Study on Identification Method of Motion States at Interface for Soil-Structure Interaction Damping System

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Soil-structure interaction, Dynamic analysis of structural systems, and Structural vibration control.

Abstract

The damping system characterizes the spatial distribution of structural energy dissipation. Identifying a damping system is the premise for determining the dynamic analysis method. Concepts of nonlinear processes and non-classical damping systems are often confused without theoretical primary and experimental verification. This paper proposes an identification method of damping systems based on motion states at the interface by analyzing the correlation between the damping system and dynamic characteristics. The relationship between the change of damping system type and relative motion states at the interface is studied by investigating multiple material properties through shaking table model tests of a large-scale soil-structure interaction (SSI) system. The results show that a nonlinear system can demonstrate the characteristics of the classical damping system as soon as there is no mutation of motion states at the interface of the system. The identification method of damping system based on motion states at the interface can reflect the change of dynamical characteristics of the system under linear and nonlinear processes.

1. Introduction

The damping system represents the spatial distribution characteristics of the energy dissipation in a dynamic process [1]. Under the assumption of the viscous damping model, the damping system can be divided into two types, namely classical damping systems and non-classical damping systems. The motion law of the classical damping system is consistent. Each part of the system moves coordinately, making the motion-solving process easy and the physical implication of the dynamic results explicit. In contrast, the system is considered a non-classical damping system if there is any mutation of the system’s energy consumption, which leads to the motion state of adjacent points being uncoordinated [2]. The motion law is discontinuous in this case, and the solution method of the system is very different from the classical damping system with the complex physical implication of the dynamic results [3,4,5]. The type of damping system is a crucial point in the research of vibration control. The reasonable identification of the damping system is the fundamental premise for reasonably choosing dynamic analysis methods and calculating motion states. It is also the key for the research on vibration control. The damping system can forebode the potential damage form or dangerous part of the system, indicating the optimized direction for seismic control and the corresponding design of the system.

The lack of in-depth research on the damping system’s identification method, especially during the nonlinear dynamic stage, induces no specific standard to identify damping systems. Traditional identification is based on the system’s material behaviors. There are two points of view about damping system identification under this simple cognition. Systems composed of different materials are non-classical damping systems [6] because the difference in system damping is the key to the identification. Another point of view is that, systems in nonlinear stages are non-classical damping systems because of their indefinite modal parameters.

Soil-structure interaction (SSI) system, comprised of reinforced concrete and foundation soil, is usually chosen as the object for the research of damping system identification due to the different material properties from inharmonic damping characteristics. In addition, foundation soil is easy to get into the nonlinear stage within an increasing kinetic effect. An SSI system is usually treated as a non-classical damping system [7]. However, experimental results of SSI damping system characterization cannot agree with traditional cognition. Shaking table tests of SSI system with soft foundation, designed by Tongji University [8], show that SSI system is classical damping system after some small initial vibrations. However, the nonlinear effect of soft soil foundation is noticeable, and the material properties of soil and structure are quite different. A series of shaking table tests designed by Xi’an Jiaotong University shows that the natural frequency of the SSI system decays gradually in increasing dynamic loads as soon as nonlinear vibration characteristics appear [9,10]. However, this SSI system can still transform into a unified system with coordinating motion states and show characteristics of a classical damping system even though it is composed of entirely different materials. These results indicate that the traditional identification depending on material characteristics is not completed yet, and nonlinear systems are not necessarily non-classical damping systems.

Non-linear processes and non-classical damping systems are quite different. The non-linear process indicates that the system’s dynamic characteristics are unsteady, so the stiffness of the system changes with time [11]. However, the characteristics of the damping system show the spatial coordination of the system motion law. Suppose there is one system entering the non-linear process, the stiffness decays with increasing loads, and its modal frequency is not steady anymore. The system can still be treated as a classical damping system as soon as the motion law stays consistent transiently with every moment of stiffness changes. This system is a non-linear classical damping system. Nonlinearity is a common problem in mechanical, aerospace, civil engineering and other engineering fields, which makes damping systems more complex. There has been a lack of research about the non-linear damping system and corresponding identification methods until now. The non-linear dynamic process is more prominent with the upsizing and complexity of the structural system. Thus, it is necessary to study the identification method of damping system in the non-linear process, which shows the optimized direction for seismic control and the corresponding design of a system.

In order to study the characteristics of damping systems of complex structure in dynamical processes and analyze the type and changing law of damping systems, especially with non-linear characteristics, motion laws of different damping systems, as well as coupling characteristics of corresponding motion states at the material interface of the system, are analyzed based on the discontinuous dynamics theory. An identification method of motion states at the interface for different damping systems is proposed by regarding the continuity of motion states as essential parameters. In addition, this paper conducts a large-scale shaking table test of the soil-structure dynamical system, investigating the coordination of motion laws among different parts of the system and the continuity at the interface. The paper also analyzes the relationship between the continuity of motion states and damping system types in linear and non-linear dynamical processes. The results can prove the rationality and correctness of the identification for damping systems proposed in this paper. They can also prove the simple identification method for damping systems with different materials and in the non-linear process.

2. Study on Damping System Type Based on Motion State

For a multiple-degree-of-freedom system, motion states can be expressed by a second-order differential equation as below [4].

$$M\ddot{u}(x,t)+\tilde{f}(u(x,t),\dot{u}(x,t))+Ku(x,t)=P(x,t)$$

(1)

where $x$ is the axis of the physical position of the system, $t$ is time, $\ddot{\mathbf{u}}(x,t)$, $\dot{u}(x,t)$ and $u(x,t)$ are acceleration, velocity and displacement with respect to coordinate $x$ and time $t$, $M$ and $K$ are the generalized mass and stiffness of the system, $P(x,t)$ is the external excitation subjected to the system, $\tilde{f}(u(x,t),\dot{u}(x,t))$ is the restoring force of the system related to the velocity and displacement.

Under the assumption of the viscous damping model, dynamical equations of the classical damping system can be decoupled by linear or nonlinear modes. Motion states can be expressed by the linear combination of shape functions and generalized coordinates [12], as shown in Equation (2).

$$\{\begin{array}{l}u\left(x,t\right)={\displaystyle \sum _{i=1}^n{\phi }_i(x)}{z}_i\left(t\right)\\ \dot{u}\left(x,t\right)={\displaystyle \sum _{i=1}^n{\phi }_i(x)}{\dot{z}}_i\left(t\right)\\ \ddot{u}\left(x,t\right)={\displaystyle \sum _{i=1}^n{\phi }_i(x)}{\ddot{z}}_i\left(t\right)\end{array}$$

(2)

where ${\phi }_i(x)$ is the ith-order shape function of the system with respect to position x, $z_i\left(t\right)$ is the ith-order generalized coordinate of the system with respect to time t.

It is often considered that the appearance of nonlinear process means the emergence of non-classical damping system characteristics in traditional analyses. However, the system’s nonlinearity indicates the unsteady change of the dynamic characteristics. On the contrary, the non-classical damping system indicates the incoordination of motion states of the system in space. Rosenberg proposed the nonlinear normal modes theory [13,14,15]. If the system keeps moving coordinately, every point of the system can reach the equilibrium position and maximal position at the same time. Motion states of the system accord with a specific spatial law and obey the nonlinear normal mode. Nonlinear normal mode is the transient mode with a time-dependent amplitude influenced by the development of the nonlinearity. In summary, the existence of modes is the necessary and sufficient condition for a classical damping system. The ith-order shape function ${\phi }_{\mathrm{i}}(x)$ and the corresponding modal frequency are constant in linear conditions. On the contrary, the motion law of the system is time-dependent. The shape function $\phi (x,t)$ changes with time t, and the corresponding modal frequency is instantaneous. The characteristics of the damping system are easy to observe under linear conditions but hard to confirm under nonlinear conditions, so a comprehensive identification via motion states is necessary for the type of damping system.

If the motion state of the system can be expressed by the linear combination of shape functions and generalized coordinates, the shape functions ${\phi }_i\left(x\right)$ of every order modal are continuous, and the acceleration ${\ddot{z}}_i\left(t\right)$ of the reference point exists, which can be deduced as follows [16,17].

$$\{ \begin{array}{l}{\phi }_{\mathrm{i}}\left(x\right)\mathrm{is} \mathrm{continuous}\\ {\ddot{z}}_{\mathrm{i}}\left(t\right)\mathrm{exist}\end{array}\iff \{\begin{array}{l}{\phi }_{\mathrm{i}}\left(x\right)\mathrm{is}\mathrm{continuous}\\ {\dot{z}}_{\mathrm{i}}\left(t\right)\mathrm{is} \mathrm{derivable}\end{array}\iff \{\begin{array}{l}{\phi }_{\mathrm{i}}\left(x\right)\mathrm{is} \mathrm{continuous}\\ {\dot{z}}_{\mathrm{i}}\left(t\right) \mathrm{is} \mathrm{smooth} \mathrm{and} \mathrm{continuous}\end{array}$$

Suppose the ith-order shape function ${\phi }_i\left(x\right)$ is continuous, and the velocity of the reference point ${\dot{z}}_i\left(t\right)$ is smooth and continuous. The real-valued models of the system exist, and the system is a classical damping system. However, the modal motion is hard to monitor in practical engineering. The synthetic motion law of the system is used to estimate whether the modal motion is smooth and continuous. The synthetic motion combined with smooth modal motions is smooth and continuous.

Furthermore, the inverse negative proposition can be concluded that one or more orders of modal motion are discontinuous as soon as the synthetic motion is not smooth or continuous. Therefore, the continuity of the synthetic motion states is related to modal motion states directly. The system is a non-classical damping system as soon as the synthetic motion states of the system are discontinuous.

3. Study on the Identification Method of Damping System Category Based on Interfacial Motion States

In the dynamical motion of the structure system, the continuous motion states belong to one continuous motion domain, and discontinuous motion states are called discontinuous motion boundaries of the domain. The interface between soil and structure is the motion boundary most likely to appear for the SSI system. The analysis of the continuity at this interface is the crucial point for identifying the SSI damping system.

Suppose there exists a material interface where motion states of every position are continuous and smooth, the system is a classical damping system with uniform modes of motion. In other words, motion states at the material interface are all smooth and continuous for the non-classical damping system. The paper takes the horizontal motion as an example. The relationship of motion states at the material interface and the damping system is established by discussing the motion laws of points at the material interface.

Suppose there is a classical damping system and the ith-order transient modal motion of any point x at the interface can be expressed as Equation (3).

$$\{\begin{array}{l}{u}_i(x,t)={\phi }_i(x)z_i(t)\\ {\dot{u}}_i(x,t)={\phi }_i(x){\dot{z}}_i(t)\\ {\ddot{u}}_i(x,t)={\phi }_i(x){\ddot{z}}_i(t)\end{array}$$

(3)

where ${\phi }_i(x)$ represents the ith-order transient modal shape of the system at point x, $z_i(t)$, ${\dot{z}}_i(t)$ and ${\ddot{z}}_i(t)$ represent the displacement, velocity and acceleration of the generalized coordinates at time t.

The synthetic motion can be expressed as Equation (4).

$$\{\begin{array}{l}u(x,t)={\displaystyle \sum u_i(x,t)}={\displaystyle \sum {\phi }_i(x)z_i(t)}\\ {\dot{u}}_i(x,t)={\displaystyle \sum {\dot{u}}_i(x,t)}={\displaystyle \sum {\phi }_i(x){\dot{z}}_i(t)}\\ \ddot{u}(x,t)={\displaystyle \sum {\ddot{u}}_i(x,t)}={\displaystyle \sum {\phi }_i(x){\ddot{z}}_i(t)}\end{array}$$

(4)

Taking $T_i(t)$ as the transient period of the ith-rode modal motion and the corresponding circular frequency is obtained as Equation (5).

$${\omega }_i(t)=\frac{2\pi }{T_i(t)}$$

(5)

The ith-order transient modal motion state of relative motion of two arbitrary points $x_1$ and $x_2$ are shown as Equation (6).

$$\{\begin{array}{l}{\overline{u}}_i(x_{1-2},t)=({\phi }_i(x_1)-{\phi }_i(x_2))z_i(t)={\overline{\phi }}_i^{x_{1-2}}{z}_i(t)\\ {\overline{\dot{u}}}_i(x_{1-2},t)=({\phi }_i(x_1)-{\phi }_i(x_2)){\dot{z}}_i(t)={\overline{\phi }}_i^{x_{1-2}}{\dot{z}}_i(t)\\ {\overline{\ddot{u}}}_i(x_{1-2},t)=({\phi }_i(x_1)-{\phi }_i(x_2)){\ddot{z}}_i(t)={\overline{\phi }}_i^{x_{1-2}}{\ddot{z}}_i(t)\end{array}$$

(6)

where ${\overline{\phi }}_i^{x_{1-2}}={\phi }_i(x_1)-{\phi }_i(x_2)$ is the deviation of modal shape functions of points $x_1$ and $x_2$.

The transient frequency of modal relative motion is determined by the motion law of the generalized coordinate because every point of the classical damping system moves with the same frequency ${\omega }_i(t)$, as shown in Equation (6), which means the transiently circular frequency of the ith-order modal relative motion is the same as the circular frequency of ith-order modal motion of the system, as shown in Equation (7).

$${\omega }_i^{’}(t)={\omega }_i(t)=\frac{2\pi }{T_i(t)}$$

(7)

The relatively synthetic motion state of two points on the interface is noted as $(\overline{u},\overline{\dot{u}},\overline{\ddot{u}})$ and used to analyze the movement rules of the interface. At time t, two sets of relatively synthetic motion states A and B at the interface are denoted by $({\overline{u}}^A,{\overline{\dot{u}}}^A,{\overline{\ddot{u}}}^A)$ and $({\overline{u}}^B,{\overline{\dot{u}}}^B,{\overline{\ddot{u}}}^B)$. Relatively synthetic motion law by two orders of modes are discussed first and relatively synthetic motion states can be expressed as Equation (8).

$$\overline{u}(x,t)={\displaystyle \sum _{i=1}^2{\overline{\phi }}_i(x)z_i(t)}$$

(8)

Two groups of relatively synthetic motion states A and B can be expressed as Equations (9) and (10), respectively.

$$\{\begin{array}{l}{\overline{u}}^A={\overline{\phi }}_1^A(x^A)z_1(t)+{\overline{\phi }}_2^A(x^A)z_2(t)\\ {\overline{\dot{u}}}^A={\overline{\phi }}_1^A(x^A){\dot{z}}_1(t)+{\overline{\phi }}_2^A(x^A){\dot{z}}_2(t)\\ {\overline{\ddot{u}}}^A={\overline{\phi }}_1^A(x^A){\ddot{z}}_1(t)+{\overline{\phi }}_2^A(x^A){\ddot{z}}_2(t)\end{array}$$

(9)

$$\{\begin{array}{l}{\overline{u}}^B={\overline{\phi }}_1^B(x^B)z_1(t)+{\overline{\phi }}_2^B(x^B)z_2(t)\\ {\overline{\dot{u}}}^B={\overline{\phi }}_1^B(x^B){\dot{z}}_1(t)+{\overline{\phi }}_2^B(x^B){\dot{z}}_2(t)\\ {\overline{\ddot{u}}}^B={\overline{\phi }}_1^B(x^B){\ddot{z}}_1(t)+{\overline{\phi }}_2^B(x^B){\ddot{z}}_2(t)\end{array}$$

(10)

where $\{\begin{array}{l}{\overline{\phi }}_i^A(x^A)={\phi }_i(x_1^A)-{\phi }_i(x_2^A)\\ {\overline{\phi }}_i^B(x^B)={\phi }_i(x_1^B)-{\phi }_i(x_2^B)\end{array}$, $i=1,2$, represents the deviation of first and second order of modal shape functions of A and B, respectively.

Suppose:

$$\{\begin{array}{l}{\overline{x}}^A={\overline{x}}_{1-2}^A=x_1^A-x_2^A\\ {\overline{x}}^B={\overline{x}}_{1-2}^B=x_1^B-x_2^B\end{array}$$

(11)

where ${\overline{x}}^A$ and ${\overline{x}}^B$ present the deviation of initial position coordinates of groups A and B, respectively, $x_1^A$ and $x_2^A$ present initial position coordinates of two points in group A, $x_1^B$ and $x_2^B$ present initial position coordinates of two points in group B.

Suppose the modal displacement under generalized coordinates is

$$z_i(t)={\alpha }_i\cos ({\omega }_{i}t)(i=1,2)$$

(12)

where ${\omega }_1$ and ${\omega }_2$ represent the first two orders of modal circular frequencies and ${\omega }_2>{\omega }_1$; ${\alpha }_i$ is a parameter for the amplitude of the i-th modal displacement.

Then the displacement of relatively synthetic motion of groups A and B can be expressed as Equation (13).

$$\{\begin{array}{cc}{\overline{u}}^A=& {\overline{\phi }}_1^A\left({\overline{x}}^A\right){\alpha }_1\cos \left({\omega }_{1}t\right)+{\overline{\phi }}_2^A\left({\overline{x}}^A\right){\alpha }_2\cos \left({\omega }_{2}t\right)\\ {\overline{u}}^B=& {\overline{\phi }}_1^B\left({\overline{x}}^B\right){\alpha }_1 \cos \left({\omega }_{1}t\right)+{\overline{\phi }}_2^B\left({\overline{x}}^B\right){\alpha }_2 \cos \left({\omega }_{2}t\right)\end{array}$$

(13)

where $\left|{\overline{\phi }}_1^A(x^A){\alpha }_1\right|$ and $\left|{\overline{\phi }}_1^B(x^B){\alpha }_1\right|$ represent the amplitude of the first order of modal relative motion of groups A and B, and can be denoted by $\left|{\overline{\phi }}_1^A{\alpha }_1\right|$ and $\left|{\overline{\phi }}_1^B{\alpha }_1\right|$, respectively.

$\left|{\overline{\phi }}_2^A(x^A){\alpha }_2\right|$ and $\left|{\overline{\phi }}_2^B(x^B){\alpha }_2\right|$ represent the amplitude of second order of modal relative motion of groups A and B, and can be denoted by $\left|{\overline{\phi }}_2^A{\alpha }_2\right|$, $\left|{\overline{\phi }}_2^B{\alpha }_2\right|$, ${\omega }_{1}t$ and ${\omega }_{2}t$ represent the anger of first and second orders of modes, respectively.

Based on the rotating vector method, the displacement amplitudes of relatively synthetic motions of groups A and B, synthesized by the first two modes, can be presented by projections of the relatively synthetic displacement vectors ${\overrightarrow{\overline{u}}}^A$ and ${\overrightarrow{\overline{u}}}^B$ along the $u$ axis, respectively, as shown in Figure 1. Besides, the first and second modal relative motion vectors of groups A and B are collinear, respectively, due to the equality of i-th order of circular modal frequencies of groups A and B.

As per the trigonometric relationship of relatively synthetic motion, the amplitudes of the relatively synthetic motion of group A and group B can be obtained as Equations (14) and (15), respectively.

$$\left|{\overline{u}}^A\right|=\sqrt{{({\overline{\phi }}_1^A{\alpha }_1)}^2+{({\overline{\phi }}_2^A{\alpha }_2)}^2+2{\alpha }_1{\alpha }_2{\overline{\phi }}_1^A{\overline{\phi }}_2^A\cos ({\omega }_2-{\omega }_1)t}$$

(14)

$$\left|{\overline{u}}^B\right|=\sqrt{{({\overline{\phi }}_1^B{\alpha }_1)}^2+{({\overline{\phi }}_2^B{\alpha }_2)}^2+2{\alpha }_1{\alpha }_2{\overline{\phi }}_1^B{\overline{\phi }}_2^B\cos ({\omega }_2-{\omega }_1)t}$$

(15)

It can be seen that the magnitude of this amplitude varies periodically with time.

The transient circular frequency of the relatively synthetic motion of groups A and B are shown as Equations (16) and (17), respectively.

$${\overline{\omega }}^A=\frac{{\omega }_1+{\omega }_2}{2}+\frac{{({\overline{\phi }}_1^A{\alpha }_1)}^2-{({\overline{\phi }}_2^A{\alpha }_2)}^2}{{({\overline{\phi }}_1^A{\alpha }_1)}^2+{({\overline{\phi }}_2^A{\alpha }_2)}^2+2{\alpha }_1{\alpha }_2{\overline{\phi }}_1^A{\overline{\phi }}_2^A\cos ({\omega }_2-{\omega }_1)t}\frac{{\omega }_2-{\omega }_1}{2}$$

(16)

$${\overline{\omega }}^B=\frac{{\omega }_1+{\omega }_2}{2}+\frac{{({\overline{\phi }}_1^B{\alpha }_1)}^2-{({\overline{\phi }}_2^B{\alpha }_2)}^2}{{({\overline{\phi }}_1^B{\alpha }_1)}^2+{({\overline{\phi }}_2^B{\alpha }_2)}^2+2{\alpha }_1{\alpha }_2{\overline{\phi }}_1^B{\overline{\phi }}_2^B\cos ({\omega }_2-{\omega }_1)t}\frac{{\omega }_2-{\omega }_1}{2}$$

(17)

If displacement vectors of relatively synthetic motion ${\overrightarrow{\overline{u}}}^A$ and ${\overrightarrow{\overline{u}}}^B$ are collinear in the diagram of the synthesis motion law, as shown in Figure 2, angles between the displacement vector of relatively synthetic motion ${\overrightarrow{\overline{u}}}^i$ and corresponding deviation of initial position coordinates $\left|{\overline{x}}^i\right|$ of groups A and B are equal, namely, proportions between the amplitude of relatively synthetic motion and its own initial position coordinates $\left|{\overline{x}}^i\right|$ of groups A and B are equal, shown as Equation (18);

$$\frac{\left|{\overline{u}}^A\right|}{\left|{\overline{x}}^A\right|}=\frac{\left|{\overline{u}}^B\right|}{\left|{\overline{x}}^B\right|}$$

(18)

Furthermore, the relationship of modal motion vectors shown as Equation (19) can be satisfied as soon as the relatively synthetic motion ${\overrightarrow{\overline{u}}}^A$ and ${\overrightarrow{\overline{u}}}^B$ are collinear.

$$\frac{{\overline{\phi }}_1^A{\alpha }_1}{{\overline{\phi }}_1^B{\alpha }_1}=\frac{{\overline{\phi }}_2^A{\alpha }_2}{{\overline{\phi }}_2^B{\alpha }_2}$$

(19)

Suppose

$$k=\frac{\left|{\overline{u}}^A\right|}{\left|{\overline{u}}^B\right|}$$

(20)

Then

$$\frac{{\overline{\phi }}_1^A{\alpha }_1}{{\overline{\phi }}_1^B{\alpha }_1}=\frac{{\overline{\phi }}_2^A{\alpha }_2}{{\overline{\phi }}_2^B{\alpha }_2}=k$$

(21)

Namely,

$$\{\begin{array}{l}{\overline{\phi }}_1^A{\alpha }_1=k{\overline{\phi }}_1^B{\alpha }_1\\ {\overline{\phi }}_2^A{\alpha }_2=k{\overline{\phi }}_2^B{\alpha }_2\end{array}$$

(22)

Taking Equation (22) into Equations (16) and (17), then Equation (22) can be obtained.

$$\frac{{({\overline{\phi }}_1^A{\alpha }_1)}^2-{({\overline{\phi }}_2^A{\alpha }_2)}^2}{{({\overline{\phi }}_1^A{\alpha }_1)}^2+{({\overline{\phi }}_2^A{\alpha }_2)}^2+2{\alpha }_1{\alpha }_2{\overline{\phi }}_1^A{\overline{\phi }}_2^A\cos ({\omega }_2-{\omega }_1)t}=\frac{{({\overline{\phi }}_1^B{\alpha }_1)}^2-{({\overline{\phi }}_2^B{\alpha }_2)}^2}{{({\overline{\phi }}_1^B{\alpha }_1)}^2+{({\overline{\phi }}_2^B{\alpha }_2)}^2+2{\alpha }_1{\alpha }_2{\overline{\phi }}_1^B{\overline{\phi }}_2^B\cos ({\omega }_2-{\omega }_1)t}$$

(23)

Namely

$${\overline{\omega }}^A={\overline{\omega }}^B$$

(24)

In conclusion, if a system belongs to a classical damping system and its synthetic motion states can be expressed by two orders of modes, transient circular frequencies of two groups of relative motions are equal as soon as displacement vectors of relatively synthetic motions at the material interface are collinear, namely motion states of these two groups satisfy Equation (18).

Similarly, if there is a classical damping system whose relatively synthetic motion is composed of three orders of modes, it can be understood that each order of circular modal frequency of the two groups of synthetic motion at the material interface are equal, respectively, namely the modal relative motion vectors of the two are collinear. Proportions among the amplitude of three orders of relatively modal motions are equal as soon as relatively synthetic motion vectors of these two groups are collinear, shown as Figure 3. It can be proved that transient circular frequencies of these two groups of relative motions are equal, similarly.

Suppose there are two groups of relatively synthetic motion composed by n-th order of modes, and displacement vectors of these synthetic motions are collinearly satisfying the identities of Equation (18), in that case the system is a classical damping system as soon as the transient circular frequencies of these two groups of relative motions are equal.

For the classical damping system, transient circular frequencies of relative motions are equal as soon as displacement vectors of these groups of relatively synthetic motions at the material interface are collinear. These relatively synthetic motions satisfying the condition of Equation (18) can be treated as the motion of a rigid body with the same transient circular frequency.

At present, relatively synthetic motions of groups A and B of any classical damping system satisfy Equation (18) are chosen, and their transient circular frequencies are supposed to be $\omega$. Their relative motion states are plotted in the phase plane. The relationship between relative displacements and relative velocities is shown in Figure 4, where the black and red lines are the trajectories of relative motion states of groups A and B, respectively. The phase angle of the motion trajectory ${\phi }_t$ indicates the proportions between the displacement and velocity at time t and can represent the characteristic of motion trajectory. Quick changing ratios of motion states of groups A and B are equal because the transient circular frequencies of both are the same. In addition, the transient changing rate $d{\phi }_t$ of phase angles of motion trajectories ${\phi }_t$ can represent the changing ratio of motion states, namely the changing speed of motion states. Equation (25) can be established as the equality of transient circular frequencies of groups A and B.

$$d{\phi }_t^A=d{\phi }_t^B$$

(25)

While

$$d{\phi }_t=\frac{d\overline{\dot{u}}}{d\overline{u}}$$

(26)

Then Equation (27) can be obtained

$$\frac{d{\overline{\dot{u}}}^A}{d{\overline{u}}^A}=\frac{d{\overline{\dot{u}}}^B}{d{\overline{u}}^B}$$

(27)

As

$$\frac{d\overline{\dot{u}}}{d\overline{u}}=\frac{\raisebox{1ex}{$d\overline{\dot{u}}$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.}{\raisebox{1ex}{$d\overline{u}$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.}=\frac{\overline{\ddot{u}}}{\overline{\dot{u}}}$$

(28)

Then Equation (28) can be obtained,

$$\frac{\overline{\ddot{u}}(x_{\mathrm{A}},t)}{\overline{\dot{u}}(x_{\mathrm{A}},t)}=\frac{\overline{\ddot{u}}(x_B,t)}{\overline{\dot{u}}(x_B,t)}$$

(29)

In conclusion, motion states at the material interface coordinate, as soon as every two groups of relative motion states satisfy Equation (18), can meet the demand of Equation (29), neighboring motion domains at the interface are coupling totally, and dynamical systems on both sides of the corresponding motion boundary can be treated as one continuous dynamical system. The system is a classical damping system at this time. Otherwise, the system is a non-classical damping system if relative motion states satisfying Equation (18) cannot meet the demand of Equation (29). Therefore, damping system identification is put forward through the law of motion states of the system at the material interface.

4. The Study of the Application of the Damping System Identification Method Based on Interfacial Motion States for the SSI Damping System

The above analysis shows that the damping system category depends on the continuity of motion states of points on the motion boundary. When motion states of each point on the boundary remain smooth and continuous, the relative motion states of the boundary satisfy Equations (18) and (28), and the system is a classical damping system. This section will take the SSI system as an example of the application of this identification method. We conducted the SSI system’s large-scale shaking table experiment to analyze the continuity of motion states of the points on the boundary. The coupling of motion states on the interface is discussed to verify the effectiveness and applicability of the identification method proposed in this paper.

4.1. The Analysis Model for SSI Damping System

There are three parts of soil, foundation (including piles) and superstructure for the soil-structure interaction (SSI) system. The foundation and superstructure are treated as a whole of structural part because they two are connected rigidly with little difference of material characteristics on the process of dynamic effect. Soil is a semi-infinite space medium, and the mainly affected part of it is selected reasonably by artificial boundary conditions in theoretical analysis. In addition, the soil is usually divided into two parts: one under the foundation and the other on the sides. The soil under the foundation is the dynamic force, and the soil on the sides impedes the motion. The schematic diagram of the simplified physical model for the structural system considering the SSI effect is shown in Figure 5. The soil on the sides of the structure is the central part undertaking active and passive earth pressure, and some parts of soil under the foundation move with the structure and bear active and passive earth pressure a little.

The characteristics of the damping system are related to the continuity of motion states at the material interface of the system, as shown in Section 2. The stick-slip motion states exist easily at the interaction interface between soil and structure because of their significant differences of physical and mechanical properties, which means the discontinuous motion states appear at the soil-structure interaction interface most likely. Therefore, the analyses of motion states’ continuity at the material interface between soil and structure become the focus of this paper. The motion states at the interface are the necessary and sufficient conditions for a classical damping system; namely, the displacement, velocity, and acceleration on both sides of the interface are all equal. Only motion states along the direction of motion are analyzed in order to highlight the problem of proof. In addition, the interaction between soil under the foundation and the structure is weak enough for ignoring. The interface of soil-structure interaction is the sidewalls of piles and the cap, shown as the red part in Figure 6.

4.2. Shaking Table Test for SSI Damping System

Shaking table tests for the SSI damping system are designed and completed by the physical model of the damping system. The test object is an eight-layer frame structure with a pile foundation and yellow soil foundation. The specific test model, test point arrangement and testing method are as follows.

4.2.1. The Introduction of Test Model

The test model is designed with basic similarity parameters of elasticity modulus of 1/4, size of 1/10, and acceleration of 2.5/1 compared with the real structure [18,19]. The sizes of model components are shown in

Table 1. Model Size and Number of Components.

Component Category		Size (m) × Number
floor	Length× width × thickness	$0.6\times 0.6\times 0.02\times 8$
beam	Length × width × thickness	$0.6\times 0.035\times 0.065\times 36$
column	Length × width × thickness	$0.065\times 0.065\times 0.3\times 32$
platform	Length × width × thickness	$0.3\times 0.9\times 0.9\times 1$
pile	Length × width × thickness	$1\times 0.065\times 0.065\times 9$
foundation	Length × width × thickness	$3\times 1.5\times 1.5\times 1$

.

Materials of superstructure and piles are micro concrete and galvanized iron wire. Micro concrete is composed of several micro aggregates whose construction method, material properties, and dynamic properties are similar to real concrete [20]. The micro concrete mix ratio chosen in this test is cement: water: coarse aggregate: fine aggregate = 1:1.3:2.4:3.7, which can meet the material property similarity ratio requirement [21]. Balancing weights are attached considering the variable load and permanent load of the roof and floor the normal working condition of the structure.

The groundsill model box is essential equipment for the shaking table test of the SSI system. The laminar shear box [22], with a three-dimensional size (length × width × height) of 3 m × 1.5 m × 1.392 m, is used considering the simulation of lateral soil pressure and shear deformation of the foundation soil under the earthquake process. In addition, this model box is composed of 13 layers steel frame of shear-type whose cross-section is rectangular. The height of each layer is 0.096 m, and the gap between every two layers is 0.012 m. Steel groove rollers are set in the gap to simulate the foundation boundary effect and shear deformation of soil in the dynamical process. Frame columns are set on both sides in the vertical direction of vibration to prevent the dropping of the steel frame, and universal spheres are set on these columns corresponding to positions of each layer to prevent the plane torsional deformation of the soil box. The steel frame is welded at the groundsill box’s bottom, which can be bolted to the shaking table. Slip bars are set at the bottom surface in order to simulate the frictional boundary condition of walls of the groundsill box. In addition, a rubber membrane of 5 mm thick, polystyrene plastic board of 5 cm thick and waterproof rubber membrane of 5 mm inch are laid from inside out for the sidewall to simulate the flexible boundary condition of the soil, shown as Figure 7.

4.2.2. Testing Points Arrangement and Testing Method

Choosing horizontal x-direction as the main direction of vibration in this test and different sensors are set to measure acceleration and displacement of each floor of the structure and soil at different depths. The SSI damping system is taken as the object of investigation. The figure of the layout of acceleration and displacement measuring points in different model parts are shown as Figure 8.

Dynamic characteristics of super structure and soil are tested first. The measurement results in their initial states are listed as: the first and second modes of natural frequencies of the frame structure are 5.14 Hz and 30.26 Hz, and the corresponding modal damping ratios are 2.79% and 3.73%, respectively. On the contrary, the soil’s first and second natural frequencies are 9.29 Hz and 17.10 Hz, and the corresponding modal damping ratios are 3.65% and 5.64%, respectively. It can be seen that dynamic characteristics of these two are obviously different at the initial state.

This large-scale shaking table test simulates experimental conditions of the frequent earthquake, primary intensity earthquakes and rare earthquakes. The overall loading configuration is: the horizontal x direction is chosen as the loading direction for multistage loadings, and the simulated earthquake intensity is increased grading by small magnitude. Different kinds of monitored seismic waves of bedrock wave, Jiangyou wave, El Centro wave and sine wave are input at each level of intensity. White noise scanning with a peak acceleration of 0.05 g is conducted before every interval of the test to determine the macroscopic changes of kinetic properties of the system. The loading program of the shaking table tests is shown in

Table 2. Test Program of the Shaking Table Tests.

No.	Working Condition	Peak Acceleration (g)	Comments	No.	Working Condition	Peak Acceleration (g)	Comments
S1	WN1	0.05		S19	WN5	0.05
S2(2) S3, S4	Seismic wave	0.125	Six degree of earthquake	S20–S22	Seismic wave	0.5	Eight degree of earthquake
S5(2)	Sine wave	0.03/0.125	S5-1, (5.5 Hz, 0.03 g) S5-2, (9.3 Hz, 0.125 g)	S23	WN6	0.05
S10	Sine wave	0.175	7.3 Hz	S28–S30	Seismic wave	1.0	Nine degree of earthquake
S11	WN3	0.05		S31	WN8	0.05
S12–S14	Seismic wave	0.25	Seven degree of earthquake	S32–S34	Seismic wave	1.2

.

4.2.3. The Continuity of Motion States of Points at the Material Interface

Motion States at the Interface of System on the Condition of White Noise Scanning

The interface between the structure and foundation soil is the critical position to identify the SSI damping system [23,24]. The continuity of measured motion states at this interface is necessary to verify the SSI damping system category. The relative phase plane method is practical and feasible to study motion states at the contact interface, and curves in the phase plane for adjacent measurement points represent actual trajectories of two sides of the contact surface. They can be used as a direct measurement for the continuity of the interface [24].

Relative phase-plane curves of “adjacent” measuring points at pile cap-soil interface under white noise scanning before each lever of simulated seismic actions are shown in Figure 9. Measuring points of AX5 for soil and AX34 for pile cap are chosen as shown in Figure 8b. There are specific distances between “adjacent” points at the test interface because of the soil thickness of sensors embedded in and the height of the sensor itself. However, these phase-plane cures can reflect the variation tendency of relative motion states at the interface. The motion states of adjacent points are uncoordinated with apparent relative displacement and velocity at the contact interface on the condition of white noise before seismic actions, while motion coordination at the contact interface appears and the relative motion amplitude of adjacent points decreases gradually after small-scale vibrations. The difference in relative velocity and relative displacement between adjacent points decreases obviously. The continuity of motion states at the system’s interface strengthens further with the process of vibrations with increasing amplitudes, shown as Figure 9c,d. The magnitude of the relative motion state of Figure 9d is caused by the specific distance between “adjacent” points. The change law of motion states on conditions of white noise entirely coincides with laws on multistage loadings, and the system behaves the development tendency of characteristics of the classical damping system.

Relative phase-plane curves of “adjacent” measuring points at pile bottom-soil interface under white noise scanning are plotted in Figure 10 and measuring points are shown as Figure 8b. It can be seen that the relative motion laws of adjacent measuring points at the soil-pile interface are almost the same as motion laws shown in Figure 9 as the vibration condition continues and the vibration intensity increases. The differences between relative displacements and velocities between soil and pile are decreasing, illustrating the coordination of motions between soil and pile is improving and a gradually coordinated trend of motion states is shown at the soil-pile interface.

Motion States at the Interface of System on the Condition of Loading Procedure

Because it is challenging for motion states to form rules in the phase plane with the uncertainties of load intensity and frequency, motion states at the interface under the same vibration intensity level are studied in this paper.

Phase plans of absolute velocities and displacements of adjacent points at the pile cap-soil interface on conditions of different seismic waves with the same mini level of 0.125 g are shown in Figure 11. In addition, Figure 11a shows the phase plane of motion states of these adjacent points under the condition of bedrock wave of 0.125 g initially. The motion states of these two adjacent points are uncoordinated, and there is a big difference between their motion trajectories. The velocities and displacements of these two adjacent points tend to match better. The difference decreases when bedrock wave, Jiangyou wave and EI Centro wave with the same level are input again with the development of working conditions of the same magnitude and growth of vibration time. That is to say, with the help of the internal interaction mechanism of the system, the uncoordinated motion states of soil and foundation tend to coordinate on the condition of a stable level of a small earthquake, and the system tends to be the classical damping system, therefore.

As the analysis of the test is complicated because of the uncertainties of loading intensity and frequency when conventional seismic wave loading modes are adopted, some groups of tests on conditions of sinusoidal excitation with specific amplitudes are taken and natural vibration frequencies of structure, soil and SSI system are used as sinusoidal excitation frequencies in order to observe the motion mechanism at the contact interface in vibration process easily.

The dynamic response of the measuring point under sinusoidal excitation is generally composed of two parts: the first is the damped free vibration response with the natural frequency, and the second is the harmonic vibration response with excitation frequency as its vibration frequency. The first part of vibration is the transient response with amplitude decaying because of the damping. In contrast, the second part is the response of sinusoidal excitation and the amplitude of the response stays constant, and the frequency is the same with the excitation. Although sinusoidal excitations with a single frequency cannot reflect the whole dynamic characteristics of the system, the consistency of motion states of the system, especially at the contact surface, can be investigated through sinusoidal excitation with different frequencies.

Phase-plane curves of “adjacent” measuring points at pile cap-soil interface under aforementioned sinusoidal excitations are shown in Figure 12. Furthermore, corresponding acceleration auto-power spectrum density curves of “adjacent” measuring points at pile cap-soil interface are shown in Figure 13.

The sinusoidal excitation causes the working condition of S5-1 shown in Figure 12a with the same frequency as the structure’s natural frequency. Motion laws for adjacent points at the pile cap-soil interface are not entirely harmonic vibration responses from the phase plane trajectory of Figure 12a. There are different frequencies than the sinusoidal frequency from acceleration auto-power spectrum density curves of Figure 13a. The response of the measuring point is not entirely the steady-state vibration response at this time, and there is a transient response with damped free vibration through the above analysis. In addition, the amplitude of this transient response decays with time. The working condition of S5-2 shown in Figure 12b is the sinusoidal excitation with the natural frequency of soil foundation. The curve of motion states presents the elliptic phase trajectory initially. The acceleration auto-power spectrum density curves of Figure 13b show that the transient response decays and steady-state vibration response are forming. The working condition of S10 shown in Figure 12c is the sinusoidal excitation with the system’s natural frequency. The curve of motion states presents the elliptic phase trajectory, and Figure 12c and Figure 13c show that the transient response disappears and the system is in the steady-state vibration response stage.

We can conclude from Figure 12a–c that motion states of adjacent measuring points at pile cap-soil interface are consistent basically on the sinusoidal excitation condition of S5-1 after several working conditions of trim levels of vibration. Namely, motor coordination mechanisms arise and motion states at the interface tend to be consistent after vibrations of several trim levels, whether the transient vibration response exists or the steady-state vibration response form. In short, motion characteristics of the SSI classical damping system can be formed at the pile cap-soil interface after vibrations of several trim levels.

Motion states of “adjacent” measuring points at pile bottom-soil interface on conditions of sinusoidal excitation are investigated, corresponding motion trajectories are shown in Figure 14 and corresponding acceleration auto-power spectrum density curves are shown in Figure 15.

We can see from Figure 14a and Figure 15a that the response at the pile bottom-soil interface is not totally a steady-state vibration response, and there is also a transient response with damped free vibration. The transient response disappears, and the system is basically in the steady-state vibration response stage, shown in Figure 14b and Figure 15b.

In Figure 12a, we can see motion states of “adjacent” points at pile bottom-soil interface by the S5-1 sinusoidal excitation is still uncoordinated. There is still some relative displacement and the relative velocity between the contact surface, different from states of points at the pile cap-soil interface. Furthermore, it results from a long-distance of “adjacent” points at the pile bottom-soil interface than those at the pile cap-soil interface, shown as Figure 8b. However, it does not affect the rules shown in the relative phase plan of the “adjacent” measuring point at the pile bottom-soil interface, roughly the same as the motion laws of points at the pile cap-soil interface. Namely, after several working conditions with small amplitude, motion states of measuring “adjacent” points are coordinated basically on the condition of S10 sinusoidal excitation, and characteristics of motion states at pile bottom-soil interface are consistent with SSI classical damping system.

4.3. Evaluation of SSI Damping System Based on Interface Motion State

The pile-soil interface is the crucial point for the classification of the SSI damping system. The coordination degree of adjacent points at the interaction surface can reflect the adjusting ability of motor coordination of the interface. It can be the direct criterion for the type of damping system. From the test data analysis in Section 4.2, we can determine that motion states at the interface of the SSI system are uncoordinated on the S5 working condition, and the system is a non-classical damping system; in contrast, motion states at the interface are coordinated on the working condition of S10 and the system is classical damping system. Then, the identification method of the damping system category mentioned in Section 2 of this paper is used for damping system identification on conditions of S5 and S10, respectively. Moreover, the validity of this identification method can be verified compared with the conclusion of Section 4.2.

Sidewalls of piles are the primary interaction interfaces of the SSI system. There are two measuring points of soil and structure, respectively, namely points AX34 and AX3 at the structure interface and points AX5 and AX23 for that of soil. The position coordinates of AX34 and AX5, and coordinates of AX3 and AX23, are the same, respectively. The relative position coordinates are the same, shown as Equation (30).

$$\left|{\overline{x}}^{\mathrm{pile}}\right|=\left|{\overline{x}}^{\mathrm{soil}}\right|$$

(30)

Displacements of relatively synthetic motions of two groups are the same because of the restraint between pile and soil, shown as Equation (31):

$$\left|{\overline{u}}^{\mathrm{pile}}\right|=\left|{\overline{u}}^{\mathrm{soil}}\right|$$

(31)

Then

$$\frac{\left|{\overline{u}}^{\mathrm{Pile}}\right|}{\left|{\overline{x}}^{\mathrm{Pile}}\right|}=\frac{\left|{\overline{u}}^{\mathrm{Soil}}\right|}{\left|{\overline{x}}^{\mathrm{Soil}}\right|}$$

(32)

Therefore, displacement vectors of relatively synthetic motions of two groups are collinear in the diagram of the synthesis motion law, and the corresponding synthetic motion law satisfies Equation (18). Relative motion states at the soil-structure interaction interface are investigated under S5 and S10 working conditions.

The relative displacement, velocity and acceleration of two measuring points of the pile are calculated through Equation (33).

$$\{\begin{array}{l}{u^{(1)}|}_{x=l_1}=u_{\mathrm{AX}34}-u_{\mathrm{AX}3}\\ {{\dot{u}}^{(1)}|}_{x=l_1}={\dot{u}}_{\mathrm{AX}34}-{\dot{u}}_{\mathrm{AX}3}\\ {{\ddot{u}}^{(1)}|}_{x=l_1}={\ddot{u}}_{\mathrm{AX}34}-{\ddot{u}}_{\mathrm{AX}3}\end{array}$$

(33)

The relative displacement, velocity and acceleration of two measuring points of soil are calculated through Equation (34).

$$\{\begin{array}{l}{u^{(2)}|}_{x=l_1}=u_{\mathrm{AX}5}-u_{\mathrm{AX}23}\\ {{\dot{u}}^{(2)}|}_{x=l_1}={\dot{u}}_{\mathrm{AX}5}-{\dot{u}}_{\mathrm{AX}23}\\ {{\ddot{u}}^{(2)}|}_{x=l_1}={\ddot{u}}_{\mathrm{AX}5}-{\ddot{u}}_{\mathrm{AX}23}\end{array}$$

(34)

Amplitude-frequency curves of relative acceleration of soil and pile are shown in Figure 16, respectively. Furthermore, the working condition of Figure 16a is S5, and Figure 16b is S10. The frequency of relative motions of two groups are not the same on the S5 condition, namely motion states of the system cannot keep coordinated; in contrast, the frequency of relative motions of two groups are the same on the S10 condition, and motion states coordinate well.

In addition, the system is a classical damping system as soon as the relative motion states between soil and pile satisfy the Equation (29), otherwise the system is a non-classical damping system. The coupling strength of motion states at the interaction interface is judged by the changing regularity of relative motion characteristics $\frac{{\ddot{u}}^{(i)}}{{\dot{u}}^{(i)}}$ at the interface of soil and pile in the time domain. The difference function of the ratio of relative acceleration and relative velocity $(\frac{{\ddot{u}}^{(1)}}{{\dot{u}}^{(1)}}-\frac{{\ddot{u}}^{(2)}}{{\dot{u}}^{(2)}})$ is chosen as the research object for comparison purposes and the result is shown as Figure 17.

Red dots represent the difference function of motion states on condition S10 and black dots represent condition S5 in Figure 17. We can see that red dots stay close to zero as time goes on, which means relatively synthetic motion states of two groups of measuring points always satisfy Equation (29), on the contrast, the high discreteness of black dots shows the un-coordination of relatively synthetic motion states at the same interface. According to the analysis conclusion in Section 4.2, motion states at the soil-structure interface stays consistent in condition S10 and the system can be treated as a classical damping system. Conversely, the condition S5 is non-classical damping because the relative synthetic motion states on soil-structure interface cannot be coordinated and unified.

5. Conclusions

Motion states of different damping systems are studied with the help of discontinuous dynamic analysis in this paper. The identification method based on relatively synthetic motion states at the interface is proposed by analyzing the coupling property of relative motion states at the interface. In addition, the large shaking table test of SSI damping system under the vibration of increasing dynamic load is done. The consistency of motion rules of different parts and the continuity of interface motion state for SSI system are investigated from linear to nonlinear development process.

Theoretical and experimental studies show that the boundary of movement in discontinuous dynamics theory is critical for identifying classical damping. It is the easiest to produce at the interface of different materials or fracture surfaces. In addition, the coupling of relatively synthetic motion states at the interface is the direct criterion for the class of damping systems. The identification method of a classical damping system based on motion states is proposed in this paper. It can reflect dynamic characteristics of multi-material systems under the linear and nonlinear processes of dynamic load soundly. This can be a reference for selecting nonlinear dynamic analysis methods of complex structural systems and further optimization of structural vibration control.