The uniform cantilever rod in Fig. P12.8 has torsional stiffness GJ, vertical bending stiffness El, mass per unit length pA, and torsional mass moment of inertia p lp. 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 u(x, f) and twists about its axis with angle of twist θ(x,t) Assume all deflections to be small (e.g., m moves approximately vertically if θ is small), and neglect gravity. Use the Extended Hamilton's Principle to derive the equations of motion and the boundary conditions for this system.
