Fundamentals of Structural Dynamics by Craig Roy R.; Kurdila Andrew J.; & Andrew J. Kurdila

Author:Craig, Roy R.; Kurdila, Andrew J.; & Andrew J. Kurdila
Language: eng
Format: epub
Publisher: Wiley
Published: 2011-08-09T16:00:00+00:00

Figure P12.1 (a) Two-material bar undergoing axial deformation (b) cross section of the bar; (c) free-body diagram of the bar segment from x to (x + Δx).

In addition to the ∑F equation in Eq. 12.5, you will need to use the free-body diagram in Fig. P12.1b and write an equation for ∑M. In these equations, include the inertia terms from the upper block and the lower block as two separate terms. (b) Give the equation of motion for axial deformation of this system. For homogeneous bars, your answer should reduce to Eq. 12.8.

12.2 A shaft consists of two identical cylindrical segments, each of length L, that are fixed to rigid “walls” at their ends and are welded in the middle to a disk of mass M and radius R (Fig. P12.2). Refer to the left-hand segment of shaft as segment 1, and the right-hand segment as segment 2. For each segment of shaft there will be a torsional equation of motion like Eq. 12.14, one in terms of θ1(x1, t) and the other in terms of θ2(x2, t). (a) State the fixed-end boundary condition for shaft 1 at x1 = 0, and state the fixed-end boundary condition for shaft 2 at x2= L. (b) State the displacement-type boundary condition where the disk is attached to the two shafts. That is, relate θ1 at x1 = L and θ2 at x2 = 0. (c) Finally, one more boundary condition is required. Sketch a free-body diagram of the disk, and derive the torque-type (natural) boundary condition of the type that is given for a single torsion rod by Eq. 3 of Example 12.2. Give special attention to the proper sign conventions for torques, rotation angles, and so on, as you draw your free-body diagram of the disk.

Figure P12.2

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