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Matlab & Mathematica

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.

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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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