Final Solution

Math and Science Report

One of the most integral parts of the capstone design project is the math and science background of each design. Each design has some backing in either Science or Math which allows it to perform its designated function. My capstone design project, a robotic arm for the MATES competition, has both mathematical and scientific backgrounds which are an important part of the design and solution choices

Technology

Before explaining the science and math concepts that are involved the arm, you must first talk about the technology aspect of it. There are two primary forms of technology in my project, the motor (Fig.1-2) and a material called Lexan ( Fig.1-5). The motor being used is a simple DC motor that is both small and energy efficient for the tasks it will have to accomplish. The other technology is a polycarbonate that has just been made in the last fifty years. By designing and creating this material engineers have developed a stronger and cheaper alternative to glass.

Science

The main category of science that my arm incorporates is physics. Physics is the science studying the concept of matter and its motion, as well as space and time the science that deals with concepts such as force, energy, experimental science, and it is the objective of physicists to understand some quality of the natural world. Although the study of physics encompasses the vast areas of study stated above, my project only focuses on the areas or force, mass, and motion. The first and possibly most important part of the arm is the torque which it produces. Torque is a measure of how much force acting on an object causes that object to rotate. In my arm this applies to the amount of torque the gears are producing when the motor is active. To find the torque that arm produces, a few factors must be known. Before the torque can be calculated the gear ratio must be found. A gear ratio (Fig.1-1) is simply the output gear number of teeth / the input gear number of teeth, calculations for gear ratio can be found further on in the report. After finding the gear ratio the torque of the arm can be found. To find the torque with regards to the arm the following equation is used: Motor Torque x gear ratio = torque at the wheel; calculations for arm torque can be found further on in report. Another important calculation regarding arm is the speed at which the gears turn; another physic related problem. To find the speed at which the gears in the arm move is a simple law. The gears speed are relative to each other, therefore the gear ratio determines the speed (seen above) of each gear in relation to each other. For example if the input gear (10 teeth) is rotating at 5 rpms, and it is connected to our output gear (50 teeth), the output gear will rotate at 1 rpms. The relationship for the gears speed is 48 to 1. This is true because for every one turn of the worm gear the worm revolves 48 times.

Another important part of the arm is the pressure the claws produce on the object that is being grabbed. Pressure is the application of continuous force by one body on another that it is touching; compression. Pressure is important because if the claws can not apply enough pressure onto the object, they will not be able to hold it; or if they apply to much pressure the object will either break( near impossible with this design) or if the pressure isn’t equally applied from each claw the object could slip out of the arms grip. To calculate the pressure that arm will produce you perform the following equation: force (torque in this case) / area of the object; (pressure calculations further in report).

The most complex component in the arm is the motor (Fig.1-2) that drives the arm. Every DC motor has six basic parts -- axle, rotor, stator, commutator, field magnet(s), and brushes. In most common DC motors, the external magnetic field is produced by high-strength permanent magnets. The stator is the stationary part of the motor -- this includes the motor casing, as well as two or more permanent magnet pole pieces. The rotor (together with the axle and attached commutator) rotates with respect to the stator. The rotor consists of windings (generally on a core), the windings being electrically connected to the commutator (Fig.1-3). The above diagram shows a common motor layout -- with the rotor inside the stator (field) magnets. Considering the motor being used was purchased rather than built, no calculations to make because the motor comes along with its numerical data. The voltage range of the motor is 1.5-4.5 volts, with the nominal voltage at 3, the rpm at nominal load is 8700, the rpm at normal load is 6400, and the current at no load is 190 milliamps.

The final piece of science that the arm incorporates is the actually gears used to make the arm work. The gear drive that is being used in the arm is called a worm drive; this drive is comprised of two parts (Fig.1-4). The first part is the worm, a screw like gear that has one tooth (Fig.1-4). The second part is the worm gear, this a circular gear that is very much like a helical gear, and these can range in tooth size, as long as the pitch remains the same between the gears. The purpose of a worm drive is to increase the torque a drive can produce, which is why this particular drive is perfect for the arm.

The newest and most advanced part of the arm design, is one of the materials being used in the construction of the arm, Lexan. Lexan is a relatively new poly carbonate, discovered in 1953, by Doctor Daniel fox. Lexan chemical structure (Fig.1-5), allows this extra strong polycarbonate to replace certain materials namely glass. What allows Lexan to be so strong is how it is produced; Lexan is produced by reacting Bisphenol A with carbonyl chloride, also known as phosgene, this is what gives the material its strength. Today lexan is used in priomarly three fields, building (glazing and domes), industry (machine protection and fabricated parts) and communication and signage. The use in indusrty as machine protection, makes it the ideal material to help house and protect the arm(Fig.1-6).

Math Calculations

Gear Ratio:

Number of teeth on worm: 1

Number of teeth on worm gear: 48

Gear Ratio: 48:1

Torque:

Motor Torque= (Hp*5252)/RPM,

HP= (V*I*Efficiency)/ (746)

HP= (3*1*.82)/ (746)

HP= .00329

Motor Torque= (.00329*5252)/8700 (nominal load rpm)

Motor Torque= .0019906

Gear Ratio= 48

Torque= Motor Torque * Gear Ratio

T= .0019906*48

T= .095 Newton’s

Pressure:

Force= Torque= .095 N

Area of Claw= L*W

A= .25inches*.25inches

A= .0625 icnhes^2

P= .095*.0625

P= .00597 PSI

Conclusion

So in conclusion, that my capstone design project encompasses a good deal of science and math. Whether the science is the gears or the motor, it all plays a crucial element in the overall effectiveness of the design. The mathematical calculations although simple have a great impact on the effectiveness of the arm. Overall the ROV arm requires a great deal of scientific evidence and mathematical calculations to make the arm successful, and the information presented in this report is the proof of that evidence.

Exploded

Orthographic

Plan Of Procedure

Plan of Procedure

1. Gather all materials listed in the above tables, and bring them to the working area
2. Gather all measuring and marking material and bring them to the working area
3. Measure, mark, and layout all sides of outer box.
4. Make sure all layout lines are within 1/16^th of the specs for each piece
5. Cut all marked pieces on band saw and cut to specifications
6. After cutting each piece number and code each piece to avoid confusion.
7. Perform a dry fit for all pieces to assure all pieces fit together
8. Make any corrections, if needed, after doing the dry fit.
9. Take bottom and top labeled pieces and mark on each the areas where holes will be drilled.
10. Take drill and drill bit and mark on drill a quarter inch from the tip of the drill bit.
11. Slowly drill marked area making sure not to exceed marked depth on drill bit.
12. After drilling first hole, change drill bit to and again mark a quarter inch up from the top of the drill bit with tape (Fig.1-1).
13. Repeat step 11 with larger drill bit, making sure to bevel the hole.
14. After drilling the holes on both the top and bottom pieces, run a dry fit with the plastic rods rods, and the drilled holes (Fig1-2).
15. Make any corrections needed after this dry run.
16. Mark all points on the labeled pieces for pilot holes.
17. After marking all points on pieces take drill with drill bit, and mark depth of screw on the drill bit.
18. Slowly drill each hole, making sure not to shatter the polycarbonate.
19. After drilling each hole take drill bit and again mark length of screw from the tip of the drill bit.
20. Using new drill bit size; continue drilling each marked area to marked depth on the drill bit, making sure not to shatter the polycarbonate, and to bevel each hole(Fig1-3).
21. After drilling each hole assemble the pieces including the plastic rods to ensure a firm fit of all the pieces.
22. Make any corrections if needed.
23. Mark on polycarbonate the specifications for the interior divider
24. Make sure all measurements are within 1/16^th of an inch, of computer drawings.
25. Cut piece using band saw, cut to specifications
26. Code piece after cutting
27. One bottom and top piece mark area in which the divider will be placed
28. After marking the top and bottom pieces do a dry fit with the divider in the box.
29. Make any corrections if needed.
30. Print out copy of full scale claw of the arm
31. Tape printed out claw onto remaining polycarbonate.
32. Using thick permanent marker trace the claw onto the polycarbonate.
33. Measure dimensions on polycarbonate, to ensure they are within 1/16^th inch, make and corrections if need.
34. Using band saw, make outline cuts of the claw, leaving space so it can be sanded down to final form.
35. Wearing a mask, and in a ventilated area sand the cut out claw pieces, starting with the coarsest sand paper and working towards the finest.
36. After sanding claws down to final form, mark on each the area where a hole will be placed for the plastic rods (Fig1-4).
37. Measure hole to make sure it is within a 1/16^th of an inch of specifications.
38. Take drill and appropriate size drill bit and drill hole, making sure not to shatter Lexan, in appropriate area (Fig1-5).
39. After drilling do a dry run with the plastic rods and claws, make corrections if needed.

40. 1-5

    Now assemble motor support by marking the dimensions of the unit on the last of the Lexan. Re-measure to assure accuracy of dimensions.
41. Cut out marked and dimensioned pieces on the band saw.
42. Run dry fit of pieces, making corrections if needed.
43. Mark and measure the area in which the hole for the shaft of the motor will be placed.
44. Using drill and appropriate drill bit, drill hole in correct area on the divider.
45. Take the largest plastic rod, and center it on the shaft of the motor.
46. After repeatedly centering it and marking each centering on the plastic rod, take a drill bit (1/16^th) and measure up from the tip of the drill bit a half an inch( length of shaft), and mark it with tape.
47. Slowly drill the hole on the center mark of the rod, making sure it does not shatter.
48. After completing the hole, run a dry fit, making corrections where needed.
49. With all components completed run a full scale dry fit, making corrections where needed.
50. After corrections have been made, start to assemble the arm.
51. First start with gluing the bottom of the box to the sides of the box, using the special polycarbonate glue, Ploy Zap.
52. After completing that, attach back to the bottom, using the same method stated above.
53. Then attach divider to the bottom using the same method as above.
54. Assemble motor support using same method as sated above and then attach it to the bottom, using same method.
55. Place motor on motor support, making sure to align shaft perfectly with hole in the divider.
56. Attach worm gears to their respective support rods.
57. Attach Connecting rod to shaft, making sure it is centered, then place connecting rod, into bore of the worm, and connect them together.
58. Place plastic rods (with gears attached), into their perspective holes, and grease the holes with WD-40(Fig1-6).
59. Carefully place top onto ROV, making sure that while screwing it in, nothing is shattered.
60. Touch any looser ends

Lists

Supply List
Item	Description	QTY	Size	Remarks
1	WD-40	1	12 oz	To lubricate metal rods
2	Tape	1	One roll	To mark drill bit
3	Sandpaper- Coarse	2	12”x 12”	Sand lexan
4	Sandpaper- Medium	2	12”x12”	Sand lexan
5	Sandpaper- Fine	2	12”x12”	Sand lexan
6	Polyzap	6	2 oz	Glue lexan

Tools and Equipment List
Item	Description	Use
1	Band Saw	Cut Lexan
2	Screws	Secure walls
3	Drill	To drill into bottom and top pieces
4	Drill Bits	Drill holes in lexan

Material List
Item	Description	QTY	Size	Remarks
1	Helical Gear	2	1.75 inches (OD)	For drive of Arm
2	Worm	1	Face Width .75 inches	Drive of Arm
3	Motor	1	Dia: 1 Inch, L: 1.25”	Drive of Arm
4	Lexan		12”x14”	Claw and casing material
5	Plastic Rod	2	H: 2.5” Dia: 1/8”	Stabilizers for gears
6	Plastic Rod	1	L: 1.25” Dia: .25”	Connection between shaft and Worm
7	Screws	6	L: .75, Phillips head, ¾ threaded	Securing top piece of lexan to rest of arm