Time Domain Damage Identification Approach of the Newly Hollow Floor Structure with TRMD

Abstract

A tuned rolling mass damper system (TRMD) is newly combined with structural hollow slabs to act as an ensemble passive damping device to mitigate structural responses. The main advantage of this controlled configuration lies in its capacity to maintain architectural integrity. To investigate damage identification of this controlled structure with TRMD, a time domain identification approach with sensitivity-based model updating is studied in this paper. The simplified model and motion equation of a controlled structure with TRMD are derived considering the interaction force between the main structure and the damper. Furthermore, the dynamic response and dynamic response derivatives with respect to elemental variations in the controlled structure are analyzed to identify damage by minimizing the response difference between the intact state and the damaged state in the sensitivity-based updating procedure. A numerical example for a six-story frame structure with a TRMD on the top is employed to verify that not only the structural dynamic responses are controlled effectively by the rolling ball damper subjected to the earthquake excitation, but also damage location and extent can be identified accurately by the proposed method even with measurement noise considered. The shaking table test of a single-story frame equipped with TRMD is additionally carried out to validate the accuracy of the proposed structural damage identification method in the laboratory.

1. Introduction

Recently, a structure system with a reinforced concrete hollow floor has been widely used in practical engineering [1,2]. It is well known that structural control techniques are intended to mitigate structure responses under ambient dynamic excitation in consideration of the safety and serviceability issues of structures [3,4,5,6]. Based on the cavity characteristics of the reinforced concrete hollow floor structural system, which is widely used in buildings at present, a tuned rolling mass damper that could be placed in the hollow floor has been proposed. The concept of the tuned rolling mass damper was first introduced as a ball vibration absorber for horizontal movement by Pirner [7]. Then, Obata and Shimazaki [8] proposed a new tuned rotary mass damper that consists of a container (outer shell) and a cylinder-type weight (rotary mass) as a passive-type vibration control system and conducted a field experiment on a lamppost for a highway bridge [9]. The tuned rolling mass damper (TRMD) proposed by Li et al. [10,11] in this paper consists of a ball rolling along an arch path located in hollow modules to absorb the structural kinetic and potential energy when subjected to ambient excitations. Furthermore, compared with the traditional vibration control system—the tuned mass damper (TMD) as an additional energy absorbing device [12]—the most significant advantage of the installation of the TRMD in hollow modules embedded in slabs lies in its capacity to maintain architectural integrity. Li and his research team [10,11] carried out theoretical and experimental research into the hollow floor structure with TRMD. The advantage of the newly combined TRMD and the hollow floor structural system, as shown in Figure 1, is that the damping device TRMD not only suppresses structural responses but also effectively solves the conflict between the setting of the control device and the function of the building, which does not occupy the use space of the building or affects the use function of the structure.

However, newly hollow floor structures with TRMD will also inevitably suffer different degrees of damage under the combined effects of natural disasters, human factors, and the aging of building materials during their long-term service. Therefore, damage identification and structural health monitoring of this newly controlled structure deserve our attention. Vibration-based structural damage identification methods have been the research focus in the field of structural damage identification [13]. The basic concept of vibration-based structural damage identification is to establish the function relation between structural response indicators and the inherent physical characteristics of the structure. Compared with the frequency domain vibration-based structural damage identification method, which adopts frequency [14], mode shapes [15], and frequency response function [16] as the structural indicators, the time domain damage identification method, which employs dynamic response as the structural indicator [17,18,19,20], has advantages for the newly hollow floor structure with TRMD. Firstly, because the time domain damage identification method directly uses measured vibration data, the signal process and transformation between the time domain and frequency domain avoid leading to no data conversion, signal distortion, or error. In addition, the dynamic response data that are employed for damage identification can be obtained directly from the sensing system and data acquisition system in the structural control system. Therefore, it is quite promising to combine with the structural control system and structural health monitoring system to control and detect the safety of the structure.

This paper presents a time domain method for structural damage detection of controlled building structures equipped with TRMD. Firstly, the simplified model and motion equation of a controlled structure with TRMD are derived considering the interaction force between the main structure and the damper. Furthermore, the dynamic response and the dynamic response derivatives with respect to the elemental variations in the controlled structure are analyzed to identify damage by minimizing the response difference between the intact state and the damaged state in the sensitivity-based updating procedure. In addition, a numerical example for a six-story frame structure with a TRMD on the top is employed to verify that damage location and extent can be identified accurately by the proposed method, even with measurement noise considered. The shaking table test of single-story frame equipped with TRMD is additionally carried out to validate the accuracy of the proposed structural damage identification method in the laboratory.

2. Theoretical Formulation

2.1. Simplified Model and Motion Equation of a Controlled Structure with TRMD

The model of the single-degree-of-freedom structure with TRMD is shown in Figure 2, where M, K, and C are the mass, stiffness, and damping of the main structure, respectively; m is the ball mass of the TRMD oscillator; and F(t) is the external ambient excitation. In the equation $F\left(t\right)=(M+x){\ddot{x}}_g$, ${\ddot{x}}_g$ is the acceleration of the external excitation. The motion model of the TRMD oscillator is shown in Figure 3, where R and r are the radii of the arc path and oscillator, respectively, θ and ψ are the rotation angles of the oscillator with respect to the center of the path and of the oscillator ball itself, and $\omega =\dot{\psi }$ represents the angular velocity of the oscillator. The equations for the motion of the controlled system with TRMD can be derived by using Lagrange’s equation, and a detailed derivation of the equation is proposed by the authors [10]:

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{\partial T}{\partial {\dot{q}}_i})-\frac{\partial T}{\partial q_i}+\frac{\partial V}{\partial q_i}=Q_i^{nc},i=1,2$$

(1)

where T and V donate the kinetic energy and potential energy of the controlled structure, respectively, q_i is the ith generalized coordinate, and Q_i^nc is the nonconservative force corresponding to the generalized coordinate q_i. The kinetic energy of the controlled structure T includes the kinetic energy of the main structure, the translational motion of the TRMD oscillator, and the kinetic energy of rotation around its own spherical center. Therefore, the kinetic energy and potential energy of the controlled structure can be expressed as the following:

$$T=\frac{1}{2}M{\dot{x}}^2+\frac{1}{2}m{(\dot{x}+\rho \dot{\theta }\cos \theta )}^2+\frac{1}{2}m{(\rho \dot{\theta }\sin \theta )}^2+\frac{1}{5}m{\left(\rho \dot{\theta }\right)}^2$$

(2)

$$V=\frac{1}{2}K{x}^2+mg\rho (1-\cos \theta )$$

(3)

where x, $\dot{x}$, and $\ddot{x}$ are the displacement, velocity, and acceleration response of the main structure. ρ = R − r represents the radius difference between the arc track and oscillator.

It is assumed that the arc track surface is smooth without considering the rolling friction of the oscillator. Based on the virtual work principle, the virtual work generated by nonconservative forces, including external forces and the damping force, can be expressed as

$$\delta W=(F(t)-C\dot{x})\delta x$$

(4)

Considering the angular motion of the oscillator θ is very small, the motion equation of the TRMD controlled structure can be written as:

$$\left(M+m\right)\ddot{x}+m\rho \ddot{\theta }+C\dot{x}+Kx=F\left(t\right)$$

(5)

$$\ddot{\theta }+\frac{5\ddot{x}}{7\rho }+\frac{5g\theta }{7\rho }=0$$

(6)

Equations (5) and (6) can be rewritten in a more compact form:

$$\left[\begin{array}{cc}M+m& m\rho \\ \frac{5}{7\rho }& 1\end{array}\right]\left\{\begin{array}{c}\ddot{x}\\ \ddot{\theta }\end{array}\right\}+\left[\begin{array}{cc}C& 0\\ 0& 0\end{array}\right]\left\{\begin{array}{c}\dot{x}\\ \dot{\theta }\end{array}\right\}+\left[\begin{array}{cc}K& 0\\ 0& \frac{5g}{7\rho }\end{array}\right]\left\{\begin{array}{c}x\\ \theta \end{array}\right\}=\left\{\begin{array}{c}F\left(t\right)\\ 0\end{array}\right\}$$

(7)

From Equation (6), it is shown that the natural frequency of the oscillator, which can be calculated as ${\omega }_d=\sqrt{\frac{5g}{7\rho }}$, depends on the radius difference between the arced track and the oscillator. In order to obtain the optimal control effect, the natural frequency of the TRMD is tuned near the first modal frequency of the main structure, which is similar to the vibration control of the traditional TMD [21].

2.2. Motion Equation for an MDOF Structure with TRMD

The motion equation for an MDOF structure with TRMD can be written in matrix form as follows:

$${\rm M}\ddot{x}\left(t\right)+C\dot{x}\left(t\right)+Kx\left(t\right)=F_{load}\left(t\right)+Hf\left(t\right)$$

(8)

where $M,\begin{array}{c}\end{array}C$, and $K$ represent the $\left(n+1\right)\times \left(n+1\right)$ mass, damping, and stiffness matrices of the MDOF controlled structure with TRMD, respectively. $\ddot{x}\left(t\right),\dot{x}\left(t\right)$, and $x\left(t\right)$ are the acceleration, velocity, and displacement vectors, respectively. $x\left(t\right)={\left[x_1,x_2,\cdots x_n,\theta \right]}^{\mathrm{T}}$ includes n degrees of freedom in the main structure and one degree of freedom for the tuned oscillator. $F_{load}\left(t\right)$ is the vector of the loading force. $f\left(t\right)=-\frac{5g}{7\rho }\theta$ donates the interaction force between the main structure and the oscillator. H indicates the location of $f\left(t\right)$. In Equation (8), $M,\begin{array}{c}\end{array}C$, and $K$ can be expressed as:

$$M=\left[\begin{array}{cc}{M}_s+\Gamma {\Gamma }^{T}m& \Gamma m\rho \\ {\Gamma }^T\frac{5}{7\rho }& 1\end{array}\right], \begin{array}{c}\end{array} C=\left[\begin{array}{cc}{C}_s& 0\\ 0& 0\end{array}\right], \begin{array}{c}\end{array}K=\left[\begin{array}{cc}{K}_s& 0\\ 0& 0\end{array}\right]$$

(9)

In Equation (9), $M_s,\begin{array}{c}\end{array}{C}_s$, and $K_s$ are the $n\times n$ mass, damping, and stiffness matrices of the main structure, respectively. Considering the TRMD is installed at the top of the structure, $\Gamma ={\left[\underset{\left(n-1\right)}{\underbrace{0,0,\cdots 0}},1\right]}^{\mathrm{T}}$ thus represents the location of the TRMD. The acceleration, velocity, and displacement responses of the MDOF structure with TRMD can be solved by the Newmark method from Equation (8).

2.3. Dynamic Response Sensitivity Analysis

It is assumed that local damage in the structure is donated as a change in the elemental stiffness factors $\Delta \alpha$, and variation in the structural stiffness matrix is represented as $\Delta K={\displaystyle \sum _{i=1}^{Ncal}\Delta {\alpha }_i}{k}_i^e$ (0 < $\Delta \alpha$ < 1). The dynamic response sensitivity matrices of the damaged controlled structure with TMRD are calculated by differentiating Equation (8) as:

$$\frac{\partial M}{\partial {\alpha }_i}\ddot{x}\left(t\right)+M\frac{\partial \ddot{x}\left(t\right)}{\partial {\alpha }_i}+\frac{\partial C}{\partial {\alpha }_i}\dot{x}\left(t\right)+C\frac{\partial \dot{x}\left(t\right)}{\partial {\alpha }_i}+\frac{\partial K}{\partial {\alpha }_i}x\left(t\right)+K\frac{\partial x\left(t\right)}{\partial {\alpha }_i}=\frac{\partial F_{load}\left(t\right)}{\partial {\alpha }_i}+\frac{\partial Hf\left(t\right)}{\partial {\alpha }_i}$$

(10)

Equation (10) can be simplified as follows:

$$M\frac{\partial \ddot{x}\left(t\right)}{\partial {\alpha }_i}+C\frac{\partial \dot{x}\left(t\right)}{\partial {\alpha }_i}+K\frac{\partial x\left(t\right)}{\partial {\alpha }_i}-\frac{\partial Hf\left(t\right)}{\partial {\alpha }_i}=-\frac{\partial K}{\partial {\alpha }_i}x\left(t\right)$$

(11)

Finally, the dynamic response derivatives with respect to the elemental stiffness variation $\frac{\partial \ddot{x}(t)}{\partial {\alpha }_i}$, $\frac{\partial \dot{x}(t)}{\partial {\alpha }_i}$, and $\frac{\partial x(t)}{\partial {\alpha }_i}$ can be calculated by the numerical Newmark method from Equation (11).

2.4. Sensitivity-Based Damage Identification

Structural damages are identified by the sensitivity-based model updating procedure in this paper [19,20]. The objective function is represented as the difference between the measured response and the analytical response from the FE model as follows:

$$\Delta \ddot{x}={\ddot{x}}_M-{\ddot{x}}_A$$

(12)

where ${\ddot{x}}_M$ and ${\ddot{x}}_A$ indicate the measured acceleration and calculated acceleration of the finite element model, respectively. Then, the dynamic response sensitivity-based identification equation is expressed by a first-order Taylor series as follows [22]:

$$\Delta \ddot{x}=S\Delta \alpha =\left[\frac{\partial \ddot{x}}{\partial {\alpha }_1},\frac{\partial \ddot{x}}{\partial {\alpha }_2},\cdots ,\frac{\partial \ddot{x}}{\partial {\alpha }_{Ncal}}\right]\left\{\begin{array}{c}\Delta {\alpha }_1\\ \Delta {\alpha }_2\\ ⋮\\ \Delta {\alpha }_{Ncal}\end{array}\right\}$$

(13)

where $\Delta \ddot{x}$ is the difference between the measured response and analytical response from the finite element model. S is the sensitivity matrix, which represents the first derivative of the dynamic response with respect to the elemental stiffness variables α. If the TRMD controlled structure has local damage, the dynamic response of the controlled structure will change compared with the intact state. The dynamic response difference between the intact and damaged states of the controlled structure is taken as the objective function. During the updating process, the dynamic sensitivity matrix is recalculated based on the iteratively updated structural matrices to make the objective function value reach the specified tolerance. Therefore, local damage, which is expressed as the elemental stiffness variation in the TRMD controlled structure, can be obtained iteratively. The damage identification flow chart based on dynamic response sensitivity analysis for the controlled structure is shown in Figure 4. The proposed time domain damage identification method is promising because the dynamic response sensitivity matrix indicates the searching direction of the optimization process, which improves iteration efficiency and converges speed. In addition, a few measured responses of the controlled structure are needed to identify the location and extent of damage.

3. Numerical Example

A six-story frame structure with a lumped mass model subject to earthquake excitation is taken as an example to verify the accuracy of the proposed identification method. In this numerical example, each lumped mass is selected to be the same at 16.32 × 10³ kg, the stiffness coefficients of each story are assumed to be 13.51, 12.86, 11.58, 9.65, 7.07, and 3.86 × 10³ kN/m, respectively, and the damping coefficients of each story are 27.9, 23.76, 21.39, 17.87, 13.07 and 10.04 kN·s/m. The first six natural frequencies of the controlled structure are 6.28, 15.38, 24.32, 33.23, 42.13, and 51.02 rad/s. In order to achieve the optimum vibration control effect, the natural frequency of the damper needs to be resonant with the natural frequencies of the structure. Therefore, the radius difference between the arc track and oscillator ρ is calculated as 0.178 m.

It is assumed that the oscillator located in the top story of the structure is made of steel with a mass density of 7850 kg/m³. The mass ratio of the tuned rolling mass damper system mass to the total mass of the main structure is selected as 5%, so the oscillator has a physical mass of 395.3 kg and a radius of 0.230 m. The external excitation, in this case, can be written as $F_i=-(M_{ii}+{\displaystyle {\sum }_{j=1}^{n_i}{m}_{ii}}){\ddot{x}}_g(t)$, where ${\ddot{x}}_g(t)$ is the El-Centro earthquake acceleration, with a peak value of 1.37 m/s². Figure 5 shows a comparison of story displacement with and without the TRMD of structures subject to El-Centro excitation. As can be observed, although the effectiveness of the TRMD at controlling the vibration for the peak responses was not significant, the responses after the peak had been reached were reduced to a large degree because of the energy dissipation mechanism of the TRMD.

In order to study the factors affecting the accuracy of the time domain damage identification approach of the controlled structure with TRMD,

Table 1. Four different cases for damage identification.

	Damage Location	Damage Severity	Noise Severity
Case 1	3rd floor	15%	No noise
Case 2	3rd floor	15%	5% noise
Case 3	3rd floor, 5th floor	15%, 10%	No noise
Case 4	3rd floor, 5th floor	15%, 10%	5% noise

shows four different cases of damage identification considering damage location, damage severity, and noise severity. It is shown in Table 1 that the stiffness of the 3rd floor is reduced by 15% in Case 1 and Case 2, and the stiffness of the 3rd floor and 5th floor are respectively reduced by 15% and 10% in Case 3 and Case 4. In Case 1 and Case 3, the measurement responses with no noise are considered, and 5% measurement noise is considered in Case 2 and Case 4. Considering the acceleration responses of the controlled structure with the TRMD directly as the damage identification index, the sensitivity of the acceleration response with respect to elemental stiffness factors is employed to identify local damage by minimizing the difference between the intact state response and the damage state response. In this numerical example, the dynamic acceleration response difference at the 3rd floor is selected as the objective function for damage identification. Figure 6 shows the dynamic acceleration response at the 3rd floor for the intact state and the damage state of Case 1.

Local damages in the six-story frame structure are identified by the iterative model updating technique based on the dynamic response sensitivity illustrated in the previous section. Figure 7 shows the identified damage results in the four different cases. It is shown that the identified damage location and extent are consistent with real damages in four cases from Figure 7a–d. The identified stiffness and stiffness variation ratio of each story in the four damage cases are listed in

Table 2. The identified stiffness and stiffness variation ratio of each story in four damage cases.

Storey Stiffness	Intact State	Case 1		Case 2		Case 3		Case 4
	Stiffness	Stiffness	Stiffness Variation Ratio	Stiffness	Stiffness Variation Ratio	Stiffness	Stiffness Variation Ratio	Stiffness	Stiffness Variation Ratio
	(N/m)	(N/m)		(N/m)		(N/m)		(N/m)
k1	13,510,000	13,502,578	−0.05%	13,442,643	−0.50%	13,499,127	−0.08%	13,512,298	0.02%
k2	12,860,000	12,889,338	0.23%	12,922,290	0.48%	12,877,745	0.14%	12,827,860	−0.25%
k3	11,580,000	9,861,830	−14.84%	9,870,298	−14.76%	9,859,303	−14.86%	9866889	−14.79%
k4	9,650,000	9,637,782	−0.13%	9,646,141	−0.04%	9,651,837	0.02%	9,662,867	0.13%
k5	7,070,000	7,065,124	−0.07%	7,048,940	−0.30%	6,367,359	−9.94%	6384208	−9.70%
k6	3,860,000	3,864,854	0.13%	3,876,070	0.42%	3,862,899	0.08%	3,857,500	−0.06%

. It is shown in Table 2 that the measurement responses without noise give better accuracy damage identification results with a relative difference of 0.23% in Case 1. In Case 2, the location and extent of the damage can also be identified with a relative difference of 0.50% under the measurement responses with 5% noise. Furthermore, Table 2 shows that the stiffness of the 3rd floor and 5th floor are respectively reduced by 14.86% and 9.94%, which is consistent with the real damage location and extent in Case 3. In Case 4, when the measurement noise is considered, the relative differences are a little larger than those without measurement noise. However, the identified damage results are not obviously sensitive to the measurement noise by the proposed method.

4. Experimental Validation

A single-story frame with the damping device TRMD at the top of the frame is conducted in the laboratory to verify the proposed method, as shown in Figure 8. The two stainless steel sheets and one aluminum alloy plate are rigidly connected as the model structure. Two stainless steel sheets, which are modeled as the frame columns, are 500 mm high, 100 mm wide, and 1 mm thick, and the aluminum alloy plate, which is modeled as the frame beam, is 300 mm long, 100 mm wide, and 10 mm thick. The natural vibration frequency of the model is measured as 1.465 Hz, and the damping ratio is 0.057. The specific parameters of the experiment model are shown in

Table 3. The parameters of the model.

Size of Stainless Steel Sheet/mm	Mass of Stainless Steel Sheet/g	Size of Aluminum Alloy Plate/mm	Mass of Aluminum Alloy Plate/g	Natural Frequency/Hz	Damping Ratio
500 × 100 × 1	333 (single sheet)	300 × 100 × 10	797	1.465	0.057

. The arched path of the TRMD is made of aluminum alloy, and the rolling ball is made of cast iron materials. The mass ratio between the ball and the main structure is selected as 0.03 so that the natural frequency of the damper will be resonant with the natural frequency of the main structure. The parameters of the damping device TRMD are listed in

Table 4. The parameters of TRMD.

Arched Path Radius R/mm	Radius of the Ball r/mm	Mass of the Ball/g
110	11	50

.

The model was fixed on the shaking table made by Canada Quanser company. Two accelerometers were mounted at the top of the frame and on the shaking table to record the acceleration response. Furthermore, two laser displacement meters were also mounted at the top of the frame and the shaking table to record the displacement response. The model was excited by the El-Centro wave with a peak acceleration of 0.2 g.

The structure was tested in three states: the uncontrolled intact state (Case 1), the controlled intact state (Case 2), and the controlled damaged state (Case 3), which are given in

Table 5. Configuration of the different cases.

	Case 1	Case 2	Case 3
Configuration	Uncontrolled	Controlled with TRMD	Controlled with TRMD
	Intact	Intact	d = 15 mm b = 100 mm

. The uncontrolled structure and controlled structure with TRMD were first tested comparatively without any damage. Afterward, the frame column was cut at the middle of the two stainless steel sheets with a depth of d = 15 mm in Case 3. The width of the cuts is b = 100 mm in this damage case. One typical cut is demonstrated in Figure 9.

Figure 10 shows the displacement response comparison between the uncontrolled and the controlled structure in the intact state. It is shown that the displacement response can be obviously reduced, especially in the latter part of the earthquake wave for the controlled structure with TRMD.

In Case 3, structural damage identification is performed by the dynamic response sensitivity-based model updating method. The displacement response difference at the top of the structure is selected as the objective function. The local damage is identified by iteratively calculating the acceleration response difference and the response sensitivity. The equivalent stiffness reduction of the frame column is identified as about 8.3% in Case 3 when the depth of the cut equals 15 mm. Based on the displacement-based finite element method, the theoretical equivalent stiffness reduction of the damaged frame column is derived as 5% in the case of cut depth d = 15 mm. It is illustrated that the location and extent of the damage can be identified accurately for the laboratory-controlled frame by the proposed method.

5. Conclusions

This paper presents a time domain damage identification approach for a controlled structure with TRMD. The equation of motion of a controlled structure with TRMD is derived. Afterward, the dynamic response and dynamic response derivatives with respect to the elemental variations are calculated by the Newmark method. A sensitivity-based model updating approach is employed to identify local damage by minimizing the acceleration response difference between the intact state and the damaged state.

A numerical example for a six-story frame structure with a TRMD on the top is employed to validate the accuracy of the proposed method. It is shown that each floor displacement of the controlled structure is reduced in the earthquake excitation because of the damping effect of the TRMD. The numerical results have shown that the damage location and extent of the controlled structure in a one-damage case or multi-damage case can be identified accurately, even with 5% measurement noise considered. In addition, not all floor responses are required for the sensitivity-based updating procedure, and only a few sensors are needed in this proposed method. The shaking table test of a single-story frame equipped with TRMD is carried out to validate the accuracy of the proposed structural damage identification method in the laboratory. The proposed method can be used to detect the damage degrees and locations for the newly hollow floor structure with TRMD only by the measured controlled displacement.