Weekly Brainware
Week 7
Part 1: Error Budget Update
Last week, I created my preliminary error budget for the concept I have been working on so far. After some peer review with Professor Slocum, I realized that I had modelled member 4 (leg) incorrectly. I treated the compliance matrix the same way I did for a cantilever beam. However, this is not entirely the case, as I need to factor in the second leg, that is a mirror image of this leg. In order to to account for both the legs in the same coordinate system, the linear and bending stiffnesses needed to be doubled while the rotational stiffness needed to be changed to the linear stiffness times the square of the distance between the legs.
The above changes were made and the final compliance matrix is seen below.
As a way of checking my compliance matrices, I made use of the default bridge machine error budget spreadsheet and changed the dimensions and properties of the members to match my own. After doing so, I obtained the same values for corresponding cells in the compliance matrix as my own error budget.
The second big change I made to my error budget was the method of calculation of moment of inertias for member 2 ( bearing rail). Previously, the rail was modeled as a solid block. The dimensions obtained were considered to be the bottom most section of the box rail. However, over the past week, I changed the CS2_Vertical_Axis tab of the error budget to accept dimensions for the box rail and a separate tab computed the correct moment of inertias for the prescribed geometry. It also computed the stiffness of the keeper rails which serves as linear stiffness for the slider in CS1.
I also included loading due to the weight of the members at their coordinate centres.
Results from Error Budget:
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Member 1: Table top support member
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Material: Wood Fir, Solid
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Dimensions(mm): 600(l) x 50(w) x 100(h)
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Member 2: Box Rail
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Material: Wood Fir, Solid
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Dimensions(mm): Length =1200mm, overall width= 6.25in, overall thickness =4in
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Member 3: Base
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Material: Wood Fir, Solid
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Dimensions(mm): Length =700mm, width= 4in, height=2in
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Member 4: Legs
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Material: Wood Fir, Solid
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Dimensions(mm): Length =500mm, width= 4in, height=2in
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The dimensions of the table top are 1000 x 600 x 25 mm resp. The weight of this table top (made of plywood) was factored in as a force load and corresponding moment load on CS1's point of stiffness.
For a point load of 300N at the tip of the table, the above displacements were obtained. The total displacement is exactly within the total apportioned error of 5mm.
Part 2: Designing actuator mount and supports
Before designing the actuator supports, I had to finalize the location of the actuator mount, ie, where and how the nut for the lead screw would be fastened. This proved to be an extremely challenging decision.
Ideally I would want to place the nut at the centre of friction ( approx middle of the slider) so as to minimize risk of jamming. Unfortunately this was not straightforward as my slider is 8.5" long. I was not sure whether drilling a hole that deep was possible. I considered drilling from both ends to meet at the middle but was still unsure if I could do this accurately.
My other option was to mount the nut, off-axis, as I had done for my slider prototype. However with this, I would need to do calculations to determine if the slider would jam and also calculate the stress in the mount to ensure that it doesn't break. Also, with an off-axis mount, the lead screw would have to pass through table top support member. I considered supporting the table top with two members on either side of the off-axis mount as well.
If possible I would definitely prefer option 2. I will seek advice from friends in class and in the machine shop to determine if this is possible. (Professor Slocum, could you advise me on this?)
In the meanwhile, I proceeded with the concept of having the actuator mounted off-axis. This first thing I determined was whether the slider would jam. From the dimensions of the nut, the minimum distance between the top surface of the slider and the centre of the nut is 20mm. Given that My slider was an inch thick, the minimum location of the actuator force, w , was 20+12.7 = 32.7mm
Given that the external load of 300 N was applied at distance of 600mm from the bearing (at the tip of the table), the load F_L = Force at tip*(length of table/ length of slider). Thus the load is multiplied due to mechanical advantage.
From the above result, the slider will not jam. The maximum value of w was determined to be 50mm.
Following this, I determined the dimensions of the mount while ensuring that it does not shear perpendicular to the grain, due to stress. I determined the length of the mount to be 85mm for a minium width and height of 30mm respectively.
I proceeded with this arrangement and my current CAD model reflects this design. However I am open to making changes if it is possible to have the lead screw pass through the centre of stiffness.
Actuator supports:
To support the lead screw, I followed Fundamental Principles and designed to have a thurst bearing and roller bearing at one end and have another roller /self- aligning bearing at the other end. The two bearings at one end will be spaced 3 shaft diameters apart to create a rigid connection (Saint Venant's Principle). These bearings would be located at the base of the rail and ideally sit back to back. The lead screw will be coupled to the NEMA17 using a flexible coupling.
At the other end, there will be a small gap between the face of the bearing and the step of the lead screw. This is to allow for any thermal expansion of the screw. For a 100deg change in temperature, the estimated expansion is 0.5mm for the 500mm steel lead screw. For a safety factor of 2, the gap will be 1mm.
Although the rail is 1.2m long, the lead screw is only 500m long. This is because my functional requirement is for the table to actuate between a comfortable sitting height and standing height only. I do not require the desk to go any lower. This also helps reduce cost.
I envisioned the bottom mount to be a fixture that I could fasten to the rail. The fixture would compose of cantilever elements with pockets for the bearings and for mounting the motor. This fixture would almost slot into the gap in the rail. The stiffness of the fixture had to be on the same order as that of the lead screw. From this, I was able to determine the dimensions of the fixture.
Carriage/Table mount:
Finally, I was ready to design the connection between the table top support member and the slider. The connection had to have a stiffness of a bolted joint ( roughly 10^7 N/mm). Else, the connector would deflect on its own.
At the time, the support member was to be made of wood. This made the connection tricky as the making an end grain to surface connection would result in a very weak joint. I explored the idea of using an L bracket to join the wooden member to the slider. (The stiffness of the member is approx. 1500N/mm.) However, I soon questioned the need for the member in the first place if I was able to find a bracket stiff enough to support the table on its own.
I switched to a steel tube in place of the wooden support member. This way, I could weld it to a plate with mounting holes and thus secure the member to the slider. Following this design change, I updated the error budget and optimized the size of the tube to achieve desired stiffness. ( 2.5"x2.5" steel tube with 0.125" thickness)
At the other end, there will be a small gap between the face of the bearing and the step of the lead screw. This is to allow for any thermal expansion of the screw. For a 100deg change in temperature, the estimated expansion is 0.5mm for the 500mm steel lead screw. For a safety factor of 2, the gap will be 1mm.
Although the rail is 1.2m long, the lead screw is only 500m long. This is because my functional requirement is for the table to actuate between a comfortable sitting height and standing height only. I do not require the desk to go any lower. This also helps reduce cost.
Final Error Budget:
For analysis on load life of bearings and lead screw, please refer to pups 7
Part 3: Seek & Geek
This past week, I spent some time learning how to weld steel tubes using the TIG welder at the Hobby Shop. The welding machine feeds the weld material ( steel wire) from a spool using a set of gears as shown below. The feed rate of the machine can be adjusted. I thought this would be a cool mechanism to analyse.
