## 5 Analysis Methods for CTG and STG Foundational Vibration Assessment  
A number of analysis methods can be proposed for the vibration assessment of the CTG and STG foundations. The critical issue is how to model soil radiation damping, and, therefore, how much the contribution of soil damping would affect the foundation's vibration response at 60 Hz. Two dynamic analysis methods are considered: modal superposition time history and direct integration time history. In the modal superposition time history analysis, five different soil damping cases are considered: (1) a direct soil damping coefficient from the elastic half-space solution plus 2% concrete material damping, (2) a direct soil damping coefficient from the elastic half-space solution with a 20% cut-off from the EPRI (Electric Power Research Institute) plus 2% concrete material damping, (3) all modal damping at 2%, (4) all modal damping at 4%, and (5) all modal damping at 10%. The modal damping of the (1) and (2) models is, therefore, computed by SAP2000 systematically depending on the modal deformation shapes, i.e., the energy is proportional at each mode and at the interface of the foundation and the soil.  
For the direct integration time history analysis, the conventional Newmark \(\beta\) method (\(\beta=0.25\) assumed) [11] is used throughout the analyses and three sets of proportional damping parameters (\(\alpha\): mass proportional damping and \(\beta\): stiffness-proportional damping) [12] are considered: (1) \(\xi=4\%\) at 1 st modal frequency and \(\xi=4\%\) at 60 Hz, (2) \(\xi=10\%\) at 1 st modal frequency and \(\xi=4\%\) at 60 Hz,  
Figure 4: STG Foundation Dynamic Unbalance Force at 60 Hz: (**a**) V60A, H60A, (**b**) V60B, H60B, (**c**) V60C, H60C, (**d**) V60D, H60D.  
and (3) \(\xi\) = 20% at the first modal frequency and \(\xi\) = 4% at 60 Hz. The general form of \(\alpha\) and \(\beta\) based on two frequency points is given by the following equations.  
\[\alpha:=2.\frac{\xi_{1}\cdot\omega_{2}-\xi_{2}\cdot\omega_{1}}{\left(\frac{ \omega_{2}}{\omega_{1}}-\frac{\omega_{1}}{\omega_{2}}\right)}.sec \tag{1}\]  
\[\beta:=2.\frac{\xi_{1}\cdot\omega_{1}-\xi_{2}\cdot\omega_{2}}{\omega_{1}{}^{2 }-\omega_{2}{}^{2}}.\frac{1}{sec} \tag{2}\]  
in which \(\xi_{1}\) and \(\xi_{2}\) are damping ratios that correspond to the two frequencies, \(\omega_{1}\) and \(\omega_{2}\), respectively.  
It should be noted that both dynamic analysis methods are approximate in terms of their solution techniques. For instance, the modal superposition solution (steady-state part only) depends on only the diagonal terms of the damping matrix but does not account for non-diagonal damping terms in the dynamic equation of motion. On the other hand, the direct integration method utilizes every term in the damping matrix. From this point of view, for the particular case where coupled modes contribute significantly to the total response, the modal superposition solution may not result in an accurate response representation. On the contrary, the direct integration method depends on the time increment, and the two frequency point proportional dampings may lead to an approximate response as well. Therefore, foundation engineers must pay close attention to the selection of damping parameters and analysis methods. A summary table of the applied analysis methods for the vibration assessment of the CTG and STG foundations is given in Table 1.