12.8 The uniform cantilever rod in Fig. P12.8 has tor- sional stiffness GJ, vertical bending stiffness EI, mass per unit

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12 8 The Uniform Cantilever Rod In Fig P12 8 Has Tor Sional Stiffness Gj Vertical Bending Stiffness Ei Mass Per Unit 1 (66.23 KiB) Viewed 50 times

12.8 The uniform cantilever rod in Fig. P12.8 has tor- sional stiffness GJ, vertical bending stiffness EI, mass per unit length pA, and torsional mass moment of iner- tia plp. At end x = L a rigid, weightless bar BC is attached parallel to the y-axis. At C a concentrated mass m is attached to bar BC, and also at C a load P(t) acts parallel to the z-axis. Assume that rod AB bends in the vertical (i.e., x-z) plane with deflection v(x, t) and twists about its axis with angle of twist 0(x, t). Assume all deflections to be small (e.g., m moves approximately vertically if 0 is small), and neglect grav- ity. Use the Extended Hamilton's Principle to derive the equations of motion and the boundary conditions for this system. v(x, t) Com L 0(x, t) Figure P12.8 B | P(t) X