ME3255_Group30

Problem 1:

The analytical solution of the Euler-Bernoulli beam equation for a constant uniform load, q, no transverse load, P, and an assumed frequency, omega, of zero is calculated by the function shape_simple_support. This function takes and input of q, the applied load, and x, the location at which deflection is to be calculated, and returns w, the deflection in the beam.

a. This function was then used by the script, PlotMaxDisplacement, to plot the maximum displacement in the beam. Since the load, q, is uniform, the maximum displacement in the beam always occurs at x = 0.5 m, the midpoint of the beam.

b. The script MonteCarlodx uses the analytical solution with a normal distribution of random variables for b and h, the width and thickness of the beam, to find the mean and standard deviation of the maximum deflection in the beam. The mean of the maximum deflection is 1.204119e-03 m. The standard deviation of the maximum deflection is 0.000316426.

Problem 2:

The central difference approximation is used to find the deflection of a beam with the assumptions of P = 0 and omega = 0 based on the Euler-Bernoulli beam equation.

a-c. The function finite_difference_method used the input of n, number of segments, and q, uniform applied load, to create a matrix, A, of the coefficients for the system of equations and b, the vector of solutions used to solve for the displacement, w.

d. Each plotter code uses the finite_difference_method function in a for loop to graph the distributed load q versus the maximum deflection. The graphs are shown below:

e. Numerically the trend seems linear for q versus deflection this data would most likely not change with additional points tested. A script was used to plot this trend by solving for the matrix coefficients and using the "\" to solve the system of linear equations. This can be seen in finite_difference_method_plot. The numerical solution is very similar to the analytical solution and shows the accuracy of the central difference approximation.

Problem 3:

The central difference approximation is used to find the deflection of a beam with the assumptions of P > 0 and omega = 0

a-c. The approach was to use the finite_difference_method_P code find the coefficients of the variable w and the vector of solutions

d. the plotter codes uses the finite_difference_method_P code and a for loop to graph the distributed load q versus the deflection for each P. The graphs are shown below:

Problem 4:

To find the natural frequency of the beam when the transverse and uniform applied loads, P and q, are both equal to zero, the central difference method was applied to develop a matrix of coefficients, as done in 2 and 3. Because P and q were set to zero, an eigen value problem could be developed from the Euler-Bernoulli equation being solved using the central difference approximation.

a-c. The function Eigen_natural_frequencies takes the number of segments to divide the beam as an input and then uses the central difference approximation to output a matrix of coefficients, A, to describe the system, and omega, a vector of natural frequencies. Omega is found from the eigen values of matrix A, which is equal to a constant times omega squared. This constant is taken from the Euler-Bernoulli equation of beam deflection. The system of equations for 6 segments is given by A-LI = 0 where A and L shown below.

A =

 5    -4     1     0     0
 -4     6    -4     1     0
 1    -4     6    -4     1
 0     1    -4     6    -4
 0     0     1    -4     5

L = 1.0e+03 *

d. The number of natural frequencies found for the beam is equal to n-1 where n is the number of segments the beam is divided into in order to use the central difference approximation. The first three natural frequencies are listed below for each number of segments.

|# of Segments|1st Nat Freq (Hz)|2nd Nat Freq (Hz)|3rd Nat Freq (Hz)|
|-------------|-----------------|-----------------|-----------------|
|     6       |     141.8       |     529.2       |     1058.3      |
|     10      |     143.9       |     561.4       |     1211.8      |
|     20      |     145         |     576         |     1282        |

e. A plot of the shape of the beam is outputted by PlotNatFreqBeamShape. This script solves for the natural frequencies for 6, 10, and 20 segments, and plots the first three modal shapes for the 20 segments. The y axis is a ratio of maximum beam deflection.

Problem 5:

For the bonus, a combination of the solutions to Problem 3 and Problem 4 were applied. The script BonusPlot uses the function Eigen_natural_frequencies_P to determine the natural frequencies of the beam under a varying transvers force of P. The script plots the lowest natural frequency for P values of 0, 100, 200, and 300. The function Eigen_natural_frequencies_P takes an input of n, the number of segments to divide the beam, and P, the transverse force applied, and outputs A, a matrix of coefficients for the system of equations used to solve the eigen value problem and lambda, a vector of natural frequencies.

ngs13002/ME3255_Group30

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