GRUEBLER'S COUNT, FORMULA, EQUATION FOR STEWART PLATFORM (6-DOF PLATFORM) MECH 1-24-21
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- เผยแพร่เมื่อ 19 ม.ค. 2021
- Using the Gruebler's Count 3D Formula along with my "Secret Sauce" rules will transform your mechanisms into "Robust" mechanisms !
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Grubler equation criterion formula mechanism design Chebychev-Grubler-Kutzbach criterion Shigley ADAMS full simulation Theory of Machines and Mechanisms DOF verification
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Another way of looking at it would be the following, I think:
Bodies:
- Platform: 1
- Ground: 1
- Cylinder: 6
- Rod: 6
Total: 14 bodies
MPDOF = 14 * 6 = 84
Joints:
U-joints (constrains 4 DOF): 12
CDOF = 1*6(due to Ground) + 12*4(due to U-joins) + 6*4(due to cylinder) = 78
SDOF = 84 - 78 = 6
Manuel, you are correct ! Great job at deciphering the substitution method ! I'm assuming you have my book? Using the substitution method (outlined on page 22 and 29) for the universal joints is a fun short cut. I ONLY include moving parts in the usage of the Gruebler's 3D formula so that I don't have to worry about adding the grounded part to the MPDOF and then subtracting the same from the CDOF. This keeps the confusion down. I never did like the term "4-bar linkage"." In reality, there are only 3 moving parts (linkages). It should be called "3-bar linkage." For those who have my book, here is a quiz: on Chapter 6, Page 38-41, I shared my design for a non- binding parallel slider mechanism. Use the substitution method (similar in flavor as what Manuel Z had used) for the Slider, P5 joint and S3 joint combo, (hint: this will reduce the moving parts count) and see if you get the same answer, SDOF =1. Cheers, Le
@@GrueblersCount Thank you for your answer, sorry for the late response. I didn't have your book at that time, I just bought it and I'm finding it enlightening. I hope I'll be able to apply it very soon for something I have at hand.
Regarding your quiz, I think you are referring to fusing a prismatic P5 and a spherical S3 to get an In-Line IL2 constraint, which produces the following count:
4 Bodies:
- 1 Slider 1
- 1 Slider 2
- 1 Slider 3+Sway (one body only)
- 1 Slide table
MPDOF = 6*4=24
Joints:
- 1 Revolute R5
- 2 Prismatic P5
- 2 Spherical S3
- 1 In-Line IL2
CDOF = 1*5 + 2*5 + 2*3 + 1*2 = 23
SDOF = 24-23 = 1, which is the same result as the original version.
Great Design. Can you tell me what type of joints you used between the upper spiders and tie rod? Appreciate it
It’s basically a revolute joint. I made it a “snap in” revolute joint. The inside of the clevis of the ears of the tie rod has a hemispherical pocket. The ends of the spider has a hemispherical ball protrusion. Then they can snap into each other since PLA plastic is flexible.
I don't understand, I'm sure you are correct, but, don't we have motion on the spiders too and not just the tie rods?
V M, the entire Mechanism has a total of 6 DOF (SDOF =6). This means only 6 motions that are needed to move the platform. If you decide to have a motion of "zero" translation on all 6 rods, it will "lock, or hold" the platform in place. You can choose the joint you want to motivate. If you choose not to translate the rods and apply only rotary motions to each of the 6 revolute (R5) joints attached to the spider, it will also move the platform. You can even mix and match rotational or translational motions to each of the 6 actuator locations and it will work. It’s just more practical to drive the rod translations than rotate the spiders. Remember, the beauty of Gruebler's Count is that you can create mechanisms that are "Smart", so that they align themselves, parts (in this case the spider) to the forces, flex, and even temperature changes they encounter in their environment. The term "Rigid Body Dynamics" (RBD) is often used in conjunction with Gruebler's Count. This is completely wrong. To me, it should be called "Flexible Body Dynamics" or (FBD). I used to work at Mechanical Dynamics Incorporated (MDI) the originators of the ADAMS software which solved RBD models using Gruebler's Count for the joint determinations, and animated them. That term is called Multi-Body Dynamics, MBD. Even better, the ADAMS software can convert the rigid body parts into flexible body parts (FEA, Modal Neutral Files, .mnf). That model would run fine without changing any of the joints. It even produced more realistic results (matching reality) using the flexible bodies instead of the rigid bodies. As a side note, if you apply 7 motions to a 6 DOF mechanism, it will fight itself. I hope this helps. Cheers, Le
@@GrueblersCount You convinced me to buy your book good sir. Wow that was such a great explanation. But allow me to re-phrase my understanding, I appreciate it you confirm my thoughts are correct:
I think I was mistaking number of moving bodies with number of systems degrees of freedom
resultant
How do you decide which bodies are better to have motion (let's say by way of an actuator)? Just gut feeling I can say here cylinders having motion is easiest, rather than having spiders rotate by way of using a robot, seems like to much effort that way. This might be another whole topic by itself.
@@QwertyCanada Thank you !! As a side note, we need to understand "Active motions" (Driven Motions) and "Passive motions." Gruebler's Count determines the "Active motions" needed to drive the mechanism. In this case, the "Active motions" are the 6 translational motions of the rods. The "Passive motions" are the motions the rest of the joints are experiencing (in this case, the spiders) as a result of the 6 "Active motions." Cheers, Le
Hi how can I contact you? needed some help regarding stewarts platform.
grueblers4thbar@gmail.com
Can I have the stl file for 3d printing ?
Yes, I created a newer model. It's on Grabcad: grabcad.com/library/le-stewart-platform-1
@@GrueblersCount Thank you so much!