ME 3255 - Final Project
Due May 1 by 11:59pm
In this project you are going to solve for the shape of a beam under different loading conditions. The shape of the beam varies along the x-axis and as a function of time.
Notes: Label the plots with legends, x- and y-axis labels and make sure the plots are easy
to read (you can use the setdefaults.m
script we have used in class). All functions
should have a help file and your README.md should describe each file in your repository
and provide a description of each problem and each solution (use #
-headings in your file
to show the start of new problems)
You will be graded both on documentation and implementation of the solutions.
We will use the Euler-Bernoulli beam equation to describe the shape of the beam, the differential equation that governs the solution is:
Where w(x,t) is the displacement of the beam away from the neutral axis as a function of
position along the beam, x, and time, t, P is the transverse loading of the beam, E is the
Young's modulus, I is the second moment of Inertia of the beam,
We can separate the function
For the simply-supported beam shown in Figure 1, the boundary conditions are:
The material is aluminum, E=70 GPa,
- Analytically solve for the shape of the beam if q(x)=cst, P=0, and
$\omega$ =0 and create a function calledshape_simple_support.m
that returns the displacement w(x) given q and x
w=shape_simple_support(x,q);
a. Plot q vs the maximum deflection,
b. Use a Monte Carlo model to determine the mean and standard deviation for the
maximum deflection
-
Now use the central difference approximation to set up a system of equations for the beam for q(x)=cst, P=0, and
$\omega=0$ . Use the boundary conditions with a numerical differentiation to determine the valuea of the end pointsa. set up the system of equations for 6 segments as a function of q
b. set up the system of equations for 10 segments as a function of q
c. set up the system of equations for 20 segments as a function of q
d. solve a-c for q=1,10,20,30,50 and plot the numerical results of q vs
$\delta x$ e. Comment on the results from the analytical and numerical approaches (if you used functions then provide help files, if you used scripts, then describe the steps used)
-
Now set up the system of equations using a central difference method if P>0 and
$\omega=0$ a. set up the system of equations for 6 segments as a function of q and P
b. set up the system of equations for 10 segments as a function of q and P
c. set up the system of equations for 20 segments as a function of q and P
d. solve a-c for q=1,10,20,30,50 and plot the numerical results of q vs
$\delta x$ for P=0, 100, 200, 300 (4 lines, labeled as P=0,P=100,...) -
Now set up an eigenvalue problem to solve for the natural frequencies of the simply supported beam if P=0 and q=0.
a. set up the system of equations for 6 segments
b. set up the system of equations for 10 segments
c. set up the system of equations for 20 segments
d. solve for the natural frequencies (
$\omega_{1}$ ,$\omega_{2}$ ,...)e. Plot the shape of the beam for the first 3 natural frequencies
-
(Bonus 5pt) Create a function to return the system of equations for the eigenvalue problem as a function of P, if P>0. Then, plot the lowest natural frequency vs the applied load P.