A recent work by Shabana [1] addressed an important issue on selecting different sets of modes for problems of elastic beams that undergo large rigid-body displacements. Extensive works on flexible bodies and rotating beams have been conducted with different assumptions of boundary conditions for the beam's ends (sample lists can be found in references [1-3]). The classical models with clamped and pinned ends are commonly used in these works. In dealing with the related topic, Shabana [1] demonstrated that different mode shapes that correspond to different sets of natural frequencies can be used to obtain the same resonance conditions by using simple co-ordinate transformations. Two classical beam models with simply supported and free-free ends were first considered [1]. The relationship between the boundary conditions and the co-ordinate systems was discussed. The equations of beam vibration were then obtained by using the following two mode shapes: f ss (x) = sin (px/l) and f mff (x) = f ff -f ff (0), where f mff = cos (lx/l) + cosh (lx/ l) -s[sin (lx/l) + sinh (lx/l)] and f ff (0) = 2. The modified mode shape f ff was generated from the classical free-free shape by defining a new co-ordinate system with a rigid translation, f ff (0). The two shape models were then applied to derive the uncoupled equation of motion in terms of modal co-ordinates: m j q¨j + k j q j = Q j , where m j = f l 0 raf 2 j dx, k j = f l 0 EI z (1 2 f j /1x 2 ) 2 dx, and Q j = f l 0 Ff j dx [4]. Assuming the beam is subjected to a harmonic force, F 0 sin v f t, acting at its center, the final equation of motion associated with the first mode of vibration was given by q¨+ v 2 q = B(F 0 /m) sin v f t, where B is a parameter defined for later discussion. The results for the simply supported and modified free-free cases were given as [1]