**1. Introduction**

With the tendency to use longer spans, the damping of modern suspension bridges is seriously reduced. Complex vibration problems follow, such as wind-induced vibration, human-induced vibration, cable-structure interactions and flutter instabilities [1–4]. It is admitted that the oversensitivity to dynamic excitation of suspension bridges is associated with the very low structural damping in the global bridge modes [5,6]. Therefore, the dissipation of the vibration energy generated by the dynamic loadings is a central issue in their design. At present, the use of damping systems such as tuned mass damper (TMD) [7], viscous dampers [8,9], or active tendon control [10] is a classical way to alleviate the vibrations in structures. This study aims at the design of multi-degree of freedom TMD for vibration damping of a suspension bridge deck.

Considering their simplicity and effectiveness, tuned mass dampers have been widely used in bridges such as the London Millenium bridge, for damping both lateral and vertical vibrations of the deck. Since Frahm et al. proposed the fundamental theory, TMDs have seen numerous applications in civil engineering [11–13]. Thus, Ben Mekki and Bourquin [14,15] proposed a new semi-active electromagnetic TMD of pendulum type to damp the torsional mode of an evolving bridge mock-up. Their studies showed that the tuned pendulum damper (TPD) is very effective in vibration damping, qualitatively and quantitatively confirming the theoretical predictions. However, the TMDs can only control single mode and easy detune, which limits their further development. For the robustness of vibration control and targeting several vibration modes, the multiple tuned mass dampers (MTMDs) has been proposed, and its performance is more e ffective as compared to the single TMD. The superior effectiveness of the MTMDs is able to control almost any type of vibration in civil structures [16]. The MTMDs is used to damp suspension bridges for several purposes. In some studies, the MTMDs are used to the suppression of bu ffeting, flutter or increasing the critical flutter wind speed [17,18]. Other studies consider MTMDs for alleviating pedestrian- and jogger-induced vibration [19–22] or tra ffic-induced vibration [23–26].

Generally, the weight of a TMD is limited to 1–3% of the structure weight. Hence, as the number of targeted modes increases, a large number of TMDs will increase the burden on the primary system and limit the damping performance (called weight penalty). To avoid such a penalty, in our previous study [27], we proposed to design a two-degree of freedom TMD, where the original mass of TMD is redistributed in such a way that the TMD has a bending mode and a torsional mode. In this design the resonance frequencies and the modal damping of the two modes can be tuned independently. In addition, Zuo and Nayfeh have proposed a multiple degrees of freedom TMD (MDOFs TMD), and experimentally demonstrated that the MDOFs TMD can damp six modes of the primary structure. They also showed that a MDOFs TMD can be used to attain better vibration suppression for single mode vibration of a primary structure [28]. Jang et al. described a novel method for selecting the parameters of a 2DOFs TMD with translational and rotational degrees of freedom [29]. Ma and Yang et al. presented a design of a multi-DOFs TMD to alleviate the dominant mode of the work piece/fixture assembly in milling [30–32].

As the DOF increases, by only selecting the appropriate DOFs and tuned frequencies, the TMDs can reach the best vibration control of the primary structure. Therefore, designing a TMD with expected DOFs and natural frequencies becomes an urgen<sup>t</sup> problem. Unfortunately, due to its complicated structure and easily detuning, the further study on the implementation of MDOFs TMD is rare.

In this paper, we propose a synthetic approach based on both the graphical approach and parameterized compliance for the concrete design of the TMDs with the expected DOFs and we verify their feasibility and performance by numerically and experimentally way on a laboratory suspension bridge mock-up. The paper is organized as follows: Section 2 describes the vibration characteristics of the bridge mock-up and builds a concept model of bridge with two 2DOFs TMDs. Based on the equations of motion, the decentralized control technique is directly used to optimize the sti ffness and damping coe fficients of the springs and dampers to obtain the optimum frequency ratios in Section 3. Section 4 presents the detailed design process of the 2DOFs TMDs based on the graphical approach and compliance analysis. Section 5 mainly focuses on evaluating the damping performance and verifying the proposed design method. Finally, findings and conclusions of the study are summarized at the end.

#### **2. Formulation of the Bridge–TMD System**

Our goal is to use two 2DOFs TMDs to control the first four vibration modes of the suspension bridge simultaneously. The 2DOFs TMD is decoupled in the physical coordinates, their mode shapes follow the physical coordinate of the mock-up, and the corresponding resonance frequencies can be tuned independently to match the desired design. The suspension bridge mock-up and its finite element modelling are detailed described in our previous studies [27,33,34]. Here, this paper only lists the vibration characteristics of the bridge mock-up, as shown in Table 1.

Since the tuning TMDs becomes increasingly complex, we cast the parameter optimization of the 2DOFs TMDs as a control problem with decentralized static output feedback for minimizing the response of the bridge system. This method has been used successfully for a single mode vibration control of a MDOFs TMD by Zuo and Nayfeh [28]. The concept model of the 2DOFs TMDs is to take the springs as local feedback elements of relative displacements and the dampers as local feedback elements of relative velocities, as shown in Figure 1.


**Table 1.** The numerical and experimental natural frequencies and mode shapes of the bridge mock-up.

B stands for bending mode, T stands for torsional mode.

**Figure 1.** Concept model of the 2DOF TMDs: the bridge system equipped with two TMDs.

In this way, the role of the springs and dampers can be replaced by a control force vector, where the control gain is composed of the spring stiffness and damping coefficients (*ki* and *ci* for *i* = 1, 2, ... , 6). The mass matrix *Mn*×*n*, stiffness matrix *Kn*×*n* and viscous damping matrix *Cn*×*n* is extracted from the numerical model of the suspension bridge mock up, respectively. The 2DOFs TMD has two planar degrees of freedom, translation *x*1 (*x*2) and rotation θ1 (θ2). Its mass is *md*1 (*md*2) and the rotational inertia about its center of mass is *Id* = *md*ρ2, where ρ is the radius of gyration.

The 2DOFs TMD is connected to the primary system at distances *d*1 (*d*2) from its center of mass via dashpots and springs. Therefore, the control force vector [*<sup>u</sup>*1, *u*2, ... , *u*6] in this case are given by:

$$\mu\_1 = k\_1 \left(\mathbf{x}\_1 - \mathbf{B}\_2^T \mathbf{X} - \theta\_1 d\_1\right) + c\_1 \left(\dot{\mathbf{x}}\_1 - \mathbf{B}\_2^T \dot{\mathbf{X}} - \dot{\theta}\_1 d\_1\right) \tag{1}$$

$$\mu\_2 = k\_2(\mathbf{x}\_1 - \mathbf{B}\_2^T \mathbf{X} + \theta\_1 d\_1) + c\_2(\dot{\mathbf{x}}\_1 - \mathbf{B}\_2^T \dot{\mathbf{X}} + \dot{\theta}\_1 d\_1),\tag{2}$$

$$
\mu\_3 = k\_3(\mathbf{x}\_1 - \mathbf{B}\_2^T \mathbf{X}) + c\_3(\dot{\mathbf{x}}\_1 - \mathbf{B}\_2^T \dot{\mathbf{X}}),
\tag{3}
$$

$$u\_4 = k\_4 \left(\mathbf{x}\_2 - \mathbf{B}\_3^T \mathbf{X} - \partial\_2 d\_2\right) + c\_4 \left(\dot{\mathbf{x}}\_2 - \mathbf{B}\_3^T \dot{\mathbf{X}} - \dot{\partial}\_2 d\_2\right),\tag{4}$$

$$\mu\_5 = k\_5(\mathbf{x}\_2 - \mathbf{B}\_3^T \mathbf{X} + \theta\_2 d\_2) + c\_5(\dot{\mathbf{x}}\_2 - \mathbf{B}\_3^T \dot{\mathbf{X}} + \dot{\theta}\_2 d\_2),\tag{5}$$

$$\mu\_2 = k\_2(\mathbf{x}\_1 - \mathbf{B}\_2^T \mathbf{X} + \theta\_1 d\_1) + c\_2(\dot{\mathbf{x}}\_1 - \mathbf{B}\_2^T \dot{\mathbf{X}} + \dot{\theta}\_1 d\_1),\tag{6}$$

where *X* is the vector of global coordinates of the finite element model, *B*2 and *B*3 are the input vector of this two TMDs, respectively.

The equations can govern the vibration of the coupled system can be decomposed into:

..

$$m\_{d1}\mathbf{x}\_1 = -\mathbf{u}\_1 - \mathbf{u}\_2 - \mathbf{u}\_3.\tag{7}$$

$$-m\_{d2}\bar{\mathbf{x}}\_2 = -\mathbf{u}\_4 - \mathbf{u}\_5 - \mathbf{u}\_6,\tag{8}$$

$$I\_{d1}\ddot{\theta}\_1 = \mu\_1 d\_1 - \mu\_2 d\_1 \tag{9}$$

$$I\_{d2}\ddot{\theta}\_2 = \mathfrak{u}\_4 d\_2 - \mathfrak{u}\_5 d\_2. \tag{10}$$

The governing equations can then be written as

> ..

$$\mathbf{M}\ddot{\mathbf{X}} + \mathbf{C}\dot{\mathbf{X}} + \mathbf{K}\mathbf{X} = \mathbf{B}\_1 f\_d + \mathbf{B}\_2(\boldsymbol{u}\_1 + \boldsymbol{u}\_2 + \boldsymbol{u}\_3) + \mathbf{B}\_3(\boldsymbol{u}\_4 + \boldsymbol{u}\_5 + \boldsymbol{u}\_6),\tag{11}$$

.

where *fd* is the external disturbances, *B*1 is the input vector of the external disturbances. We can express Equations (7)–(11) in matrix form as:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *M* 0 0 00 0 *md*1 0 00 0 0 *md*2 0 0 00 0 *Id*1 0 00 00 *Id*2 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *X*..*x*1..*x*2..θ1 ..θ2 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *C* 0000 00000 00000 00000 00000 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *X*.*x*1.*x*2.θ1.θ2 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *K* 0000 0 0000 0 0000 0 0000 0 0000 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *Xx*1 *x*2 θ1 θ2 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *B*10000 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ *fd* + ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *B*2 *B*2 *B*2 *B*3 *B*3 *B*3 −1 −1 −10 0 0 000 −1 −1 −1 *d*1 −*d*<sup>1</sup> 00 0 0 000 *d*2 −*d*<sup>2</sup> 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *u*1 *u*2 *u*3 *u*4 *u*5 *u*6 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , (12)

or

$$\mathcal{M}\_p \ddot{p} + \mathcal{C}\_p \dot{p} + \mathcal{K}\_p p = \mathcal{B}\_f f\_d + \mathcal{B}\_u u\_\prime \tag{13}$$

where *p* = [*X*, *x*1, *x*2, θ1, θ2]<sup>T</sup> and *u* = [*<sup>u</sup>*1, *u*2, ... , *u*6] and T denote the complex conjugate matrix transpose. The matrices *<sup>M</sup>p*, *<sup>C</sup>p*, *<sup>K</sup>p*, *<sup>B</sup>f*, and *Bu* can be obtained from Equation (12) directly.

Defining the state variables of the system as:

$$\mathbf{x} = \begin{bmatrix} p \\ \dot{p} \end{bmatrix} \tag{14}$$

The governing equations are written in first-order form as:

$$
\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\_{11}\mathbf{w} + \mathbf{B}\_{12}\mathbf{u},\tag{15}
$$

where *w* = *fd* and:

$$A = \begin{bmatrix} 0 & I \\ -\mathcal{M}\_p^{-1} & \mathcal{K}\_p \end{bmatrix} \quad \mathcal{B}\_{11} = \begin{bmatrix} 0 \\ \mathcal{M}\_p^{-1} \mathcal{B}\_f \end{bmatrix} \quad \mathcal{B}\_{12} = \begin{bmatrix} 0 \\ \mathcal{M}\_p^{-1} \mathcal{B}\_u \end{bmatrix} \tag{16}$$

The cost output can be taken as the absolute or relative displacement, velocity, or acceleration of the primary system, which can be expressed in the form:

$$\mathbf{z} = \mathbf{C}\_1 \mathbf{x} + D\_{11} \mathbf{w} + D\_{12} \mathbf{u}\_\prime \tag{17}$$

For the displacement response of the primary system, the cost output can be written as:

$$
\mathbf{z} = \mathbf{X} = \mathbf{C}\_1 \mathbf{x},\tag{18}
$$

where:

$$\mathbf{C}\_{1} = \begin{bmatrix} I\_{n \times n} & \mathbf{O}\_{n \times 4} & \mathbf{O}\_{n \times n} & \mathbf{O}\_{n \times 4} \end{bmatrix} \tag{19}$$

To complete the state-space description, we rewrite the control force given by Equations (1)–(6) as a static feedback gain *F* multiplied by the "measurement output" *y*:

$$\mathbf{u} = \mathbf{F}\mathbf{y} = \begin{bmatrix} k\_1 & c\_1 \\ & & k\_2 & c\_2 \\ & & & \dots & \dots \\ & & & & k\_6 & c\_6 \end{bmatrix} \mathbf{y}\_{\prime} \tag{20}$$

where *y* is given by:

$$\begin{aligned} y &= [\mathbf{x}\_1 - \mathbf{B}\_2^T \mathbf{X} - \partial\_1 d\_1, \dot{\mathbf{x}}\_1 - \mathbf{B}\_2^T \dot{\mathbf{X}} - \dot{\partial}\_1 d\_1, \mathbf{x}\_1 - \mathbf{B}\_2^T \mathbf{X} + \partial\_1 d\_1, \dot{\mathbf{x}}\_1 - \mathbf{B}\_2^T \dot{\mathbf{X}} + \dot{\partial}\_1 d\_1] \\ &\ge [\mathbf{x}\_1 - \mathbf{B}\_2^T \dot{\mathbf{X}}, \dot{\mathbf{x}}\_1 - \mathbf{B}\_2^T \dot{\mathbf{X}}, \dot{\mathbf{x}}\_2 - \mathbf{B}\_3^T \dot{\mathbf{X}} - \partial\_2 d\_2, \dot{\mathbf{x}}\_2 - \mathbf{B}\_3^T \dot{\mathbf{X}} - \dot{\partial}\_2 d\_2, \mathbf{x}\_2 - \mathbf{B}\_3^T \dot{\mathbf{X}} + \dot{\partial}\_2 d\_2] \\ &\dot{\mathbf{x}}\_2 - \mathbf{B}\_3^T \dot{\mathbf{X}} + \dot{\partial}\_2 d\_2, \mathbf{x}\_2 - \mathbf{B}\_3^T \dot{\mathbf{X}}, \dot{\mathbf{x}}\_2 - \mathbf{B}\_3^T \dot{\mathbf{X}}]^T = \mathbf{C}\_2 \mathbf{x} + \mathbf{D}\_{21} \mathbf{w} + \mathbf{D}\_{22} \mathbf{u} \end{aligned} \tag{21}$$

where *C*2 can be obtained from the definition of the state given by Equation (14) and the matrices *D*21 = 0 and *D*22 = 0. Equations (15), (17), and (21) cast the design of the two 2DOFs TMDs system as a decentralized control problem in the block diagram of Figure 2.

**Figure 2.** Block diagram of the bridge—2DOFs TMDs system with decentralized control.

#### **3. Numerical Optimization and Simulation**

According to above equations, the decentralized control techniques can be directly used to optimize the damping and stiffness coefficients of the dampers and springs to achieve performance (measured by *z*) under the disturbance *w*. The minimax numerical method [28] is utilized to minimize the response magnitude of the primary system.

## *3.1. Optimization Criteria*

Traditionally, based on the Den Hartog method, the optimized frequency ratio and TMD damping ratio are aimed to minimize the structural response by the minimization of the structural dynamic magnification function. This classic procedure consists of two separate steps: tuning of the frequency of the damper, and selection of the optimal level of the TMD damping ratio. This classic procedure has been used in our previous research [27]. The goal of this study is to design the parameters (*ki* and *ci* for *i* = 1, 2, ... , 6) in order to determine the optimum frequency ratio ν*j* (*j* = 1, ... , 4) and the optimum TMD damping ratio ξ*j* for minimizing the response of the bridge system.

Due to the fact that the maximum amplitude of the bridge system should be controlled in a reasonable range, herein the damping performance of TMDs is evaluated by the *H*∞ criterion. *H* is the selected FRF value of the bridge system under the excitation *fd*. The value range of *Hj* is constrained by the value range of <sup>ω</sup>*j* (<sup>ω</sup>*j* [0.7<sup>ω</sup>s*j*,1.3<sup>ω</sup>s*j*]). The goal of the optimization is to minimize the maximum value of each value range of *Hj*. χ = [*k*1, *c*2, ... , *k*6, *c*6]<sup>T</sup> is selected as the design parameter vector of the TMDs. The optimization problem can be written as:

$$\begin{array}{ll}\text{Find:} & \chi = \left[k\_1, c\_1, k\_2, c\_2, k\_3, c\_3, k\_4, c\_4, k\_5, c\_5, k\_6, c\_6\right]^\top\\\text{Minimize:} & \sum\_{j=1}^4 W\_j \Big(\max\left|H\_j(\chi, \omega\_j)\right|\Big) \\\text{Subject to:} & I = \left\{j \middle|0.7\omega\_{\text{sj}} \le \omega\_j \le 1.3\omega\_{\text{sj}}, \omega\_j = \left|\text{eig}(A + B\_{12}F C\_2)\right|\right\} \end{array} \tag{22}$$

where *j* (*j* = 1, ... , 4) is the mode number considered and <sup>ω</sup>s*j* is the *j*-th natural frequency of the primary system. <sup>ω</sup>*j* is the evaluation of the eigenvalues, which corresponding to the modal frequencies of the entire system inside the specified frequency band. *Wj* = 0.25, which is the weight coefficient. For each TMD, 2% of the total mass of the structure.

#### *3.2. Numerical Optimization Results*

Two TMD devices are used to damp the mode pairs (1B,2T) and (2B,1T); one TMD is placed at the quarter length of the deck (TMD1), the second is located at the center of the deck (TMD2). A detailed description is shown in Section 4.2. A disturbance force *fd* is applied at one fixed point of the deck edge and the displacement *z* is measured at another fixed point, as shown in a small graph of Figure 3. According to Equation (22), the frequency response functions of the primary system with two 2DOFs TMDs are optimized, and the optimum frequency ratios and TMD damping ratios are obtained, as listed in Table 2.

**Figure 3.** Numerical results: FRFs of the deck without and with the TMDs. The TMDs are targeted for the damping of the mode pair (2B,1T) and (1B,2T).


**Table 2.** Optimum parameters of 2DOFs TMDs.

Figure 3 shows the frequency responses of the deck *<sup>z</sup>*/*fd*, when: (i) without any TMD; (ii) the four classical TMDs are targeting to damp the two mode pairs (2B,1T) and (1B,2T), which is achieved by Den Hartog criterion; and (iii) the two 2DOFs TMDs are targeting to damp the two mode pairs (2B,1T) and (1B,2T), which is optimized by that introduced in the present paper. The results indicate that the 2DOFs TMD concept model is e ffective to suppress both the bending modes and torsional modes of the bridge system at the same time. But, if using the classical configuration of the TMD, we need four TMDs, each of them is tuned on a single mode at the same time: two translation TMDs, with a lumped mass identical to that used in the 2DOFs TMD, and two other TMDs with moment of inertia identical to that of the 2DOFs TMD too. Hence, comparing with the classical configuration of TMD, the two 2DOFs TMDs can reduce the weight penalty.

Figure 4 shows the FRF for di fferent sensor locations, with the same TMD design as in Figure 3. This figure demonstrates the robustness with respect to the FRF used in the TMD design. Figure 5 plots the frequency response curve of the bridge equipped with TMD1, for di fferent values of the TMD1 damping ratio ξ. Here, the TMD1 damping ratio ξ is selected as 10% ξ*opt*, 25% ξ*opt*, 50% ξ*op<sup>t</sup>* and ξ*opt*, respectively, and the ξ*op<sup>t</sup>* is the optimum damping ratio of TMD1, which is listed in Table 2. From this figure, we see that the dynamic responses of the bridge deck always tend to decrease on increasing the damping ratio of TMD1. Furthermore, the frequency ratio is insensitive to the TMD damping ratio, but if the action is not perfectly resonant, the performance of TMD may decay seriously even though the value of the TMD damping ratio is very high [35]. For TMD structure design, unlike the damping ratio ξ*j* which is di fficult to quantify, the frequency ratio ν*j* is important parameter which can be used to guide the following TMD structural parameters design.

**Figure 4.** *Cont.*

**Figure 4.** Numerical results: FRFs of the deck respect to different measuring position (**<sup>a</sup>**–**d**). The excitation point is fixed.

**Figure 5.** Influence of the damping of TMD1 on the FRF of the system. ξ = 10% ξ*opt*, 25% ξ*opt*, 50% ξ*op<sup>t</sup>* and ξ*opt*, respectively.

#### **4. Structural Design of 2DOFs TMD**

Due to the lack of an effective theoretical guidance, deterministic structure design of multi-DOFs TMDs is still a challenge in the field of TMD design. The main problems are: (1) kinematic constraints design of the multi-DOFs, it ensures that the TMDs have the expected DOFs; and (2) parametric modeling of the multi-DOFs, that contributes to design the TMD and ensure it has the expected natural frequencies. To solve the above problems, this study presents a synthetic approach based on both the graphical approach [36] and parameterized compliance for the concrete design of the TMD with the expected DOFs.
