Derive the equations of motion for the problem of Example 6.5 using

(a) Newton's second law,

(b) Lagrange's equation, and

(c) Hamilton's principle, and then solve for the response.

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Example 6.5 Equations of Motion with Mass and Stiffness Coupling Consider the two degree-of-freedom rigid-body model of Figure 6.10. For the rigid beam, modeled as a lumped mass, two coordinates are required to specify its transla- tion and rotation. If the force is applied at the center of mass C and if k =k2, then the beam will only translate. The location of C depends on the distribution of mass in the beam. If the mass is uniformly distributed, then C is at the geometric center. Derive the governing equations of motion, assuming small motion, for the case in which a force F(t) is applied at an arbitrary point O and the mass is distributed in an arbitrary manner.