**4. Mode Shapes**

The mode shapes are calculated for the two different sections of the beam corresponding to the clamped-pinned and pinned-free sections. To calculate the mode shapes, the boundary condition (Equations (6)–(13)) are used to find a relationship between the coefficients. The method used here consists of solving all coefficients in terms of *a*4. Note, there are not enough equations to determine a unique solution for each coefficient. The solutions for the *a* coefficients are:

$$a\_1 = -a\_3\tag{24}$$

$$a\_2 = -a\_4\tag{25}$$

$$a\_3 = \frac{\cos\left(\beta L\frac{\theta}{L}\right) - \cosh\left(\beta L\frac{\theta}{L}\right)}{\left(\sinh\left(\beta L\frac{a}{L}\right) - \sin\left(\beta L\frac{a}{L}\right)\right)^4} a\_4\tag{26}$$

and for the *b* coefficients:

$$b\_1 = -b\_2 \cot(\beta L) + b\_3 \frac{\sinh(\beta L)}{\sin(\beta L)} + b\_4 \frac{\cosh(\beta L)}{\sin(\beta L)}\tag{27}$$

$$b\_2 = b\_3 \frac{z\_1}{z\_3} + b\_4 \frac{z\_2}{z\_3} \tag{28}$$

$$b\mathfrak{z} = \frac{\frac{z\_2 z\_4}{z\_3} + z\_6}{\frac{z\_1 z\_4}{z\_3} + z\_5} \tag{29}$$

where

$$z\_1 = \frac{\sinh(\beta L)}{\sin(\beta L)} (\cos(\beta L) + \beta L \frac{m\_{\text{attached}}}{m\_{\text{beam}}} \sin(\beta L)) - (\cosh(\beta L) + \beta L \frac{m\_{\text{attached}}}{m\_{\text{beam}}} \sinh(\beta L)) \tag{30}$$

$$z\_2 = \frac{\cosh(\beta L)}{\sin(\beta L)} (\cos(\beta L) + \beta L \frac{m\_{\text{attached}}}{m\_{\text{beam}}} \sin(\beta L)) - (\sinh(\beta L) + \beta L \frac{m\_{\text{attached}}}{m\_{\text{beam}}} \cosh(\beta L)) \tag{31}$$

$$z\_3 = \cot(\beta L)(\cos(\beta L) + \beta L \frac{m\_{\text{attached}}}{m\_{\text{beam}}} \sin(\beta L)) + (\sin(\beta L) + \beta L \frac{m\_{\text{attached}}}{m\_{\text{beam}}} \cos(\beta L))\tag{32}$$

$$z\_4 = \cos(\beta L \frac{a}{L}) - \cot(\beta L)\sin(\beta L \frac{a}{L}) \tag{33}$$

$$z\_5 = \frac{\sinh(\beta L)}{\sin(\beta L)} \sin(\beta L \frac{a}{L}) + \sinh(\beta L \frac{a}{L}) \tag{34}$$

$$z\_6 = \frac{\cosh(\beta L)}{\sin(\beta L)} \sin(\beta L \frac{a}{L}) + \cosh(\beta L \frac{a}{L}) \tag{35}$$

Substituting the equations for the coefficients (Equations (24)–(35)) into the boundary condition from Equation (12), a relationship between *a*<sup>4</sup> and *b*<sup>4</sup> is obtained. For brevity, this expression is not presented here. The mode shapes are determined for the multi-span beam by substituting all coefficient expressions in terms of *a*<sup>4</sup> into Equation (3).

Normalizing at *a*<sup>4</sup> = 1, the first five mode shapes for the clamped-pinned-free beam with a mass at the free end are plotted in Figures 7–10 for *a* = 100, 200, 300, and 400 mm with *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2. The red triangle on the plots denotes the pin location. Note that for *a* = 100 (Figure 7), the mode shapes are as expected for a fixed-pinned-free cantilever beam with a mass on the free end. However, when *a* = 200 (Figure 8), mode shape 4 is highly non-symmetric because the constraint point (pin) is just past the node and in combination with the effect of the mass this mode shape flattens out for the remainder of the beam. For *a* = 300 (Figure 9) and *a* = 400 (Figure 10) the more expected sinusoidal shape dominates the mode shapes.

**Figure 7.** Mode shapes for pinned at *a* = 100 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.

**Figure 8.** Mode shapes for pinned at *a* = 200 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.

**Figure 9.** Mode shapes for pinned at *a* = 300 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.

**Figure 10.** Mode shapes for pinned at *a* = 400 mm and *<sup>m</sup>*attached *<sup>m</sup>*beam = 0.2.
