Linear Motion Slide (LMS)

ANALYSIS

From the Kalpakjian, Manufacturing Engineering and Technology I derived the cutting forces on a lathe for aluminum. I applied beam bending theory, beam shearing, and beam axial compression to do the stiffness analysis of the LMS (see design spreadsheet):

• In more depth, for the angular stiffnesses, I assumed a linear load distribution on the contact areas, found the pressure magnitude and computed the angle deflection assuming a distributed stiffness on the contact areas.
• Next, I did an HTM analysis on the LMS, the carriage and the tool to find the errors gains from the LMS deflection on the tool tip, putting each reference frame at the center of stiffness of each element considered.
• For the accuracy of the system, I used the stiffness calculations, HTM analysis and cutting loads to compute the tool deflection under roughing cutting loads and geometric errors (flatness from manufacturing process).
• For the repeatability, I estimated it from the wear equation and PV analysis as well as the change in tool deflection when the cutting loads change up to 50% (using the same error gain analysis from the HTM analysis).
• From the analysis, I was able to size each component in the dovetail system: rail size, carriage, bearing pads, gibs, preload mechanism (Belleville washers) to fulfill all the functional requirements.

CONCEPT EXPLORATION

I then explored different concepts that would be suitable for this application including, twin rails, dovetails, box ways, roller/rails as well as ball/groove designs. I then selected the dovetail rails and twin rails concept to do the analytic model and predict their performance.

From the analysis, I was able to understand the trade-offs of each design.

• The dovetail design is very stiff but will be very sensitive to the manufacturing since any “bump” on the rail of the carriage will impose a large normal force, thus the force required to move the linear slide will increase drastically. Also preloading the system with gibs will not be an easy task.
• The twin rail design is much more compliant especially in roll unless the footprint (size) of the LMS is increased significantly (diameter of the rails and spacing between the rails). However, the compliance of the system and the quasi-kinematic design make it easier to manufacture, assemble and less sensitive to geometric errors of the LMS. In addition, having the tool closer to the center of stiffness (and increasing the tool height to give some clearance for cutting the part) will help alleviate the error motions from the roll stiffness of the LMS.

I then went on and made a sketch models of each design and tested them in terms of stiffness and geometric errors to validate my analytical model. From the analysis and the sketch models, I selected the dovetail LMS design for the ease of preloading the rails, the increased overall stiffness of the system.

FABRICATION

Most of the fabrication was done using a mill. The carriage and the rail were made from aluminum stock, using a dovetail cutter. The gibs were made from brass stock. The teflon pads were made on the waterjet and press-fitted in the carriage. For the dovetail brass gib, I first made it using angle blocks but during assembly, there was an angular mismatch from the dovetail brass gib and the rail (line contact). To fix that mismatch, I fixtured the gib in the carriage and used the dovetail cutter to get the right angle. For the next iteration, I will definitely machine the dovetail gib having it bolted on the carriage.

TESTING

The LMS was tested for angular and translation stiffness, as well as angular accuracy to ensure that it satisfied all the functional requirements (see design spreadsheet that includes all the experimental results).

STIFFNESS TESTS:
For the Y and Z stiffness the LMS was positioned on the mill and the deflection was measured using a dial indicator. For the Y stiffness a digital scale was used to load the LMS but for the vertical stiffness body weight was used to load the LMS and a weight scale was used to record the body weight load on the LMS.

The measure Z-stiffness (vertical) was of 7.7 N/um which matches the predicted value of 11 N/um (stiffness mainly from Teflon bearing pads). On the contrary, the measured Y-stiffness of 24.3 N/um was almost an order of magnitude less from the predicted value of 423 N/um. The measured value is also several orders of magnitude higher than the preload springs stiffness. I added some oil on the rails to get a sense on the contact area between the dovetail gib and the rail, which turned out to be a small patch rather than the entire area. This contact area difference explains the reduced stiffness value.

For the angular stiffnesses, I had incorporated a mount to apply the loads at a set distance from the carriage and measured the motion of the laser pointer on a sheet 16.8m away from the carriage.The predicted angular stiffnesses were derived using the translation stiffnesses so measured the pitch, and roll stiffness 5.9 and 4.7 Nm/mrad matched the predicted values of 7 Nm/mrad. Similarly, since the yaw stiffness is derived from the y-stiffness there was a discrepancy between the measure yaw stiffness 13.7 Nm/mrad and the predicted yaw stiffness of 173 Nm/mrad. However, if we use the measured y-stiffness to derive the yaw-stiffness we get a value of 11 Nm/mrad which is much closer to the measured yaw stiffness.

ANGULAR GEOMETIC ACCURACY OF THE LMS:
To test the angular geometric errors of the LMS, it was fixtured on the mill, mounted with a laser and the LMS was moved back and forth and the laser motion was marked on a sheet 16.8 m away from the carriage. The measured angular errors were:

• Yaw Error: 0.5mrad --> 27.5um tool tip associated error (predicted 56 um)
• Pitch Error: 0.9mrad --> 31.5um tool tip associated error (predicted 25 um)
• Roll Error: 0.1mrad --> 3.5um tool tip associated error (predicted 25 um)

The predicted values were computed assuming an accuracy of 0.001” on the mill (flatness error). The manufactured LMS fulfilled the functional requirements in terms of geometric errors.

DISCUSSION

I am very satisfied with the dovetail LMS and its performance, knowing that it only requires less than 10N when preloaded to move back and forth. Oiling the rails and using Teflon pads definitely helped getting such low friction on the slide. Using more compliant gibs such as delrin might be easier to manufacture and improve the lateral and yaw stiffnesses since it will conform the the rail after being preloaded.

Leadscrew

ANALYSIS

For the analysis, I looked at (see design spreadsheet):

• The required driving torque using the screw equation, cutting loads ,friction of the carriage on the rails and friction at the bearing blocks on the leadscrew.
• The axial and bending stiffness of the leadscrew (beam under compression and pin-pin beam), as well as the bushings and bearings (hertz contact).
• Look at the backdriveability (screw equation condition)
• Stress analysis on the leadscrew from cutting loads and misalignment.
• Geometric errors due to misalignment of the leadscrew (force displacement on a pin-pin beam, resulting in a load on the carriage --> displacement at the tool tip)
• Load induced errors at the tool tip due to leadscrew’s assembly axial stiffness and cutting loads.

FABRICATION

Most of the fabrication was done using the lathe. The bushings were made on user a delrin rod and the leadscrew was also turned down on the lathe. The leadscrew was indicated using a dial indicator to find the high points so that when it had to be rechucked for the second operation we could place the high points at the same location (minimizing radial error from 3-jaw chuck). The threads on the leadscrew where then added using a M6x1.0 die.

TESTING

For this module, I tested the following characteristics: Axial stiffness of the leadscrew assembly, Driving torque to move the carriage, Carriage error motions Z- Axis and Y-axis, Lead Error (see design spreadsheet that includes all the experimental results).

STIFFNESS TESTS:
For this measurement, I used a digital load scale, a dial indicator (0.0002” resolution) with the assembly fixtured in the mill. The measured stiffness was 2.6 N/um which matches the predicted stiffness of 5.6 N/um.

DRIVING TORQUE
For this measurement, I used a spring scale on the handle (52 mm in diameter) and recorded the load at which the handle starts to move. This measurement actually considers the static friction and not the dynamic one making it a more conservative value. The measured torque 128 Nmm matched the predicted one of 145 Nmm. The LMS is very easy to move!

GEOMETRIC ERRORS
For the lead error as well as the carriage error motions measurements, I used the dial indicator with the LMS fixtured on the mill. I had the carriage travel and measured the error on the dial indicator. For the lead error, I couldn’t measure any error on the dial indicator for half a revolution. Similarly for the geometric errors, they were on the order of the rail flatness errors measured last week ( about 30 um) showing that any misalignment or bowing in the leadscrew didn’t affect the motion of the carriage significantly.

DISCUSSION

The bearing layout and modified leadscrew fulfill the design requirements. The custom anti-backlash system enables the cutting tool to cut in both direction withtout losing the preload nor increasing the driving torque.