## 2.1 Equations of Motion

_{b}is the mass of the sliding base; u

_{g}(t), u

_{s}(t), and u

_{r}(t) are the ground displacement, sliding displacement, and relative displacement between the top mass and sliding base, respectively; \(\dot{u}_{g}\), \(\dot{u}_{s}\), and \(\dot{u}_{r}\) are the corresponding velocities; \(\ddot{u}_{g}\), \(\ddot{u}_{s}\), and \(\ddot{u}_{r}\) are the corresponding; and f is the friction force between the sliding base and foundation. The first equation of Eq. (2.1) denotes the dynamic equilibrium of the top mass, while the second equation is the dynamic equilibrium of the entire system.

_{i}is the moment when sliding starts, \(\tau\) = t − t

_{i}, and

## 2.2 Typical Response Histories

_{g}and \(\omega_{g}\) are the amplitude and frequency of the sinusoidal wave, respectively.

_{g}= 2\(\pi /\omega_{g}\) = 1 s, and a

_{g}= 0.4g, in which T represents the natural period of the superstructure, and T

_{g}indicates the period of the ground acceleration. As shown in Fig. 2.3, the responses of the system converge to steady periodic responses with the same period as the ground acceleration after several cycles. The sliding base system pauses briefly before sliding in the opposite direction in this instance. This is referred to as the stick-sliding case. After the amplitude of the ground acceleration exceeds a certain value, the sliding base system will continue to slide incessantly during the steady periodic state, as shown in Fig. 2.4, where the ground acceleration amplitude, a

_{g}, is increased to 1.2g. This is referred to as the sliding-sliding case. Sliding will not occur if the ground acceleration is sufficiently low, which is known as stick-stick case. The sliding base structure and the fixed base structure have no difference in this case. The following analyses focus on the steady periodic responses.

## 2.3 Occurrence Conditions of the Three Types of Periodic Responses

### 2.3.1 Boundaries Between the Stick-Stick and Stick-Sliding Cases

_{g}; thus, we have

### 2.3.2 Boundaries Between the Stick-Sliding and Sliding-Sliding Cases

_{st}, is given in Eq. (2.12), which has the same sign as the sliding direction. As shown in Fig. 2.4, for the sliding-sliding case, the sliding base system slides in one direction for half period of the ground motion, 0.5T

_{g}, and then slides in the opposite direction for another 0.5T

_{g}, and so on. Therefore, the equivalent step force also changes direction in 0.5T

_{g}when the sliding direction changes, as shown in Fig. 2.6. Thus, the relative displacements of the two opposite half sliding cycles have the same absolute values but opposite signs.

_{r0}and \(\dot{u}_{r0}\) are the relative displacement and velocity at the moment when this half sliding cycle starts, and this moment is denoted as t

_{i}; \(\tau\) = t − t

_{i}is the local time during this half sliding cycle, so \(0 \le \tau \le \pi /\omega_{g}\). When \(\tau = \pi /\omega_{g}\), the opposite half sliding cycle starts, the relative displacement and velocity at this moment are equal to −u

_{r0}and \(- \dot{u}_{r0}\), respectively. Therefore,

_{i},

## 2.4 Parametric Study for the Maximum Responses

### 2.4.1 Maximum Pseudo Acceleration of the Top Mass

_{d}, for a fixed base structure:

### 2.4.2 Amplitude of the Sliding Displacement

_{s,ap}, is a more appropriate response quantity compared to the maximum sliding displacement. It is defined as the difference between the maximum and minimum sliding displacements during the steady state; therefore, the value of it is exclusively linked to the responses of the steady state.

_{s,ap}is equal to the vibration amplitude of the ground displacement, \(2a_{g} /\omega_{g}^{2}\).

## 2.5 Theoretical Solutions for the Responses of the Sliding-Sliding Case

### 2.5.1 Solutions for the Maximum Pseudo Acceleration

### 2.5.2 Interpretation of the Solutions for the Maximum Pseudo Acceleration

_{st}. If the damping is sufficiently small, the relative displacement can approach − 3u

_{st}during the first oscillation cycle; if the damping is very large, the oscillation of the relative displacement diminishes rapidly and is only capable of attaining the new static equilibrium displacement, −u

_{st}. Figure 2.13 shows the steady state response of the normalized pseudo acceleration for \(\alpha\) = 0.8, \(\xi\) = 5%, T = 1 s and T

_{g}= 10 s (\(\omega_{g} /\omega_{1}\) = 0.045).

_{g}is a large value so that sliding can occur during the whole time of investigation. Since \(\omega_{1d} = \omega_{1} = (2n - 1)\omega_{g}\) for \(\xi\) = 0, the relative displacement reaches the peak value of a certain half sliding cycle when the direction of sliding changes, which is the furthest location from the static equilibrium displacement of the next half sliding cycle. From Eq. (2.13), the relative displacement at the end of one half sliding cycle for \(\xi\) = 0 is

_{g}, the maximum relative displacement can increase by \(4\mu g/\omega^{2}\), and can continue increasing until the ground acceleration is not large enough anymore to start sliding. For actual structures, the maximum relative displacement will be reached after several cycles of ground motion because of the damping, as shown in Fig. 2.15 for \(\xi\) = 5%, \(\alpha\) = 0.8, T = 1 s and T

_{g}= 0.45 s \(\left( {\omega_{g} /\omega_{1d} = 1} \right)\). When \(\omega_{g} /\omega_{1} \to 0\), \(\theta_{1} \to + \infty\) from Eq. (2.31), so Eq. (2.86) tends toward Eq. (2.84). This means that when \(\omega_{g} /\omega\) decreases, the resonance will slowly disappear, as shown in Fig. 2.12b.

### 2.5.3 Solutions for the Sliding Displacement Amplitude

_{g}, for the sliding-sliding case. When \(a_{g} /\mu g \to + \infty\), \(\cos \left( {\omega_{g} t_{i} } \right) \to 0\) as revealed by Eq. (2.90), so the normalized sliding displacement amplitude, \(u_{s,ap} /\left( {a_{g} /\omega_{g}^{2} } \right)\), in Eq. (2.92) tends toward to 2, which is consistent with the results presented in Sect. 2.5.2.