Vibrations and Dynamics of Aerospace Systems
As we learned in the lecture, a continuous systems is an extension of multi-degree of freedom system when the number of DOF increases to infinity.
In this exercise, you will develop a system of large degrees of freedom that mimics the behavior of a continuous string under vibration.
To create this model, we are assuming that each element of the string can be modeled by a tiny mass, and it is attached to the neighboring elements by a rigid massless bar, as shown in the figure
For our example, we assume that the string has a total length of L = 10 m, total mass of M = 3 kg, and is under a constant tension of P = 100 N . Also, consider that the string was horizontal at t = 0 but the element on the middle, element (n + 1)/2th, is given a sudden vertical velocity of 1 m/s (for simplicity, assume n is always an odd number).
Using the discrete model show in the figure above, find out the equivalent value for mi, and the the stiffness force fi−1 and fi+1 on each element as functions of x and i.
Write the equation of motion for each element. Assume that element i is attached with a rigid massless bar with tension of P to elements i − 1 and i + So, fi−1 and f1+1 should be functions of position of neighboring elements. Rearrange the equations in the form of an MDOF system.
Write a MATLAB code that considers n = 9 mass elements and uses ODE45 to integrates the motion of the n-DOF Find the position of each element as a function of time.
Plot the vertical position of each element at t = 1 s, t = 2 s, t = 6 s vs the location of the center of each element (plot of w x), all on the same figure. Use different color to distinguish the different time.
use the analytical solution for the motion of the string with the same properties and IC, and solve for the analytical solution of the vibration of a continuous string. Plot the analytical response at t = 1 s, t = 2 s, t = 6 s similar to previous
compare the results and write your
increase n to 999 elements and repeat section c, d, and f.
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