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hw31solutions - AAE 221 Aerospace Structures II Spring 2004...

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AAE 221 – Aerospace Structures II – Spring 2004 Chapter 3.1 – Work and Potential Energy Principles – Solution 1. Consider the free body diagrams below. The circular object on the left represents the upper mass while the circular object on the right represents the lower mass. We have 2 DOFs: the position of the two masses, u 1 and u 2 . The forces associated with the springs are 1 1 2 ku F = : If 0 1 > u , the top spring will pull back on mass 1. ( ) 1 2 2 u u k F = : If 0 1 2 > u u , the bottom spring will pull the 2 adjacent masses toward each other. If 0 1 2 < u u , the bottom spring will push the two masses apart. Applying virtual work, ( ) ( ) 2 2 1 2 1 2 2 2 2 1 2 1 1 1 u F F mg u F F mg u F u mg u F u F u mg u F W δ δ δ δ δ δ δ δ δ + + = + + + = Setting 0 = W δ and substituting the values from above result in two simultaneous linear equations. ( ) ( ) = = + 0 0 2 1 2 1 2 1 F u u k mg u u k ku mg eq eq eq eq eq k F mg u eq 2 2 1 = and k F mg u eq 2 3 4 2 = 2a. We have 2 DOFs: let us choose the displacement of the attachment points of the left spring ( q 1 ) and the middle spring ( q 2 ). The displacement of the other 2 forces (the gravitational force Mg and the right spring F 3 ) can be expressed in terms of q 1 and q 2 as follows.

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