## 5 Parameterisation Studies
### Material Pairs for Bar Support  
The force transmission of a steel beam supported with the isotropic bars is investigated. The axial bar used to support the steel beam is formed isotropically from the materials _St_, _Al_, _Br_, _Ag_, _Cu_ and _Pb_ (Table 1). The maxima of the frequency and force transmission values are given in Table 4 and Figure 11. The effects of the materials on the force transmission are analysed for the isotropic case and an idea is gained for the material formation of FG pairs. To better understand the variations in frequency parameters and force transmission, the bar materials are ranked by Young's modulus and internal damping coefficient (\(c_{A}\)) in Table 4.  
Due to the harmonic force acting from the centre of the beam, only _SS_ mode vibrations occur in the frequency response. The frequency response of the system can be considered to be two parts. Some frequency components belong to the beam's dominant motion; the others belong to the bars' dominant motion. With the addition of bars, side frequencies appear depending on the bars' material. For the _Pb_ material bar, after all _SS_ modes, side frequency peaks are observed. The side frequency peaks appear at _3.Region_ for _St_ and _Al_ material; _2.Region_ for _Cu_ and _Br_ material, _2.Region_; and _4.Region_ for _Ag_ material. The  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Bar Material**} & \multicolumn{2}{c}{**Pb**} & \multicolumn{2}{c}{**Cu**} & \multicolumn{2}{c}{**Ag**} & \multicolumn{2}{c}{**Br**} & \multicolumn{2}{c}{**St**} & \multicolumn{2}{c}{**Al**} \\
**Mode Number** & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) \\ \hline SS1 & 3.14128 & 243,782 & 3.14156 & 14,412,284 & 3.14154 & 1,789,396 & 3.14155 & 2,004,509 & 3.14158 & 8.340,324 & 3.14153 & 8.287,499 \\ SS2 & 9.41013 & 644 & 9.42390 & 651,465 & 9.42313 & 271,842 & 9.42349 & 983,524 & 9.42441 & 3.849,090 & 9.42304 & 6.501,443 \\ _1.Region_ & 11.33643 & 61 & & & & & & & & & \\ SS3 & 15.71125 & 4056 & 15.70121 & 108,121 & 15.68292 & 24,358 & 15.69666 & 143,805 & 15.70562 & 1,432,198 & 15.69806 & 1,460,498 \\ _2.Region_ & 19.67222 & 18 & 19.79233 & 558 & 17.01995 & 544 & 18.70356 & 998 & & & & \\ SS4 & 22.20275 & 1412 & 22.01536 & 45,711 & 22.00044 & 46,618 & 22.00860 & 115,014 & 21.95969 & 183,227 & 21.88989 & 186,157 \\ _3.Region_ & 25.36540 & 10 & & & & & & & & & & \\ SSS & 28.27558 & 619 & 28.28267 & 48,419 & 28.25448 & 7200 & 28.27770 & 61,126 & 28.29214 & 476,127 & 28.30824 & 499,885 \\ _4.Region_ & & & & & & & & & & & & \\ \hline \multicolumn{1}{c}{**(b)**} & & & & & & & & & & & & \\ \hline \multicolumn{1}{c}{**Bar Material**} & \multicolumn{2}{c}{**St**} & \multicolumn{2}{c}{**Cu**} & \multicolumn{2}{c}{**Br**} & \multicolumn{2}{c}{**Ag**} & \multicolumn{2}{c}{**Al**} & \multicolumn{2}{c}{**Pb**} \\
**Mode Number** & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) & \(\mathbf{\lambda}\) & \(\mathbf{T_{R}}\) \\ \hline SS1 & 3.14158 & 8,340,324 & 3.14156 & 14,412,2  
location of the side peaks in the frequency response depends on the specific mass and Young's modulus of the materials. Since \(Pb\) is the material with the lowest Young's modulus and the highest specific mass, the first side peak appears in the frequency range at the lead bar (Figure 11). Table 4a,b show the frequency values and maximum forces transmitted to the ground according to the internal damping coefficients (\(c_{A}\)) and Young's modulus of the bar materials. The values of the force transmitted to the ground (\(T_{R}\)) decrease as the frequency parameter (\(\lambda\)) increases. The \(Pb\), \(Ag\) and \(Br\) materials provide this for all modes.