Thomas Edison State College a Bead Sliding Along a Rod MATLAB Project This assignment follows the standard form for a project submission. You need to inclu

Thomas Edison State College a Bead Sliding Along a Rod MATLAB Project This assignment follows the standard form for a project submission. You need to include an introduction, primary discussion, and summary. Include graphs, tables, and images, as necessary, to improve the clarity of your discussion. Your project needs to be both correct and well written. Communication remains a critical component of our modern, technological society. A few notes about format: you MUST use MS Word for your project and use Equation Editor for all mathematical symbols, e.g. (??)=sin(??)+1ln(??). This project addresses modeling with Ordinary Differential Equations and solutions to those equations. You will solve a problem analytically and program an Improved Euler’s method numerical solver. You are required to write your own numerical methods in MATLAB. The physical model and problems are provided in the MS PowerPoint presentation entitled: Project2Description.pptx. it is attached Open the PowerPoint file and hit the F5 button on your keyboard to activate the animation. As you review the required problems, ask questions if you encounter anything that is not clear. A *.pdf file has been provided if you cannot open the *.pptx file. Please provide MATLAB Code and graphs separately too. Project 2: A bead sliding along a rod
Simple Harmonic Motion
Free Sliding Bead
A bead is constrained to slide along a rod of length ??. The rod is rotating in a vertical plane with a
constant angular speed, ??, about a pivot in the middle of the rod. The pivot allows the bead to freely slide
along the rod, i.e. the pivot does not impede the movement of the bead.
Let ?? ?? denote the distance of the bead away from the pivot where ?? ?? can be positive or negative.
??
Equation of Motion
Applying Newton’s second law provides a balance of forces due to gravity, friction, centripetal
acceleration, and linear acceleration. The equation resulting from these forces is
??2??
????
?? 2 + ?? ? ????2 ?? = ????? sin ????
????
????
where ?? is the mass of the bead, ?? is the coefficient of viscous damping, ?? is the constant speed of
angular rotation, ?? = 9.81 ??/?? 2 is the acceleration due to gravity, and ?? is the distance between the pivot
and the bead.
The rod is initially horizontal, and the initial conditions for the bead are ?? 0 = ??0 and ?? ? 0 = ??0 .
??0
??0
Problem 1
Consider the frictionless rod, i.e. ?? = 0. The equation of motion becomes
??2??
?? 2 ? ????2 ?? = ????? sin ????
????
with ?? = 9.81 ??/?? 2 and a constant angular speed ??.
The rod is initially horizontal, and the initial conditions for the bead are ?? 0 = ??0 and ?? ? 0 = ??0 .
A) Analytically solve this initial value problem for ?? ??
B) Consider the initial position to be zero, i.e. ??0 = 0. Find the initial velocity, ??0 , that results in a solution,
?? ?? , which displays simple harmonic motion, i.e. a solution that does not tend toward infinity.
C) Explain why any initial velocity besides the one you found in part B) causes the bead to fly off the rod.
D) Given ?? ?? displays simple harmonic motion, i.e. part B), find the minimum required length of the rod,
??, as a function of the angular speed, ??.
E) Suppose ?? = 2, graph the solutions, ?? ?? , for the initial conditions given here: ??0 = 0 and initial
velocities of ??0 = 2.40, 2.45, 2.50, and the initial velocity you found in part B). Use 0 ? ?? ? 5
Problem 2
Consider the frictionless rod, i.e. ?? = 0. The equation of motion becomes
??2??
?? 2 ? ????2 ?? = ????? sin ????
????
with ?? = 9.81 ??/?? 2 and a constant angular speed ??.
The rod is initially horizontal, and the initial conditions for the bead are ?? 0 = ??0 and ?? ? 0 = ??0 .
You will need to write an Improved Euler Method system solver to find ?? ?? and ??(??)
A) Numerically solve for ?? ?? when ?? = 2, ??0 = 0, and ??0 = 2.40, 2.45, 2.50. Solve in the time interval
1
1
1
?? ? 0,5 . Use step sizes ? = 32 , 128 , 512 and compare your results. Also, compare your best
numerical answers with your analytic answers from Problem 1 part E).
B) Numerically solve for ?? ?? when ?? = 2, ??0 = 0, and ??0 is selected to give simple harmonic motion, i.e.
1
1
1
Problem 1 part B. Use small step sizes, e.g. ? = 512 , 2048 , 8192 , etc. Solve for the longest time interval
that provides reasonable values for ?? ?? . Compare your results to the analytic solution that gives
simple harmonic motion. What does this demonstrate about numerical solutions?

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