# Study of wind-induced vibrations in tall buildings with tuned mass dampers taking into account vortices effects

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## Abstract

In recent years, construction of tall buildings has been of great interest. Use of lightweight materials in such structures reduces stiffness and damping, making the building more influenced by wind loads. Moreover, tall buildings of more than 30 to 40 stories, depending on the geographical location, the wind effects are more influential than earthquakes. In addition, the complexity of the effects of wind flow on the structure due to the interaction of the fluid flow and solid body results in serious damages to the structure by eliminating them. Considering the importance of the issue, the present study investigates the phenomenon of wind-induced vibration on high-rise buildings, taking into account the effects of vortices created by the fluid flow and the control of this phenomenon. To this end, the governing equations of the structure, the fluid flow and the tuned mass damper (TMD) are first introduced, and their coefficient values are extracted according to the characteristics of ACT skyscraper in Japan. Then, these three coupled equations are solved using a program coded in MATLAB. After validation of the results, the effects of wind loads are analyzed and considered with regard to the effects of vortices and the use of TMD, and are compared with the results of the state where no vortices are considered. Generally, the results of this study point out the significance of vibrations caused by vortices in construction of engineering structures as well as the appropriate performance of a TMD in reducing oscillations in tall buildings.

## Keywords

Vibration Vortices Tall building Tuned mass damper## Introduction

Assuming the body is secured by an elastic support of relatively low mass and damping, it starts vibrating due to the oscillatory forces, which, if not controlled, may lead to damages to the structure or even its destruction. This is of great concern especially in high-rise buildings housing a significant number of people or in oil extraction pipes that are installed deep into the ocean at high cost for extraction of petroleum products. The destruction of the cooling towers of Ferrybridge power plant in England in 1960 and the Tacoma Bridge in USA, in 1940, are practical examples where neglecting vortex effects resulted in disasters (Sarpkaya 2004). The Tacoma Bridge was designed to withstand winds at a speed of 100 mph; however, due to neglecting the vibrations caused by the vortex effects, the wind loads caused torsional instability at a traveling speed of 42 mph and destroyed the bridge.

Tuned mass dampers (TMDs) have been widely used in practice, among other devices and configurations for supplemental damping, for vibration mitigation in wind-excited tall buildings to meet occupants’ comfort performance criteria prescribed by building codes and guidelines (Giaralis and Petrini 2017). In its simplest form, the linear passive TMD comprises a mass attached toward the top of the building (primary structure), via linear stiffeners, or hangers in case of pendulum-like TMD implementations, and supplemental damping devices (dampers). The effectiveness of the TMD relies on “tuning” its stiffness and damping properties for a given primary structure and attached mass, such that significant kinetic energy is transferred from the vibrating primary structure to the TMD mass and eventually dissipated through the dampers. Focusing on the suppression of lateral wind-induced vibrations in (tall) buildings, the TMD is tuned to the first natural frequency of the primary structure aiming to control the fundamental (translational) lateral mode shape. As an example, TMD was used in Taipei 101, formerly known as Taipei World Financial Center, a skyscraper built in 2004 with 101 stories located in Taipei, Taiwan (Alex and Tuan Shang 2014). The major part of this TMD system is a 6-m-diameter sphere that is made of steel and weighs 600 tons. It is the largest of its kind in the world. It is installed at the center of the 87th floor and suspended from the 91st floor by cables. There are eight dampers around the mass, to prevent it from moving excessively. This TMD is essentially a pendulum that spans five floors (88–92th), whose primary function is to suppress wind-induced vibration in this building and reduces the vibrations in the building up to 40%.

## The governing differential equations

*w*denotes the width of the solution domain. Since the effect of the walls around the domain on the wind flow and structure is neglected, this parameter was considered sufficiently large so that it has no effects on the results.)

*m*

_{s}and

*m*

_{d}are the masses of structure and TMD), \(\xi_{\text{d}}\) and \(\omega_{\text{d}}\) are the TMD damping ratio and frequency. As suggested by the figure, the planar motion along the body streamlines,

*X*, is described using the linear oscillatory Eq. 4:

*k*are damping coefficient and the stiffness of the body, respectively, when the fluid is not present. The hydrodynamic effects of the fluid on the structure are identified as the main effects of the fluid added in the form of added fluid mass and damper according to the equations \(m_{\text{f}} = \frac{1}{4}\pi C_{\text{m}} \rho D^{2}\) and \(c_{\text{f}} = \varOmega \gamma \rho D^{2}\), respectively. Additionally, the vortex effects are modeled as the external force

*E*defined as \(E = \frac{1}{2}\rho U^{2} DC_{\text{D}}\). In these equations,

*ρ*is the fluid density,

*C*

_{m}is the added mass coefficient (usually considered 1),

*C*

_{D}is the drag coefficient,

*U*is the wind velocity and \(\gamma\) is the damping coefficient of the added fluid flow (Facchinetti et al. 2004). Moreover, \(\varOmega\) is the vortex formation frequency defined as \(\varOmega = \varOmega_{\text{f}} = \frac{2\pi StU}{D}\), where

*St*is the Strouhal number. By defining the natural frequency of the oscillations as \(\varOmega_{\text{s}} = \left( {\frac{k}{{m_{\text{s}} }}} \right)^{0.5}\), the reduced-damping coefficient of the structure as \(\xi = \frac{{c_{\text{s}} }}{{2m_{\text{s}} \varOmega }}\), the total mass as

*m*=

*m*

_{ s }+

*m*

_{ f }and the dimensionless mass ratio as \(\mu = \frac{{m_{\text{s}} + m_{\text{f}} }}{{\rho D^{2} }}\), we may rewrite Eq. 3 to obtain Eq. 4:

*q*, which associated with the drag coefficient on the structure, is defined as \(q\left( t \right) = \frac{{2C_{\text{D}} (T)}}{{C_{{{\text{D}}_{0} }} }}\), where

*C*

_{D}(

*T*) is the drag coefficient and

*C*

_{D0 }is the reference drag coefficient related to a stationary body exposed to vortices. The amplification factor \(\frac{q}{2} = \frac{{C_{\text{D}} }}{{C_{{{\text{D}}0}} }}\) is multiplied by the drag coefficient of the stationary body to give the amplified drag coefficient of the oscillatory body. The parameter

*F*indicates the effect of cylinder motion in the vortex formation region. Hence, the coupled dimensionless equations of the fluid–structure system are obtained as follows:

*U*

_{r}is the reduced flow velocity defined as \(U_{\text{r}} = \frac{2\pi U}{{\varOmega_{\text{s}} D}}\). Moreover, the coupling parameters are defined as \(e = \frac{E}{{D\varOmega_{f}^{2} m}} = E\frac{D}{{4\pi^{2} {\text{St}}^{2} U^{2} m}}\) and \(f = \frac{F}{{D\varOmega_{f}^{2} }} = F\frac{D}{{4\pi^{2} {\text{St}}^{2} U^{2} }}\). Using these parameters and by substituting them in the above equations, the coefficient

*e*is defined as \(e = \frac{{C_{D} }}{{8\pi^{2} {\text{St}}^{2} \mu }}\), and by definition of the coefficient

*M*as the relation \(M = \frac{1}{2}\frac{{C_{{{\text{D}}_{0} }} }}{{8\pi^{2} {\text{St}}^{2} \mu }}\), we may rewrite

*e*as

*e*=

*Mq*.

*M*is obtained from Eq. 8 as follows:

The added damping coefficient \(\gamma = \frac{{C_{\text{D}} }}{4\pi \text{St}}\) is the last parameter to be determined, which is proportional to the drag coefficient. In case of an oscillating body, the drag coefficient can be approximated to 2, so that the value of \(\gamma\) is obtained as 1.32. Different studies show that the applied force on the body can be a function of the displacement, velocity or acceleration of the body. However, a review of other similar studies suggests that the coupled acceleration model can produce better results. Moreover, following (Srinil et al. 2013), the values *A* = 12 and *ε* = 0.3 are assumed for the coupling model.

*t*is defined, and the values of acceleration, velocity and displacement are calculated in each step using the rectangle (midpoint) method and solved by MATLAB code. Unconditional stability is the main advantage of this method. According to (Julien 2012), the one DOF system’s stiffness is given by \(k = \frac{\kappa }{n}\). The typical floor value according to the equivalent formula for fixed columns is:

*β*is a factor relating to the number of columns and their contribution and equal to 19/6,

*h*is the height of a floor (typical value is 4 m), EI is the stiffness of one column (typical value is \(15 \times 10^{6} \;{\text{Nm}}^{ - 1} )\) and \(n = \frac{{H_{\text{total}} }}{h}\), where

*H*

_{total}has been picked as the ACT Tower height. Using these values, yields to

*k*= 168,000 Nm

^{−1}. The other specifications of ACT Tower are given in Table 1. Moreover, the optimum values for the TMD are adapted from (Chopra 1995). According to (Kawai 1992), at height to width ratios of higher than eight for the structure, the vortices formed in the flow have a great influence on the oscillations of the structure. Hence, this ratio was considered ten for the structure considered in this study. Moreover, the wind flow velocity was considered 25 m/s.

Mass of the ACT tower | |

Stiffness of the ACT tower | |

Values in civil engineering range from 0 to 5%, in rare case to 10% | \(\xi = 2\%\) |

Damping | \(c = 2\xi \omega = 2m\xi \sqrt {\frac{k}{{m_{\text{s}} }}}\) |

## Results and discussion

### Validation of results

In what follows, the structure responses to various wind power profiles like (Julien 2012), in the presence of a TMD, are addressed for the cases where the vortex effects are once considered and once neglected.

### Free vibrations

Comparison of the maximum vibration amplitudes during free vibrations, once considering the vortex effects and once neglecting them

Maximum vibration amplitudes neglecting the vortex effects (m) | Maximum vibration amplitudes taking into account the vortex effects (m) | |
---|---|---|

The structure with a TMD | 0.2361 | 0.2367 |

The structure without a TMD | 0.2360 | 0.2365 |

### Harmonic loading taking into the vortex effects

#### Without any effects

Comparison of maximum vibration amplitudes at \(\alpha = 0.2\) in two cases where vortex effects are once considered and once neglected

Maximum vibration amplitudes neglecting the vortex effects (m) | Maximum vibration amplitudes taking into account the vortex effects (m) | |
---|---|---|

The structure with a TMD | 1.08 × 10 | 3.24 × 10 |

The structure without a TMD | 1.09 × 10 | 3.46 × 10 |

#### Inverse effects

Comparison of the maximum amplitudes of vibrations at \(\alpha = 0.92\) with and without considering vortex effects

Maximum vibration amplitudes neglecting the vortex effects (m) | Maximum vibration amplitudes taking into account the vortex effects (m) | |
---|---|---|

The structure with a TMD | 4.75 × 10 | 7.21 × 10 |

The structure without a TMD | 5.56 × 10 | 8.13 × 10 |

#### Optimum effects

Comparison of the maximum amplitudes of vibrations at α = 0.92 with and without considering vortex effects

Maximum vibration amplitudes neglecting the vortex effects (m) | Maximum vibration amplitudes taking into account the vortex effects (m) | |
---|---|---|

The structure with a TMD | 1.50 × 10 | 1.70 × 10 |

The structure without a TMD | 5.57 × 10 | 8.73 × 10 |

## External random sawtooth wave by taking vortex effects into consideration

*rand*function used in Eq. 12 indicates a random value ranging from zero to one. This load is very similar to real-world wind loads, which consequently results in structure responses close to the reality.

Comparison of the maximum amplitudes of vibrations with and without considering vortex effects when random external loads are applied

Maximum vibration amplitudes neglecting the vortex effects (m) | Maximum vibration amplitudes taking into account the vortex effects (m) | |
---|---|---|

The structure with a TMD | 2.59 × 10 | 3 × 10 |

The structure without a TMD | 2.11 × 10 | 3.16 × 10 |

## Conclusions

- 1.
The wind forces exerted on the high-rise structures can induce large vibrations to the buildings. The results suggest that in addition to earthquake forces, which are among the most important design parameters, the effect of wind forces should also be accurately taken into consideration in such buildings.

- 2.
The TMD performs optimally when the wind blows at frequencies close to that of the natural frequency of the structure. This optimal performance can considerably reduce the amplitude of oscillations (up to 70%) and hence prevents extreme structural vibrations. In other frequencies, the effect of the TMD is reduced and, even in some cases, it has reverse effects and the amplitude of structural oscillations is increased.

- 3.
The vortices formed around the structure increase the amplitude of building oscillations. Comparison of the amplitude of vibrations indicates that structural oscillations increase up to 40% by taking into account the effects of vortices. Therefore, neglecting the effects of vortices in high-rise buildings can lead to serious damages to the structure.

- 4.
TMD performs favorably against both harmonic and non-harmonic excitations. In other words, use of TMD is one of the effective methods to reduce the amplitude of oscillations in high-rise buildings subject to wind flow, especially at frequencies close to the natural frequency of the structure.

## Notes

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