MATLAB Live Scripts

Description

Note: Use MATLAB Live Scripts for writing the solution to these problems. A template is uploaded on Blackboard that includes the guideline for writing your code. Please follow the instruction carefully. Then, upload all your MATLAB files into Blackboard. For receiving full credit, your code should run correctly without any issue.

1. 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).

(a) 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.

(b) 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 + 1. So, fi−1 and f1+1 should be functions of position of neighboring elements. Rearrange the equations in the form of an MDOF system.

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