Control efficiency optimization and Sobol’s sensitivity indices of MTMDs design parameters for buffeting and flutter vibrations in a cable stayed bridge

Abstract

This paper studies optimization of three design parameters (mass ratio, frequency ratio and damping ratio) of multiple tuned mass dampers MTMDs that are applied in a cable stayed bridge excited by a strong wind using minimax optimization technique. ABAQUS finite element program is utilized to run numerical simulations with the support of MATLAB codes and Fast Fourier Transform FFT technique. The optimum values of these three parameters are validated with two benchmarks from the literature, first with Wang and coauthors and then with Lin and coauthors. The validation procedure detected a good agreement between the results. Box-Behnken experimental method is dedicated to formulate the surrogate models to represent the control efficiency of the vertical and torsional vibrations. Sobol’s sensitivity indices are calculated for the design parameters in addition to their interaction orders. The optimization results revealed better performance of the MTMDs in controlling the vertical and the torsional vibrations for higher mode shapes. Furthermore, the calculated rational effects of each design parameter facilitate to increase the control efficiency of the MTMDs in conjunction with the support of the surrogate models.

Keywords

MTMDs / power spectral density / fast Fourier transform / minimax optimization technique / Sobol’s sensitivity indices / Box-Behnken method

Introduction

Long-span bridges are very sensitive to wind effects. The large dimensions and flexibility result in experiencing very long fundamental periods, which affect their dynamic behavior. The Tacoma incident is the best example case of a wind induced failure. Therefore, numerous computational methods have been designed which can effectively handle material and structural failure [ 1– 54]. However significant numbers of bridges have experienced extreme responses which were of sufficiently large amplitudes to be considered at serious condition. The aerodynamic instability which is often a classical two-degree of freedom heave pitch flutter with onset at a critical wind speed, aerodynamic buffet by lift and moment forces induced by turbulence in the oncoming wind is the most important wind induced phenomenon. When the free span of a bridge increases, the aerodynamic instability increases. The critical wind speed would step down due to the resulting reduced torsional stiffness particularly and the reduced bending stiffness. Generation of a negative aero-elastic damping becomes a matter of fact if further increase in the wind speed occurs, as a result a steady increase in the vibration with unlimited amplitudes until the failure of structure. The flutter stability of bridge structures is due to the critical wind speed [ 55– 70, 148– 150].

Fast increase of bridge spans led to undertake research on controlling wind-induced vibration in long span bridges. Many research efforts have been done to improve aerodynamic stabilities and to suppress excessive buffeting vibrations in long span bridges both at the construction and at service time. The solution for the buffeting and flutter vibrations control in long-span bridges is mainly related to the use of passive devices, dynamic energy absorbers such as TMDs, which have been studied to mitigate serious dynamic buffeting vibration or to enhance the flutter stability of long span bridges. These control devices that are called dynamic energy absorbers, dissipate external energy through supplying damping to the designated mode shapes of vibrations. Increasing the lengths of the bridge span and adopting slender decks tend to make the frequencies of the mode shapes of vibrations close to each other, which results in increasing the modal coupling effects via aero-elastic effects in strong wind cases. The effects of modal coupling resulted from a strong wind may lead to a significant additional component to the buffeting vibration of each certain mode, compared with the modal coupling effects resulted from a weak wind. There is a limitation imposed on the application of TMD because it is effective in suppressing vibrations in one mode only, usually it is the first mode. Furthermore, a TMD is efficient in a narrow frequency range only, this exactly when it is tuned to a certain natural frequency of the structural system and it doesn’t act efficiently if the system manifest many narrow natural frequencies [ 1– 32, 62– 66].

Many researchers have studied the application of TMD system in long span bridges. Jain et al. (1998) analyzed the effects of modal damping on bridge performance of aero-elasticity. It was found that supplemental damping provided through appropriate external dampers could certainly increase the flutter stability and reduce the buffeting response of long-span bridges. Nobuto et al. (1998) made a study on flutter control using a couple of TMDs, and the numerical example indicated its efficiency. On this basis, a more advanced parametric study was performed by Gu et al. (1998) through a theoretical analysis and a wind tunnel test on the Tiger-gate bridge model. Lin, et al. (2000) studied the effect of TMD system in the reduction of torsional and vertical responses of suspension bridges subjected to wind loading. The important parameters involved are the natural frequency ratio and the mass ratio of damping device to the structure. They used a TMD system with two degrees of freedom, vertical and torsional. They obtained a TMD mass ratio of 2% for getting a reduction of 25% and 33% in vertical and torsional responses of the bridges, respectively. Considerable effort has been directed toward the reduction of the mass ratio to an acceptable level, say less than 1%. Weight penalty and precise tuning of frequency are major considerations in their application. Wang et al. (2014) studied the optimum control of buffeting displacement in the Sutong Bridge using MTMDs. They discovered that the mass ratio and damping ratio parameters have a significant effect in controlling the vertical vibration of the deck. They obtained a mass ratio of 2% reduces 29% and a damping ratio of 3% reduces 27.5% of the vertical response of the Sutong Bridge [ 71– 85].

The most of previous researches have studied the application of MTMDs to optimize the design parameters considering the first lower natural frequencies of the long span bridges, in the same time the performance of the MTMDs is still limited and hasn't controlled the vibrations of the deck totally. It is necessary to search this sensitive case for higher mode shapes of vibrations so that to optimize and to enhance the performance of the MTMDs in controlling both the vertical and the torsional vibrations of the deck, in addition to the need of formulating surrogate models to represent the vibrations control operation, also to calculate the rational effects of each design parameter comprising the main and the interaction effects between these parameters. Thus in this study, the application of MTMDs to control the vertical and torsional vibrations of the deck in a cable stayed bridge for higher mode shapes of vibrations is studied by using a model of the cable stayed bridge with the MTMDs generated in ABAQUS finite element program to run numerical simulations dedicated to commence the optimization and the sensitivity analysis procedures. Optimization and sensitivity analysis including stochastic optimization has been carried out in many areas [ 86– 96] and an excellent open source program for sensitivity analysis and uncertainty quantification has been made available in Ref. [ 97].

TMDs

The concept of TMD application dates back to the 1940s (Den Hartog 1947). This device is a linear dynamic vibration absorber system, consists of a secondary mass suspended via a viscous damper and a spring from a point on the main structure. It is tuned to a certain structural frequency of the structure such as a bridge in order to, when excited, the damper will resonate out of phase with the motion of the bridge, i.e., the TMD mass oscillates in the opposite direction of the main structure. Nowadays, establishment of this system succeeded in reducing wind-induced vibrations in many structural systems due to their particularity, where they need less construction and maintenance cost, in addition to that their operation is easy when used in the mechanical and civil engineering systems. These devices are most effective when positioned at places of highest amplitudes. Therefore, considering this feature supplies the opportunity for energy dissipation through the damper inertia force acting on the main system. There are two main functions carried out by the TMD system, first it decreases the resonant response of the main structure, and secondly, increases the overall damping of the structure by the dashpot attached to, supplying additional source of energy dissipation. The modal damping is affected by the mass ratio within low frequency ratio range, and this effect almost finishes in high frequency ratio cases. The damping ratio has a great effect on the response of cable-stayed bridges, and it is emphasized that under a particular combination of mass ratio, frequency ratio and the difference in the modal damping ratios, the effect of non-classical damping is highly important [ 63– 64, 80, 98– 116].

Energy dissipation mechanism

Fluid-structure interaction between the exciting wind and the long span bridge generates aerodynamic instabilities. As a result the bridge deck would be prone to vertical and torsional vibrations. The deck motion is due to feeding energy into the bridge system by the the wind forces during one cycle of oscillation. The energy is exchanged by the phase shift between the vertical and the torsional vibrations and it counteracts the energy absorption by structural damping of the bridge. When the structural damping of the bridge is sufficiently low which can't dissipate these vibrations, the application of TMD becomes one of the solutions to suppress these vibrations and boost the structural damping of the bridge system. The energy dissipation mechanism starts when the bridge deck vibrates, with the presence of the TMD, this device would be excited by the motion of the deck. When the mass of the damper moves in a certain direction, as the movement of the deck would be toward the opposite direction, hence the damping of the deck vibration is accomplished. The kinetic energy of the bridge structure transfers into the TMD system where its viscous damper absorbs it. To attain the most efficient energy absorbing rate, the natural period of the TMD is tuned to the natural period of the bridge system. This is why it is called (Tuned Mass Dampers), so the strategy of TMD application is attempting to dissipate the energy responsible of feeding the vibration. The viscous damper of the TMD should be adjusted to optimum value so that to maximize absorption of the energy [ 29– 42, 61, 67, 117].

MTMDs

The performance of TMD application in long span bridges is sensitive to the difference of frequency ratios between the TMD and the bridge structure, even when optimally designed, leads to serious performance deterioration this when the dynamic characteristics of the structure are different from those used to accomplish its optimum design. This can be overcome by the use of MTMDs. One of the important parameter for MTMDs is the mass distribution to reduce effectively the dynamic response of a main system. When the mass distribution is controlled along with other design parameters like damping ratio, frequency range, number of dampers, the control of the main system response is achieved. To increase the robustness of the vibration control system to various uncertainties in the structure, the use of more than one TMD is an effective solution, where MTMDs have distributed natural frequencies around the controlled mode of vibration. Adopting wide frequency band width makes it more robust and less sensitive to the change of the frequency ratio than using TMD (Igusa and Xu, 1994), in addition they are suggested to mitigate the vertical and torsional vibrations and to boost the aerodynamic stability of the long span bridges, which permits to control of more than one mode of a multi degree of freedom (MDOF) structure by tuning each TMD to the corresponding vibration mode of the main structure, or by tuning the TMDs to frequencies close to a certain mode of the system. Though, in spite of their higher effectiveness and robustness compared to single TMDs, the use of MTMDs encounters lack capabilities of real time retuning, consequently their adaptation to frequency varying excitation is very hard. The optimum TMD system for a certain structure is that one which minimizes the expected vibrations amplitudes. Practically, the design of a TMD begins by setting the value of the mass ratio, usually set within 1% to 10% (Warburtun and Ayorinde, 1980). The selection of this parameter value depends on the space availability to place the TMDs. Despite the fact that use of high mass ratio values aid to further decrease the vibration amplitudes, it may not be reasonable or economic because additional weight of the TMDs necessitates strengthening other structural elements. As soon as the value of the mass ratio has been elected, other design parameters such as frequency ratio and damping coefficients of the viscous dampers of the TMDs require to be assigned [ 65, 67, 107, 118– 125].

Concept of TMD using two-mass system

The equation of motion for primary mass as shown in Fig. 1 is:

(1) (\mathrm{1}-\mu )\ddot{u}+\mathrm{2}\xi \omega m\dot{u}+{\omega }^{\mathrm{2}}u={\displaystyle \frac{p}{m}}-\mu {\ddot{u}}_d,

\mu is defined as the mass ratio,

(2) \mu ={\displaystyle \frac{m_d}{m}},

where m_d is the mass of the damper and m is the primary mass.

(3) {\omega }^{\mathrm{2}}={\displaystyle \frac{k}{m}},

(4) c=\mathrm{2}\xi \omega m,

(5) c_d=\mathrm{2}{\xi }_d{w}_d{m}_d,

where \dot{u} is the velocity, \ddot{u} is the acceleration, {\xi }_d is the damping ratio of the mass damper {\omega }_d is the natural frequency of the mass damper, ξ, ω, c, k are the damping ratio, natural frequency, damping coefficient and stiffness of the primary mass respectively.

The equation of motion for tuned mass is given by:

(6) {\ddot{u}}_d+\mathrm{2}{\xi }_d{\omega }_d{\dot{u}}_d+{\omega }_d^{\mathrm{2}}{u}_d=-\ddot{u}.

The purpose of adding the mass damper is to control the vibration of the structure when it is subjected to a particular excitation. The mass damper is having the parameters; the mass m_d , stiffness k_d, and damping coefficient c_d. The damper is tuned to the fundamental frequency of the structure such that:

(7) {\omega }_d=\omega ,

(8) k_d=\mu k.

The primary mass is subjected to the following periodic sinusoidal excitation

(9) p=\widehat{p}\sin \Omega t,

then the response is given by

(10) u=\widehat{u}\sin (\Omega t+{\delta }_{\mathrm{1}}),

(11) u_d={\widehat{u}}_d\sin (\Omega t+{\delta }_{\mathrm{1}}+{\delta }_{\mathrm{2}}),

where \widehat{u} and \delta denote the displacement amplitude and phase shift, respectively. The critical loading scenario is the resonant condition. The solution for this case has the following form:

(12) \widehat{u}={\displaystyle \frac{\widehat{p}}{\overline{m}\overline{k}}}\sqrt{\mathrm{1}/(\mathrm{1}+(\mathrm{2}\xi /\mu +\mathrm{1}/\mathrm{2}{\xi }_d\left)\mathrm{2}\right)},

(13) {\widehat{u}}_d=(\mathrm{1}+\mathrm{2}{\xi }_d)\widehat{u},

(14) \tan {\delta }_{\mathrm{1}}=-(\mathrm{2}{\displaystyle \frac{\xi }{\mu }}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\xi }_d}}),

(15) \tan {\delta }_{\mathrm{2}}=-{\displaystyle \frac{\pi }{\mathrm{2}}}.

The above expression shows that the response of the tuned mass is 90° out of phase with the response of the primary mass. This difference in phase produces the energy dissipation contributed by the damper inertia:

(16) \widehat{u}={\displaystyle \frac{\widehat{p}}{k}}\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}\xi }}\right),

(17) {\delta }_{\mathrm{1}}=-{\displaystyle \frac{\pi }{\mathrm{2}}}.

To compare these two cases, we can express Eq. (12) in terms of an equivalent damping ratio:

(18) \widehat{u}={\displaystyle \frac{\widehat{p}}{k}}\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\xi }_e}}\right),

where

(19) {\xi }_e={\displaystyle \frac{\mu }{\mathrm{2}}}\sqrt{{\displaystyle \frac{\mathrm{1}}{\mathrm{1}}}+{(\mathrm{2}{\displaystyle \frac{\xi }{\mu }}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\xi }_d}})}^{\mathrm{2}}}.

Equation (19) shows the relative contribution of the damper parameters to the total damping. Increasing the mass ratio magnifies the damping. However, since the added mass also increases, so there is a practical limit on it.

Equation of motion

Based on the classical method of flutter analysis and relating to the vertical bending and torsion of a bridge, the frequencies of the TMDs are tuned to the neighborhood of the flutter frequency so as to increase the critical flutter wind speed as shown in Fig. 2.

The equation of motion of multi-degree of freedom structure TMDs which consists of structure, TMD1 and TMD2 motion attached to h^th and k^th degree of freedom subjected to wind load is given in Eq. (20). Besides, Eq. (21) presents the modal analysis expressions.

(20) m\ddot{u}+c\dot{u}+ku=p,

(21) u=\left\{\begin{array}{c}\hfill u_s\hfill \\ \hfill u_{t\mathrm{1}}\hfill \\ \hfill u_{t\mathrm{2}}\hfill \end{array}\right\}=Ax=\left[\begin{array}{ccccc}\hfill {\phi }_{\mathrm{1}}\hfill & \cdots & {\phi }_n& \mathrm{0}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{1}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right]\left\{\begin{array}{c}\hfill x_s\hfill \\ \hfill x_{t\mathrm{1}}\hfill \\ \hfill x_{t\mathrm{2}}\hfill \end{array}\right\},

where u_s is a vector of absolute displacement of the main structure; u_t1 and u_t2 are absolute displacement of TMD1 and TMD2 respectively; x_s is a vector of generalized-coordinate displacement of main structure; x_t1 and x_t2 are generalized displacement of TMD1 and TMD2 respectively; and {\phi }_n is an n^th natural mode shape of the main structure without TMD, normalizing such that modal mass= 1.

Substitute Eq. (21) into Eq. (20) and pre-multiply with transpose of A gives the equation of motion in modal coordinates, as shown in in Eq. (22).

(22) K\ddot{x}+C\dot{x}+Kx=P,

where

(23) M=\left[\begin{array}{ccccc}\hfill \mathrm{1}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill \mathrm{0}\hfill & \cdots & {\mathrm{1}}_n& \mathrm{0}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& m_{t\mathrm{1}}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{0}& m_{t\mathrm{1}}\end{array}\right]=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{1},\mathrm{1},\mathrm{...},{\mathrm{1}}_n,m_{t\mathrm{1}}{m}_{t\mathrm{2}}),

(24) C=A^{\mathrm{T}}cA=\left[\begin{array}{ccccc}\hfill {\phi }_{\mathrm{1}}^{\mathrm{T}}{C}_s{\phi }_{\mathrm{1}}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill \mathrm{0}\hfill & \cdots & {\phi }_n^{\mathrm{T}}{C}_s{\phi }_n& \mathrm{0}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right]+C_{t\mathrm{1}}\left[\begin{array}{ccccc}\hfill {\phi }_{\mathrm{1}}{\phi }_{\mathrm{1}h}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill {\phi }_{nh}{\phi }_{nh}\hfill & \cdots & {\phi }_{nh}{\phi }_{nh}& -{\phi }_{nh}& \mathrm{0}\\ \hfill -{\phi }_{nh}\hfill & \cdots & {\phi }_{nh}& \mathrm{1}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right]+C_{t\mathrm{1}}\left[\begin{array}{ccccc}\hfill {\phi }_{\mathrm{1}k}{\phi }_{\mathrm{1}k}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill {\phi }_{nk}{\phi }_{nk}\hfill & \cdots & {\phi }_{nk}{\phi }_{nk}& -{\phi }_{nk}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill -{\phi }_{nk}\hfill & \cdots & {\phi }_{nh}& \mathrm{1}& \mathrm{0}\end{array}\right],

(25) K=A^{\mathrm{T}}kA=\left[\begin{array}{ccccc}\hfill {\phi }_{\mathrm{1}}^{\mathrm{T}}{k}_s{\phi }_{\mathrm{1}}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill \mathrm{0}\hfill & \cdots & {\phi }_n^{\mathrm{T}}{k}_s{\phi }_n& \mathrm{0}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right]+k_{t\mathrm{1}}\left[\begin{array}{ccccc}\hfill {\phi }_{\mathrm{1}}{\phi }_{\mathrm{1}h}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill {\phi }_{nh}{\phi }_{nh}\hfill & \cdots & {\phi }_{nh}{\phi }_{nh}& -{\phi }_{nh}& \mathrm{0}\\ \hfill -{\phi }_{\mathrm{1}h}\hfill & \cdots & {\phi }_{nh}& \mathrm{1}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right]+k_{t\mathrm{1}}\left[\begin{array}{ccccc}\hfill {\phi }_{\mathrm{1}k}{\phi }_{\mathrm{1}k}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill {\phi }_{nk}{\phi }_{nk}\hfill & \cdots & {\phi }_{nk}{\phi }_{nk}& -{\phi }_{nk}& \mathrm{0}\\ \hfill \mathrm{0}\hfill & \cdots & \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \hfill -{\phi }_{nk}\hfill & \cdots & {\phi }_{nk}& \mathrm{1}& \mathrm{0}\end{array}\right],

where m_ti,c_ti, k_ti (i=1,2) represent mass, damping coefficient and stiffness of TMD1 and TMD2 respectively.

To achieve the most efficient response suppression of the Bridge deck, the optimal damping ratio of TMD should be considered and to simplify the analysis, structural damping to be neglected. The optimum damping ratio and optimum tuned parameter of randomly vibration can be calculated by Eq. (28) and Eq. (29) respectively (Ayorinde and Warburton, 1980), where \mu is ratio of TMD mass by structure mass Eq. (26) and tuned parameter Eq. (27).

(26) \mu ={\displaystyle \frac{m_t}{m_s}},

(27) f_{t,opt}={\displaystyle \frac{{\omega }_t}{{\omega }_s}},

(28) {\xi }_{t,opt}=\sqrt{{\displaystyle \frac{\mu (\mathrm{3}\mu +\mathrm{4})}{\mathrm{8}(\mu +\mathrm{1})(\mu +\mathrm{2})}}},

(29) f_{t,opt}=\sqrt{{\displaystyle \frac{\mu +\mathrm{2}}{\mathrm{2}{(\mu +\mathrm{1})}^{\mathrm{2}}}}}.

Finite element model of the TMD

The model of the TMD is created in ABAQUS as a steel mass structure having material properties of mass density 7800 kg/m3 and elastic properties of Young’s modulus 200+ E9 Pa and Poisson’s ratio 0.3, with a rectangular shape and dimensions (4×4×0.24) m. The TMDs are meshed with standard linear 3D stress C3D8R element type. They are installed and attached to the slab of the segmental deck of the cable stayed bridge inside the hollows, and they are connected to the bridge deck on both sides and in the center of the deck so as to provide both point-wise control forces and control torques. This distribution pattern allows controlling both the vertical and torsional motions. The TMD is attached to the slab of the deck by a spring with a variable stiffness and a dashpot having variable damping coefficient depending on the TMD design parameters.

The cable stayed bridge model is 324 m length and 22 m width, the main parts of the bridge is the deck which is consist of connected reinforced concrete deck segments with 2.6 m height. Four reinforced concrete pylons with square shapes 4×4 m dimensions and 103 m height, 80 stay cables are connecting the deck to the pylons in a fan shape arrangement, each cable with cross section area 0.00785 m². The main steel bar diameter is 0.06 m and the diameter of the temperature steel bars in addition to the stirrups are 0.04 m. The pylons are fixed at the bottom and each two pylons are connected by six reinforced concrete ties with 4×4 m dimensions and 22 m length, Fig. 3shows the cable stayed bridge model with the TMDs.

The wind forces on TMDs are assumed to be negligible because the TMDs are installed inside the segmental deck. It is assumed that the mass ratio of the TMDs to the equivalent bridge mass is very small, therefore, the attachment of TMDs does not introduce a meaningful change to the static equilibrium of the bridge, and the structural mode shapes remain the same as those of the original cable stayed bridge without the TMDs.

Wind load and vibration mode shapes

A strong design wind speed used in all simulations which was 54 m/s and the frequency of the excitation is in such away so that to produce a frequency matching higher mode shapes of vibrations in the cable stayed bridge model with coupling effects. The wind load assigned in the simulation is with duration of 30 s (see Fig. 4). The wind fluctuation data has been prepared depending on exact data from the literature and modification of the frequency of excitation [ 126].

The cable stayed bridge model is divided into five regions along its height considering wind pressure variation. The wind attack angle is 25° perpendicular to the longitudinal axis of the cable stayed bridge. Frequency analysis has been undertaken to identify the first 20 mode shapes of vibrations for the model. In this work we concentrate on the vertical and the torsional vibrations only. Table 1 contains the frequency magnitude and the type of vibration for eight mode shapes for analysis.

Mode shapes analysis

The process of designing the TMDs to control the vertical and torsional vibrations of the deck in the cable stayed bridge needs to investigate the frequency analysis to identify the mode shapes of vibrations, in particular the natural frequency of a certain mode shape that is equal or is near to the frequency of the exciting wind. The natural frequency of the dominant mode of vibration is considered in the design process to investigate the optimization of TMDs parameters. Figure 5 shows eight mode shapes of vibrations.

The vertical and torsional displacement time-histories at the mid span both for the center and the outer edge are transformed into frequency domain signals using the Fast Fourier Transform technique by utilizing MATLAB codes, and the corresponding power spectral density PSDs of the time-histories are shown in Fig. 6. The energy of the vertical vibrations of the bridge deck are concentrated on 0.7018 Hz, which is close to the sixth mode vibration natural frequency of the cable stayed model; and the energy of the torsional vibrations is centered on 0.7985 Hz, which is close to the seventh mode vibration natural frequency. Results indicate that the vertical responses of the cable stayed bridge model are dominated by the sixth mode, and the torsional responses are dominated by the seventh mode, and it is clear that the two dominant modes are similar in their natural frequencies to a big content. Thus, in this paper the sixth mode shape coupled (vertical-torsional) with natural frequency of 0.776 Hz would be the dominant vibration which would be based on to assign the TMDs design data so that to get the optimum effect in suppressing the vertical and torsional vibrations.

Optimization of TMD parameters

TMD parameters are found using minimax optimization technique proposed by Tsai and Lin. This technique to calculate the optimum values of frequency ratio and damping ratio for the specified mass ratio is an iterative numerical search. For a fixed value of frequency ratio, the maximum displacements for different values of damping ratios are found. Then the minimum values are selected from the maximum displacements of response, which is the minimax displacement for that value of frequency ratio. Then the above procedure is repeated for different values of frequency ratio to find the minimax value of each frequency ratio. Finally, the smallest minimizes are selected and corresponding frequency ratio and damping ratio are the optimum parameters of the system having specified mass ratio. The numerical simulations for all cases are implemented using ABAQUS finite element program.

Optimum mass ratio

When TMDs are designed, the mass ratio is an important parameter that is very sensitive to design the overall weight and the number of these TMDs in order to get the most optimized results for vibration mitigation. This parameter is the ratio of the TMDs total weight to the total weight of the cable stayed bridge. Many mass ratios are considered which are (0.25%, 0.75%, 1.25%, 1.75% and 2.25%). The dimensions of each TMD are designed (4×4×0.24) m, they are distributed inside the three hollows of the segmental deck, and due to restriction of place availability they are positioned parallel along the center line of the cable stayed bridge model in a symmetric pattern starting from the mid span region in the left and right directions, so every mass ratio situation will take a certain number of TMDs (3 numbers, 9 numbers, 15 numbers, 21 numbers and 27 numbers) respectively for each mass ratio case. Table 2 details the design data of the TMDs considering the mode shape 6 with natural frequency of 0.776 Hz and constant frequency ratios of 1.0 and constant damping ratios of 0.05 which have been used in the simulations. These situations are compared to the situation of the cable stayed bridge model excited by the wind without the application of TMDs so that to compare the result to get the effect of mass ratio on the vertical and torsional vibrations of the deck.

Results and discussion

Considering the vertical vibration, the results confirm that the mass ratio of 2.25% for designing the TMDs is the optimum case to control the vertical vibration in this model. The data are calculated for the point at the center above the deck at the mid span. Despite the fact that the TMDs are in the beginning of the wind excitation are increasing the vertical vibration, it needed approximately 10 s in order to start damping the vertical vibration (buffeting). This case has an obvious effect in reducing the vertical vibration that reached 34.76% of its magnitude without adding TMDs (see Fig. 7). The case of minimum mass ratio 0.25% increased 33.20% of the vertical vibration, while the mass ratio of 0.75% reduced 12.77% of the vertical vibration, and the mass ratio 1.25% increased 9.24% of the vertical vibration. Finally the mass ratio 1.75% has reduced the vertical vibration 15.12% of the vertical vibration (see Table 3). This means that there is a nonlinear relation in the vertical vibration mitigation due to the increase of TMDs mass ratio, and due to lack of place availability inside the segmental deck and load effect in generating stresses due to the application of huge amounts of masses of the TMDs, the mass ratio application should be limited to a desired quantity.

Relating to the torsional vibration of the deck, it was detected that the increase of the TMDs mass ratio decreases the torsional (flutter) vibration to a significant amount as follows: for mass ratio 0.25% decreased 13.72% and for mass ratio 0.75% decreased 33.80%, and for mass ratio 1.25% decreased 39.59% while for mass ratio 1.75% decreased 50.81% and finally for mass ratio 2.25% decreased 58.47%, (see Fig. 8 and Table 4). This is an indication that the weight of the TMDs helps to balance the torsional vibration because the TMDs are opposing the torsional vibration by their mass moments in the opposite direction. This means that increasing mass ratio has a significant effect in mitigating the torsional vibration of the deck.

From the results above, it is clear the importance of using TMDs with the mass ratio 2.25% to secure both the mitigation of the vertical vibration to 34.76% and the torsional vibration to 58.47%.

Simulation of the models

The mass ratio effect on the vertical and torsional vibrations of the cable stayed bridge is simulated in ABAQUS and the following screen shots (see Fig. 9) show the situations of vibrations due to wind excitation at time history of 30 s. The location of data collection is at the mid span of the deck first at the center point and the second at the right and left outer edges, where the first point is a reference to calculate the change in the vertical displacement in each case and the second two points are used to calculate the change in the torsional displacement.

Optimum frequency ratio

Another parameter of designing TMDs is the frequency ratio, which is the ratio of the TMD frequency and the natural frequency of the structure. The frequencies of the TMDs are tuned to the natural frequency of the cable stayed bridge or near this frequency that belongs to the dominant mode shape of vibration. In this paper as mentioned in the previous section, the dominant mode of vibration is the mode shape 6, so the frequencies of the TMDs are tuned to 0.776 Hz which has been calculated from the frequency analysis. Five frequency ratios cases which are (0.8, 0.9, 1.0, 1.1 and 1.2) would be utilized to detect the optimum reduction of the vertical and torsional vibrations. Table 5 shows the magnitudes of the stiffness and the damping coefficient of the TMDs for constant damping ratios of 0.05 and constant mass ratios of 2.25% which was the best case to reduce the vertical and torsional vibrations.

Results and discussion

To consider the vertical vibration, the TMDs in the beginning of the wind excitation are increasing the vertical vibration, and it needed approximately 10 s in order to start mitigating the vertical vibration. It was detected that the case of frequency ratio 1.2 has the largest effect in reducing the vertical vibration that reached 50.47% of its magnitude without applying TMDs (see Fig. 10). The case of frequency ratio 0.8 increased 21.76% of the vertical vibration, while the frequency ratio of 0.9 decreased 6.16% of the vertical vibration and the frequency ratio of 1.0 decreased 34.76% of the vertical vibration, but in other hand the frequency ratio of 1.1 has decreased 46.18% of the vertical vibration (see Table 6). There is linearity in the vertical vibration mitigation due to the change in frequency ratio design parameter till frequency ratio 1.0, after that the mitigation is less and tend to be stable or with a very small change. The reduction in the vertical vibration between the two frequency ratios 1.2 and 1.1 are near to each other, so the frequency ratio 1.1 can be considered the suitable ratio to assign the TMDs frequencies which is near to the natural frequency of the cable stayed bridge to optimize mitigating the vertical vibration of the deck.

Related to torsional vibration, all the frequency ratio cases have shown significant reduction of the torsional vibration of the deck (see Fig. 11 and Table 7). The change in the ratios has a very little effect on the results. Considering the frequency ratio 0.8, the torsional vibration was reduced to 58.74% and for frequency ratio 0.9 decreased 58.25%, and for frequency ratio 1.0 decreased 58.47% while for frequency ratio 1.1 was decreased to 58.36% and finally for frequency ratio 1.2 decreased 58.12%. This means that changing the frequency ratio of the TMDs has an appreciable effect in mitigating the torsional vibration of the deck in the cable stayed bridge till reaching the frequency ratio 1.0 or near the natural frequency of the cable stayed bridge but after that it has a small effect especially for large TMD mass ratios.

From the results discussed above, it is essential to use the frequency ratio 1.1 to tune the TMDs near to the natural frequency of the cable stayed bridge for the dominant mode shape 6 so that to control the vertical vibration to 46.18% and to reduce the torsional vibration to 58.36%.

Optimum damping ratio

The damping ratio parameter is considered for designing TMDs in vibration control, and it is the ratio of the TMD damping coefficient to the critical damping coefficient of the main structure. Five damping ratios (0.01, 0.05, 0.1, 0.15 and 0.2) are used to calculate the optimum reduction in the vertical and torsional vibrations. Table 8 shows the magnitudes of the stiffness and the damping coefficient of the TMDs considering these five damping ratio cases with constant frequency ratios of 1.0 and constant mass ratios of 2.25%.

Results and discussion

Considering the vertical vibration, the results confirm that the damping ratio of 0.2 for designing the TMDs is the optimum case to control the vertical vibration in this model. The case of damping ratio 0.2 has the largest effect in reducing the vertical vibration which reached 46.00% of its magnitude without applying TMDs (see Fig. 12 and Table 9). The case of damping ratio 0.01 decreased 33.03% of the vertical vibration, while the damping ratio of 0.05 decreased 34.76% of the vertical vibration and the damping ratio of 0.1 decreased 38.93% of the vertical vibration, but the damping ratio of 0.15 has decreased 43.74% of the vertical vibration. Approximately a linear relation is apparent in the vertical vibration mitigation due to the change of damping ratio.

Considering the reduction of the torsional vibration of the deck (see Fig. 13 and Table 10). For damping ratio 0.01 the torsional vibration was reduced to 41.52% and for damping ratio 0.05 decreased 41.54%, and for damping ratio 0.1 decreased 41.52%, and for damping ratio 0.15 was increased to 41.53% and finally for damping ratio 0.2 decreased 41.56%. This means that changing the damping ratio of the TMDs has no effect on mitigating the torsional vibration of the deck in the cable stayed bridge for this range of damping ratios.

Through the evaluation of the results, it is recommended to use the damping ratio 0.2 for assigning TMDs damping coefficient so that to control the vertical vibration to 46.00% and to reduce the torsional vibration to 41.56%.

From the minimax optimization technique results done for the three design parameters, it was found that the use of TMDs with mass ratio 2.25%, frequency ratio of 1.1 and damping ratio 0.2 is the optimal case to guarantee the most efficient control of vertical and torsional vibrations of the deck.

Validation

The validation of the results for the TMDs design parameters is based on the comparison with the benchmarks from the literature which have been considered in the introduction section of this paper. The first benchmark is the results obtained by Wang and coauthors for two TMDs design parameters mass ratio and damping ratio effects on the vertical vibration of the deck for Sutong Bridge. The second benchmark is the results obtained by Lin and coauthors for TMDs mass ratio effect on the vertical and torsional vibrations of the deck.

Wang and coauthors benchmark

The following Fig. 14 is showing the curves of the results of the vertical vibration control efficiency considering mass ratio obtained by Wang and coauthors and the results obtained from the finite element model in this paper.

The control efficiency of the vertical vibration obtained from the results of Wang and coauthors starts from 19.25% for mass ratio of 0.25% while it starts from -33.20% for the results obtained from the finite element model in this paper. The curve of results of this paper reaches near the curve of the results of Wang and coauthors at mass ratio 0.75% with a control efficiency of 12.73% and 25% for the latter. The nearest position is at mass ratio 2% with a value of 25% and 29% for the latter. The results obtained from this paper showed a good agreement with the results of Wang and coauthors at mass ratio range of (2% – 2.25%).

The curves of the results for the vertical vibration control efficiency considering the effect of damping ratio for both Wang and coauthors and the finite element model are shown in Fig. 15. The control efficiency of the vertical vibration obtained by Wang and coauthors start with 26.70% for damping ratio of 0.01, while this output starts with 33.03% for the same damping ratio obtained from the finite element model and the two response curves are coinciding till the damping ratio 0.03. The control efficiency curve for Wang and coauthors ends with 22.25% for damping ratio 0.08 and the curve of the finite element model has a value of 36% at the same damping ratio. Supporting on the comparison between the two results, a good agreement exists between them in the damping ratio range (0.01–0.03).

Lin and coauthors benchmark

Considering the aerodynamic coupling between the vertical and torsional vibrations for the dominant mode shape of vibration, and when the mass ratio of the TMDs is 2%, the control efficiency of the vertical vibration of the deck obtained from the results of the finite element model in this paper was 25%, and it was the same value 25% obtained from Lin and coauthors. This indicates that the two results have an excellent agreement, which pours to the validity of the obtained results in this paper.

Global sensitivity analysis

Global sensitivity analysis studies how the uncertainty in the output of a model numerical or otherwise can be apportioned to different sources of uncertainty in the model input. A good sensitivity analysis should run analyses over the full range of plausible values of key parameters and their interactions, to assess how impacts change in response to changes in key parameters. Global sensitivity analysis methods such as Sobol’s sensitivity indices have been widely applied recently. Global sensitivity analysis explores the parameter space so that they provide robust sensitivity measures in the presence of nonlinearity and interactions among the parameters. GSAs, however, can be computationally intensive, since they require sampling parameter sets. Several approximation methods have been developed to reduce the computational cost. Global sensitivity analysis can provide additional information for improving the system understanding.

Sobol’s sensitivity indices

Sobol sensitivity analysis is intended to find out how much of the variability in model output is dependent upon each of the input parameters, either upon a single parameter or upon an interaction between different parameters. The decomposition of the output variance in a Sobol sensitivity analysis employs the same principal as the classical analysis of variance in a factorial design. Sobol sensitivity analysis is not intended to identify the cause of the input variability. It only indicates what impact and to what extent it will have on model output. As a result, it cannot be used to identify the source(s) of variance. The Sobol sensitivity indices for the three design parameters (mass ratio, frequency ratio and damping ratio) are utilized in this paper to obtain the control efficiency of vertical and torsional vibrations of the deck after application of MTMDs. The Sobol's sensitivity indices are ratios of partial variances to total variance, and for independent variables satisfy the relationship:

(30) \mathrm{1}={\displaystyle \sum _i{\mathit{S}}_i}+{\displaystyle \sum _i{\displaystyle \sum _{j>\mathrm{1}}{\mathit{S}}_{ij}+}}{\displaystyle \sum _i{\displaystyle \sum _{j>\mathrm{1}}{\displaystyle \sum _{k>\mathrm{1}}{\mathit{S}}_{ijk}}+}}\cdots .

The Sobol’ sensitivity indices are:

First order sensitivity index

(31) {\mathit{S}}_i={\displaystyle \frac{{\mathit{V}}_i}{\mathit{V}}}.

Second order sensitivity index

(32) {\mathit{S}}_{ij}={\displaystyle \frac{{\mathit{V}}_{ij}}{\mathit{V}}}.

Total sensitivity index

(33) {\mathit{S}}_{Ti}={\mathit{S}}_i+{\displaystyle \sum _{j\ne \mathrm{1}}{\mathit{S}}_{ij}+\cdots }.

The first order index, {\mathit{S}}_i , is a measure for the variance contribution of the individual parameter {\mathit{X}}_i to the total model variance. The partial variance {\mathit{V}}_i in Eq. (31) is given by the variance of the conditional expectation {\mathit{V}}_i=\mathit{V}\left[E\right(Y/X_i\left)\right] and is also called the main effect’ of X_i on Y. The impact on the model output variance of the interaction between parameters {\mathit{X}}_i and {\mathit{X}}_j is given by {\mathit{S}}_{ij} and {\mathit{S}}_{Ti} is the result of the main effect of {\mathit{X}}_i and all its interactions with the other parameters (up to the p^th order) [ 127– 131].

Experimental design

Experimental design is widely used to control the effects of parameters in many problems. The use of this method decreases the number of experiments, using time and material resources. Furthermore, the experimental errors are minimized and the analysis performed on the results is easily realized. Utilizing experimental design, the statistical methods measure the effects of change that are resulting from operating parameters and their mutual interactions on the process.

Box-Behnken method

Box–Behnken experimental design was based on to identify the relationship between the response functions (vertical vibration and torsional vibration) of the deck in the cable stayed bridge finite element model and the three design variables of the TMDs (mass ratio, frequency ratio and damping ratio). Box-Behnken design is rotatable second-order designs based on three-level incomplete factorial designs. The special arrangement of the Box–Behnken design levels allows the number of design points to increase at the same rate as the number of polynomial coefficients. For three factors, for example, the design can be constructed as three blocks of four experiments consisting of a full two-factor factorial design with the level of the third factor set at zero.

Box–Behnken design requires an experiment number according to N=k² + k + C_p, where (k) is the factor number and (C_p) is the replicate number of the central point. Box–Behnken is a spherical, revolving design. Viewed as a cube (Fig. 16(a)), it consists of a central point and the middle points of the edges. However, it can also be viewed as consisting of three interlocking 2² factorial design and a central point (Fig. 16(b)). It has been applied for optimization of several chemical and physical processes.

For the three-level three-factorial Box–Behnken experimental design, a total of 15 experimental runs, shown in Table 11 are needed. The model is of the following form:

(34) \mathit{y}={\mathit{\beta }}_{\mathrm{0}}+{\mathit{\beta }}_{\mathrm{1}}{\mathit{x}}_{\mathrm{1}}+{\mathit{\beta }}_{\mathrm{2}}{\mathit{x}}_{\mathrm{2}}+{\mathit{\beta }}_{\mathrm{3}}{\mathit{x}}_{\mathrm{3}}+{\mathit{\beta }}_{\mathrm{11}}{\mathit{x}}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{\beta }}_{\mathrm{22}}{\mathit{x}}_{\mathrm{2}}^{\mathrm{2}}+{\mathit{\beta }}_{\mathrm{33}}{\mathit{x}}_{\mathrm{3}}^{\mathrm{2}}+{\mathit{\beta }}_{\mathrm{12}}{\mathit{x}}_{\mathrm{1}}{\mathit{x}}_{\mathrm{2}}+{\mathit{\beta }}_{\mathrm{13}}{\mathit{x}}_{\mathrm{1}}{\mathit{x}}_{\mathrm{3}}+{\mathit{\beta }}_{\mathrm{23}}{\mathit{x}}_{\mathrm{2}}{\mathit{x}}_{\mathrm{3}},

where \mathit{y} is the predicted response, β₀ model constant; x₁, x₂ and x₃ independent variables; β₁, β₂ and β₃ are linear coefficients; β₁₂, β₁₃ and β₂₃ are cross product coefficients and β₁₁, β₂₂ and β₃₃ are the quadratic coefficients. The coefficients, i.e., the main effect (β_i) and two factors interactions (β_ij) have been estimated from the experimental results by computer simulation programming applying least square method using MATLAB [ 132– 142].

The following Table 12 is showing both the coded and actual variable values that are related to the 15 runs of the simulation of wind excitation with application of TMDs to reduce the vertical and torsional responses of the deck utilizing variation in the three design parameters mentioned above.

Response surface methodology

Response surface methodology is a set of statistical and mathematical methods being used for the analysis and modeling of many problems in the engineering field. The main purpose of this methodology is to optimize the response surface which is being influenced by various parameters. This methodology quantifies the relationship between the obtained response surfaces and the controllable input parameters.

If all variables are assumed to be measurable, the response surface can be expressed as follows:

(35) \mathit{y}=\mathit{f}({\mathit{x}}_{\mathrm{1}}+{\mathit{x}}_{\mathrm{2}}+{\mathit{x}}_{\mathrm{3}},\mathrm{...},{\mathit{x}}_k),

where y is the answer of the system, and x_i the variables of action called factors. The goal is to optimize the response variable y. It is assumed that the independent variables are continuous and controllable by experiments with negligible errors. It is required to find a suitable approximation for the true functional relationship between independent variables and the response surface. Two important models are utilized in response surface methodology RSM, the first-degree model which is represented as follows:

(36) \mathit{y}={\mathit{\beta }}_{\mathrm{0}}+{\displaystyle \sum _{i=\mathrm{1}}^k{\mathit{\beta }}_i{\mathit{x}}_i+\mathit{\epsilon }},

and the second-degree model is formulated in the following equation:

(37) \mathit{y}={\mathit{\beta }}_{\mathrm{0}}+{\displaystyle \sum _{i=\mathrm{1}}^k{\mathit{\beta }}_i{\mathit{x}}_i+{\displaystyle \sum _{i=\mathrm{1}}^k{\mathit{\beta }}_{ii}{\mathit{x}}_i^{\mathrm{2}}+}}{\displaystyle \sum _{i=\mathrm{1}}^{k-\mathrm{1}}{\displaystyle \sum _{j=\mathrm{2}}^k{\mathit{\beta }}_{ij}{\mathit{x}}_i{\mathit{x}}_j}+}\mathit{\epsilon },

where x₁, x₂,..., x_k are the input factors which influence the response y; {\mathit{\beta }}_{\mathrm{0}} , {\mathit{\beta }}_{ii} (i = 1,2,...,k), {\mathit{\beta }}_{ij} (i = 1,2, ...,k; j = 1, 2,..., k) are unknown parameters and \epsilon is a random error. The \mathit{\beta } coefficients are obtained by the least square method.

A series of n experiments should first be carried out, in each of which the response y is measured (or observed) for specified settings of the control variables. The totality of these settings constitutes the so-called response surface design, or just design, which can be represented by a matrix, denoted by D, of order n × k called the design matrix,

(38) \mathit{D}=\left(\begin{array}{cccc}\hfill {\mathit{x}}_{\mathrm{11}}\hfill & {\mathit{x}}_{\mathrm{12}}& \cdots & {\mathit{x}}_{\mathrm{1}k}\\ \hfill {\mathit{x}}_{\mathrm{21}}\hfill & {\mathit{x}}_{\mathrm{22}}& \cdots & {\mathit{x}}_{\mathrm{2}k}\\ \hfill ⋮\hfill & \hfill \hfill & & \hfill ⋮\hfill \\ \hfill {\mathit{x}}_{n\mathrm{1}}\hfill & {\mathit{x}}_{n\mathrm{2}}& \cdots & {\mathit{x}}_{nk}\end{array}\right),

where x_ui denotes the u th design setting of x_i (i = 1,2, . . .,k; u = 1,2, . . .,n). Each row of D represents a point, referred to as a design point, in a k-dimensional Euclidean space. Let y_u denote the response value obtained as a result of applying the u th setting of x, namely x_u = (x_u1, x_u2, . . ., x_uk), (u = 1,2, . . .,n).

In general Eq. (32) can be written in matrix form:

(39) \mathit{Y}=\mathit{X}\mathit{\beta }+\mathit{\epsilon },

where Y=(y₁, y₂, …, y_n)which is defined as a matrix of measured values, X is a matrix of independent variables of order n× p whose u th row is f^{\prime }\left(x_u\right) , and ϵ = (ϵ₁, ϵ₂,…, ϵ_n). Note that the first column of X is the column of ones 1n.

Assuming that \mathit{\epsilon } has a zero mean, the so called ordinary least-squares estimator of β is:

(40) \mathit{\beta }={\left({\mathit{X}}^{\prime }\mathit{X}\right)}^{-\mathrm{1}}{\mathit{X}}^{\prime }\mathit{Y},

where {{X}}^{\prime } is the transpose of the matrix X and {\left({{X}}^{\prime }{X}\right)}^{-\mathrm{1}} is the inverse of the matrix {{X}}^{\prime }{X} [ 134, 135, 143– 147].

Surrogate models results

The regression coefficients calculated from the second-degree model both for vertical and torsional actual responses obtained using Box-Behnken method. Total of 15 runs are used to formulate the surrogate models for the predicted vertical and torsional vibrations efficiency control based on quadratic orders. The surrogate model for the case of vertical vibration control efficiency has a coefficient of determination R² of 99.45% (see Fig. 17) which is a very good representation of the predicted system response and just 0.55% of the system response still unexplained.

The surrogate model for the case of torsional vibration control efficiency has a coefficient of determination R² of 99.99% (see Fig. 18) which is an excellent approximation for the prediction of the system response and just 0.01% of the system response still unexplained.

Sensitivity indices results and discussion

The main orders of sensitivity indices for each design parameter in addition to their interaction orders were calculated considering the convergence process recommending the use of 250 samples to calculate the vertical vibration control efficiency (see Fig. 19). Furthermore, the main and interaction orders of sensitivity indices for each design parameter were calculated supporting on 100 samples to calculate the torsional vibration control efficiency (see Fig. 20). Supporting on the calculated results, the total sensitivity indices for each design parameter have been calculated (see Table 13).

In relation with the vertical vibration control efficiency, the total order sensitivity index of design parameter mass ratio X1 is 0.6979, this value is bigger than the total order sensitivity index of design parameter frequency ratio X2 which is 0.2279, also it is much bigger than the total order sensitivity index of design parameter damping ratio X3 which is 0.0883, this means that the vertical vibration control efficiency is 69.79% due to variation in mass ratio between the TMDs and the cable stayed bridge, and it is 22.79% due to the variation in frequency ratio between the natural frequencies of the TMDs and the cable stayed bridge, also it is 8.83% due to the variation in the damping ratio between them. While the interaction index between the design parameter X1 and X2 is 0.0091 and between X1 and X3 is 0.0030, while between X2 and X3 is 0.006, which means that there is a small interaction between the input factors taking part in the variation of the vertical vibration control efficiency.

While considering the torsional vibration control efficiency, the total order sensitivity index of X1 is 0.9999, which can be named the controlling design parameter which is much bigger than the total order sensitivity index of X2 which is 0.0002, also it is much bigger than the total order sensitivity index of X3 which is 0.0002, this means that the torsional vibration control efficiency is dependable 99.99% on the mass ratio variation, while it is 0.02% due to the frequency ratio variation, and it is 0.02% due to the variation in damping ratio. While the interaction index between X1 and X2 is 0.0001 and between X1 and X3 is 0.0001, while between X2 and X3 is 0.0000, this proofs that the surrogate model is additive approximately which means that there is no interaction between design parameters take part in the variation of the torsional vibration control efficiency.

Convergence of the results

The process of global sensitivity analysis supporting on Sobol’s sensitivity indices requires certain or adequate samples of experiments to find out the predicted effect of the design parameters on the response of the system. The most efficient number of samples is being identified through the convergence of the sum of first orders and total sensitivity indices of the design parameters. All the sensitivity indices (first orders, interaction orders and total orders) for each design parameter have been calculated using m MATLAB codes. Two outputs have been utilized in the process of converges, the vertical vibration and torsional vibration control efficiency. Figures 19 and 20 show the relation between the number of samples and the sum of first orders sensitivity indices, in the same time between the number of samples and the sum of total sensitivity indices of all design parameters.

For the case of vertical vibration control efficiency (see Fig. 19), the two curves of the sum of first orders and total orders of sensitivity indices at the beginning are not coinciding to reach convergence till 250 samples. After this stage the two curves are starting to converge at the 250 number of samples, where the two curves continue to remain in a stable position after many times of changing the number of samples starting from 250 samples.

While for the torsional vibration control efficiency (see Fig. 20) the two curves are trying to reach convergence earlier than the previous case, the convergence starts at 100 number of samples and the process continues in the stability pattern.

The convergence results of the two cases necessitate utilizing 250 samples of experiments to efficiently get the predicted rational effects of each design parameter on the variation of the vertical vibration control efficiency, and to consider 100 samples to obtain the predicted rational effects of the design parameters on the variation of the torsional vibration control efficiency.

Conclusions

The following conclusions have been drawn:

1) The mass ratio 0.02 resulted in better performance of the MTMDs by obtaining control efficiencies 25% and 56% for vertical and torsional vibrations. While the frequency ratio1.1 obtained control efficiencies 46.18% and 58.36% for vertical and torsional vibrations. Furthermore, the damping ratio 0.2 led to control efficiencies 46% and 41.5% for vertical and torsional vibrations, which means that the performance of the MTMDs has been increased for higher mode shapes of vibrations.

2) Box-Behnken experimental method formulated excellent surrogate models representing the actual response of the system which is used to easily and efficiently calculate the performance of the MTMDs in suppressing both the vertical and torsional vibrations of the deck.

3) The mass ratio was the most effective parameter that has the maximum rational effects 68.51% and 99.97% on the vertical and torsional vibrations reduction simultaneously. While the frequency ratio has 21.76% and damping ratio has 8.4% rational effect on the vertical vibration reduction only. But the interactions between all the design parameters have a small rational effect on the vertical vibration only which does not exceed 0.91% for the highest. These results greatly help to optimize the performance of the MTMDs.

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