*4.1. Graphical Approach*

The screw theory as the theoretical foundation of graphical approach have been widely applied to design and analysis the compliant mechanisms [37,38]. For object behavior design, adding constraints is the most important step to reach the specific motion. Thus, finding the relationship between the constraints and movements is indispensable in TMD design. Currently, the screw theory is the most popular way to describe this relationship. In the screw theory, a unit screw *\$* is defined by a straight line with an associated pitch and is represented as a pair of three-dimensional vectors:

$$\mathbf{S} = \begin{bmatrix} \mathbf{s} \\ \mathbf{s}\_0 + h\mathbf{s} \end{bmatrix} = \begin{bmatrix} \mathbf{s} \\ \hline r \times \mathbf{s} + h\mathbf{s} \end{bmatrix} \tag{23}$$

where *s*0 = *r* × *s* defines the moment of the screw axis about the origin of a coordinate system, *s* is a unit vector parallel the screw axis, *r* is the position vector of any point on the screw axis with respect to the origin of a coordinate system, and *h* is the pitch of the screw. If *h* is equal to zero, the screw reduces to a line quantity (Figure 6a):

$$\mathbf{S}\_{t} = \begin{bmatrix} \mathbf{s} \\ \mathbf{s}\_{0} \end{bmatrix} = \begin{bmatrix} \mathbf{s} \\ \mathbf{r} \times \mathbf{s} \end{bmatrix} \tag{24}$$

If *h* is infinite, the screw reduces to:

$$\mathbf{S}\_w = \begin{bmatrix} 0 \\ \mathbf{s} \end{bmatrix} \tag{25}$$

In addition, an infinite-pitch screw can be considered as a line located at infinity, as shown in Figure 6b.

For a better understanding and applicability, the two special cases of unit screw (*\$<sup>t</sup>*, *\$w*) are visualized by geometric patterns in Figure 7. The unit screw of zero pitch (*\$t*) stands for a pure rotation in freedom space (rotational freedom line) or a unit pure force in static along the line in constraint space (constraint force line). A unit screw of infinite pitch represents a pure translation in freedom space (translational freedom line) or a pure couple in constraint space (constraint couple line). It is worth noting that the rotational freedom line represents the axis of rotational movement, and the constraint couple line stands for the axis of couple imposed on a rigid body.

**Figure 7.** Geometric patterns representing screws.

Based on Maxwell's principles of constraints, the freedoms and constraints in a mechanical system can be defined as:

$$N = \mathfrak{G} - n,\tag{26}$$

where *N* is the number of DOFs, *n* is the number of non-redundant constraints. When a rigid body (e.g., TMD) is constrained by several mechanical connections providing *n* constraints, while *N* DOFs of the body will remains. In this regard, based on the reciprocal screw theory, *n* non-redundant constraints form a wrench *\$*1 in constraint space, and the remained DOFs constitute a twist *\$*2 in freedom space [39]. Based on the definition, the reciprocity of these two screw systems is expressed as:

$$\begin{aligned} \mathbf{s}\_1^T \Delta \mathbf{s}\_2 &= \mathbf{s}\_1 \cdot (r\_2 \times \mathbf{s}\_2 + h\_2 \mathbf{s}\_2) + \mathbf{s}\_2 \cdot (r\_1 \times \mathbf{s}\_1 + h\_1 \mathbf{s}\_1) \\ &= (h\_1 + h\_2)(\mathbf{s}\_1 \cdot \mathbf{s}\_2) + (r\_2 - r\_1) \cdot (\mathbf{s}\_2 \times \mathbf{s}\_1) \\ &= (h\_1 + h\_2) \cos \alpha\_{12} - a\_{12} \sin \alpha\_{12} \\ &= 0 \end{aligned} \tag{27}$$

where Δ = 0 *I I* 0 , *a*12 is the normal distance of the two screw axes and α12 is the twist angle between thetwoscrews.*h*1and*h*2denotethepitchof*\$*1and*\$*2,respectively.

 Thus, according to Equation (27), the relationship between the freedom lines and the constraint lines can be written as a brief form in Table 3.


**Table 3.** Geometric relationship between the freedom and constraint lines.

The 2DOFs TMD with the expected DOFs and mode shapes can be designed by the above geometric relationship.

#### *4.2. Conceptual Design of the TMDs*

By studying the characteristics of human-induced vibration, it is found that the first four modes of suspension bridge are easy to be stimulated to produce resonance phenomenon, which is the primary target of vibration reduction. The targeted mode shapes are shown in Figure 8.

The optimal location of a TMD is at the maximum modal displacement. Hence, the center of the deck (point *A*) is the optimal location for the second bending and the first torsional modes of the deck (2B,1T), while for the first bending and the second torsional modes (1B,2T), it is at the quarter length of the deck (point *B*). The motion of the TMDs only along the vibration direction of the two pairs can reach the best vibration control. Two TMDs have the same geometric relationship as shown in Figure 9. Therefore, the dimension of the freedom space *N* is two, and the dimension of the constraint space is four, according to Equation (26).

The translational freedom line and rotational freedom lines are orthogonal in Figure 9a. For the constraint space, the four constraint force lines intersect with the rotational freedom line and are orthogonal with the translational freedom line. Here the constraint couple lines can be ignored due to the fact that four constraint force lines have already formed the expected constraints. Thus, the corresponding constraint space can be divided into two pairs, each pair contains two parallel constraint force lines, and these two pairs are orthogonal (Figure 9b). Then constraint force lines are implemented by flexure elements, each of them can provide single DOF constraint along its axial direction (Figure 9c). Therefore, the exact constraints are formed on the TMDs and the expected DOFs are defined.

**Figure 8.** Shapes of the targeted modes and the corresponding optimal location of the TMD: the optimal location of the modes (2B,1T) at point A, while it is point B for modes (1B,2T); the freedom space of the TMDs.

**Figure 9.** Conceptual design of 2DOFs TMDs: (**a**) freedom space of 2DOFs TMD; (**b**) constraint space of 2DOFs TMD; (**c**) physical model of 2DOFs TMD.

#### *4.3. Parametric Design of 2DOFs TMDs*

The conceptual model of TMDs only have two expected DOFs, but in actual design, due to the material and geometric properties, the TMD may have more DOFs than expected. The corresponding redundant modes of TMD may affect the performance of TMD, even lead to TMD failure. This is a key problem that has been perplexing TMD design. In general, the redundant modes which are far from the targeted modes can be ignored, and the empirical design has always been the major tool to achieve this goal. However, as the number of DOF increases, the empirical method gradually fails to realize the complex design of TMDs. Furthermore, the empirical design may cause the increase in time and cost. Therefore, the theoretical guidance has become particularly important in the multi-DOFs TMD design. In this section, we introduce a parameterized compliance approach for parametric design of the 2DOFs TMD.

Figure 10 shows the configuration of the TMD. It can be seen that the TMD is formed by eight slender beams in parallel distributed on mass block, which is transformed from physical models (Figure 9c). Each slender beam is considered as cantilever beam (Figure 10a) with length *L*, width *w*, and thickness *t*. Then, according to the Bernoulli-Euler model, the compliance matrix *Ccp* (*p* = 1, ... , *n*) for each slender beam at location coordinate system *Oxyz* is given as follows:

$$\mathbf{C} \mathbf{c}\_p = \frac{L\_p}{EI\_y} \text{diag} \left\{ \begin{array}{cccc} \alpha & 1 & \frac{1}{\lambda \overline{\gamma}} & \frac{L\_p^2}{12} & \frac{L\_p^2}{12} \alpha & \frac{L\_p^2}{12} \beta \end{array} \right\} \tag{28}$$

where:

$$
\alpha = \left(\frac{t}{w}\right)^2, \beta\_p = \left(\frac{t}{L\_p}\right)^2, \chi = \frac{G}{E} = \frac{1}{2(1+\nu)}, \gamma = \frac{I}{I\_y} \tag{29}
$$

and

$$I\_x = \frac{w^3 t}{12}, I\_y = \frac{wt^3}{12}, I = I\_x + I\_y,\tag{30}$$

where *G*, *E* is the shear and Young's modulus, respectively. γ is the ratio of torsion constant over moment of inertia, ν is the Poisson's ratio.

**Figure 10.** Coordinate systems of the slender beams: (**a**) local coordinate; (**b**) upper view of global coordinate; (**c**) front view of global coordinate.

Further, to simplify the following non-dimensionalization, we choose *t* = *w*, and according to Equations (29) and (30), γ = 2. Due to the different length of slender beam (*L*1, *L*2 and *s* = *<sup>L</sup>*1/*L*2), the compliance matrix of two type slender beams can be written as:

$$\mathbf{C} \mathbf{c}\_A = \frac{L\_2}{EI\_y} \text{diag} \left[ \begin{array}{cccc} \text{s} & \text{s} & \frac{\varepsilon}{2\chi} & \frac{s^3 L\_2^2}{12} & \frac{s^3 L\_2^2}{12} & \frac{s L\_2^2}{12} \beta\_2 \end{array} \right] \tag{31}$$

$$\mathbf{C} \mathbf{c}\_{\overline{B}} = \frac{L\_2}{EI\_y} \text{diag} \left[ \begin{array}{cccc} 1 & 1 & 1 & \frac{1}{12} & \frac{L\_2^2}{12} & \frac{L\_2^2}{12} & \frac{L\_2^2}{12} \beta\_2 \\ \end{array} \right] \tag{32}$$

In order to combine the local compliance matrix *Ccp* of the eight slender beams, they should be transformed from the local to global coordinate system. The origin *O'* of the global coordinate system *O'XYZ* is defined in the centroid of the mass block (Figure 10b). For the parallel flexure mechanism, the global compliance matrix can be given as:

$$\mathbf{C}\_{s} = \left(\sum\_{p=1}^{m} \left(\boldsymbol{\mathcal{A}} \boldsymbol{d}\_{p} \boldsymbol{\mathcal{C}} \boldsymbol{c}\_{p} \boldsymbol{\mathcal{A}} \boldsymbol{d}\_{p}^{T}\right)^{-1}\right)^{-1},\tag{33}$$

where *m* is the number of slender beams; *Adp* is the adjoint transformation matrix from the *p*th element to the global system:

$$\mathbf{Ad}\_p = \begin{bmatrix} \mathbf{R}\_{\mathbf{x},y,z}(\theta) & 0\\ \mathbf{T} \mathbf{R}\_{\mathbf{x},y,z}(\theta) & \mathbf{R}\_{\mathbf{x},y,z}(\theta) \end{bmatrix} \tag{34}$$

where *T* is the translation matrix. *<sup>R</sup><sup>x</sup>*,*y*,*<sup>z</sup>*(θ) = *<sup>R</sup>x*(θ)*Ry*(θ)*Rz*(θ), which is the multiplication of rotation matrices. *Rx*(θ), *<sup>R</sup>y*(θ), and *Rz*(θ) stand for the rotation matrices by an angle θ about the *x*, *y*, and *z* axis, respectively. They are given in Equation (35):

$$\begin{aligned} \mathcal{R}\_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \end{bmatrix}, \quad \mathcal{R}\_y(\theta) = \begin{bmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \end{bmatrix}, \quad \mathcal{R}\_z(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{35}$$

For the vector (Δ*x*, Δ*y*, <sup>Δ</sup>*z*)*<sup>T</sup>* between two position, the translation matrix can be expressed as

$$T = \begin{bmatrix} 0 & -\Delta z & \Delta y \\ \Delta z & 0 & -\Delta x \\ -\Delta y & \Delta x & 0 \end{bmatrix} \tag{36}$$

Based on Equations (33)–(36), the global compliance matrix of TMD is computed by:

$$\mathbf{C}\_{s} = \frac{L\_{2}}{EI\_{y}} \begin{bmatrix} c\_{11} \\ & c\_{22} \\ & & c\_{33} \\ & & & c\_{44} \\ & & & & c\_{55} \\ & & & & & c\_{66} \end{bmatrix} \tag{37}$$

The principal diagonal elements of *Cs* are selected as follows:

$$\begin{aligned} c\_{11} &= \frac{L\_2^2 \beta\_2 s^3}{16} \left( \frac{\rho\_2 L\_2^2 s^3 + \beta\_2 L\_2^2 s^2 + 3 \rho\_2 L\_2 d\_1 s + 3 \rho\_2 L\_2 d\_1 s^3 + 3 \rho\_2 d\_1^2 + 3 \rho\_2 d\_2^2 s^3 + 3 d\_4^2 s^2}{16} \right) \\ c\_{22} &= \frac{L\_2^2 s^3}{16} \left( \frac{1}{0.5 \chi L\_2^2 s^3 + L\_2^2 s^2 + 3 L\_2 d\_1 s + 3 d\_1^2 + 3 d\_3^2 s^3} \right) \\ c\_{33} &= \frac{L\_2^2 \beta\_2 s^3}{16} \left( \frac{1}{\rho\_2 L\_2^2 s^3 + 0.5 \rho\_2 L\_2^2 s^2 + 3 \rho\_2 L\_2 d\_2 s^3 + 3 \rho\_2 d\_2^2 s^3 + 3 d\_3^2 s^3 + 3 \rho\_2 d\_4^2} \right) \\ c\_{44} &= \frac{L\_2^2 s^3}{48} \left( \frac{1}{s^3 + 1} \right) c\_{55} = \frac{L\_2^2 \beta s^3}{48} \left( \frac{1}{s^3 + \rho\_2} \right) \\ c\_{66} &= \frac{L\_2^2 \beta s}{16} \left( \frac{1}{\beta \rho s + 1} \right) \end{aligned} \tag{38}$$

where *c*11, *c*22, and *c*33 are the rotational compliance/stiffness about the *x*, *y* and *z* axis while *c*44, *c*55, and *c*66 are the translational compliance/stiffness along the *x*, *y* and *z* axis, respectively. In the end, the natural frequencies of 2DOFs TMDs can be approximate calculated by Equations (39)–(41).

The bending mode:

$$
\omega\_1 = \frac{1}{2\pi} \sqrt{\frac{k\_{44}}{m\_d}} = \frac{1}{2\pi} \sqrt{\frac{1}{m\_d} \left(\frac{L\_2 c\_{44}}{EI\_y}\right)^{-1}}\tag{39}
$$

The torsional mode:

$$
\omega\_2 = \frac{1}{2\pi} \sqrt{\frac{k\_{22}}{I\_Y}} = \frac{1}{2\pi} \sqrt{\frac{1}{I\_Y} \left(\frac{L\_2 c\_{22}}{EI\_Y}\right)^{-1}}\tag{40}
$$

and:

$$I\chi = \frac{1}{3}m\_d(d\_1^2 + d\_5^2),\tag{41}$$

where *d*5 is the height of mass block (Figure 10c).

#### *4.4. Results and Discussion*

According to the optimum frequency ratios ν*i* (Table 2) and the target modes of bridge, the expected modes of TMDs are obtained and listed in Table 4.


**Table 4.** Expected modes of 2DOFs TMDs.

In general, the size of TMD should be smaller and not occupy the space of bridge as much as possible. Thus, considering the processing conditions and the size of deck, we choose *t* = *w* = 1mm. The mass block is iron and *md* = 0.18 kg. The material of slender beams is Acrylonitrile Butadiene Styrene (ABS), and the elastic modulus *E* = 2 GPa, χ = 0.37, the Poisson's ratio ν is 0.394. In addition, in order to simplify calculation and TMD design, let *d*4 = *d*2.

The calculation results of 2DOFs TMDs are summarized in Table 5. It is noticed that the three-order natural frequency ω3 are about 7 times greater than ω2, and ω4 is much larger than ω1 and ω2. Therefore, the undesired DOFs can be neglected, and the expected 2DOFs TMDs are obtained. According to the dimension parameters, FE model of 2DOFs TMDs are built as shown in Figure 11. The mode shapes of two expected TMDs are shown in Figures 12 and 13, respectively.

**Table 5.** Calculation results of 2DOFs TMDs.


(**a**) (**b**) **Figure**

**11.**CADviewofTMD. (**a**)TMD1;(**b**)TMD2.

**Figure 12.** Mode shapes of TMD1. (**a**) The 1st mode; (**b**) the 2nd mode.

**Figure 13.** Mode shapes of TMD2. (**a**) The 1st mode; (**b**) the 2nd mode.
