Chapter31 - AAE 221 – Structures II – Spring 2004...

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Unformatted text preview: AAE 221 – Structures II – Spring 2004 Chapter 3.1 – Work and potential energy principles 1. Using PVW, find the equilibrium position of the two masses subjected to the effect of an external force F and of gravity (see fig. 1). fig. 1 2. Using PVW, solve the following problem involving a rigid bar of length L and mass M subjected to the gravitational field and to the action of various linear springs (see fig. 2). fig. 2 Next, solve the same problem assuming that the rigid bar rotates about its (fixed) right end. 3. Use the PVW method to obtain the BVP (GDE and BC) describing the deflections of the following cantilever homogenous non-symmetric beam. Assume that there is no loading in the x-y plane (but there is a displacement in both y- and z- directions (since the beam is not symmetric), that the beam is homogenous (with Young’s modulus E) and that the beam is simply cantilever in the x-y plane (no elastic boundary condition in that plane). fig. 3 4. One of the simplest (conservative) mechanical systems consists of a rope of initial length L under tension T and subjected to a lateral force f(x) (figure 4). fig. 4 We will use the following assumptions: - the initial tension T is not affected by the elongation o the rope associated with the lateral loading (i.e., the additional tensions created by the lateral deflection of the rope are negligible); du << 1 ; - the deflection of the rope is small, and we also assume dx - the two ends of the rope are fixed; (i) show that the total potential energy P for this system is given by T Ê du ˆ P = U + V = Ú Á ˜ dx - Ú f ( x)u ( x)dx 2 0 Ë dx ¯ 0 (Hint: the internal strain energy U can be written as T(l-L) where l is the deformed length of the rope and L the original length.) (ii) Use the PVW (or, since the system is conservative, the PMPE) to deduce from (i) the differential equation describing the deflection u(x) of the rope. (iii) Obtain the same differential equation of equilibrium by performing a free body diagram on a piece of the rope. L 2 L ...
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