## 2 Rigid-Body Model of an Earthquake Excited Tower-Like Structure
Its linearized version has the form  
\[\ddot{\varphi}=\Omega_{\varphi}^{2}\bigg{[}-\frac{1}{\kappa}\frac{a_{\rm H}}{ g}+\bigg{(}1-\frac{1}{\kappa}\frac{a_{\rm V}}{g}\bigg{)}\varphi\bigg{]}, \tag{7}\]  
where the squared linear frequency of the pendulum with a linear restoring spring is given by  
\[\Omega_{\varphi}^{2}=\Omega_{0}^{2}\kappa,\quad\kappa=\frac{k_{\rm L}}{mgL}-1. \tag{8}\]  
The primary instability regime for vertical parametric resonance, see, e.g., ([17], Chapter 23) and ([23], Chapter 10), extends from the following frequency  
\[\Omega_{\rm V}=2\Omega_{\varphi}=2\Omega_{0}\sqrt{\kappa}. \tag{9}\]  
For the harmonic excitation at this parametric resonance frequency, we obtain the following non-linear second-order ODE, assuming the horizontal ground excitation frequency also to be in resonance, i.e., to be equal to the vertical parametric resonance frequency:  
\[\ddot{\varphi}=\Omega_{0}^{2}\bigg{[}a_{\rm H}(t)\bigg{(}1-\frac {1}{2}\varphi^{2}\bigg{)}+(\bar{a}_{\rm V}(t)+1)\bigg{(}\varphi-\frac{1}{6} \varphi^{3}\bigg{)}-\frac{k_{\rm L}}{mgL}\varphi+\frac{k_{\rm N}}{mgL}\varphi^ {3}\bigg{]}=f_{2}(\varphi,t),\] \[\bar{a}_{\rm H}(t)=\frac{a_{\rm H0}\sin(\Omega_{\rm V}t)}{g}, \tag{10}\] Figure 1:0  
Amplitudes of horizontal and vertical earthquake accelerations are denoted as \(a_{\rm H0}\) and \(a_{\rm V0}\), respectively. It is convenient to convert Equation (10) into the normal form  
\[\bigg{(} \begin{array}{c}\dot{\varphi}\\ \dot{\omega}\end{array} \bigg{)}=\bigg{(} \begin{array}{c}f_{1}(\omega)\\ f_{2}(\varphi,t)\end{array} \bigg{)}, \tag{11}\] \[f_{1}(\omega)=\omega.\]  
We now present a discussion of the computational method published in [11] and how to apply it in the present context. Our method is a time-stepping procedure; in the following, we consider time intervals of equal length \(T\). The normal form of the balance equation in Equation (11) is brought into its integral form by integrating over the finite interval (time step) \(\tau\in[0,T]\), the local time in the time interval being defined as \(\tau=t-(n-1)T\), where \(n=1,2,\dots\) is the number of the time step under consideration, and \(t\) denotes the physical time, starting at the beginning of the observation period. We thus obtain the following formal representation of the angular velocity for any local time \(\tau\) in the interval  
\[\omega(\tau)=\int\limits_{0}^{\tau}\tilde{f}_{2}(\varphi(\tau),\tau)d\tau+ \omega_{0}. \tag{12}\]  
The reason for assigning \(f_{2}\) formally with a superimposed tilde is explained later, after Equation (15). The angle of the tower from the vertical direction then follows as  
\[\varphi(\tau)=\int\limits_{0}^{\tau}f_{1}(\omega(\tau))d\tau+\varphi_{0}. \tag{13}\]  
The generally inhomogeneous initial conditions at the beginning of the time interval under consideration are:  
\[\varphi(0)=\varphi_{0},\ \ \ \varphi(0)=\omega_{0}. \tag{14}\]  
The two integral Equations (12) and (13) are solved in an iterative manner, motivated by the work of [PERSON] (1981). We however perform the first iterative step by substituting into the integral in Equation (12) the result obtained for \(\varphi\) from suitable explicit discrete-mechanics-type scheme. As a reference one, we here take the explicit fourth-order Runge-Kutta time-integration scheme, see (1981):  
\[ \begin{array}{l}\varphi(\bar{\tau})=\varphi_{0}+(k_{11}+2k_{21}+2k_{31}+k_{41 })\bar{\tau}/6,\\ k_{11}=f_{1}(\omega_{0}),\ k_{21}=f_{1}(\omega_{0}+k_{12}\tau/2),\ k_{31}=f_{1 }(\omega_{0}+k_{22}\tau/2),\ k_{41}=f_{1}(\omega_{0}+k_{32}\tau),\\ k_{12}=f_{2}(\varphi_{0},0),\ k_{22}=f_{2}(\varphi_{0}+k_{11}\tau/2,\tau/2), \ k_{32}=f_{2}(\varphi_{0}+k_{21}\tau/2,\tau/2),\\ k_{42}=f_{2}(\varphi_{0}+k_{31}\tau,\tau).\end{array}  \tag{15}\]  
An analogous result is obtained for \(\omega(\bar{\tau})\), which is not repeated for the sake of brevity.