Brachistochrone curve

thingiverse

Is the straight line always the shortest way? Back in 2013 I visited the "Museo Galileo" in Florence, Italy. I was amazed on what I saw there and specially one object caught my attention. This wooden object made me think about the question asked at the begining of this lines... When I saw this new version of the Maker Ed Challenge my mind went back to that object called The Brachistochrone. Students can work different aspects of this item starting with a bit of History, following with some Technical Drawing and Maths and finishing with some Physics. I hope you enjoy creating your own Brachistochrone, here you will find all you need to do it. Good trip! Print Settings Printer Brand: RepRap Printer: BCN3D+ Rafts: No Supports: No Resolution: 0.2 is ok Infill: 20% Notes: I used a 40% infill for the main body with no horizontal layers, this is optional of course! Post-Printing After printing all the parts you need to glue them together. Maybe your students will find better solutions for the final assembly? How I Designed This Drawing the Cycloid The key point of the Brachistochrone is the cycloid that forms the curved surface where the ball (in our case a marble) will slide. In order to draw the cycloid the fist thing we must decide is its size. In the following example we are drawing a cycloid starting with a 50 mm radius circle. First of all we must draw the starting circle and an horizontal line with the lenght of the circle's perimeter as show in the image below. The starting circle and its perimeter Then we must divide the circle in equal parts (in this example 12 parts). As many divisions you use the better the definition of the cycloid will be. The horizontal line must be as well divided in the same amount of parts used for the circle. We will also draw a new horizontal line through the center of the circle. Now vertical lines must be drawn starting in those divisions. After that we will draw circles with the same radius than the starting one in every intersection between the upper horizontal line and the vertical ones. We will also draw horizonal lines through the end of the lines we used to divide the circle at the begining. The final and most important step is to draw a line (in my case a spline) joining the points where de circles intersect the horizonal lines as shown in the picture below. This step is not easy to explain with words, please pay attention to the picture! We have our cycloid ready for the next step. Great job! Doubts? There is a step by step explanation on how to draw a cycloid in this link:http://edpstuff.blogspot.com.es/2010/07/cycloids-and-their-construction.html Completing the design Once we have the cycloid drawing now we can finish the design in different ways. I show the solution I choose for mine. The holes are important to let the Brachistochrone work with different slopes. Designing the straight way Now we need a straight path that will run paralel to our cycloid. Using both, cycloid and straight paths, we will compare the time it takes for the marbles to reach the crossing point between the two tracks and so we will see which one is faster. The design of the straight track is quite simple. It can be done for example as shown below. As you can see it has a semi circular path where the marble will slide. I also included a hole for a M4 bolt that will let the part rotate in order to get diferent experiments (with different inclinations). Designing the feet To hold the whole object some feet will be needed. They can be designed in many different ways, the important thing is that they fit well and give stability to the assembly. My solution in shown below. Final details To finish the design process we will need a pin that will allow the straight track take different positions for diferent inclinations. Another interesting part will let us release both marbles at the same time in the cycloid and in the straight tracks. Assembly and other considerations In order to get a Brachistochrone big enough to experiment the phenomenom I decided to split the main parts in two so we will need to glue them together once printed. As I wanted to give my Brachistochrone an special look, I printed the main body with no horizontal layers, this of course is optional an can be adapted to your own liking. Project:Brachistochrone Curve Project Name This project is called Brachistochrone Curve as this is the name this kind of contraption takes from the ancient Brachistochrone Problem that tries to find the shortest time between two points. The word Brachistochrone cames from Ancient Greek βράχιστος χρόνος (brakhistos khrónos), meaning "shortest time". The solution to the Brachistochrone Problem is the cycloid. Brachistochrone Problem Overview & Background This project can be taken form different points of view. It can be done from the Maths, Physics, Technical Drawing or History point of view. So the subject involved in this project can be one, some or all of them depending on the context and the kind of students we are teaching to. Students can learn the theoretical part related to this problem with all its Maths concepts, the Physics behind the solution of this problem, how to design and draw the Cycloid and the Brachistochrone and the Historical part from Galileo to Johann Bernoulli solutions to this problem. Objectives Students will learn: Brachistochrone Problem, meaning and solution Drawing a cycloid curve Designing an object composed of several pieces Why the cycloid is the shortest way form point A to B Who Galileo and Bernouilli where and how they contributed to our present knowledge 3D printing techniques (of course!!) Audiences This project can be adapted depending on the age and skills os students but it would be optimum for students from 16 to 18 years old as they will understand better the whole process. Subjects Maths Physics Technical Drawing History Design Skills Learned Students will learn all (or at least some) of this skills: Critical thinking, problem solving, reasoning, analysis, interpretation, synthesizing information Research skills and practices, interrogative questioning Creativity, artistry, curiosity, imagination, innovation, personal expression Scientific literacy and reasoning, the scientific method Lesson/Activity To complet this project student will need some computers with internet conexion in order to design and develop the project. A design software will also be needed, any software will work as the design itself is not complicated. There is an step by step explanation on how to design and build the project prototype. A 3D printer will also be needed. The project can be started by posing the Brachistochrone problem to students and letting them some time to search for information and to think different solutions and designs for the prototipe they will finally build. Then the design process will start and student will work in teams (for example) so each team will finally get their own design ready for 3D printing. After 3D printing it will be time for experiments and to really see if theory becomes reality... A piece of advice: use slow motion videos to compare both marble positions as it all happen so fast!;) Examples:https://youtu.be/_1DhpiR-xL4https://youtu.be/NEINB9l4TYU Duration This project can be completed in 12 lesson hours (more or less) Preparation Students should already have some basic knowledge of 3D designing and 3D printing. A good background in Maths will also help. References Some references students and teacher may find useful: https://en.wikipedia.org/wiki/Brachistochrone_curve http://edpstuff.blogspot.com.es/2010/07/cycloids-and-their-construction.html http://catalogue.museogalileo.it/object/BrachistocronousFall.html https://www.google.es/search?q=Brachistochrone+Problem&client=firefox-b&source=lnms&tbm=isch&sa=X&ved=0ahUKEwjCsLjH1JTOAhVMJMAKHe5rAbAQ_AUICSgC&biw=1920&bih=970#imgrc=_ The Brachistochrone you can find in Museo Galileo Rubric & Assessment Students should have designed a complete 3D printed Brachistochrone prototipe according to the standards the teacher gave them. They should also make a report explaining their experiments and the results they got from them. Accuracy, originality and design quality will be taken into account.