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.