable set of specifications was chosen and the design effort was focused on maximizing the potential of the available hardware.

6 Specifications and Features Footprint 3' x 4'6" + computer Work Volume 2'6" x 5' x 10" Router 3 hp, 8-22 krpm, '/2" collet Accuracy ±0.005" for the Y and X Axes, ±0.010 for the Z-Axis Feed Rate 20 fpm

Vacuum Clamping Vacuum Chip Removal Joinery Capable Force Controlled Cutting Table 1 The specifications and features were developed in consultation with Ken Stone, Director of the MIT Hobby Shop. The last two features were added midway through the project and will be described in detail later. See the Appendix for competitive benchmarking data.

7 Now

Configuration An optical table was selected as the machine's base with the intention of mounting the table nearly vertically to meet the footprint specification. The optical table provides an inexpensive, lightweight, precision surface that is essentially pre-configured for vacuum clamping (see Vacuum Clamping section). Three initial configurations were considered, one open (Configuration A, Figure 1) and two closed (Configurations B and C, Figures 3 and 2). A structure is closed if its components form a loop, and open if they do not. For example, a "U"shape is an open structure, while an "0"shape is closed. Of the closed configurations, the one with the longer axis fully supported (B) was clearly superior. Open structures are inherently more compliant than closed structures, but are also simpler and less expensive, so once a preliminary spreadsheet analysis of the structure's stiffness established that it could be stiff enough, an open structure was pursued.

Figure 1: Configuration A (Open) Figure 2: Configuration

Fon

Figure 3: Configuration B (Closed)

8 -1

A-Axis --...

Fixed Supports to elevate the Y-Axis -~

Figure 4: More complete view of configuration A Reflection on the strengths of configuration A created configurations D and E (See Figures 5 and 6). Configurations D and E placed the linear guide for the long axis directly on the optical table that serves as the machine's base. This change eliminated the need to transfer a reference surface more than a foot up from the optical table. Placing the long axis directly on the optical table also removed any problems from the axis twisting along the unsupported span.

9 Optical Table

Figure 5: Configuration D

10 Figure 6: Configuration E The names of the axes will now be defined (See Figure 5 and 6). The Y-Axis is the longest axis (over 60") and its linear guide is mounted directly to the optical table. The X-Axis (30") extends across the plane of the optical table orthogonally to the Y-Axis. The Z-Axis (over 10") is orthogonal to the plane of the optical table. The rotation axes are named A, B, and C in order based on which linear axis their rotation axis is parallel to. For example, an A-Axis will revolve around a rotational axis that is parallel with the X-Axis, while a B-Axis will revolve around an axis parallel to the Y-Axis, and so on.

11 The essential difference between Configurations D and E is that Configuration D places the Z-Axis at the end of the X-Axis instead of placing the X-Axis on the Z-Axis as in Configuration E. Configuration D thus enables the router to go into and cut deep cavities. By cantilevering so much mass, however, Configuration D will have a lower first natural frequency. It is important that the first natural frequency be above 30 Hz., or the machine will tend to ring. Ringing can damage the work piece and make the machine uncontrollable. Further consultation with the Director of the MIT Hobby Shop revealed that Configuration D had few advantages over configuration E because woodworking rarely requires cutting into deep cavities. Such activity is largely confined to mold and pattern making specialties, which represent only a tiny fraction of all woodworking activity. Consultation also revealed the great utility of having a joinery cutting capability. Joinery is cut on the edges of boards (See Figure 7), however, and neither configuration had the ability to work edges longer than the Z-Axis.

Figure 7: Router cutting joinery Two methods were developed for relieving this edge length limitation. One was to create a manual B-Axis that could cut joinery on boards mounted to a second optical table that was perpendicular to the primary machine base (See Figure 6). The other option was to create a manual A-Axis. The router motor would be rotated 90* and driven to an extension to the end of the Y-Axis (See Figure 4). With either of these options, joinery could be cut in the edges of pieces as long as the Y-Axis. Using an A-Axis had the obvious advantages of involving fewer parts, requiring less assembly work, and being able to cut joinery into longer edges. There were two possible locations for the rotational joint of the A-Axis. The joint could be integrated with the casting for connecting the X-Axis with the Z-Axis (the entire X-Axis linear module would rotate along with the router head) or the joint could be built into the mount for the router. The tradeoff between the two options was rather complex. The choice affected the length of the Y-Axis, the weight of the router mount (and thus the machine's natural frequency), how constant the machine's stiffness was, and the ease with which the router bit could be driven beneath the level of the optical table when making joinery. Because of these considerations, as well as various geometric and manufacturing constraints, the A-Axis was integrated with the router mount.

12 Experiments

Cutting Forces Before any analysis could be done to determine the structure's deflection during cutting, the magnitude of the cutting forces had to be determined. To this end, several experiments were done with the apparatus diagramed in Figure 8.

Wood

Scale Router Spring

Figure 8: Cutting force experimental apparatus cross section A number of different cuts were made in different types of wood, with different size bits, and at different feed rates. No force over 31 N was ever observed, and cuts with forces greater than 27 N resulted in an unacceptably bad surface finish. With this in mind, the displacements due to cutting forces were evaluated by simultaneously applying 50 N to each axis while the machine was in a worst case configuration. The worst case was determined to be when the Y-Axis was set to mid-travel, the Z-Axis was at the high extreme travel, and the X-Axis was set to the far extreme travel. This configuration maximized the applied moments and minimized the structure's ability to resist them.

Friction With Clamping Surface Because 3 = F,(1) it was necessary to determine the static coefficient of friction between the aluminum top of the optical table and the wood work piece. Machinery's Handbook gave a range of 0.2 to 0.6 for "wood on metals (clean)"', but prudence demanded experimental verification. To that end, a block of wood was placed on a sheet of steel (the surface material for the optical table), and the angle of the sheet with respect to the horizontal steadily increased until the block was just about to slip (See Figure 9). The test was performed with the wood grain both parallel and perpendicular to the direction of travel, but the difference proved to be negligible.

Green, Robert E., ed. Machinery's Handbook. 25 ed. New York: Industrial Press, 1996. p. 190.

13 The block will start to slip mg cos(O) when 3 = mg sin(9), which implies that mg Psmg cos(9) = mg sin(O)

Figure 9: Block on an incline with force vectors

P, = mgsin(9) = tan(O) (2) mg cos(9) The result after averaging several runs was that p,=0.38 (See Table 2), which is essentially at the midpoint in the range provided by Machinery's Handbook. Run Orientation Angle Ps 1 19 deg 0.344 2 Parallel 20 deg 0.364 3 21 deg 0.384 4 20 deg 0.364

5 22 deg 0.404 6 21 deg 0.384 7 21 deg 0.384 8 22 deg 0.404 Average: 0.379 Table 2

14 Linear Guides Because it was the fastest way to obtain a rough estimate of the necessary size, the linear guides were selected first based on expected bearing life. Estimates of the likely forces and moments were input into a spreadsheet along with the relevant parameters for a number of different manufacturers' rails. Due to the unusually complex loading the trucks were to undergo, the standard equivalent forces definition component of the bearing life equation,

equivaent Fvertical I + IFhorizontalI + C roI Fr (3) quzvan (~roll-capacity) 0 1 was inadequate, so after consulting with the manufacturer, a slightly modified version was used:

Irpitch ~yaw~ > FI~v - Iroll I + Ia Fequivalent =|Fvertica I+ |Fhorizontal + (4) I C i +roll yaw-capacity q roll-capacity Fpitch-capacity Fyaw

rpitch roll

Truck

Figure 10: Definition of torques and forces 10 roller Life = (equivale" L n = 3 (5) 3 ball Bearings with sufficient life (5 years of 8-hour days at maximum speed with a 50% duty cycle) were then checked to see if they could withstand the occasional shock of a 200 lbf. impact load at the spindle. Combinations with two trucks on one rail, two trucks on two rails, and four trucks on two rails were also examined.

15 dacross

daiong

Figure 11: Diagram view of two rail - four truck

J~roll I + +tc IF.yawl 6 Fequivalent = Fverticai I+ IFhorizontal )+ " (+) 4 2 dacross dalong dalong

dacross

Figure 12: Diagram of two rail - two truck configuration

roll Ipitch yaw

equivalent =- \Fvertical + - + (7) 2 racross 2 tch-capacity yaw-capacity

dalong

Figure 13: Diagram of one rail - two truck configuration

+1pitchI +1yaw C yIrollI Fequivalent = 2 (Fvertical I+ Fhorizontali)j) + + 2 (8) 2daong rol-aact The results of the bearing life analysis were used to size the rails that went into the first solid model. Finite element Analysis (FEA) of that model suggested that the dominant constraint would be stiffness. The cornerstone to successfully determining the stiffness of the machine's structure was a simple method of modeling the linear guide trucks. The truck's torsional compliance would be the principal source of the Abbe (angular) errors, which tend to dominate in an open structure. Successive iterations (See Figure 14) of a finite element model of just a truck and a rail were used in conjunction with roll data (roll being the most compliant

16 mode) supplied by STAR Linear Systems to find an equivalent Young's Modulus for the necessary trucks. The Young's Modulus could then be input into the FEA program. 0.18

0.16 Iterative Steps 0.14

0.12 0 0.10

0.08

8 0.06

0.04 Desired Output based on data supplied by STAR Linear Systems 0.02L

0.00 2000 2500 3000 3500 4000 4500 Young's Modulus (Pa)

Figure 14: Iteration towards an effective equivalent Young's modulus In the future, it might be better to model the truck as two parts (See Figure 15). After all, the trucks themselves are by no means monolithic. The center section would be "mushy" to approximate the compliance of the balls or rollers, while the outer section would be one to one and half times the stiffness of steel. The stiff outer sections would ensure that the loads were transmitted evenly to the compliant interior of the block. If, as is now the case, the entire block is given a low Young's modulus adjacent parts must provide the stiffness necessary to spread the loads throughout the block, a stiffness, which is in fact already present in the block. Using the stiffness of adjacent parts is not always desirable or feasible, and it is a great frustration to accommodate a design to the analysis tool.

Figure 15: Future block with a compliant inside and a stiff outer shell

2 Bamberg, Eberhard. Ph.D. Thesis MIT Mechanical Engineering Department

17 The Router Motor Consultation with the Director of the MIT Hobby Shop established the requirement for a variable speed (8-22 krpm) 3 hp. router motor with a " collet. It was tentatively assumed that an existing router would be bought off the shelf, and after being stripped of various unnecessary elements, bolted to the rest of the machine. There are two major varieties of routers available today: pneumatic and electric. Pneumatic routers were attractive because they don't burn out and because they are lighter than electric routers. Unfortunately, after consulting manufacturers of pneumatic tools such as Sioux, Campbell-Hausfeld, and Beaver, it was found that pneumatic routers over 1'/2hp do not exist. The higher power would require excessive flow rates at standard shop air pressures. The catalogs of major electric router manufacturers Sears, Porter-Cable, Bosch, and DeWalt were therefore surveyed to determine reasonable characteristics for 3 hp. routers. It was conservatively estimated that such a router would weight 8 kg. and require a cylindrical envelope 5 " in diameter and 6" high. These parameters were used in the solid model to fix associated dimensions and to accurately assess the machine's natural frequencies.

18 Concept Refinement Perhaps it has already become obvious, but it is very difficult to write a linear narrative about a non-linear design process. Version One Version One is the first in a series of solid model implementations of Configuration E with a manual A-Axis (See Figure 5). A single size 55 roller rail with a long style runner block creates the Y-Axis. The Z-Axis is composed of a single size 55 ball rail with a single truck. An integrated linear module was chosen for the X-Axis. The decision which rails to use for the first two (Y and Z-Axes) was made principally on the basis of bearing life and impact loading requirements. An integrated linear module was used for the X-Axis despite its expense because it was unlikely that a better combination of stiffness, compactness, and light weight could be designed in a short period of time. The specific model chosen was a STAR Linear Systems MKK 2080. The MKK 2080 had sufficient life and was the largest model before a substantial step-up in size and weight.

X-Axis

Z-Axis

Truck

Y -Axis

Shoulder

Figure 16: Version 1, note the shoulders for mounting the linear guides.

19 For the finite element analysis, the runner blocks were represented by materials optimized to have the same torsional stiffness (See the Linear Guides section). The integrated linear module was approximated by a rectangular cross-section beam contrived to have the same second moments of inertia that the manufacturer supplied for the selected linear module. Until a bracket could be designed, the part that serves as a proxy for the router was given the stiffness of steel, but a high enough density to ensure it would have twice the mass of the expected router. The factor of two was used to provide a safety margin and to account for the weight of the mount. This version showed promise, but was extremely short lived. The natural frequency of the first mode, 23 Hz., was unacceptable but not by so much as to be without hope. Furthermore, the casting to which the Y-Axis runner block and the Z-Axis rail were mounted would be quite difficult to manufacture because of the required interior grinding. Version Two Two approaches were taken to overcoming the deficiencies of Version One. The first was to replace the single Y-Axis rail with two smaller rails (See Figure 17). On the basis of bearing life, size 35 ball rails were selected. The single rail for the Z-Axis was also replaced with two rails to see if doing so had any advantages.

Figure 17: Version Two, multiple rails and runner blocks used for both the Y and Z-axes

20 Despite the cost, complexity, and other penalties, two small rails (in this configuration) proved to have no structural advantage over one larger rail. The natural frequency actually dropped to 20 Hz. In hindsight, this test was probably a little unfair. The Young's Modulus of the blocks was calibrated for roll moment loading (the blocks' weakest mode), while they were actually loaded in compression.

Version Three The second approach to solving Version One's problems was putting two trucks on a single rail for the Y-Axis. To ease manufacture, the casting was simplified by removing the troublesome overhang and using special assembly fixtures instead of shoulders for mounting the linear guides. To increase its stiffness, the casting was closed by attaching a plate to the back side.

Figure 18: Version Three, note the inclusion of the ball screws and the cleaner Y-to-Z casting This version had an acceptably high natural frequency, but the displacements due to anticipated cutting forces were too high. The major sources of these displacements were torsional deflection of the Z-Axis truck and bending of the X-Axis linear module. Because the ball screw sizes had been determined by this point (See Ball Screw Selection Section), a more complete model was created to include them, while the finite element analysis was still done on a more primitive model to reduce the computing time. The ball screws were mounted directly "above" the linear guides in an effort to minimize the torque they would apply to the block and to keep the castings as simple as possible.

21 The more complete model revealed an even larger problem than the unacceptably high displacements, however. When the Z-Axis was driven to the bottom of its range, the X-Axis linear module interfered with the Y-Axis ball screw. Fixing these problems resulted in a complete redesign of the main casting in Version 4.

X-Axis linear module interference with the Y-Axis Ball Screw Y-Axis ball screw

X-Axis Linear Module

Figure 19: Version Three, note how X-Axis linear module interferes with the Y-Axis ball screw

Version Four The main casting was completely redesigned, resulting in a number of improvements. Solving the interference problem while keeping the height (and thus the moment of inertia) of the casting to a minimum meant placing the Z-Axis rail on the side of the casting directly above the Y-Axis rail (See Figure 21). This change had numerous advantages. It reduced the distance from the center of the Y-Rail to the point of closest approach to the router. That distance plus the 30" that the router must be able to travel to meet the specification determines how much moment is applied to both the Y and Z- rails, so reducing the length was advantageous. The natural frequency improved as well. This change also simplified the casting connecting the Z-Axis to the X-Axis to a plate. FEA revealed that the structure's displacements due to the expected cutting forces were still unacceptably high. The primary sources of this problem were bending in the X-Axis linear module and torsion about the Z-Axis. Several attempts were made to solve this problem. The roller block on the Z-Axis was changed into a long style block, and steel stiffeners were added to the linear module. These measures were partially successful. The deflection in the Y direction, which had always been the most troublesome, was reduced to only 103% of the allowable accuracy. A high price was paid for this rather limited success, however. The additional mass of the stiffeners pushed the structure's first natural frequency down to a dubiously acceptable 26 Hz.

22 Figure 20: Version Four, view 1, note the redesigned casting

Figure 21: Version Four, view 2, note the stiffeners on the X-Axis linear module

23 Version Five Stronger measures were taken in the hopes of finally resolving the natural frequency and cutting force displacement issues. Slightly increasing the width of the casting connecting the Y-Axis to the Z-Axis and moving the Z-Axis rail all the way down created enough space to replace the long truck on the Z-Axis with two standard length slimline trucks, while increasing the height of the casting only slightly. The shape of the stiffeners was optimized and aluminum was used instead of steel because of its lower density. The new stiffener shape came from tapering a triangle two- thirds (See Figure 22) down the length of the linear module. The mounting structure for the router motor was also replaced with a lighter structure made of aluminum in an effort to create more natural frequency margin.

Redesigned Stiffener -

Z-Rail moved all the way down

Figure 22: Version 5 The changes were effective. The displacements were reduced to acceptable levels and the natural frequency was increased to 41 Hz. Because the model is a simplification, the

24 calculation of the first natural frequency was done without accounting for the mass of several components (e.g. the motors for the X-Axis and Z-Axis as well as other pieces of transmission hardware). The addition of these components will effect the natural frequency, but due to their light weight and proximity to the Y-Axis, the effect should not overwhelm the available margin.

25 Force Control During the experiments to measure the cutting loads the router would experience, it was discovered that a person naturally performs force control when working with a router. As the cutting forces vary due to the heterogeneity of the wood, the operator compensates to maintain surface finish. Torque (and thus force) sensing can be accomplished at almost no additional cost by measuring motor current. The drive system's losses due to friction are minimal so they can be neglected, and motor current is already an output from the servo controller. Using force control has advantages other than improving surface finish. Because the machine is monitoring the cutting forces, it could alert the operator when the specified tolerance can no longer be met. With this in mind, the operator could plan a roughing cut with high forces, and then come back in to finish the work with a less aggressive cut that applied lower forces to the machine. There are safety advantages as well. With knowledge of the size of the work piece's clamping face (either input by the operator or obtained by other means) and the available clamping pressure, the controller could set a hard limit on cutting force in order to prevent work piece slippage.

26 Ball Screw Selection The leads and the diameters of the ball screws were selected by matching the inertia of the transmission (the ball screw being the primary component) with the equivalent inertia of the relevant stage and load forces while simultaneously applying the shaft whip constraint.

NN2 2 t Mla3M ++r ~,(9) o trans where F 2 t2 r= !"" " (10) r M load and 1 1 (1 C =-+ a 1-b is the equation for minimizing the heat dissipation. Naturally, when there is no external load,

N2t oad, (12) which results in inertia matching.

whip E (13) s, A~px4 is the shaft whip constraint. Combining (9) with the shaft whip constraint, the definition of N,

No,, = ,z (14) the definition of lead, 2 -(15) =- the second moment of inertia,

I= , (16) 64 the cross-sectional area,

AX = 2 ,(17) *4 and the moment of inertia,

3 J. Park and S. Kim, "Optimum Speed Reduction Ratio for D.C. Servo Drive Systems," International Journal of Machine Tools Manufacturers, Volume 29, Number 2, 1989.

27 Jtrans 32 (18) yields the equations: d= 512M 22 d 7&dXE Sw ii7 (19) whip and

2 = p7rxd (20) 8 Mioad The results of applying these-equations with a shaft whip safety factor (sw) of 0.8 to the machine's three axes are show in Table 3. Y-Axis Z-Axis X-Axis diameter (d) 46.92 mm 16.12 mm 17.83 mm lead (1) 48.59 mm/rev 2.83 mm/rev 12.49 mm/rev Table 3 The buckling criterion,

Fuke, Cbuckle 2 (21) 'bukle 64sbx was then applied to these results. All met the constraint, but ball screws are naturally not available in exactly these sizes, so after checking against packaging constraints and availability, the following values were settled on: Y-Axis Z-Axis X-Axis diameter (d) 50 mm 16 mm 16 mm lead (e) 20 mm/rev 5 mm/rev 10 mm/rev Table 4 In the future, a better way to do this would be to plot dissipated heat,

WcR a il 2m2lv load 2 r 2 N2 am + 12 + , (22) Wd K t [ Moa N N2 over a reasonable range of diameter and lead values using a 3-D plotting program. Then, after applying the buckling and shaft whip constraints, the available lead and diameter combinations could be plotted as points on the graph so that the power minimizing combination could be selected more graphically and intuitively. Finally, the candidates were tested to ensure sufficient life. The relevant life equation is:

Life[hours]= 0 . (23) Fnlen, 2c o When all the candidates proved to have sufficient life, ball screw selection was complete.

28 Vacuum Clamping Effective work piece clamping is naturally a must for this application. The safety and accuracy of the machine depend upon it. At the same time, the clamping mechanism must be flexible to accommodate a variety of work pieces and have a low profile so that it does not interfere with the machine's operation. The clamping must also be quickly and easily done. Clamping a work piece is not a value adding process. Vacuum clamping offered a number of advantages. The operator need only set the piece on the machine's base and open a few valves. Accommodating irregular parts would not be a problem. There need only be one flat surface of sufficient size. This concept fit well with using an optical table for the machine's base. Optical tables already have holes through which the vacuum could be drawn. The optical table could also be easily partitioned internally (See Figure 23). By sub-dividing the table, a manifold of valves can limit how much of table vacuum is drawn through.

Optical Table .- .- .- .- .-. . . -...... - -. Internal partitions

Holes to draw vacuum through )0 X Figure 23: Partitioned optical table base The power necessary for the vacuum pump was calculated with the equation Power = (P, - P )Q. (24) The flow was determined using the orifice equation:

P0 d T=293K

t I

Pi Figure 24: Orifice diagram and variable definitions

29 Q = A.,Cd P- p I d9i n <86.78

2.23745+64 t (25) 1 d9" > 86.78

1.5+13.740 t t

where

9, = pvd (26) and

P + P P= ' . (27) while the pressure was set by the magnitude of the cutting forces as determined experimentally (gravitational loads are supported by the fences). Because of the piecewise nature of the orifice equation, the problem was solved iteratively using a seed velocity from the Bernoulli Equation (See Figure 25).

Bernou:iill veioty 93 -- Cd velcQt

Figure 25: Iterative process for determining flow rate After setting the internal pressure to 40% of atmospheric pressure, the maximum flow (for input into equation 24) was determined by multiplying the flow through one orifice by the number of holes in one sixth of the optical table (the standard hole pattern is 00.25" on 1"centers). The clamping force on any particular work piece is calculated using the equations:

F, = N ,, Ahoe (atm - i)Aclamp face Aoptical table Famp = 3 = pF, (29) Parts with clamping faces as small as 10" x 10" can be held, and assuming that the maximum open area on the board would be one sixth of the total area, the necessary horsepower of the pump, neglecting the pump efficiency is 9.5 hp. One weakness of using vacuum clamping is that while the clamping force scales with work piece size, the magnitude of the cutting forces which the clamping force must resist do not. One solution to this problem is to make lighter, lower force, cuts on smaller work pieces. Another is to provide fixtures to aid in the clamping of small work pieces. The

30 fixtures would have large clamping faces (possibly coated with a high friction material) that could in effect increase the size of the small part's clamping face (See Figure 26).

Work Piece Fixture Fence - Friction Material Optical Table

Figure 26: Clamping fixtures for small work pieces

31 Servo Motors The input torque has several different components. The largest is the torque required to overcome gravitational and cutting forces. Once those forces were estimated by experimentation and calculation, they were input into the equation:

Fexternal - (Fgravity + F,,,,,,g )(30)

After consultation with manufacturer's catalogs, an efficiency (q) of 90% was used. The next major torque component is that due to the friction from the pre-loaded ball nut.

£Fpre-load oad - F = (31) 404~ain,8)

tan(p)= (32) ball-circle

Fpre-load - (33) 10 The sum of the external torque and the torque due to pre-loading is the required continuous torque of the motor. The intermittent torque requirement is the sum of the continuous torque and the acceleration torque. The acceleration torque is found with the equation:

Facceleration = equivalent a , (34) where

Jequivalent = Mload 2 + Jball scre (35)

The maximum operating speed of the motor was calculated with the equation 2 -.= (36)

These calculations were performed for the three axes, and the results are summarized in Table 5. Y-Axis Z-Axis X-Axis External Torque 4.95 Nm 0.18 Nm 0.11 Nm Pre-Load Torque 1.69 Nm 0.08 Nm 0.09 Nm Continuous Torque 6.64 Nm 0.25 Nm 0.20 Nm

Acceleration Torque 1.26 Nm 0.05 Nm 0.02 Nm Intermittent Torque 8.61 Nm 0.30 Nm 0.21 Nm

Maximum Operating Speed 952.2 rpm 3808.8 rpm 1904.4 rpm Table 5 These results were used to select motors in Table 6 for the three axes.

32 Continuous Intermittent Speed-Torque Curve Torque Torque I TORQUE DDM-030/F-4050 TORQUE 15.8 (Nm)

13.6

11.3

9.0 Y-Axis 6.90 Nm 13.60 Nm 6.8

4.5

2.3

0 0 1000 2000 3000 4000 SPEED (R.P.M.) DDM-005N-1003 TORQUE (Nm) _0.9

Z-Axis 0.7 & 0.35 Nm 1.00 Nm X-Axis 0-5

-0.2

0 SPEED (R.P.M.) = Intermittent Operating Region Legend: = Continuous Operating Region = Drive Operation with 115 VAC RMS Input Voltage Table 6

33 .... W

Error Budget X Y Z Comments Structure (Gravity) 0.68 pm 7.70 pm 0.42 pm Structure (Cutting) 35.91 pm 70.08 pm 31.62 pm Ball Screw Lead 7.00 pm 15.40 pm 5.00 pm Accuracy grade 5 Ball Screw Windup (Cutting) 0.42 pm 0.04 pm 0.04 pm Ball Screw Windup (Gravity) 0.00 pm 1.09 pm 0.09 pm Ball Screw Axial (Cutting) 1.29 pm 0.35 pm 0.47 pm Ball Screw Axial (Gravity) 0.00 pm 8.69 pm 1.04 pm Ball Nut (Cutting) 0.14 pm 0.04 pm 0.14 pm Ball Nut (Gravity) 0.00 pm 0.97 pm 0.32 pm Ball Screw Fixed Support (Cutting) 0.15 pm 0.04 pm 0.15 pm Ball Screw Fixed Support (Gravity) 0.00 pm 0.97 pm 0.34 pm Parallelism P_ (Y-Axis) 22.00 pm 0.00 pm 22.00 pm HAccuracy Class Parallelism P_1 (Z-Axis) 8.00 pm 8.00 pm 0.00 pm H Accuracy Class Parallelism P_1 (X-Axis) 0.00 pm 12.50 pm 12.50 pm H Accuracy Class A_3 (Y-Axis) 15.00 pm 0.00 pm 0.00 pm H Accuracy Class A_3 (Z-Axis) 15.00 pm 0.00 pm 0.00 pm H Accuracy Class A_3 (X-Axis) 0.00 pm 15.00 pm 0.00 pm H Accuracy Class H (Y-Axis) 0.00 pm 0.00 pm 15.00 pm H Accuracy Class H (Z-Axis) 0.00 pm 15.00 pm 0.00 pm H Accuracy Class H (X-Axis) 0.00 pm 0.00 pm 15.00 pm H Accuracy Class Y-to-Z Casting 2.00 pm 10.00 pm 2.00 pm Z-to-X Plate 1.00 pm 20.00 pm 1.00 pm Optical Table 0.00 pm 0.00 pm 101.60 pm 0.004" optical table Total 108.59 pm 185.85 pm 208.74 pm RMS (of non-zero values) 13.42 pm 18.76 pm 26.29 pm

Average of Total and RMS 61.01 pm 102.31 pm 117.51 pm

Goal 127.00 pm 127.00 pm 254.00 pm Table 7 The errors due to structure, both gravity and cutting, were determined using FEA. Software compensation is used to filter out 95% of the gravity loading and 80% of the ball screw lead error. Budget items "Ball Screw Lead," "Parallelism," "A3," "H," and "Optical Table Flatness" were taken from the relevant manufacturers' catalogs. Ball screw windup was calculated with equation

(37) gshaft windup = ,xdG , which reflects the axial stiffness of the ball screw back through the encoder. Ball screw axial deflection was calculated with equation 4xF axial = 2 . (38) rrd E The errors due to ball nut and ball screw mount deflections were accounted for using stiffness data supplied by the manufacturer. The errors due to the Y-to-Z and Z-to-X

34 transition castings were calculated by programming a 10 arc second per meter grinding error on each surface and evaluating the displacement of the router bit.

Finally, to convert all of the worst-case assumptions to a more "typical" series of errors, the root-mean-square of all the non-zero errors was averaged with the sum. These results suggest that it might even be possible to tighten the specification for the X and Z- Axes.

35 Conclusion Going forward, there are still several open issues. The linear guides and ball screws need contamination protection. The X-Axis linear module has integrated seals, but the ball screws and linear guides for the Y and Z-Axes are currently unshielded. Because the ball screws are directly above the linear guides, both might be covered by a single set of bellows. The addition of vacuum chip removal should not be a problem. Ducting will have to be run from the router bit area back to the base of the machine, but surplus vacuum can be easily drawn from the clamping system. The biggest remaining challenge is probably the pivot for the A-Axis. The pivot mount must be both lightweight and stiff. Lightweight so that it does not unnecessarily erode the natural frequency margin, and stiff so that it does not add to the error budget significantly. Naturally, the pivot will have to be adjustable, but the locking mechanism will have to be designed with vibration in mind. It would probably also be a good idea to include a potentiometer (or equivalent) so that the controller can measure the angle of the pivot. The controller must know the set angle so that it can match it when cuffing at miter angles other than 00 and 90*.

36 Appendix: Competitive Benchmarking Name The AutoRout Router Series Company AutoRout Inc. Work Volume 5'x5', 5'x8', 5'xlO',5'x12' Feed Rate Repeatability Accuracy Spindle Multiple, 7-12 hp Clamping Vacuum, 10-30 hp URL www.autorout.com Comment drills, saws, 4th Axis Price $53,296 Base

Name AXYZ Series Company AXYZ Automation Inc. Work Volume 39"x39"x6" to 150"x99.5"x6" Feed Rate 300 ipm Repeatability Accuracy Spindle 1.5-7.5 hp Clamping URL www.axyz.com Comment Price

Name RT Series Company BobCAD CAM Inc. Work Volume 8'x4'xlO" Feed Rate 200-1000 ipm Repeatability Accuracy 0.005" Spindle Multiple, 3.25 hp Clamping Vacuum optional URL www.bobcadcam.com Comment THK or STAR Linear Guides Price $15,000-$47,000

Name CNT 1000/900 Series Company CNT Motion Systems Work Volume 60"x120"x5" Feed Rate 1000 ipm Repeatability Accuracy Spindle Multiple, 0.5-3 hp Clamping Vacuum, 10 hp URL www.cntmotion.com Comment Price $13,000-$45,000

37 Name Thermwood Router Series Company Thermwood Inc. Work Volume 5'x5' to 5'x20' Feed Rate 1500 ipm Repeatability 0.002" Accuracy Spindle 4-15 hp Clamping Vacuum URL www.thermwood.com Comment Turret head, 5 Axis available Price $40,000-$120,000

Name 200-600 Series Company Gerber Scientific Products Work Volume 35.4"x31.4"-6'8"x10' Feed Rate 600-54 ipm Repeatability Accuracy Spindle Clamping Vacuum URL www.gspinc.com Comment Price

Name CNC Mini-Router Company Minitech Work Volume Feed Rate Repeatability Accuracy Spindle Clamping URL www.minitech.com Comment "Portable" Price $9,500

Name Camtool Company Larken Automation Work Volume 24.5"x24.5"x5" to 100"x150"x6" Feed Rate 200 ipm Repeatability ±0.001 Accuracy Spindle 1.5-3 hp Clamping T-slot or vacuum URL www.storm.ca/l-arken/ Comment Price $7,000 - $30,000

38 Name Shoda NC-1 Company Shoda USA Work Volume 0.5m x 1.Om x 0.4m Feed Rate Repeatability Accuracy Spindle Up to 3, 12 hp Clamping Vacuum clamping or T-Slot URL www.shodausa.com Comment tilting spindle, very industrial Price

Name ShopBot Personal Robotic 96 Company ShopBot Tools Inc. Work Volume 8'x4'x7" Feed Rate 200 ipm Repeatability Accuracy 0.01" Spindle bracket for any standard router Clamping URL Comment user built from parts Price www.shopbottools.com

Name Company Super Tech and Associates Work Volume 36"x23.5"x2.375" Feed Rate 150 ipm Repeatability Accuracy Spindle DeWalt Router Clamping URL www.super-tech.com/root/i Comment Price

Name DaVinci Production Center Company Techno-Isel Work Volume 8"x8"x6.8" to 59"x96"xlO.8" Feed Rate Repeatability Accuracy Spindle 0.75 hp to 7.5 hp Clamping Vacuum or Clamp bar URL www.techno-isel.com Comment Price

39