**3. Calculations of Eigenvalues**

The beam vibration eigenvalues are calculated in terms of *βL* for different mass ratios, *<sup>m</sup>*attached *<sup>m</sup>*beam . The eigenvalues are plotted as a function of the normalized pinned location, *<sup>a</sup> <sup>L</sup>* in Figures 2–6. Note, the *βL* values corresponding to *<sup>a</sup> <sup>L</sup>* = 0 is equivalent to the clamped-free system with a mass at the free.

**Figure 2.** Eigenvalues of first 5 modes, *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.

**Figure 3.** Eigenvalues of first 5 modes, *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.4.

**Figure 4.** Eigenvalues of first 5 modes, *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.6.

**Figure 5.** Eigenvalues of first 5 modes, *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.8.

**Figure 6.** Eigenvalues of first 5 modes, *<sup>m</sup>*attached *<sup>m</sup>*beam = 1.
