Note that the equations for the slow-time amplitudes have been modified to include modal damping terms (ζ_{1} and ζ_{2}). The modal damping coefficients (ζ_{1} and ζ_{2}) were set equal to 0.1 arbitrarily as no experiments were conducted to estimate the real damping coefficients. Equations (17a)–(17d) are similar to the equations for modal amplitudes obtained in Refs. [6,7,12] for the case of 1:2 internal resonance in a system with two degrees of freedom. These equations can be solved for steady-state solutions to give single-mode (only second modal amplitude is nonzero) and coupled-mode solutions (both first and second mode amplitudes are nonzero). Figure 11 illustrates some important aspects of the nonlinear dynamic response of structure 1. In a linear system, when a particular mode is excited at resonance, it acquires a large amplitude which theoretically, in the absence of damping, can even go up to infinity; correspondingly, the contributions of the other modes to the system response are zero or minimal at that particular modal frequency. This is what is observed in the second mode's linear response in Fig. 11, where it can be clearly seen that the second mode has a large amplitude when excited at its natural frequency. However, in the presence of nonlinearities and when the frequencies of the first two modes of the system are in a specific ratio, an energy transfer takes place between the second and first modes which again can be observed from Fig. 11, wherein the second mode's amplitude in the nonlinear response is attenuated (as compared to the linear response) and the first mode appears with a finite, nontrivial amplitude even when the system is being excited near its second natural frequency. The coupled-mode solution is of interest here as it represents the true solution with modal coupling resulting from internal resonance. Despite the structure being excited close to its second natural frequency, a nonzero response of the first mode at almost half the excitation frequency is obtained due to quadratic nonlinearities in the system. Figure 12 shows the magnitude of the first-mode amplitude (defined by $a1=p12+q12$) obtained for structures 1,2,3, and 4 shown in Fig. 8. The responses are shown all together in Fig. 13 for a quantitative comparison as well. For another quantitative comparison of the candidate structures developed, the point of maximum deflection for the first mode for each of the structures 1, 2, 3, and 4 is taken and its total deflection is computed using Eq. (9). Figure 14 shows the location of the point of maximum deflection for the first mode for structures 1, 2, 3, and 4. As can be observed from Fig. 14, the point of maximum deflection for the first mode occurs at the end of one of the constituent beams and thus its deflection can also be referred to as the tip deflection. These tip deflections will vary periodically with time with a constant amplitude during steady state. Figure 15 shows the comparison between the tip amplitudes for the structures 1, 2, 3, and 4 when they are subjected to the same level of base excitation. It can be observed from Fig. 15 that different structures have different magnitudes of deflection for the same level of excitation and thus from an application standpoint, some structures may be more appropriate than others.