Desk Design Update: Twin-Column Design

Revelation about structural analysis

During our peer review meeting last Saturday, Yadu pointed out that we have been focused on the deflection over the cantilevered tabletop and neglecting the compliance in the bracket used to attach the desktop to the slider in a single-column design. This was something I had not thought about previously, but I pointed out that it would be relatively easy to stiffen up the bracket using gussets or heavier gauge material.

After checking my error budget to take this into account, however, I realized that this was a bigger problem than I thought. My friends who are building standalone desks will likely be fine because a standalone desk gives you a large amount of clearance below the tabletop to add stiffening members. I, on the other hand, need to keep the structure below the tabletop as thin as possible so that the “sit” position is not excessively elevated above the original height of the supporting table. I estimated that I could get away with elevating the work surface in the “sit” position by at most 2.5 inches, considering the amount of vertical adjustment allowed by the typical office chair.

Change of direction: dual-column design

Preliminary rendering of dual-column desk design

The limited space below the tabletop means I would have to either resort to unsightly gussets above the work surface to make my bracket sufficiently stiff, or use comically thick angle stock to make my bracket. A quick calculation showed that a 2 inch-wide bracket made our of steel would have to be 7 mm-thick just to be stiff enough for me to blow the entire 4 mm of allowable load-induced deflection on that one member. At this point, I decided that the chunky hardware required to realize the single-column cantilevered desk design would obviate the visual lightness that made such a design appealing in the first place. Therefore, I will pick up my fall-back twin column design from this point on.

Error budget and CAD

Most of this week was spent on populating and refining my error budget for the dual-column design. In a way, I am glad I pursued the single-column design initially — this helped me gain some very valuable experience preparing error budgets. Especially interesting was figuring out how to approximate a closed-loop structure in the error budget spreadsheet.

Based on the error budget, which of course is still a work in-progress as I work on better ways to model interface stiffnesses, the twin-column design looks set to satisfy my total error allowance of 5 mm (80:20 split between load-induced and geometric). In fact, I have been able to cut back on the size of many structural members, which improves the appearance of the desk and allows me to build using actual furniture-grade boards instead of heavy structural timber.

CAD models for my desk can be accessed here. The error budget is available here.

Bearing Preload System Design

Excessive radial clearance in my linear motion axis, necessary because of the tolerances I could achieve using conventional woodworking techniques, was the major cause of inaccuracies and non-repeatability in my system. We learned recently in class that appropriate preload is a feasible approach to eliminate radial clearance and improve geometric repeatability without compromising overall bearing stiffness.

There is a large variety of possible ways to build preload into a sliding contact linear bearing. I decided to explore the use of an elastomeric compliant layer in the slider since it seemed like a concept I could eventually incorporate into my desk. Few suitable compliant materials have the wear and frictional properties to be good bearing materials, so this compliant layer will be sandwiched between the original wooden slider and a new bearing plate made of a wear-resistant material.

All of the repeatability and stiffness benefits of preloaded bearings only hold if the preload is never completely relieved under normal operation. I estimated that the primary source of aberration in the boxway is due to local imperfections in the side rails due to the poor surface finish achievable in the oriented strand board I used to build them. To the first order, I estimated that these imperfections have a characteristic dimension of approximately 0.1 mm.  In order to ensure some preload is retained through the entire travel, I invoked St. Venant’s Principle and designed for a preload displacement of 10 mm.

Simple design spreadsheet for bearing preload system

Using these values, I built a simple spreadsheet that predicts the marginal amount of actuation force due to the preload and specifies the geometry of individual parts to help support fabrication. For a hypothetical material with a modulus of 0.001 GPa (based off of linoleum), the appropriate displacement would result in an additional 140 N of actuation force required. Since my toy linear motion axis does not drive anything other than its internal resistance, I figured my stepper motor would be able to move it without too much trouble.

Desk Concept Exploration Overview

Overall Form

Alternative concepts for overall structure

After ruminating on the overall architecture of my desk for the past couple of weeks, I have decided to commit to a desk-mount design with a single linear motion axis. My reason for pursuing a desk-mounted design is so that the new device would fit into current and future spaces that I would live in. I anticipate living in pre-furnished spaces in the near future, so an adjustable desk that complements an existing desk is preferable to an additional piece of free-standing furniture.

I have also decided to limit myself to a single axis of motion in order to control costs and mechanical complexity. In my experience using desks, I have had little occasion to desire adjustment other than height. I am currently considering a single-column cantilever design as well as a twin-column design. Read more about these concepts below.

Error Apportionment

Error apportionment summary for adjustable desk

I am setting a total allowable error of 5 mm along the height dimension for my desk. 10% of this is allocated to the actuator (I think cheap bearings and structural members would contribute significantly more error than purchased leadscrews), and the remainder is evenly split between bearings and structure. As I mentioned previously, user experienced is degraded far more significantly by “free wiggle” geometric errors than load-induced deflections that are associated with significant resistance.

Single-Column Design

This is an early concept that I started off with, and it has stayed around through the process largely because of its simplicity. Additionally, there is something visually appealing (to me) about a cantilevered desk with a substantial support column made from wood.

The drawbacks are primarily associated with the relatively long cantilever length, which natural translates to larger deflections and higher forces at the bearing.

Twin-Column Design

This alternative design has the tabletop simply supported (an idealization) between two bearings each running in a column. These columns are located centrally at each of the sides of the desk. The advantage of this design is that it halves the moment arm for the worst-case scenario where all of the design load is concentrated at one corner of the desk. This translates to lower forces felt at the bearing and smaller deflections in the tabletop.

However, the need to double up on structural material and drive mechanisms (need to somehow actuate both sides to prevent racking) would increase cost. Additionally, the visually interesting bearing columns are not moved to the sides and out of direct view for the user. And finally, the columns may turn into a nuisance by reducing elbow room.

In the interest of time, I am going to focus on the single-column design for my exploration this week. The twin-column design will serve as a backup that I can come back to if I run into unanticipated and terminal problems with the single-column design.

See my first-order error budget for the single-column design here.

Quick Note on Self-Locking Bearing Concept

Avid readers (there are not that many…) of this site will remember reading about an idea I had to exploit the binding characteristics of sliding bearings to achieve self-locking. The primary reason for doing that was to allow the use of a more compact and lighter belt or chain drive system. Unlike leadscrews, belt or chain drive systems generally don’t have sufficient friction in the drivechain to resist substantial forces when powered off.

More in-depth analysis on self-locking slider bearings, leading to abandonment of the idea

I spent some time this week analyzing a concept that has a chain attached some distance away from the center-of-stiffness of the bearing. The idea was to design the bearing so that the weight of the table and things piled on top would cause the bearing to bind up and resist moving downwards. When the actuator is switched on, it pulls on the chain to relieve part of the binding moment, thereby releasing the table to move upwards. Unfortunately, my analysis showed that the reciprocal action to this — when the user wants to move the table downwards — actually increases the binding moment and causes even more friction. I did some what-ifs scenario in a spreadsheet and found that my actuator would have to exert thousands of Newtons to overcome the friction and move the table downwards. This is not only difficult to achieve using a small stepper motor, but also very very difficult to achieve without breaking the structure or bearings due to the large forces involved. Ultimately, I think there are too many drawbacks to this idea for it to be worth pursuing further, so back to leadscrews it is.

First-Order Error Budget: Single-Column Design

The single-column design is made up of 4 structural elements:

  • Tabletop
  • Bearing column
  • Crossbeam
  • Legs

Since this is a desk, the sensitive direction is vertically up and down.

I am making this error budget assuming my entire design load (20kg) is concentrated at one corner of the tabletop. This is a worst-case scenario that is unlikely to happen in use, but gives conservative error estimates that would guarantee performance. I can back off on the worst-case scenario for further design refinement if it turns out to demand excessive material or cost.

Tabletop

The cantilevered tabletop is under bending both across the depth of the table and across half of the width. We can approximate their joint contribution using superposition, although this may not be exactly accurate due to shears in the tabletop.

Bearing column

The bearing column can also be modeled as a cantilever undergoing bending in both “pitch” and “roll” directions, since the point load is placed at a corner. The actual cross-section of this member would depend on what bearing design I use. Here, I am assuming a solid rectangular cross-section to build the spreadsheet. The correct second moment of area can be substituted once I nail down a bearing design.

Crossbeam

The crossbeam undergoes torsion and bending simultaneously. The torsional component is relatively self-explanatory, but I took a while to realize that the corner loading meant that one leg would be (at least partially) unweighted, allowing half of the crossbeam to behave like a cantilever. This first-order model neglects the dead load from structural components, leading to a relatively large contribution by this term. The symmetrical design means that the weight of the structural members will partially cancel out this contribution.

Legs

The legs act as cantilevers under bending. However, after realizing that the crossbeam bends and transfers load differentially between the two legs, I became a bit confused about how to formulate the deflection contribution from these members properly. I plan to sleep on it and revisit in the near future.

First-order Spreadsheet

First-order error budget

The total load-induced error from the structure alone was about 5 mm, which is approximately twice as high as I have apportioned. On the bright side, the vast majority of this error came from the tabletop. My current model assumes a flat sheet of plywood without any bracing, trusses, or composite panels. There is a great deal of room for improvement in that area at little cost, so my focus for next week will be improving the stiffness of the tabletop module.

The geometric error, which in this case comes entirely from the bearing, is around 2 mm. This comes from assuming a 0.25 mm radial clearance in plain sliding bearings. As we discussed in class, appropriate use of preload can all but eliminate this sort of error, so I am not excessively worried about this either.

PUPS 3: Machine Specifications

The PUPS this week forced me to take a step back and look at the desk design problem from a high level. I concluded that I have prematurely focused in on a desk-mounted design previously. Although I still think that strategy would be the easiest way to achieve my functional requirements surrounding compactness, I will be revisiting alternative strategies and concepts in the next couple of weeks to make sure I am giving each idea the due consideration before selecting one.

Deterministic Design of Simple Linear Motion Axis

Overview

FRDPARRC table: linear motion axis design

I plan to ultimately use this linear motion axis design in my standing desk, so the design here will be driven by broader design requirements I have in mind for the desk. I have narrowed the concepts I am considering down to some form of sliding-contact bearing, partly to minimize system cost and mechanical complexity, but more importantly to try to exploit the binding characteristics of slider bearings in creating a self-locking axis.

Error Apportionment

As seen in the FRDPARRC table, the maximum deflection I am willing to tolerate at the extreme corners of the worktop is 0.5″. This is a perceptible amount of movement, but not uncommon in portable furniture. While the first-order analysis we did in class simply summed the error contributions from bearing slop and worktop flexural deflection, I decided to distinguish between the two sources of error.

The error motion due to radial clearance in the slider bearing takes place against negligible stiffness — i.e., a tiny load applied by the user in a new direction can almost instantaneously cause the system to take up all the backlash, going “clunk” in the process. On the other hand, the load-induced deflection is resisted by the flexural stiffness of the worktop structure. There is a more or less linear mapping between load applied by the user and deflection. In my experience, this form of motion has a far smaller impact on user experience since small amounts of flexing under load are expected by most people, while the sudden “backlash” type motion creates an unnerving sense of instability. To account for this different weighting from a usability standpoint, I assigned a 80:20 split between load-induced and geometric errors.

My error apportionment spreadsheet can be viewed here.

Self-Locking Axis Analysis

One of the most important functional requirements for my standing desk is that the mechanism is self-locking, even when the drive system is de-energized. This is critical for safe and efficient operation of the system. In a screw-driven system, one way to achieve self-locking is to increase friction in the threads. The cost of increasing friction (by using a tighter thread fit class, oversizing balls for ballscrews etc.) is that a more powerful and more expensive motor is required to overcome the friction whenever the system is actuated. An alternative (reciprocity!) way to achieve self-locking is to reduce the torque required to support the design loads by specifying a leadscrew with a finer pitch. Unfortunately, there is also a tradeoff here with actuation speed, since a finer-pitched leadscrew will have to rotate through more revolutions to move the carriage by a given amount.

I have decided to focus my efforts this week on utilizing the usually undesirable binding behavior of sliding linear bearings to produce a linear motion axis with self-locking capabilities. Intuitively, the jamming of sliding bearings is associated with rotation caused by loads acting away from the center-of-friction, which means that it can be “switched” on an off by an applied moment. Since the linear motion axis will most likely be placed along an edge of the tabletop (as opposed to through the middle), the dead load associated with the tabletop structure and things piled on top is inherently off-center. There is a chance, then, that this concept will work. The above pages from my notebook contain my analysis to verify this intuition. From this analysis, it appears that it is workable. My next step is to consolidate this analysis in a spreadsheet and try out several what-if scenarios to determine whether (and if so, by how much?) it improves upon a conventional self-locking design that relies on friction torque in the leadscrew.

Three Groove Kinematic Coupling Design

Concept Selection

One of the tasks for the course this week is to design and build a three groove kinematic coupling. You can read more about kinematic couplings here, but they are primarily used to repeatably position objects relative to each other. I don’t yet have a specific application in mind for this coupling, apart from a vague notion that I would like to eventually use it to fixture some sort of camera in a future project. Therefore, I will be building my coupling to a 2.5″ coupling circle diameter — a reasonable size to accommodate most hand-held cameras.

In the interest of controlling cost, I have chosen to use plywood for the top and bottom plates of the coupling. To select the material for ball and groove contact elements, I estimated the Hertzian contact stresses resulting from anticipated loads (from a ~1 kg camera) using Prof. Slocum’s design spreadsheet and found that a wood-on-wood interface would have been feasible. However, I ultimately decided to use steel for the contact elements in order to avoid potential issues that may result from material and geometric irregularities found in biological materials. Additionally, gluing in steel balls and dowel pins would be easier and faster than machining them out of wood, which would require either a time-consuming workholding setup or a CNC machine.

Analysis

I adapted the template spreadsheet to predict the stiffness and error motions specific to my design. You can view the full spreadsheet here. Some of the key insights from this analysis are:

  • With a preload of 1.48 N per ball, as would be the best case scenario with the 3 lb max.-pull magnets I am intending to use, and a 2-kg object (reasonably large SLR or medium format camera) on the coupling, the maximal lateral force it can tolerate is approximately 13 N when applied in line with a vee groove. Lateral loads above this threshold would cause the coupling to come unseated.
  • An RMS stiffness of 62 N/micron. This means that the Z-position should be capable of achieving sub-micron repeatability in the face of minor variations in the preload or weight of the upper piece of the coupling. This is important as it would facilitate interchangeability of the part being fixtured.

Detailed Design and CAD

After convincing myself of the feasibility of my design, I built a solid model to check that I haven’t made any mistakes in my geometric calculations and to generate toolpaths for the CNC router I plan to use to machine the plywood boards that will receive the steel contact elements. I also decided to use a pair of neodymium magnets inset into the plywood plates to provide a more consistent preload for the coupling. These magnets will be glued into circular pockets positioned at the center of each half of the coupling. My CAD models can be downloaded here.

PUPS 1: Pliers

For this problem set, I considered a basic pair of pliers and emphasized durability and robustness in use for the design.

Free-body diagrams and error motion identification for pliers
FRDPARRC Table for a pair of pliers

One piece of feedback that I received from the peer review discussion (see below) was that the clamping force exerted by the jaws and the amount of torque that can be transmitted are related functional requirements. While this is true, my original rationale for separating the two was to try to capture the different things people do with pliers. In the ensuing discussion, I thought of another possible loading configuration when one tries to pull a nail or staple by gripping and twisting. This induces bending and torsional loads on the jaws and introduces error into jaw alignment. In fact, I now remember that I have done this to a pair of cheap pliers I own (see below). This is a reminder that we need to consider not just intended loading configurations, but also configurations that result when the machine is abused/misused!

Permanently sprung plier jaws; this permanent deformation adds to the misalignment caused by pivot clearance! (Photo by Shien Yang Lee, CC-BY-SA 4.0)

Planar Exact Constraint Toy Design

The idea behind exact constraint design is to build things that are simple to analyze. When an object is constrained by exactly the same number of elements as there are degrees of freedom, it occupies a tranquil middle ground between the complexities of motion and elasticity. This obviously makes the design engineer’s life easier, and hopefully encourages more analysis and less “let’s just build it and see how it turns out”.


“DSCN9828” by mtneer_man is licensed under CC BY-ND

The design of this planar exact constraint toy is inspired by the pegboards used to organize tools. Most pegboards are mounted securely in workshops and hipster lairs, where the walls don’t move appreciably. But what if you wanted to design a pegboard for use on a boat? or just someplace prone to earthquakes? This toy allows you to explore how far a particular peg configuration would allow you to tip the board before your object falls off.

I plan to make my toy from a small sheet of plywood in which I will drill a 1″-pitch grid of 3/8″ holes. The user tests out different support configurations by inserting supplied dowel pins into these holes. A square block of wood will serve as the planar object to be fixtured. The board can be suspended by the attached screw eye and rotated about the pivot point to vary the direction of the gravitational force.

I wrote a simple MATLAB script to model the stability of a planar object supported by three pins. The script takes the coordinates of the pin contact points as well as the orientation of the weight vector (theta) as inputs and returns the reaction force developed at each contact point. The values of theta that correspond to sign changes in the pin reaction forces should be interpreted as the limits of stability for the support configuration. My MATLAB model neglects friction, both between the object and support pins, as well as between the object and the underlying board surface, so one can expect the analysis results to vary slightly from actual system behavior.

Standing Desk White Paper

Coke can standing desk (Photo by Shien Yang Lee, CC-BY-SA 4.0)

This is a (non-adjustable) standing desk I built my sophomore year of college from empty soda cans and a shelf borrowed from my kitchen cabinet. It cost me virtually nothing to built and was extremely portable. When it came time to move out of my apartment, I simply removed the gaffer tape, recycled the cans, and returned the shelf. It, therefore, addresses two major problems I have with commercially available standing desks — cost and portability. However, its non-adjustable nature meant I was subjected to the tyranny of being forced to stand all the time. Healthy and fun when I was working; less so when I was trying to watch the latest edition of Top Gear or call my friends on Skype.

I am going to take Precision Product Design as an opportunity to build a standing desk that is not only adjustable, but programmable, so that it can (sometimes) have a mind of its own and gently enforce some healthy proportion of standing time.

Another interesting functional requirement I have specified is portability. I will be moving back to Singapore after completing my program at MIT in August, and I would like to be able to bring this machine with me. My vision is for the core drive and locating elements of the mechanism to be separable from bulky but easily replaceable structural elements like the tabletop. This requirement is also pushing me towards building a table-top appliance (e.g., Varidesk Pro Plus) rather than a standalone desk.

 

FRDPARRC table in notebook (Photo by Shien Yang Lee, CC-BY-SA 4.0)

Here is a FRDPARRC table outlining some additional functional requirements. The plan is to keep evolving this document for the next few weeks as I get feedback, and eventually to consolidate it in an electronic spreadsheet.