PUPS 5 & 6 | Concept Details Exploration & Error Budget
To see all the calculations and work below as well as the problem and description head to my Transforming Table page.
Small apartments need furniture to be multifunctional and flexible. Who has space for a coffee table and dinner table and work table? What if the same table can transform between the three?
The first step in designing my table mechanism is to design a single precision linear motion axis. For this transforming table I considered several options including a wood based actuator and an aluminum actuator shown in Figure 1 below. The aluminum actuator and bearings were more expensive, but also more precise with a low profile. Igus Inc., the company that manufactures these particular parts, was generous to donate the parts I needed as free samples for this class project, which allowed me to meet my functional requirements.
Figure 1, the left image, shows the rail and bearings with a coordinate system located at the end of the rail. The 'Y' direction is most sensitive, because movement of the bearing along the rail determines motion of the linkage and the height of the table.
A key risk of a bearing/rail design is applying a moment about the x-axis in the y-direction from the leg linkage that would cause the bearings to jam on the rails. Using four bearings instead of two and spacing them in the y-direction would resist a moment, but an alternative countermeasure could be eliminating the moment by aligning the center of friction with the center of rotation of the pin such that there is no moment because there is no moment arm on which to act. As I design the slider I'll work to use this strategy even though it is constrained by a low profile.
Figure 1: The linear actuator and bearings I will use for table movement from IGUS. Part drawings and spec sheets can be found here
The total allowable error in the height adjustability for this transforming table is a generous 4mm (4000 microns). Based on my error apportionment calculations based on a 3 axis design (one axis for each linear actuator for each leg) gives an allowable portions of the error to geometric and load-induced sources of error and excludes thermal and process errors, which are not significant in this design. The total allowable error is split evenly between geometric and load-induced sources, each with 1.150 mm per axis. For both, bearings have 0.705 mm of error, the structure has 0.705 mm of allowable error, the actuator and cables both have 0.141 mm of allowable error. Sensors are not used in this design and thus do not contribute.
The spreadsheet and calculations for this apportionment can be found in the excel spreadsheet link to this button, and are summarized in Table 1.
As discussed in the risks and countermeasures, a long slider such as the one shown on the left is not necessary if the center of friction is aligned with the center of rotation of the pin joint. This allows for cost savings using fewer bearings as well as shorter rails, which saves cost and space on the underside of the table.
Table 1: Error apportionment calculations (Spreadsheet made by Professor Alex Slocum)
The most sensitive direction for this table is the up and down z height, which is directly determined by the x displacement of the slider, shown by the arrow in the upper left sketch.
To test the leg concept, I built an early SolidWorks model, scaled all the parts to 1/5 the full size and laser cut the prototype out of acrylic. As you can see in the CAD image, the slots are larger than the sliders allowing the legs to wobble out of place. Furthermore, the hot glue attachment I used for the legs quickly broke broke off due to moments about the connection. This janky prototype with larger than normal errors and wiggle helped me to exaggerate and understand some of the key areas I needed to focus on designing precisely and what the major risks for failure are. I discussed this prototype with my peer reviewers Maha and Abbas, to better understand how I need to design for lateral stability and robust pin connections. We also identified the moment acting in the pitch direction on the "slider." Summarizing my learning, going forwards I want to focus on designing moment connections between the short legs and the linear actuator and a bracket that resists moment for a pin joint connection between the fixed long leg and the bottom of the table.
Figure 2: CAD Model of 1/5 scale laser cut prototype and preliminary sketches
Attached is the error budget spreadsheet I initially used to wiggle lengths, stiffnesses, and materials to iterate my design.
At the start, using error budget thinking to do my own calculations was more useful that using the actual full error budget, because I wasn't confident in the way I set up my error budget. Much later on I realized that using the error budget above won't work for linkage systems like my desk legs that have multiple structural and open loops, so I wrote my own linkage error budget.
Some design decisions that were made using the error budget and auxiliary calculations include:
1. Changed the long leg material from 2x4 lumber to 2x2 lumber - 2x2 demonstrated enough stiffness for the expected forces
2. Chose to use 2 bearings instead of 4 for each carriage - enough pitch stiffness with 2 (shown above)
3. Chose to use shoulder bolts for the leg joints, other than the carriage to short leg pin connection
- Shoulder bolts have more friction than using bushings or bearings, but the increase in friction is not significant enough to
justify the added cost
- Chose to use 3/4" plywood for the desk top - met necessary stiffness, from a SolidWorks static simulation I found
1.5x10^-3 mm of deflection in the center when a 90 lbf. was applied
From the error budget, I identified that alignment of the rails with the pulley and the long leg mount all in plane was important to minimizing error, most importantly by reducing friction on the joints and avoiding jamming of the structural members. I was able to achieve the best alignment by drilling all the pilot holes in the bottom of the table at the same time using the CNC router. To the right is an image from my CAD showing the locations of the drilled holes. Furthermore, I also cut the circular shape with the same cutting program, so that they were both cut using the same zero position. From building my planar exact constraint mechanism with the CNC router I learned that the center placement of a hole is highly accurate, but the size of a hole is highly variable, thus I knew that drilling pilot holes, which are allowed to have +/- 2mm tolerance was an ideal operation accounting for router inaccuracy.
Adding onto my analysis for PUPS 6, I changed some of my error apportionment, increasing the total allowable error from 4000 microns (4mm) to 0.25 inches (approximately 6mm) Since the rest of my desk is measured in inches I changed my error calculations accordingly.
Up until now, in my analysis I primarily considered z-height as the sensitive direction and the primary axis in which I was concerned about error. Lateral forces acting on the desk are a different challenge and require their own analysis of the distribution of lateral forces on the 3 legs spaced kinematically and the wiggle of the pin joint leg mechanisms.
One important environmental error that would be of concern in highly precise assemblies would be the temperature gradients experienced by the object which might cause expansion or contraction of the materials.
The following pages lay out the coordinate systems and points of interest for an error budget in the form of the spreadsheet above. First, a point of interest is selected and the points of attachment and coordinate systems are successively assigned walking from one structural member to it's connecting member through a single structural loop. Linkages such as this design cannot be solved in that closed form method because they include multiple and open loops in their structures. Even though this method didn't end up working out, it was helpful to try to work through it because I really learned how to identify points of interest and attachment, calculate the stiffnesses of the members, and identify areas of compliance in the system.
Notice that the sketches below primarily focus on an unstable single leg system - which was one of the points of discontinuing error budgeting. In my new error budget I still focus on calculating errors in height for one leg, and then using kinematic equations coupling the three legs.