Parametric control of structural responses using an optimal passive tuned mass damper under stationary Gaussian white noise excitations

Abstract

In this study, the structural control strategy utilizing a passive tuned mass damper (TMD) system as a seismic damping device is outlined, highlighting the parametric optimization approach for displacement and acceleration control. The theory of stationary random processes and complex frequency response functions are explained and adopted. For the vibration control of an undamped structure, the optimal parameters of a TMD, such as the optimal tuning frequency and optimal damping ratio, to stationary Gaussian white noise acceleration are investigated by using a parametric optimization procedure. For damped structures, a numerical searching technique is used to obtain the optimal parameters of the TMD, and then the explicit formulae for these optimal parameters are derived through a sequence of curve-fitting schemes. Using these specified optimal parameters, several different controlled responses are examined, and then the displacement and acceleration based control effectiveness indices of the TMD are examined from the view point of RMS values. From the viewpoint of the RMS values of displacement and acceleration, the optimal TMDs adopted in this study shows clear performance improvements for the simplified model examined, and this means that the effective optimization of the TMD has a good potential as a customized target response-based structural strategy.

Keywords

tuned mass damper / parametric optimization / passive control / white noise / earthquake excitation

Introduction

Tuned mass damper (TMD) system is one of the practical and well accepted structural control methodologies for flexible structures, especially for tall buildings. By use of additional sub-mass with properly tuned spring and damping elements, the TMD system derives a frequency-dependent hysteresis that supplies additional damping in the main structure. The mechanical operation of suppressing structural vibrations by attaching a TMD to the main structure is to transfer the vibration energy of the main structure to the TMD and then to dissipate the energy in the damper of the TMD.

Most TMD applications have been made toward mitigation of the building response under wind excitation [1-4]. For seismic applications, as of today, many numerical and experimental studies have been carried out on the effectiveness of TMDs in reducing seismic response of structures [5-9]. However, there has not been a general agreement on the efficiency of TMD systems to reduce the structural response. Many results show that the seismic effectiveness of TMDs for the same structure during some earthquakes or for the different structures during the same earthquake is significantly different. Some cases give good performance and in others there are little or even no effect. This implies that there is a dependency of the attained reduction in response on the characteristics of the ground motion that excites the structures [10-12]. By the result of the uncertainty related to the effectiveness of the TMD under various earthquake intensities, the seismic design of using TMD system remains an important issue for the further study.

While the basic principles of TMDs on reducing structural response have been well established, optimal TMD design is a quite a different issue. This issue is also related to the uncertainty of the effectiveness of the TMD controlling the seismic response of the structures as mentioned above. In the design of any control device for the suppression of undesirable vibrations, the aim would be to provide optimal damper parameters to maximize its effectiveness. The chief design parameters of the TMD are its tuning ratio (the ratio of the damper frequency to the natural frequency of the structure) and TMD damping ratio. The other important design parameter is the mass ratio (the ratio of the damper mass to the mass of the structure). In the classical work by Den Hartog [13], simple expressions for the optimal tuning ratio and damping ratio of a mass damper were derived. These expressions were based on minimizing the displacement response of an undamped primary system subjected to sinusoidal excitation. To obtain the analytical solution for the optimal values of the parameters that minimize the dynamic response of the building measured, Crandall and Mark [14] obtained the mean square response of the stationary process when the spectral density is known. When the input spectrum is assumed to be ideally white, the mean square of the response was determined. For the case of damped primary structure, it is difficult to obtain closed-form solutions for the optimal damper parameters. Randall et al. [15] used numerical optimization procedures for evaluating the optimal TMD parameters while considering damping in the structure. Warburton [16] carried out a detailed numerical study for a lightly damped structure with TMD, subjected to both harmonic and random excitations. As a result of the continuous relevant studies, the optimal damper parameters of tuning ratio and damping ratio for various values of mass ratio and structural (internal) damping ratio have been well established in the form of design tables [6]. Recently, thus, considerable researches have been devoted to enabling proper selection of large TMD parameters [17-20].

In this study, a parametric searching procedure to find the optimal parameters of a TMD that can reduce the response of structures to a satisfactory level of displacement and acceleration is proposed. For this procedure, zero-mean white noise excitation ground acceleration is selected and the acceleration is supposed to be stationary Gaussian and the value of its spectral density S₀ is to give random properties of earthquake loading. A series of parametric-based TMD operations for the controlling of the main systems are performed with the searched optimal parameters, from which closed-form formulas of the TMD parameters are derived. From the view point of displacement and acceleration, the response properties of the TMD controlled main systems are presented and, finally, the effectiveness of the TMDs are compared when the inherent parameters of the main systems are changed.

Equations of motion and complex frequency response functions

Equations of motion

From the schematic of single degree-of-freedom (SDOF) model for the analysis of main system with a TMD (Fig. 1), the equation of motion of the system subjected to the earthquake load, {\ddot{\mathit{x}}}_{\mathrm{g}}, can be derived in terms of absolute and relative displacements respectively as follows,

(1)\left[\begin{array}{mm}{m}_{\mathrm{1}}& \mathrm{0}\\ \mathrm{0}& m_{\mathrm{2}}\end{array}\right]\left\{\begin{array}{m}{\ddot{x}}_{\mathrm{1}}\\ {\ddot{x}}_{\mathrm{2}}\end{array}\right\}+\left[\begin{array}{mm}{c}_{\mathrm{1}}+c_{\mathrm{2}}& -c_{\mathrm{2}}\\ -c_{\mathrm{2}}& c_{\mathrm{2}}\end{array}\right]\left\{\begin{array}{m}{\dot{x}}_{\mathrm{1}}\\ {\dot{x}}_{\mathrm{2}}\end{array}\right\}+\left[\begin{array}{mm}{k}_{\mathrm{1}}+k_{\mathrm{2}}& -k_{\mathrm{2}}\\ -k_{\mathrm{2}}& k_{\mathrm{2}}\end{array}\right]\left\{\begin{array}{l}{x}_{\mathrm{1}}\\ x_{\mathrm{2}}\end{array}\right\}=\left\{\begin{array}{l}{k}_{\mathrm{1}}\\ \mathrm{0}\end{array}\right\}{x}_g+\left\{\begin{array}{l}{c}_{\mathrm{1}}\\ \mathrm{0}\end{array}\right\}{\dot{x}}_g,

(2)\left[\begin{array}{mm}{m}_{\mathrm{1}}& \mathrm{0}\\ m_{\mathrm{2}}& m_{\mathrm{2}}\end{array}\right]\left\{\begin{array}{m}{\ddot{y}}_{\mathrm{1}}\\ {\ddot{y}}_{\mathrm{2}}\end{array}\right\}+\left[\begin{array}{mm}{c}_{\mathrm{1}}& -c_{\mathrm{2}}\\ \mathrm{0}& c_{\mathrm{2}}\end{array}\right]\left\{\begin{array}{m}{\dot{y}}_{\mathrm{1}}\\ {\dot{y}}_{\mathrm{2}}\end{array}\right\}+\left[\begin{array}{mm}{k}_{\mathrm{1}}& -k_{\mathrm{2}}\\ \mathrm{0}& k_{\mathrm{2}}\end{array}\right]\left\{\begin{array}{m}{y}_{\mathrm{1}}\\ y_{\mathrm{2}}\end{array}\right\}=\left\{\begin{array}{m}-m_{\mathrm{1}}\\ -m_{\mathrm{2}}\end{array}\right\}{\ddot{x}}_g,

in which

• m₁ = mass of the main system

• m₂ = mass of the TMD

• k₁ = stiffness of the main system

• k₂ = stiffness of the TMD

• c₁ = damping coefficient of the main system

• c₂ = damping coefficient of the TMD

• x₁ = absolute displacement of the main system

• x₂ = absolute displacement of the TMD

• x_g = absolute displacement of the ground

• y₁ = x₁ - x_g, relative displacement between the main system and the ground

• y₂ = x₂ - x₁, relative displacement between the TMD and the main system

• {\ddot{\mathit{x}}}_{\mathrm{1}} = {\ddot{\mathit{x}}}_{\mathrm{g}} + {\ddot{\mathit{x}}}_{\mathrm{1}}, absolute acceleration of the main system

• {\ddot{\mathit{x}}}_{\mathrm{2}} ={\ddot{\mathit{x}}}_{\mathrm{g}} + {\mathit{ÿ}}_1 + {\mathit{ÿ}}_2, absolute acceleration of the TMD

and one dot (·) and two dots (··) mean the first and second derivatives for each variable and in order to standardize the subsequent treatment, the following notations are used.

(3)\begin{array}{llll}\mu ={\displaystyle \frac{m_{\mathrm{2}}}{m_{\mathrm{1}}}},\hfill & f_{\mathrm{2}}={\displaystyle \frac{{\omega }_{\mathrm{2}}}{{\omega }_{\mathrm{1}}}},\hfill & {\omega }_i^{\mathrm{2}}={\displaystyle \frac{k_i}{m_i}},\hfill & c_i=\mathrm{2}{\xi }_i{\omega }_i{m}_i(i=\mathrm{1},\mathrm{2}),\hfill \end{array}

in which

• ω₁ = frequency of the main system

• ω₂ = frequency of the TMD

• ξ₁ = damping ratio of the main system

• ξ₂ = damping ratio of the TMD

• μ₁ = mass ratio of the TMD to the main system

• f₂ = frequency tuning ratio of the TMD to the main system

Complex frequency response functions

To calculate the complex frequency response functions with respect to the relative displacement, the form of {\ddot{x}}_{\mathrm{g}}=e^{i\omega t} can be used as input with y₁ in the form of

(4)\begin{array}{ll}{y}_{\mathrm{1}}=H_{y_{\mathrm{1}}}(\omega )e^{i\omega t},& y_{\mathrm{2}}=H_{y_{\mathrm{2}}}(\omega )e^{i\omega t},\end{array}

and can be substituted into Eq. (2).

(5)\left[\begin{array}{mm}{k}_{\mathrm{1}}+i\omega c_{\mathrm{1}}-m_{\mathrm{1}}{\omega }^{\mathrm{2}}& -k_{\mathrm{2}}-i\omega c_{\mathrm{2}}\\ -\mu m_{\mathrm{1}}{\omega }^{\mathrm{2}}& k_{\mathrm{2}}+i\omega c_{\mathrm{2}}-\mu m_{\mathrm{1}}{\omega }^{\mathrm{2}}\end{array}\right]\left\{\begin{array}{l}{H}_{y_{\mathrm{1}}}(\omega )\\ H_{y_{\mathrm{2}}}(\omega )\end{array}\right\}=\left\{\begin{array}{l}-m_{\mathrm{1}}\\ -m_{\mathrm{2}}\end{array}\right\}\mathrm{.}

The complex frequency response function, H_y1(ω), H_y2(ω) can be solved by using standard matrix techniques.

(6)H_{y_{\mathrm{1}}}(\omega )={\displaystyle \frac{{\omega }^{\mathrm{2}}(\mu m_{\mathrm{1}}^{\mathrm{2}})-i\omega (\mathrm{1}+\mu )c_{\mathrm{2}}{m}_{\mathrm{1}}-k_{\mathrm{2}}{m}_{\mathrm{1}}(\mathrm{1}+\mu )}{\mathit{\Delta }}},

(7)H_{y_{\mathrm{2}}}(\omega )={\displaystyle \frac{-i\omega (\mu c_{\mathrm{1}}{m}_{\mathrm{1}})-\mu k_{\mathrm{1}}{m}_{\mathrm{1}}}{\mathrm{\Delta }}},

in which

\mathrm{\Delta }={\omega }^4\left(\mu m_1^2\right)-i{\omega }^3\left(c_2+c_1\mu +c_2\mu \right)m_1-{\omega }^2\left(c_1{c}_2+k_2{m}_1+k_1\mu m_1+k_2\mu m_1\right)+i\omega \left(c_2{k}_1+c_1{k}_2\right)+k_1{k}_2.

The complex frequency response function for the system without a TMD can also be solved using the same method.

(8)H_y(\omega )={\displaystyle \frac{m_{\mathrm{1}}}{{\omega }^{\mathrm{2}}{m}_{\mathrm{1}}-i\omega c_{\mathrm{1}}-k_{\mathrm{1}}}}.

Substituting Eq. (4) into Eq. (2) provides the complex frequency response functions in terms of acceleration.

(9)H_{{\ddot{x}}_1}\left(\omega \right)=1-{\omega }^2{H}_{y_1}\left(\omega \right)={\displaystyle \frac{\left(-i{\omega }^3\left(\mu c_1{m}_1\right)-{\omega }^2\left(c_1{c}_2+\mu k_1{m}_1\right)+i\omega \left(c_1{k}_2+c_2{k}_1\right)+k_1{k}_2\right)}{\mathit{\Delta }}},

(10)H_{{\ddot{x}}_2}\left(\omega \right)=1-{\omega }^2{H}_{y_1}\left(\omega \right)-{\omega }^2{H}_{y_2}\left(\omega \right)={\displaystyle \frac{-{\omega }^2\left(c_1{c}_2\right)+i\omega \left(c_1{k}_2+c_2{k}_1\right)+k_1{k}_2}{\mathit{\Delta }}},

in which

\mathit{\Delta }={\omega }^4\left(\mu m_1^2\right)-i{\omega }^3\left(c_2+c_1\mu +c_2\mu \right)m_1-{\omega }^2\left(c_1{c}_2+k_2{m}_1+k_1\mu m_1+k_2\mu m_1\right)+i\omega \left(c_2{k}_1+c_1{k}_2\right)+k_1{k}_2.

The frequency function for the system without TMD is as follows:

(11)H_{\ddot{x}}(\omega )={\displaystyle \frac{-i\omega c_{\mathrm{1}}-k_{\mathrm{1}}}{{\omega }^{\mathrm{2}}{m}_{\mathrm{1}}-i\omega c_{\mathrm{1}}-k_{\mathrm{1}}}}.

To predict the responses of the main system and the TMD, it is useful to display the complex frequency functions as effective functions of responses. Figure 2(a) shows the squared complex frequency response functions for the main system and the TMD in terms of displacement for the case where the mass ratio is 0.02. The sharp peaks in the functions, {\left|\mathrm{H}{(\omega )}_{y_{\mathrm{1}}}\right|}^{\mathrm{2}}, {\left|\mathrm{H}{(\omega )}_{y_{\mathrm{2}}}\right|}^{\mathrm{2}} and {\left|\mathrm{H}{(\omega )}_y\right|}^{\mathrm{2}} will mainly influence the RMS displacement responses related by the ordinates of spectral density S(ω) at the location of these peaks. Also, the RMS acceleration responses can be predicted through the plots of {\left|\mathrm{H}{(\omega )}_{x_{\mathrm{1}}}\right|}^{\mathrm{2}}, {\left|\mathrm{H}{(\omega )}_{x_{\mathrm{2}}}\right|}^{\mathrm{2}} and {\left|\mathrm{H}{(\omega )}_x\right|}^{\mathrm{2}} as shown in Fig. 2(b).

Parametric optimization of TMD

Optimal TMD configurations

The performance of tuned mass damper systems in buildings and other structures can be readily assessed by parametric studies. It is common practice to study the structure-damper system under sinusoidal and white noise excitations which are regarded as the two extremes of narrow-band and wide-band excitations, respectively. In general, the optimal parameters such as the “damping ratio” and the “frequency” of the TMD need to be determined to achieve the optimal structural performance. The optimal parameters can be derived for the required dynamic load depending on the control criteria.

Previously suggested control criteria [13] were used by minimizing the displacement of the structure. Displacement essentially determines safety and integrity of a structure under external excitations. Meanwhile, large accelerations of a structure under excitations produce detrimental effects in functionality of nonstructural components, base shear, and occupant comfort. Thus, minimizing structural acceleration can also be a viable control criterion. The TMD travel relative to the building is another important design criterion. However, the large movements of the TMD often need to be accommodated for a reasonable response reduction for the building.

Generally speaking, if a building is subjected to a far-field earthquake of long duration, the absolute acceleration of the TMD needs to be reduced to improve the comfort of occupants. However, for a near-field earthquake of strong intensity, the priority of the control objective changes to the reduction of the structure displacement to protect the structure itself. It is clear that some of the above criteria overlap with each other and have similarities. In this study, four different control criteria, the relative displacements of the main system and the TMD (y₁ and y₂) and the absolute accelerations of the main system and the TMD ({\ddot{\mathit{x}}}_{\mathrm{1}} and {\ddot{\mathit{x}}}_{\mathrm{2}}) will be examined to observe the effects of the TMD.

To obtain the analytical solution for the optimal parameters which minimize the dynamic responses (the mean square values of the displacement and acceleration) of the building measured, it is assumed that the earthquake ground motion is stationary white noise which may be a reasonable idealization for the fluctuating component of the earthquake excitations.

Optimal parameters for displacement control

The mean square response E[y²] of the stationary process y(t) can be obtained when the spectral density S_y(ω) is known. When the input spectrum is assumed to be ideally white, i.e., S_x(ω) = S₀, a constant for all frequencies, the mean square E[y_{\mathrm{1}}^{\mathrm{2}}] of the response y(t) is determined [14] as,

(12)E[y^{\mathrm{2}}]={\displaystyle {\int }_{-\infty }^{\infty }{S}_y(\omega )\mathrm{d}\omega }={\displaystyle {\int }_{-\infty }^{\infty }{\left|H(\omega )\right|}^{\mathrm{2}}{S}_x\mathrm{d}\omega }=S_{\mathrm{0}}{\displaystyle {\int }_{-\infty }^{\infty }{\left|\mathrm{H}(\omega )\right|}^{\mathrm{2}}\mathrm{d}\omega }.

According to the above process, the mean square responses E[y_{\mathrm{1}}^{\mathrm{2}}] and E[y_{\mathrm{2}}^{\mathrm{2}}]can be derived as follows,

(13)$$\begin{gathered} {\sigma }_{y_1}^2=E\left[y_1^2\right]=S_0{\displaystyle {\int }_{-\infty }^{\infty }{\left|H_{y_1}\left(\omega \right)\right|}^2\mathrm{d}\omega }=\mathrm{\pi }{S}_0{\displaystyle \frac{m_1\left(c_2^2{k}_1+3c_2^2{k}_1\mu +3c_2^2{k}_1{\mu }^2+c_2^2{k}_1{\mu }^3 \\ +k_2^2{m}_1-2k_1{k}_2\mu m_1+4k_2^2\mu m_1 \\ +k_1^2{\mu }^2{m}_1-3k_1{k}_2{\mu }^2{m}_1+6k_2^2{\mu }^2{m}_1 \\ +4k_2^2{\mu }^3{m}_1+k_1{k}_2{\mu }^4{m}_1+k_2^2{\mu }^4{m}_1\right)}{c_2{k}_1^3{\mu }^2}}, \end{gathered}$$

(14){\sigma }_{y_2}^2=E\left[y_2^2\right]=S_0{\displaystyle {\int }_{-\infty }^{\infty }{\left|H_{y_2}\left(\omega \right)\right|}^2\mathrm{d}\omega }=\mathrm{\pi }{S}_0{\displaystyle \frac{m_1^2{\left(k_2+2k_2\mu +k_1{\mu }^2+k_2{\mu }^2\right)}^2}{c_2{k}_1{k}_2}}.

As a white noise acceleration of the base of a structure is often used as an approximate input in earthquake engineering, simple expressions for the optimal TMD parameters for an undamped main system are useful. The optimizing conditions are as follows,

(15)\begin{array}{ll}{\displaystyle \frac{\partial E[y^{\mathrm{2}}]}{\partial f_{\mathrm{2}}}}=\mathrm{0},\hfill & {\displaystyle \frac{\partial E[y^{\mathrm{2}}]}{\partial {\xi }_{\mathrm{2}}}}=\mathrm{0}\hfill \end{array}.

Applying these conditions, simple expressions for the optimal values of the frequency ratio (f_2dopt) and damping ratio (ξ_2dopt) in terms of displacement are obtained for different responses to be optimized under different sources of excitation. To decouple the equations for optimization, the damping ratio ξ₁ is assumed to be zero. The optimal parameters to minimize the relative displacement of the main system are as follows [16],

(16)f_{\mathrm{2}\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}={\left({\displaystyle \frac{k_{\mathrm{2}\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}}{\mu k_{\mathrm{1}}}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}={\displaystyle \frac{{\left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mu \right)}^{\mathrm{2}}}{\mathrm{1}+\mu }},

(17){\xi }_{\mathrm{2}\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}={\left({\displaystyle \frac{c_{\mathrm{2}\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}{\xi }_{\mathrm{1}}}{f_{\mathrm{2}\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}\mu c_{\mathrm{1}}}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}={\left({\displaystyle \frac{\mu {\left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}\mu \right)}^{\mathrm{2}}}{\mathrm{4}(\mathrm{1}+\mu )\left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mu \right)}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}.

For practical application to a real system, it is necessary to obtain practical parameters for the TMD such as the optimal TMD damping stiffness (k_2dopt) and optimal damping coefficient (c_2dopt). These practical parameters can be derived using the parametric relationships between the above optimal parameters derived as

(18)k_{2\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}=f_{2\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}^2\mu k_1={\displaystyle \frac{\mu \left(1-\frac{1}{2}\mu \right)}{{\left(1+\mu \right)}^2}}{k}_1,

(19)c_{2}dopt={\displaystyle \frac{{\zeta }_{2dopt}^2{f}_{2dopt}\mu c_1}{\zeta 1}}=\left({\displaystyle \frac{{\mu }^{{\displaystyle \frac{3}{2}}}{\left(1-\frac{1}{4}\mu \right)}^{{\displaystyle \frac{1}{2}}}}{2{\zeta }_1{\left(1+\mu \right)}^{{\displaystyle \frac{3}{2}}}}}\right)c_1.

Optimal parameters for acceleration control

For the optimal parameters in terms of acceleration, the same process as that for displacement control can be used. The mean square responses for the main system and the TMD become

(20)$$\begin{gathered} {\sigma }_{{\ddot{x}}_1}^2=E\left[{\ddot{x}}_1^2\right]=S_0{\displaystyle {\int }_{-\infty }^{\infty }{\left|H_{{\ddot{x}}_1}\left(\omega \right)\right|}^2\mathrm{d}\omega }=\mathrm{\pi }{S}_0{\displaystyle \frac{\left(c_2^2{k}_1+c_2^2{k}_1\mu +k_2^2{m}_1 \\ -2k_1{k}_2\mu m_1+2k_2^2\mu m_1+k_1^2{\mu }^2{m}_1 \\ -k_1{k}_2{\mu }^2{m}_1+k_2^2{\mu }^2{m}_1\right)}{c_2{k}_1{\mu }^2{m}_1}}, \end{gathered}$$

(21)$$\begin{gathered} {\sigma }_{{\ddot{x}}_2}^2=E\left[{\ddot{x}}_2^2\right]=S_0{\displaystyle {\int }_{-\infty }^{\infty }{\left|H_{{\ddot{x}}_2}\left(\omega \right)\right|}^2\mathrm{d}\omega }=\mathrm{\pi }{S}_0{\displaystyle \frac{\left(c_2^2{k}_1+c_2^2{k}_1\mu +k_2^2{m}_1 \\ +2k_2^2\mu m_1+k_1{k}_2{\mu }^2{m}_1+k_2^2{\mu }^2{m}_1\right)}{c_2{k}_1{\mu }^2{m}_1}}. \end{gathered}$$

The optimizing conditions in terms of accelerations are

(22)\begin{array}{ll}{\displaystyle \frac{\partial {\sigma }_{{\ddot{x}}_{\mathrm{1}}}^{\mathrm{2}}}{\partial f_{\mathrm{2}}}}=\mathrm{0},\hfill & {\displaystyle \frac{\partial {\sigma }_{{\ddot{x}}_{\mathrm{1}}}^{\mathrm{2}}}{\partial {\xi }_{\mathrm{2}}}}=\mathrm{0}\hfill \end{array}.

The optimal frequency tuning ratio (f_2aopt) and optimal damping ratio (ξ_2aopt) can be obtained [16]. Thus, the practical parameters of the optimal TMD damping stiffness (k_2aopt) and optimal damping coefficient (c_2aopt) in terms of acceleration are as follows,

(23)f_{\mathrm{2}\mathrm{a}\mathrm{o}\mathrm{p}\mathrm{t}}={\displaystyle \frac{{\left(\mathrm{1}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mu \right)}^{\mathrm{2}}}{\mathrm{1}+\mu }},

(24){\xi }_{\mathrm{2}\mathrm{a}\mathrm{o}\mathrm{p}\mathrm{t}}={\left({\displaystyle \frac{\mu {\left(\mathrm{1}+{\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}\mu \right)}^{\mathrm{2}}}{\mathrm{4}(\mathrm{1}+\mu )\left(\mathrm{1}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mu \right)}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}},

(25)k_{\mathrm{2}\mathrm{a}\mathrm{o}\mathrm{p}\mathrm{t}}={\displaystyle \frac{\mu \left(\mathrm{1}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mu \right)}{{(\mathrm{1}+\mu )}^{\mathrm{2}}}}{k}_{\mathrm{1}},

(26)c_{\mathrm{2}\mathrm{a}\mathrm{o}\mathrm{p}\mathrm{t}}=\left({\displaystyle \frac{{\mu }^{\frac{\mathrm{3}}{\mathrm{2}}}{\left(\mathrm{1}+{\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}\mu \right)}^{\frac{\mathrm{1}}{\mathrm{2}}}}{\mathrm{2}{\xi }_{\mathrm{1}}{(\mathrm{1}+\mu )}^{\frac{\mathrm{3}}{\mathrm{2}}}}}\right)c_{\mathrm{1}},

Comparison of optimal parameters

For structural design where both displacement and acceleration are restricted to certain levels, the equations of f_2dopt and ξ_2dopt are at odds. To evaluate their inconsistencies, the values for f_2dopt, f_2aopt, ξ_2dopt and ξ_2aopt are compared in Fig. 3 when μ varies. It can be observed that the optimal TMD damping ratios for displacement and acceleration reduction are consistent while the frequency ratios show a large discrepancy.

TMD parameters and structural performance for damped systems

Optimal TMD parameters and curve-fitting scheme

Applying the above search procedure to systems with five different critical damping ratios, ξ₁ = 0, 0.01, 0.02, 0.03 and 0.05, the optimal parameters and the corresponding responses for different mass ratios, varying from 0.0 to 0.1, are determined and listed in Tables 1 and 2. Meanwhile, for convenience in future applications, explicit mathematical expressions that correspond to the computed optimal values are determined. From the numerical data of Tables 1 and 2, four parametric closed form formulae are obtained using curve-fitting methods as shown in Eqs. (27) to (30) (for displacement) and Eqs. (31) to (34) (for acceleration), respectively.

(27)\begin{array}{r}{f}_{\mathrm{2}\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}=\mathrm{0.999}-\mathrm{1.238}\mu +\mathrm{0.996}{\mu }^{\mathrm{2}}\\ +(-\mathrm{0.125}-\mathrm{11.36}\mu +\mathrm{55.84}{\mu }^{\mathrm{2}}){\xi }_{\mathrm{1}}\\ +(-\mathrm{2.035}+\mathrm{11.26}\mu -\mathrm{72.89}{\mu }^{\mathrm{2}}){\xi }_{\mathrm{1}}^{\mathrm{2}}\end{array},

(28)\begin{array}{r}{\xi }_{\mathrm{2}\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}=\mathrm{0.0253}+\mathrm{2.236}\mu -\mathrm{9.683}{\mu }^{\mathrm{2}}\\ +(-\mathrm{0.0017}+\mathrm{0.153}\mu -\mathrm{1.08}{\mu }^{\mathrm{2}}){\xi }_{\mathrm{1}}\\ +(\mathrm{0.0258}-\mathrm{2.28}\mu +\mathrm{20.88}{\mu }^{\mathrm{2}}){\xi }_{\mathrm{1}}^{\mathrm{2}}\end{array},

(29)\begin{array}{r}{k}_{\mathrm{2}\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}=\mathrm{1.684}+\mathrm{16630}\mu -\mathrm{33653.5}{\mu }^{\mathrm{2}}\\ +(\mathrm{52.82}-\mathrm{13972.3}\mu -\mathrm{77802}{\mu }^{\mathrm{2}}){\xi }_{\mathrm{1}}\\ +(-\mathrm{116.59}-\mathrm{50078.9}\mu +\mathrm{135929}{\mu }^{\mathrm{2}}){\xi }_{\mathrm{1}}^{\mathrm{2}}\end{array},

(30)\begin{array}{r}{c}_{\mathrm{2}\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{t}}=-\mathrm{2.667}+\mathrm{762.58}\mu +\mathrm{6355.45}{\mu }^{\mathrm{2}}\\ +(\mathrm{0.217}-\mathrm{59.829}\mu -\mathrm{10083}{\mu }^{\mathrm{2}}){\xi }_{\mathrm{1}}\\ +(\mathrm{11.15}-\mathrm{2157.9}\mu -\mathrm{4521}{\mu }^{\mathrm{2}}){\xi }_{\mathrm{1}}^{\mathrm{2}}\end{array}.

The optimal tuning frequency ratios (f_2dopt) and optimal TMD damping ratios (ξ_2dopt) in terms of the displacement response for the five different damped systems are plotted in Fig. 4 with the curves of the closed-form expressions. These figures show that the curves for an undamped system correspond very well with the curves of an undamped system depicted in Fig. 3. It is seen that the higher the system’s damping, the more the optimal tuning frequencies deviate from those in the undamped systems (Fig. 4(a)). However, the values for ξ_2dopt are not so affected by the system damping (Fig. 4(b)). The optimal TMD damping stiffness, k_2dopt and optimal TMD damping coefficient, ξ_2dopt are plotted in Fig. 5.

The optimal parameters and the corresponding absolute acceleration responses for a damped main system and the TMD are determined by the same search procedure described for the case of the displacement response, and these results are listed in Table 2. The optimal frequency tuning ratios (f_2aopt) having different system damping values are displayed in Fig. 6(a), which shows that the deviation due to the system’s damping is less pronounced than that in the case of the displacement and the values are larger than the corresponding values for the displacement case. In the case of an optimal TMD damping ratio (ξ_2aopt), the values are the same as the values for displacement as shown in Fig. 6(b). The optimal parameters of the TMD damping stiffness (k_2aopt) and the TMD damping coefficient (c_2aopt) have larger values than those for the displacement case and are not as influenced by system’s damping (Fig. 7). Using the same curve-fitting method, the four parametric closed-form formulae in terms of acceleration response are derived. These formulae are explicit mathematical expressions suitable for general application.

(31)$$\begin{gathered} f_{2\mathrm{a}\mathrm{o}\mathrm{p}\mathrm{t}}=1.21415-38.5413\mu +279.561{\mu }^2 \\ +\left(489.704-13691.3\mu +96636.7{\mu }^2\right){\xi }_1 \\ +\left(-7114.59+212517\mu -1.51574\times {10}^6{\mu }^2\right){\xi }_1^2, \end{gathered}$$

(32)$$\begin{gathered} {\xi }_{2\mathrm{a}\mathrm{o}\mathrm{p}\mathrm{t}}=0.999699-0.71009\mu +0.295481{\mu }^2 \\ +\left(-0.005514-4.83619\mu +34.8772{\mu }^2\right){\xi }_1 \\ +\left(-0.279593+42.7006\mu -368.949{\mu }^2\right){\xi }_1^2, \end{gathered}$$

(33)$$\begin{gathered} k_{2\mathrm{a}\mathrm{o}\mathrm{p}\mathrm{t}}=0.0253664+2.22923\mu -9.63656{\mu }^2 \\ +\left(-0.00569217+0.708789\mu -6.45248{\mu }^2\right){\xi }_1 \\ +\left(0.0860452-10.2099\mu +97.4772{\mu }^2\right){\xi }_1^2, \end{gathered}$$

(34)$$\begin{gathered} c_{2\mathrm{a}\mathrm{o}\mathrm{p}\mathrm{t}}=0.424251+16824.6\mu -21473{\mu }^2 \\ +\left(44.4569-7028.6\mu +21731.1{\mu }^2\right){\xi }_1 \\ +\left(-532.188+67748\mu -532455{\mu }^2\right){\xi }_1^2. \end{gathered}$$

Structural performance and control effectiveness

To demonstrate the responses of the main system and the TMD, the root-mean-square (RMS) responses with respect to relative displacement ({\mathrm{\sigma }}_{y_{\mathrm{1}}} and {\mathrm{\sigma }}_{y_{\mathrm{2}}}) and absolute acceleration ({\sigma }_{{\ddot{\mathit{x}}}_1} and {\sigma }_{{\ddot{\mathit{x}}}_2}) are computed. These values are calculated as functions of the uncoupled natural frequency ratio (f₂) and the TMD damping ratio (ξ₂). To observe the control effectiveness in reducing the response of the main system and the TMD, four non-dimensional indices of the normalized RMS responses are defined as {\sigma }_{y_1}/{\sigma }_y and {\sigma }_{y_2}/{\sigma }_y (for displacement control) and {\sigma }_{{\ddot{\mathit{x}}}_1}/{\sigma }_{\ddot{\mathit{x}}} and {\mathrm{\sigma }}_{{\ddot{\mathit{x}}}_{\mathrm{2}}}/{\mathrm{\sigma }}_x (for acceleration control). These RMS and the normalized RMS response values of the damped main systems are computed as functions of the five different system damping and mass ratio, as listed in Table 1. The damping ratio and circular frequency of the main system used are ξ₁ = 0.05 and ω₁ = 3.343 rad/s, respectively. It is assumed that the system is subject to a stationary Gaussian white noise excitation with spectral density S₀ = 3.62 × 10^-4m²/s³. Considerable information can be gleaned from the analytical procedure by observing the above responses.

Displacement performance results

Figures 8(a–c) represent the effects of two parameters, f₂ and ξ₂, on the proposed system which has a mass ratio (μ) of 0.02. The figures show that the parameter range in which the system can be effective is relatively narrow for this mass ratio. As can be seen in these figures, good reductions of the main system can be obtained at 0.9-1 for f₂ and 0.05-0.14 for ξ₂. It is clear that there are certain values of f₂ and ξ₂ that lead to a maximum displacement reduction. When f₂ is small, the RMS response of the main system is not greatly affected by the presence of the TMD and it has a peak around the value of one for f₂. From the result of numerical analysis for five different system damping values, the maximum reduction of RMS and normalized RMS values are 0.0143 and 0.821 (18% reduction), respectively for f_2dopt of 0.9534 and ξ_2dopt of 0.0702 (Fig. 9 and Table 1). In the case of the main system, increasing the mass ratio reduces the system response, but for the same mass ratio increment, the heavier optimal damper is less successful in reducing the response (Fig. 9(a)). To investigate the influence of a system’s damping on the effectiveness of the TMD, especially, Fig. 9(b) shows the ratios of the amplitudes of the systems with the optimal TMD dampers as compared to those without the TMD. This figure represents that the TMD is more effective in less damped system. For the system with 2% critical damping with a mass ratio of 0.05, the response is only 55% of the response without the TMD, whereas for the system of 5% critical damping, the response is increased to 75% of that without the TMD.

Another optimization criterion can be the travel of the TMD relative to the structure. The stroke or relative displacement between the main system and the TMD can be described as a function of f₂ and ξ₂. Figures 10(a–c) show the properties of the TMD responses when the mass ratio is 0.02. Around a value of one for f₂, the RMS value of the TMD stroke is increased while the RMS value of the main system is decreased. Also, it can be noted that when ξ₂ is greater than 0.2, a good reduction of the main system response can be attained over a relatively wide range of f₂. When the two parameters have optimal values of 0.9534 and 0.0702 for the maximum reduction of the main system, the RMS stroke is 0.0665 and the normalized RMS stroke is 3.8090 (Fig. 11 and Table 1). The TMD relative response is about 4.7 times the response of the controlled main system.

As another important design parameter, the magnitudes of the TMD stroke and normalized TMD stroke (this mormalizaion ratio is to the response of the main system) are shown from the view point of the RMS responses (Figs. 10 and 11). It can be seen that the effectiveness of the system damping on the reduction of TMD responses is very sensitive for small mass ratios. When the mass ratio is 0.01, the normalized RMS response of the TMD with the system having 2% (ξ₁ = 0.02) critical damping is 2.5 times the response of the TMD with a 0.05 mass ratio (Fig. 11(b)).

Acceleration performance results

In the design of tall buildings, a TMD is usually called upon to improve the serviceability performance of a building which is affected by excessive accelerations experienced at the top floors of the building under the most unfavorable dynamic excitations. This may cause discomfort to the building occupants. Also, due to the fact that high frequency components are involved more in acceleration than in displacement effects, the optimal parameters of the TMD are expected to be different from the displacement control parameters. Therefore, a parametric study of the system with a TMD should be based not only on building displacement responses but also on acceleration responses. However, in the conventional design of a TMD, the parameter selection is usually based on the displacement responses alone. An examination of the acceleration response effect is therefore necessary as well.

The optimal parameters can also be determined to minimize the acceleration response of the main system. When the mass ratio (μ) is 0.02 and the system damping (ξ₂) is 0.05, the effects of the various parameters on the responses of the RMS response and normalized RMS response in terms of the main system acceleration look similar to the case of the displacement as shown in Figs. 12(a) and (b). The parameter range for the greater reduction is 0.93-1 for f₂ and 0.05-0.12 for ξ₂. The shape of the frequency tuning diagram is relatively sharper and ξ₂ has a optimal value between 0.05 and 0.1. The numerial results for five different mass ratios show that the maximum reductions of the RMS and the normalized RMS response are 0.1587 and 0.81 (19% reduction) at the optimal parameters of 0.9820 and 0.072 respectively (Fig. 13 and Table 2).

Table 2 shows that the results of several optimized acceleration responses using optimal f₂ and ξ₂ parameters for the five mass ratios. As expected, the mass ratio of the TMD to the main system (μ) increases the RMS displacement of the main system and the TMD decreases with each of the two optimal parameters, f_2dopt and ξ_2dopt. This means that a greater effectiveness of the TMD requires a larger mass ratio. To reduce the acceleration of the main system, the TMD acceleration should be increased using the corresponding optimal parameters with respect to acceleration. These properties for the case of the mass ratio of 0.02 are shown in Figs. 14(a–c). Using the above optimal parameters, the RMS and the normalized RMS acceleration for the damped main system and the TMD are computed and displayed as functions of the system damping and mass ratio in Table 2 and Figs. 15(a) and (b). The figures indicate that the optimal tuning frequency ratio (f_2aopt) is strongly influenced by the damping level of the system with regard to the system’s response. The numerical results show that when the system has a high damping, the TMD becomes less effective in reducing the system’s acceleration. For a system with 2% critical damping with a mass ratio of 0.05, the response is only 53% of the response without the TMD, whereas for the system with 5% critical damping, the response is increased to 72% of that without the TMD (Fig. 15(a) and Table 2). It can be seen that the influence of system’s damping on the reduction of the TMD response is very sensitive for small mass ratios. When the mass ratio is 0.01, the normalized RMS response of a TMD with 2% critical system damping is 2.7 times the response of the TMD with a 0.05 mass ratio (Fig. 15(b)). Overall, the acceleration results indicate that a TMD is more effective in controlling acceleration responses than displacement responses for the same mass ratio.

Conclusion

This study presented a process for designing optimal TMD systems, highlighting the optimized parametric configuration, to enable optimal displacement and acceleration control of structures. From a series of parametric studies examining several response criteria, the following numerical and quantitative results were obtained.

• The optimal parameters for an undamped system can be determined by analytical solution, whereas a numerical search procedure is needed for a damped system and closed-form expressions can be obtained from the above numerical results for future practical applications.

• The optimal parameters with respect to displacement and acceleration can be derived independently, and the optimal frequency tuning ratio (f₂) is strongly influenced by the level of critical damping in the main system, whereas the optimal TMD damping ratio (ξ₂) is insensitive to the level of critical damping in the system.

• The TMD is less effective in a damped system with regard to displacement responses, so when the system has a high fraction of critical damping, the TMD becomes less effective in reducing the system’s response.

• For a given level of critical damping in the structure, increasing the mass ratio makes the TMD more effective in reducing structural response. This implies that a system with higher intrinsic damping requires a TMD with a larger mass ratio to provide similar reduction to that needed for a system with lower damping.

• For the SDOF system, the TMD is more effective in reducing the acceleration response than the displacement response, particularly for the high mass ratio TMD.

For the advanced application of TMD principle, the concept of TMD can be extended to convert a structural system into a TMD system by specially designing the structural system. To overcome the limitation of the added mass, as an alternative approach, a structure’s partial portion itself can be considered to be the tuned mass saving excessive non-functional added weight. This configuration supplies reaction forces to the main structure, generated by the relative motion between the structure and the isolated partial portion. Some structural control studies by using or changing structural configuration have been developed [22-25], thus the systematic and parametric results of this paper could supply the useful information for the effective optimal design for the relevant modified TMD principle-based structures.

References

[1] Kwok K C S, MacDonald P A. Wind-induced response of Sydney tower. In: Proceedings of First National Structural Engineering Conference, Sydney, Austrailia, 1987, 19-24

[2] Kwok K C S, Samali B. Performance of tuned mass dampers under wind loads. Engineering Structures, 1995, 17(9): 655-667

[3] Lu X, Chen J. Mitigation of wind-induced response of Shanghai Center Tower by tuned mass damper. Structural Design of Tall and Special Buildings, 2011, 20(4): 435-452

[4] Chung L L, Wu L U, Yang C S W, Lien K U, Lin M C, Huang H H. Optimal design formulas for viscous tuned mass dampers in wind-excited structures. Structural Control and Health Monitoring, 2012, (Accepted)

[5] Villaverde R, Koyama L A. Damped resonant appendages to increase inherent damping in buildings. Earthquake Engineering & Structural Dynamics, 1993, 22(6): 491-507

[6] Sadek F, Mohraz B, Taylor A W, Chung R M. A method of estimating the parameters of tuned mass dampers for seismic applicatins. Earthquake Engineering & Structural Dynamics, 1997, 26(6): 617-635

[7] Wong K K F, Chee Y L. Energy Dissipation of Tuned Mass Dampers during Earthquake Excitations. Structural Design of Tall and Special Buildings, 2004, 13(2): 105-121

[8] Ueng J M, Lin C C, Wang J F. Practical design issues of tuned mass dampers for torsionally coupled builings under earthquake loadings. Structural Design of Tall and Special Buildings, 2008, 17(1): 133-165

[9] Ikago K, Saito K, Inoue N. Seismic control of single-degree-of-freedom structure using tuned viscous mass damper. Earthquake Engineering & Structural Dynamics, 2012, 41(3): 453-474

[10] Bernal D. Influence of ground motion characteristics on the effectiveness of tuned mass dampers. 11th World Conference on Earthquake Engineering, 1996, Paper No. 1455

[11] Soto-Brito R, Ruiz S E. Influence of ground motion intensity on the effectiveness of tuned mass dampers. Earthquake Engineering & Structural Dynamics, 1999, 28(11): 1255-1271

[12] Murudi M M, Mane S M. Seismic effectiveness of tuned mass damper (TMD) for different ground motion parameters. 13th World Conference on Earthquake Engineering, 2004, Paper No. 2325

[13] Den Hartog J P. Mechanical Vibrations. New York: McGraw-Hill, 1956

[14] Crandall S H, Mark W D. Random Vibration in Mechanical Systems. Academic Press, 1963

[15] Randall S E, Halsted D M, Taylor D L. Optimum vibration absorbers for linear damped systems. Journal of Mechanical Design- Transactions of the ASME, 1981, 103(4): 908-913

[16] Warburton G B. Optimum absorber parameters for various combinations of response and excitation parameters. Earthquake Engineering & Structural Dynamics, 1982, 10(3): 381-401

[17] Hoang N, Fujino Y, Warnitchai P. Optimal tuned mass damper for seismic applications and practical design formulas. Engineering Structures, 2008, 30(3): 707-715

[18] De Angelis M, Perno S, Reggio A. Dynamic response and optimal design of structures with large mass ratio TMD. Earthquake Engineering & Structural Dynamics, 2012, 41(1): 41-60

[19] Ziyaeifar M, Gidfar S, Nekooei M. A model for mass isolation study in seismic design of structures, Structural Control and Health Monitoring, 2012 (first published online)

[20] Moutinho C. An alternative methodology for designing tuned mass dampers to reduce seismic vibrations in building structures, Earthquake Engineering & Structural Dynamics, 2012 (first published online)

[21] Wolfram S. Mathematica — A System for Doing Mathematics by Computer. 2nd ed. Addison-Weley Publishing Company Ltd, 1991

[22] Charng P H. Base isolation for multistory building structures. <DissertationTip/> Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, 1998

[23] Pan T C, Cui W. Response of segmental buildings to random seismic motions. ISET Journal of Engineering Technology, 1998, 35(4): 105-112

[24] Kobayashi M, Koh, T. Earthquake response prediction and rationalization of dynamic design of mid-story isolated buildings. Journal of Structural Construction Engineering, Architecture Institute of Japan, 2005, No. 592: 31-57 (in Japanese)

[25] Chey M H, Chase J G, Mander J B, Carr A J. Semi-active tuned mass damper building systems: application. Earthquake Engineering & Structural Dynamics, 2010, 39(1): 69-89

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