Study on identification of damping system based on motion state

Abstract

A correct identification for damping system is the foundation to select the dynamic analysis method. In this study, the differences between motion states in different damping systems are investigated initially. A theoretical identification method based on motion state is proposed with the spatial continuity of the motion, and the coordination of the motion state is used as the objective function. In addition, dynamic simulation results of a set of plates are compared to clarify the physical interpretation of the identification method. The method is applied to three practical examples with recognized damping system, including the motion of seismic isolation bearings, whiplash effect, semi-active controlling systems, and soil–structure interaction system. The analysis results verify that the proposed identification method is feasible in practical engineering.

Keywords

acceleration magnified effect damping system motion state shaking table test stick–slip motion whiplash effect

Introduction

Damping system represents the distribution characteristics of energy dissipation in the dynamical structure, which also refers to the distribution characteristics of overall energy dissipation in the dynamical system (Aprile et al., 1994; Clough and Penzien, 1975; Widodo, 1995). For a multi-degree-of-freedom (MDOF) system, the damping system can be divided into classical damping systems and non-classical damping systems based on different overall energy dissipation characteristics of the system. The classical damping system is characterized as the uniform energy dissipation. The motion state of every point is consistent (Adhikari, 2005; Clough and Penzien, 1975; Widodo, 1995). As for the non-classical damping system, the motion state of every point in the system is inconsistent (Adhikari, 2005; Clough and Penzien, 1975). The motion equations are coupled in its principal modal space. The identification of damping system is the foundation for further selection of dynamic analysis method. According to the identification results, the different dynamic analysis methods (Nguyen et al., 2012; Wang et al., 2011, 2016, 2019) will be used, which can lead to dynamic analysis results varying greatly for different damping systems (Alan and Kim, 2014; Cong, 2019; Mohammad et al., 2016; Nguyen et al., 2012; Niranjan, 2017; Shrestha et al., 2015; Yan et al., 2012). Therefore, a reasonable identification of damping system is significant for the selection of analysis method and the credible dynamic analysis results.

There has been no specific standard to identify damping systems in practical projects. Traditional identification is based on the materials behavior of the system. On one hand, systems composed of materials with similar properties are identified as the classical damping systems. On the other hand, systems composed of materials with different properties are identified as the non-classical damping system, such as soil–structure interaction (SSI) system and steel-reinforced concrete structure system (Liang et al., 2014; Zhang et al., 2017, 2018; Zhou, 2013). However, current studies stated that traditional identification is not in a good agreement to experiment results and practical observation in engineering (Zhang et al., 2017, 2018). For the system composed of different materials, when the interface between each material is intact, the classical damping systems’ characteristics are observed in the course of oscillation. The dynamic analysis based on the classical damping system is in good agreement with the experimental results. As for the SSI system, though the material properties of the upper structure and soil are different, yet the classical damping system characteristics are observed in existing experiments (Zhang et al., 2017, 2018). Therefore, traditional identification based on the materials behavior lacks effective investigation and practical validation. It is urgent to have fundamental and substantial investigation on the identification of damping system.

In this study, according to the theory on discontinuous dynamical system, a preliminary identification of damping system is investigated and established based on the continuity and coordination of the motion state of the system. The completeness and feasibility of the developed method is verified and validated by a series of finite element analysis and shaking table tests. It is expected to provide a preliminary theoretical foundation for the identification of damping system and the selection of dynamic analysis methods in the seismic analysis in practical engineering.

Study on damping system category based on motion state

Damping system represents the distribution characteristics of energy dissipation in the structure under dynamical excitation. The distribution characteristics of energy dissipation are directly related to the motion state of each point in the system. The motion state is the basic manifestation of the damping system. Starting with analyzing the differences between motion states in different damping systems, the article then represents the differences of motion states between classical and non-classical damping systems.

The motion state is used to describe the variation of trajectory of displacement, velocity, and acceleration for an oscillator along with time. It can also be interpreted as a set of manifolds in a geometric space coordinate for describing how future state will be affected by the present state.

In state space, the motion state of one point is described by its velocity and displacement. If the motion state of one point is continuous, the displacement and velocity are continuous and the acceleration exists. The acceleration can be expressed as

\overset{\cdot\cdot}{u} = \frac{d \overset{\cdot}{u}}{d t} = \frac{d \overset{\cdot}{u}}{d u} \cdot \frac{d u}{d t} = \overset{\cdot}{u} \frac{d \overset{\cdot}{u}}{d u}

(1)

According to equation (1), if the acceleration of one point exists, not only should the velocity exist but also the derivative of velocity with respect to displacement should exist. In another words, both the displacement–time curve and the velocity–displacement curve of the point should be smooth and continuous. Likewise, if the motion state of the system is continuous, the velocity of each point should exist at any time and its derivative with respect to displacement should be smooth and continuous. For a frequently used viscous damping model, the motion of a system can be expressed as below

[M] {\overset{\cdot\cdot}{u} (x, t)} + [C] {\overset{\cdot}{u} (x, t)} + [K] {u (x, t)} = {p (x, t)}

(2)

For the solution of equation (2), the dynamic analysis methods for classical damping system and non-classical damping system are quite different. The main concern is whether the dynamic equations can be decoupled. If the motion equation can be decoupled, the motion of the system can be expressed by a linear combination of several decoupled modals. As for an arbitrary decoupled modal, it is obtained by assuming that the system has one and only one deflection curve. If the deflection curve is considered as shape function and the modal motion of the reference point is expressed by generalized coordinates, the decoupled modal motion in any order can be expressed by shape function and generalized coordinates as

{\begin{matrix} u_{i} (x, t) = φ_{i} (x) z_{i} (t) \\ {\overset{\cdot}{u}}_{i} (x, t) = φ_{i} (x) {\overset{\cdot}{z}}_{i} (t) \\ {\overset{\cdot\cdot}{u}}_{i} (x, t) = φ_{i} (x) {\overset{\cdot\cdot}{z}}_{i} (t) \end{matrix}

(3)

where $u_{i} (x, t)$ , ${\overset{\cdot}{u}}_{i} (x, t)$ , and ${\overset{\cdot\cdot}{u}}_{i} (x, t)$ are the ith-order modal displacement, velocity, and acceleration with respect to coordinate x and time t, respectively; $φ_{i} (x)$ stands for the shape function, which is correlated with coordinate x; $z_{i} (t)$ , ${\overset{\cdot}{z}}_{i} (t)$ , and ${\overset{\cdot\cdot}{z}}_{i} (t)$ are the displacement, velocity, and acceleration in the generalized coordinate, respectively, which are correlated with t.

According to equation (3), the necessary conditions for classical damping system are obtained: the shape functions $φ_{i} (x)$ of every order modal are continuous, and the acceleration ${\overset{\cdot\cdot}{z}}_{i} (t)$ of the reference point exists. The necessary condition can also be deduced as follows

{\begin{matrix} φ (x) is continuous \\ \overset{\cdot\cdot}{z} (t) exists \end{matrix} \Leftrightarrow {\begin{matrix} φ (x) is continuous \\ \overset{\cdot}{z} (t) is derivable \end{matrix} \Leftrightarrow {\begin{matrix} φ (x) is continuous \\ z (t) is smooth and continuous \end{matrix}

(4)

Namely, if the shape function $φ (x)$ is continuous, the velocity of the reference point is smooth and continuous; the real-valued modals of the system exist. In this scenario, the motion equation can be analyzed by modal superposition method. Based on the results of modal superposition method, the superposed motion state of each point in the system is continuous during the oscillation. Therefore, equation (3) shows the necessary and sufficient conditions for the classical damping system. The identification process can be initiated by investigating on the motion state of a dynamical system. When the system becomes non-classical damping system, any of the following circumstances will appear:

The shape function of the modal is discontinuous with respect to x at time t. When the shape function of one or more modes is discontinuous, the corresponding modal displacement and the modal superposition displacement change sharply. However, the discontinuity of the modal motion displacement means that the macroscopic motion state of the whole system is discontinuous within the structure. At this time, the motion state of the system belongs to two different continuous motion domains. The dynamic behavior of different continuous motion domains will be different. Therefore, the modal shape function of the system is discontinuous with respect to the coordinates, and the system is a non-classical damping system.

The velocity curve of modal is unsmooth or discontinuous with respect to time t for one point at height x. Although the macroscopic motion displacement of the system can still be continuous, a singularity or discontinuity appears at height x for the distribution function of the motion velocity. It is indicated that the velocity of the upper and lower points at height x no longer conforms to the modal superposition assumption of the same classical damping system. If the analysis is conducted on the basis of the modal theory hypothesis of the same classical damping system, the modal velocity on the both side of height x will be quite different. In another word, there must be one order or more than one-order modal of the velocity function that is not smooth or discontinuous with respect to time t, either above the height x or below the height x. This cause a significant difference in the velocity of the system along the height. In fact, it is the singularity or discontinuity of the modal velocity along the height of the structure that makes overall macroscopic motion state of the system being divided into two continuous motion domains (upper and lower domains). The motion superposition will be conducted according to different modal motions. The whole system is in a non-classical damping system. Therefore, the identification can be accomplished by studying the motion state of a dynamical system.

There is a necessary relationship between the continuity of modal motion and the continuity of macroscopic motion of modal superposition. In engineering, macroscopic motion is more easily observed than modal motion. To visualize the relationship between the continuity of two motions, the dynamic response of a set of thin plates with different soft interlayers at the center are simulated by finite element method. Three thin plates with the same dimension are adopted, where the length, width, and thickness of the plate are 2.0, 1.0, and 0.2 m, respectively, as shown in Figure 1. Plate A is a steel plate, where each part of the plate is the same material. In Plates B and C, the bottom and top plate are same as that of Plate A, but the middle is sandwiched with a plate of different materials of 0.2 m in x-direction. The middle plate is closely connected to the upper and lower parts. In the dynamic analysis, the bottom plate is fixed. The material property of the middle plate in each model is shown in Table 1. It should be mentioned that the simulation is performed on commercial software ABAQUS. To ensure the computation precision and efficiency, 11,400 elements are adopted, and the mesh convergence has been verified. The type of element is C3D8R.

Figure 1.

Finite element analysis model for motion continuity: (a) Plate A, (b) Plate B, and (c) Plate C.

Table 1.

Material property of the middle plate.

Material No.	Young’s modulus (GPa)	Poisson’s ratio	Density (kg/m³)
Material I	210	0.3	7850
Material II	126	0.3	4710
Material III	3.8	0.4	950

The same sinusoidal waves with continuously increasing amplitude along the y-direction are adopted as the input excitation, as shown in Figure 2. And the motion response of the different plates at t = 0.001 s is shown in Figure 3.

Figure 2.

The input wave was applied at a time increment of 0.001 s and the total duration is 58 s.

Figure 3.

The (a) displacement, (b) velocity, and (c) acceleration of each plate along Path 1.

The distribution of the displacement, velocity, and acceleration of the three plates along the height at a certain time t is given in Figure 3. From Figure 3, it can be seen that the displacement, velocity, and acceleration along the height of Plate A are continuous and smooth because the material in each part of Plate A is same and the whole plate has only one continuous motion domain. Although the middle part of Plate B is soft, the displacement, velocity, and acceleration of Plate B are also continuous and smooth, as shown in Figure 3. However, the middle part of Plate C is the softest; the singularity is observed in the displacement, velocity, and acceleration of Plate C.

The motion state of Plate B and Plate C is investigated based on the analysis of Plate A. It can be found that although the middle part is weakened and the section stiffness decreases, the weakened property of material is not enough to change the coordinated motion characteristics of the system under a certain dynamic load. The motion state of each point along the height remains smooth and continuous. The macroscopic motion state of the middle plate remains smooth and continuous. Thus, each point in Plate B belongs to the same continuous motion domain. And the motion state of each point in the same continuous domain can be expressed by the unified generalized coordinate and shape function, which makes equation (3) satisfied. Thus, Plate B is a classical damping system. However, the material of the middle plate in Plate C is further weakened and its section stiffness is reduced greatly. From Figure 3, it is shown that the motion state of each part remains coordinated only within their corresponding material segment. However, the singularity appears in the velocity distribution curve at the interface where different materials are bonded. The velocity distribution curve is no longer smooth and continuous along the height.

According to the theory of modal motion continuity, if there is a singularity appears in the velocity distribution curve along the height, there must be one order or more than one-order modal velocity function with respect to time t is not smooth. And for the middle part of Plate C, the modal motion characteristics at the material interface cannot make equation (3) satisfied. At this time, the motion state of the whole Plate C has been divided into three continuous motion domains. The superposition motion of each point cannot be expressed by the unified shape function and generalized coordinates. In different motion domains, the superposition motion will be conducted according to different modal motions. Thus, the motion of the system is no longer coordinated. At this time, Plate C is a non-classical damping system.

From the above analysis, it is indicated that the traditional method for damping system identification based on the material properties is flawed. In this article, the proposed identification method that is based on motion state of the system is reasonable and correct.

Engineering examples of damping system identification

According to the identification method proposed in section “Study on damping system category based on motion state,” three practical examples in engineering, which have recognized damping system, are exemplified to illustrate the relationship between the continuity of the motion and damping system.

Stick–slip motion

Stick–slip motion is a universal phenomenon in engineering. The stick–slip motion refers to two motion states, namely stick motion or slip motion, which appears as relatively static or sliding motion at the interface under some conditions (Albert, 2009; Luo, 2013; Thapa, 2009). For example, the motion of the structure with a friction sliding isolation system under the earthquake can be simplified as stick–slip motion, as shown in Figure 4. The system is assumed as non-classical damping system as soon as the sliding isolation bearing starts to work. And the damping system is changed from classical to non-classical in this scenario. Herein, the damping systems of the sliding isolation bearing in engineering are analyzed in this section.

Figure 4.

Schematic diagram of systems with the sliding isolation bearing.

The stick–slip motion system is assumed as two sets of sub-structures with a connection at the interface. $m_{1}$ and $m_{2}$ represents the mass of upper components (including upper bearing and column) and lower components (including support and concave round tray), respectively. $k_{1}$ and $k_{2}$ represents the stiffness of two components respectively. The damping behaviors of both sides of the interface are consistent. Once shear strength of the interface is reached; stick motion or slip motion will appear in the system under the effect of seismic acceleration ${\overset{\cdot\cdot}{u}}_{g}$ , as shown in Figure 5.

Figure 5.

Schematic diagram of stick–slip motion: (a) stick motion and (b) slip motion.

Under dynamic excitation, since the motion states of both sides of the interface show the classical damping system characteristics, the motion of each side can be expressed by corresponding shape function and generalized coordinates. To investigate the damping characteristics of the system, the bottom of $m_{1}$ is considered as the reference point of the coordinates. The ith-order shape function of two components is $φ_{1 i} (x)$ , $x \in (0, l_{1})$ and $φ_{2 i} (x)$ , $x \in (l_{1}, l_{1} + l_{2})$ , respectively. The generalized coordinates of $m_{1}$ and $m_{2}$ at time t are $z_{1 i} (t)$ and $z_{2 i} (t)$ , respectively. Thus, the modal displacements can be expressed as

u_{1 i} (x, t) = φ_{1 i} (x) z_{1 i} (t)

(5)

u_{2 i} (x, t) = φ_{2 i} (x) z_{2 i} (t)

(6)

where $u_{1 i} (x, t)$ and $u_{2 i} (x, t)$ are the ith-order modal displacements of $m_{1}$ and $m_{2}$ at time t . The damping system of two different motions will be discussed in two different conditions as follows.

Stick motion appears in the system, as shown in Figure 5(a). Motions are continuous at the interfaces of the system. The displacement, velocity, and acceleration of the ith-order modal are same on both sides of the interface at any time, namely

{\begin{matrix} φ_{1 i} (l_{1}) z_{1 i} (t) = φ_{2 i} (l_{1}) z_{2 i} (t) \\ φ_{1 i} (l_{1}) {\overset{\cdot}{z}}_{1 i} (t) = φ_{2 i} (l_{1}) {\overset{\cdot}{z}}_{2 i} (t) \\ φ_{1 i} (l_{1}) {\overset{\cdot\cdot}{z}}_{1 i} (t) = φ_{2 i} (l_{1}) {\overset{\cdot\cdot}{z}}_{2 i} (t) \end{matrix}

(7)

The displacement, velocity, and acceleration in the generalized coordinate can be expressed as

{\begin{matrix} z_{2 i} (t) = ψ_{i} z_{1 i} (t) \\ {\overset{\cdot}{z}}_{2 i} (t) = ψ_{i} {\overset{\cdot}{z}}_{1 i} (t) \\ {\overset{\cdot\cdot}{z}}_{2 i} (t) = ψ_{i} {\overset{\cdot\cdot}{z}}_{1 i} (t) \end{matrix}

(8)

where

ψ_{i} = \frac{φ_{1 i} (l_{1})}{φ_{2 i} (l_{1})}

(9)

Therefore, we have

z_{2 i} (t) = ψ_{i} z_{1 i} (t)

(10)

The displacement of the ith-order modal in the whole system can be expressed as

u_{i} (x, t) = {\begin{matrix} u_{1 i} (x, t) \\ u_{2 i} (x, t) \end{matrix}} = {\begin{matrix} φ_{1 i} (x) z_{1 i} (t) \\ φ_{2 i} (x) z_{2 i} (t) \end{matrix}} = {\begin{matrix} φ_{1 i} (x) z_{1 i} (t) \\ ψ_{i} φ_{2 i} (x) z_{1 i} (t) \end{matrix}} = {\begin{matrix} φ_{1 i} (x) \\ ψ_{i} φ_{2 i} (x) \end{matrix}} z_{1 i} (t)

(11)

Define the notation $φ_{i} (x)$ as

φ_{i} (x) = {\begin{matrix} φ_{1 i} (x), x \in (0, l_{1}) \\ ψ_{i} φ_{2 i} (x), x \in (l_{1}, l_{2} + l_{1}) \end{matrix}

(12)

then $φ_{i} (x)$ becomes a piecewise function. Combined with equation (8), we have

φ_{i} (l_{1}) = ψ_{i} φ_{2 i} (l_{1})

(13)

Since the function is continuous at the segment point, the ith-order modal motion of the system can be expressed as

{\begin{matrix} u_{i} (x, t) = φ_{i} (x) z_{1 i} (t) \\ {\overset{\cdot}{u}}_{i} (x, t) = φ_{i} (x) {\overset{\cdot}{z}}_{1 i} (t) \\ {\overset{\cdot\cdot}{u}}_{i} (x, t) = φ_{i} (x) {\overset{\cdot\cdot}{z}}_{1 i} (t) \end{matrix}

(14)

Equation (14) shows that when stick motion occurs, the system has a continuous function $φ_{i} (x)$ and unified generalized coordinates $z_{1 i} (t)$ . Since ${\overset{\cdot}{z}}_{1 i} (t)$ is smooth and continuous, the necessary and sufficient conditions in equation (3) are satisfied in stick motion. The motion equation of the system can be decoupled. Thus, modal displacement can be described by the combination of continuous shape function $φ_{i} (x)$ and generalized coordinates $z_{1 i} (t)$ whose first-order derivative is smooth and continuous. To sum up, when stick motion occurs, the motion state of the system keeps continuous. The system is a classical damping system.

2. Slip motion occurs at the interface in the system, as shown in Figure 5(b). The motion displacement is discontinuous at the interface. Although both two sides of the interface can still be assumed to have classical damping characteristics, respectively, and their modal displacements can still be expressed by equations (4) and (5), the displacement is discontinuous at the interface and difference of modal displacement exists. It is assumed that the difference of modal displacement of an arbitrary order at the interface is $Δ_{i} (t)$ , we have

φ_{1 i} (l_{1}) z_{1 i} (t) = φ_{2 i} (l_{1}) z_{2 i} (t) + Δ_{i} (t)

(15)

z_{2 i} (t) = \frac{φ_{1 i} (l_{1})}{φ_{2 i} (l_{1})} z_{1 i} (t) - \frac{Δ_{i} (t)}{φ_{2 i} (l_{1})} = ψ_{i} z_{1 i} (t) - \frac{Δ_{i} (t)}{φ_{2 i} (l_{1})}

(16)

Therefore, the modal displacements of the upper and lower parts of the system are

{\begin{matrix} u_{1 i} (x, t) \\ u_{2 i} (x, t) \end{matrix}} = {\begin{matrix} φ_{1 i} (x) z_{1 i} (t) \\ φ_{2 i} (x) z_{2 i} (t) \end{matrix}} = {\begin{matrix} φ_{1 i} (x) z_{1 i} (t) \\ ψ_{i} φ_{2 i} (x) z_{1 i} (t) - Δ_{i} (t) / φ_{2 i} (l_{1}) \end{matrix}}

(17)

When slip motion occurs, the shape function $φ_{i} (x)$ is discontinuous. The system does not meet necessary and sufficient conditions of the classical damping system in equation (3). The system does not have decoupled modals, which feature a non-classical damping system.

Whiplash effect in high-rise buildings

For buildings with a slender tower on the top, the vibration amplitude of the tower increases dramatically during earthquake (Tao and Tong, 2019; Tian et al., 2019; Zhang et al., 2017; Zhou et al., 2010; Zhang and Zhang, 2017; Li et al., 2017), which is referred as whiplash effect (Tyler and Alain, 2003). It has been found that whiplash effect appears easily in high-rise buildings under dynamic loads. Owing to rigidity and mass distributed unevenly along the height, especially at locations like connecting point between the main body and tower, where rigidity and mass mutates, whiplash effect can be arisen easily, stating that the vibration response of the tower can be more than 10 times higher than that of the main body when subjected to wind. It should be mentioned that the system belongs to classical damping system according to traditional identification method. Herein, in terms of studying the motion state of whiplash effect, a series of shaking table tests for a high-rise building with a slender tower are conducted. The motion state and the corresponding damping system with whiplash effect are analyzed. The shaking table test system is manufactured by MTS Company and located at the Key Laboratory of Structure and Earthquake Resistance of Xian University of Architecture and Technology. The test model of structure with a slender tower on the top is illustrated in Figure 6.

Figure 6.

Structure with a slender tower on the top: (a) shaking table test model, (b) analytical model, and (c) schematic diagram.

As Figure 6 shows, the system is assumed as two mass $m_{1}$ and $m_{2}$ ( $m_{1} >> m_{2}$ ). The stiffness is $k_{1}$ and $k_{2}$ ( $k_{1} >> k_{2}$ ) and the damping coefficient is $c_{1}$ and $c_{2}$ , respectively. It is supposed that the motion state of each part shows the characteristics of classical damping system and the strength is enough at the interface of the system. Thus, the displacement at the interface keeps continuing and the motion state at the interface is continuous. In other words, the ith-order modal displacement exists on the interface at time t. It can be expressed as

u_{i} (x, t) = φ_{i} (x) z_{1 i} (t)

(18)

where

φ_{i} (x) = {\begin{matrix} φ_{1 i} (x), x \in (0, l_{1}) \\ ψ_{i} φ_{2 i} (x), x \in (l_{1}, l_{2} + l_{1}) \end{matrix}

(19)

But when whiplash effect occurs, kinematic velocity may be changed sharply. The displacement response time histories of each floor of the tower and the main structure under different earthquake magnitudes are shown in Figure 7(a) and (b), respectively. Figure 7(a) and (b) illustrate the displacement time histories of the main body and tower under different earthquake inputs. Figure 8(a) and (b) illustrate the velocity time histories of the eighth floor of the main body and the first floor of the tower under different earthquake inputs.

Figure 7.

Displacement response time histories of main body and tower under El Centro wave: (a) A_max = 0.125 g and (b) A_max = 0.75 g.

Figure 8.

Velocity response time histories of main body and tower under El Centro wave: (a) A_max = 0.125 g and (b) A_max = 0.75 g.

From Figures 7 and 8, it can be found that all points of main body and tower pass through their maxima at the same instant time, and the motion state of main body and tower is identical. The displacement of the first floor of the tower is 1.58 times greater than that of the eighth floor of the main body under the small earthquake, as shown in Figure 7(a). Meanwhile, the velocity of the first floor of the tower is 1.99 times greater than that of the eighth floor of the main body, as shown in Figure 8(a). At this time, no obvious whiplash effect is observed. When the magnitude of the input earthquake increases, the displacement of the first floor of the tower is 1.65 times greater than that of the eighth floor of the main body, but the velocity of the first floor of the tower is 18 times greater than that of the eighth floor of the main body. Figure 7(b) shows that all points do not pass through their maxima at the same instant time—points appear to lag behind other points. And the motion state of main body and tower is not coordinated. The whiplash effect is observed.

To determine the frequencies in the time histories of the main body and tower, a small time interval of 5 s in time histories is used for fast Fourier transformation (FFT). FFT spectrum of velocity time histories under different magnitude of earthquake is shown in Figures 9 and 10, in which the accordance of the frequency between main body and tower can be read more clearly. From Figure 9, it can be found that the velocity histories of main body and tower contain the same frequencies under small earthquake. It illustrates that the system is classical damping system. However, in Figure 10, it can be found that the main body and tower show the very different frequencies under large earthquake.

Figure 9.

FFT spectrum of velocity time histories under small earthquake at different time intervals: (a) 7.0–7.5 s and (b) 23.0–23.5 s.

Figure 10.

FFT spectrum of velocity time histories under large earthquake at different time intervals: (a) 7.0–7.5 s and (b) 23.0–23.5 s.

It should be noted that no slip motion is observed in the experiment because the bond strength at the interface is sufficient. Therefore, the displacement of the upper tower and main body at the interface is continuous; the gradient of velocity in the upper tower is different from that in the main body, namely

lim_{x \to l_{1}^{+}} {\overset{\cdot}{u}}_{2 i} (x, t) = lim_{x \to l_{1}^{+}} φ_{i} (x) {\overset{\cdot}{z}}_{2 i} (t) \neq lim_{x \to l_{1}^{+}} φ_{i} (x) {\overset{\cdot}{z}}_{1 i} (t)

(20)

where ${\overset{\cdot}{u}}_{2 i} (x, t)$ represents the velocity at the interface in modal space at any time t.

The shape function of ith-order mode is continuous at the interface of the tower and main body because the displacement at the interface is continuous. So, the equation below will be satisfied

lim_{x \to l_{1}^{+}} φ_{i} (x) = lim_{x \to l_{1}^{+}} φ_{i} (x)

(21)

Thus, the generalized velocity at the interface is discontinuous, namely

z_{1 i} \neq z_{2 i}

(22)

where $z_{1 i}$ and $z_{2 i}$ represents the generalized coordinates of the main body and the upper tower, respectively.

It should be noted that the $φ_{i} (x)$ curve of the system is continuous, but ${\overset{\cdot}{z}}_{1 i} (t)$ of the upper tower is different from the main body. The necessary and sufficient conditions of classical damping system in equation (3) cannot be satisfied. To sum up, the system with the whiplash effect happening is a non-classical damping system.

Acceleration amplification effect in semi-active control system and SSI system

Semi-active control has been found to have an advantageous controlling effect and become one of the hotspots in oscillation controlling. On the one hand, semi-active control can be as effective as active control. On the other hand, there is only a little energy sources required as the input (James et al., 1991; Luo et al., 2014; Qi et al., 2007). For a typical bridge with semi-active controllers, where semi-active controlling dampers are installed at A, B, E and F and C and D are fixed supports, a pinpoint-like magnified effect around the dampers is observed during the earthquake, as shown in Figure 11 (Qi et al., 2007). When dampers are installed at the rigid supports (C and D), no amplified effect is observed. This acceleration amplification effect in semi-active control system has been identified as non-classical damping system.

Figure 11.

Acceleration response time histories of the bridge with the semi-active controller dampers under different earthquakes (Qi et al., 2007).

Note that if a semi-active damper is installed at a non-rigid support to adjust the motion of system, the system is a classical damping system before the damper comes into effect. The motion state can be expressed by linear combination of several modes. The displacement, velocity, and acceleration function can be expressed as

{\begin{matrix} u (x, t) = \sum φ_{i} (x) z_{i} (t) \\ \overset{\cdot}{u} (x, t) = \sum φ_{i} (x) {\overset{\cdot}{z}}_{i} (t) \\ \overset{\cdot\cdot}{u} (x, t) = \sum φ_{i} (x) {\overset{\cdot\cdot}{z}}_{i} (t) \end{matrix}

(23)

After the semi-active damper comes into effect at the non-rigid support, a force is applied to the support at this instant. According to Newton’s second law, the acceleration on this point mutates under effect of this force. To address the description of the problem, the vertical displacement of the bearing is neglected. And because the force is applied to the support, it has no influence on the mid-span bending moment and no obvious mutation appears in the bending moment. Furthermore, since all the points are closely bonded to each other, their displacements are continuous. And there is no mutation appears in the displacement.

Another practical example is SSI system. It is found that a gap could be possibly formed between the pile and soil in Loma Prieta earthquake, which can lead to pinpoint-like acceleration magnified effect (Chau et al., 2009). To make further validation, a series of shaking table tests have been conducted previously. The experimental model is shown in Figure 12, and detailed information of the shaking table test is given in Chau et al., 2009. Pinpoint-like acceleration magnified effect is observed in the acceleration time history of the pile in the shaking table test, as shown in Figure 13. Note that the enlargements of each acceleration response are given to make a better comparison. $a_{p}$ stands for the acceleration of the pile and structure. $a_{s}$ represents the acceleration of the soil around the pile.

Figure 12.

Soil-structure interaction system: (a) shaking table test model, and (b) schematic diagram (Chau et al., 2009).

Figure 13.

Acceleration response time histories of SSI system under different magnitude of sinusoidal waves: (a) A_max = 0.02 g of sinusoidal wave input, f = 4.8 Hz, (b) A_max = 0.05 g of sinusoidal wave input, f = 4.4 Hz, and (c) A_max = 0.116 g of sinusoidal wave input, f = 4.3 Hz.

From Figure 13(a), it can be found that the responses of soil and pile are almost the same, and the magnitude of soil is larger than that of the structure. When the input increases from 0.02 g to 0.05 g, several spikes are observed in the acceleration response of the structure, as shown in Figure 13(b). When the input increases to 0.116 g, the response magnitude of the structure increases by more than 34% due to the increased input acceleration. More spikes are observed in the acceleration response of the structure, as shown in Figure 13(c). The spikes in the acceleration responses of structure suggest that the impact–pounding mode interaction exists in SSI system.

It should be noted that acceleration amplification effect in semi-active control system and SSI system has been identified as non-classical damping system (Ghahari et al., 2013). The common feature is that the system is affected by impact out of sudden in the process of vibration. And a pinpoint is observed in the acceleration time history. When semi-active damping controlling force is applied or the impact–pounding between soil and structure occurs, the mutation of acceleration appears in the acceleration response time histories. It implies that velocity function becomes unsmooth. In modal space, the shape function and the generalized coordinate become different from its original ones. And the acceleration function cannot be expressed as the linear combination of the original shape function and the generalized coordinate. The modal acceleration of the system is expressed as

{\begin{matrix} u (x, t) continuous \\ \overset{\cdot}{u} (x, t) continuous \\ \overset{\cdot\cdot}{u} (x, t) discontinuous \end{matrix} \Rightarrow {\begin{matrix} u (x, t) = \sum φ_{i} (x) z_{i} (t) \\ \overset{\cdot}{u} (x, t) = \sum φ_{i} (x) {\overset{\cdot}{z}}_{i} (t) \\ \overset{\cdot\cdot}{u} (x, t) \neq \sum φ_{i} (x) {\overset{\cdot\cdot}{z}}_{i} (t) \end{matrix} \Rightarrow {\begin{matrix} φ_{i} (x) is continuous \\ {\overset{\cdot}{z}}_{i} (t) is unsmooth \end{matrix}

To sum up, as for the SSI system or the system with semi-active controllers, the mutation of acceleration in the system appears due to the instantaneity of applied force. The inflection point appears in the velocity–displacement curve. The modal velocity ${\overset{\cdot}{z}}_{1 i} (t)$ is not smooth. So, the necessary and sufficient conditions of classical damping system in equation (3) cannot be satisfied. The SSI system and semi-active control system are non-classical damping system.

Conclusion

The identification of damping system is the foundation for further selection of dynamic analysis method. Traditional identification based on the materials behavior lacks practical investigation and validation, and the motion state of each point in the system is the basic manifestation of the damping system. According to the theory on discontinuous dynamical system, a preliminary identification of damping system is investigated and established based on the continuity and coordination of the motion state of the system. The study has illustrated that the continuity of shape function, the smoothness, and the continuity of velocity are the necessary and sufficient conditions for classical damping system. Through the dynamical analysis for different composite plates, the relationship between the continuity of the macroscopic motion state in the whole system and the identification of damping system is discussed. The identification method is proposed based on the motion state near the interface. The proposed identification method is validated by a series of shaking table tests. The physical interpretation of the identification method is clarified. The method is applied to three practical examples with recognized damping system, including the motion of seismic isolation bearings, whiplash effect, the bridge with semi-active controllers, and SSI system. The analysis results verify that the identification method is feasible in practical engineering. It is expected to provide a preliminary theoretical foundation for the identification of damping system and the selection of dynamic analysis methods in the seismic analysis in practical engineering.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research was supported by the National Natural Science Foundation of China (Grant No. 51478387).

ORCID iD

Hongyang Wei

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