Performance of TMDI for Tall Building Damping

Abstract

This study investigates the vibration reduction of tall wind-excited buildings using a tuned mass damper (TMD) with an inerter (TMDI). The performance of the TMDI is computed as a function of the floor to which the inerter is grounded as this parameter strongly influences the vibration reduction of the building and for the case when the inerter is grounded to the earth whereby the absolute acceleration of the corresponding inerter terminal is zero. Simulations are made for broadband and harmonic excitations of the first three bending modes, and the conventional TMD is used as a benchmark. It is found that the inerter performs best when grounded to the earth because, then, the inerter force is in proportion to the absolute acceleration of only the pendulum mass, but not to the relative acceleration of the two inerter terminals, which is demonstrated by the mass matrix. However, if the inerter is grounded to a floor below the pendulum mass, the TMDI only outperforms the TMD if the inerter is grounded to a floor within approximately the first third of the building’s height. For the most realistic case, where the inerter is grounded to a floor in the vicinity of the pendulum mass, the TMDI performs far worse than the classical TMD.

1. Introduction

Tall buildings may be susceptible to wind-induced structural vibrations due to their slenderness; the corresponding fundamental period is typically within the range of 3 to 12 s. The resulting peak structural acceleration may be inacceptable from the vibration comfort perspectives defined by ISO 10137:2007 [1] for the one-year return period wind and by the CTBUH (Council on Tall Buildings and Urban Habitats) for the ten-year return period wind. The acceptable peak structural acceleration defined by ISO 10137:2007 depends on the eigenfrequency of the fundamental bending mode and the use of the building, i.e., the acceleration limit for a residential building is lower than that for an office building. To get a feeling of typical acceleration limits, an example is given here. For a tall residential building with the typical fundamental eigenfrequency of 0.14 Hz, the acceptable peak structural acceleration for the one-year wind becomes 9.8 mg, which is approximately the average of the range of the perceptible acceleration from 5 to 15 mg. The ten-year wind CTBUH recommendation for residential and office buildings is that structural accelerations should not be greater than 10 to 15 mg, of which the upper bound is adopted more often and sometimes extended to 18 mg.

The common method for reducing structural accelerations to the above-mentioned acceptable limits is to install a tuned mass damper (TMD, [2]). The TMD parameters, such as natural frequency and damping, are optimized to the most susceptible modes, which—for most cases—are the fundamental bending modes in both main directions and the first torsional mode [3,4]. As bending modes and torsional modes show their anti-nodes at or close to the top of the building, the TMD is usually installed just below roof level.

The optimum TMD design aims to guarantee the acceleration limits according to ISO 10137:2007 and the CTBUH with minimum pendulum mass, as this will minimize TMD costs. The required footprint of the entire TMD is then given by the dimensions of the pendulum mass and its relative motion amplitude. From the building economics point of view, the footprint must be minimized as well. Hence, novel TMD systems are sought that guarantee the required acceleration limits with less pendulum mass and without increased relative motion in order to minimize both TMD costs and required TMD space. This need has led to the development of real-time-controlled TMDs that evoke the same vibration reduction as passive TMDs, but with reduced pendulum mass [5,6,7,8,9]. In recent years, another approach has gained attention: the TMD with inerter (TMDI) [10,11,12,13,14,15,16,17,18,19,20,21,22]. The TMDI is a conventional TMD that is enriched by an inerter whose force is defined to be in proportion to the difference of the accelerations of both inerter terminals, i.e., it works as an ideal (frictionless) gyroscope [23]. Hence, the basic idea is that the total TMDI inertia is produced by the inertia of the physical pendulum mass and by the so-called inertance of the inerter. Different inerter configurations were investigated; the basic configurations are when the inerter is used to replace the TMD mass [12] and when the inerter is envisaged to augment the TMD mass [15]. The latter case is the topic of this research. Then, the inerter is connected to the pendulum mass and grounded to any position on the vibrating structure or the inerter is grounded to the earth. These two cases are not equivalent. In the former case, the inerter force is in proportion to the relative acceleration between the pendulum mass and its terminal on the vibrating structure, while in the latter case, the inerter force is in proportion to the absolute acceleration of the pendulum mass. It is understood that for the latter case, the inerter force increases the pendulum mass according to its inertance, whereby the TMDI performs exactly the same as the TMD with the sum of pendulum mass and inertance. However, for the former case, the impact of the TMDI on the vibration reduction of the primary structure strongly depends on the location of the inerter terminal on the structure.

The goal of this research is to investigate all possible configurations where the inerter of the TMDI is grounded to any position on the vibrating structure. For this, a building model with 20 lumped masses is used. The pendulum mass is hung from the roof, which is represented by the mass with number 20 of the building model, and the inerter is grounded to any of the masses below, i.e., to any of the first 19 masses. The configuration where the inerter is grounded to the earth is computed as well. The building with a classical TMD is considered as a benchmark. The simulations are performed for both broadband and harmonic excitations of the first three bending modes, considering that the wind load exciting tall buildings is of a broadband nature, while it is of a harmonic nature in the case of chimneys, antennas, etc.

The paper is structured as follows. First, the modeling of the structure with TMDI and TMD is described (Section 2). Relevant information on the simulation procedure is given in Section 3. The numerical results are presented and discussed in Section 4. The paper finishes with concluding remarks.

2. Modeling

2.1. Structural Model and TMDI Configuration

The structure is modeled as a homogenous shear frame with of $n$ = 20 masses and degrees of freedom, respectively (Figure 1). A typical tall building that is susceptible to wind-induced vibrations may have 60 stories. Thus, the mass of one degree of freedom can be interpreted as the lumped mass of three stories of such a building. Typical parameters of the first horizontal bending mode are assumed: eigenfrequency $f_1$ = 0.14 Hz, modal mass $m_1$ = 50,000 t, inherent damping ratio ${\zeta }_1$ = 1.5%. Considering sinusoidal modeshape functions, the mass of the building is $2\times m_1$ [24,25], i.e., 100,000 t, whereby the mass of one degree of freedom becomes $m$ = 5000 t. The excitation force $f_w$ is assumed to excite all masses, as the wind load acts on the entire façade. There is no excitation force on the TMD mass because the TMD is installed inside of the building.

The pendulum mass of the TMDI is assumed to be hung from the roof of the building, which is mass 20 of the structural model under consideration. One terminal of the inerter is connected to the pendulum mass, while the other terminal is connected either to an arbitrary structural mass with number $gr$ ($gr$ = 1…19) or to the earth denoted as $gr$ = earth (Figure 1a,b). The pendulum mass of the classical TMD is connected to structural mass 20 as well (Figure 1c) to ensure comparability with the results of the TMDI.

2.2. TMDI Properties

Like the conventional TMD, the TMDI consists of a pendulum mass and viscous damper, but is enriched with the so-called inerter, which, per definition, produces a force $f_I$ that is in proportion to the difference of the absolute accelerations of both inerter terminals:

$$f_I=b\hspace{0.17em}\left({\ddot{x}}_{gr}-{\ddot{x}}_{21}\right)$$

(1)

where $b$ is the so-called inertance with units of mass, ${\ddot{x}}_{gr}$ denotes the absolute acceleration of the structural mass with number $gr$ ($gr$ = 1…19) to which the inerter is grounded, and ${\ddot{x}}_{21}$ describes the absolute acceleration of the pendulum mass. The crucial feature of the inerter is that the inertance $b$ represents a virtual mass that may be considerably big when, e.g., a frictionless flywheel mechanism is adopted with a small wheel mass but a great gear ratio. Similarly to the published works on the TMDI, all results of this research are based on an ideal inerter, i.e., a frictionless inerter according to (1).

The design parameters of the TMDI are the mass ratio $\mu$ that relates the pendulum mass $m_2$ to the target modal mass $m_1$ and the inertance ratio $\beta$, which describes the virtual mass $b$ due to the inerter compared to the target modal mass $m_1$:

$$\mu =\frac{m_2}{m_1}$$

(2)

$$\hspace{0.17em}\beta =\frac{b}{m_1}$$

(3)

Due to the inertance, the optimum designs of the natural frequency and damping ratio of the TMDI are different from those of the classical TMD. In the literature, several design rules are available [15,16,17,18]. For this study, the natural frequency, stiffness, damping ratio, and viscous damping coefficient of the TMDI were designed according to the solutions given in [15].

$$f_2=f_1\hspace{0.17em}\hspace{0.17em}\frac{\sqrt{1+0.5\hspace{0.17em}\left(\beta +\mu \right)}}{1+\beta +\mu }$$

(4)

$$k_2=\left(b+m_2\right)\hspace{0.17em}{\left(2\hspace{0.17em}\pi \hspace{0.17em}{f}_2\right)}^2$$

(5)

$${\zeta }_2=\sqrt{\frac{\left(\beta +\hspace{0.17em}\mu \right)\hspace{0.17em}\left(1+0.75\hspace{0.17em}\left(\beta +\hspace{0.17em}\mu \right)\right)}{4\hspace{0.17em}\left(1+\beta +\mu \right)\hspace{0.17em}\left(1+0.5\hspace{0.17em}\left(\beta +\hspace{0.17em}\mu \right)\right)}}$$

(6)

$$c_2=2\hspace{0.17em}{\zeta }_2\hspace{0.17em}\left(b+m_2\right)\hspace{0.17em}\left(2\hspace{0.17em}\pi \hspace{0.17em}{f}_2\right)$$

(7)

These closed-form solutions are derived for a classical TMD with total mass ratio $\mu +\beta$, are valid for white noise excitation, and minimize the structural displacement variance [26]. As stated in [15], this does not mean that (4)–(7) lead to the optimum TMDI, but that (4)–(7) represent a reasonable suboptimal solution. Two remarks are added:

• The authors of this article investigated the performance of the TMDI with numerically optimized parameters as well. The main findings of this study in terms of TMDI vibration reduction efficiency are the same as for the TMDI with parameters (4)–(7).
• The authors are aware of the fact that tall building damping aims at minimizing the structural acceleration response. Nevertheless, the design rules for minimum structural displacement are adopted here because the resulting acceleration response is almost minimized as well, which will be shown by the numerical results of this study.

2.3. TMD Properties

As the TMDI is designed for minimum displacement response, the same design target is applied to the classical TMD, which, according to Den Hartog [3], leads to the TMD parameters:

$$f_2=\frac{f_1}{1+\mu }$$

(8)

$$k_2=m_2\hspace{0.17em}{\left(2\hspace{0.17em}\pi \hspace{0.17em}{f}_2\right)}^2$$

(9)

$${\zeta }_2=\sqrt{\frac{3\hspace{0.17em}\mu }{8\hspace{0.17em}\left(1+\mu \right)}}$$

(10)

$$c_2=2\hspace{0.17em}{\zeta }_2\hspace{0.17em}{m}_2\hspace{0.17em}\left(2\hspace{0.17em}\pi \hspace{0.17em}{f}_2\right)$$

(11)

2.4. Equations of Motion with TMDI Grounded to Any Structural Mass

For the formulation of the equations of motion of the building with TMDI, it is assumed that the pendulum mass is hung from the roof mass, i.e., mass 20, while the inerter is grounded to one of the masses below the roof mass (mass 19, mass 18, …, mass 1), but not to the earth (Section 2.5.). The resulting model of the building with TMDI therefore has 21 degrees of freedom, of which the index 21 stands for the TMDI. The according equations of motion with excitation of all structural masses become

$$M\hspace{0.17em}\ddot{x}+C\hspace{0.17em}\dot{x}+K\hspace{0.17em}x=F_w$$

(12)

where $\ddot{x}$ denotes the vector of the absolute accelerations of the 20 structural masses and the TMDI mass, $\dot{x}$ and $x$ represent the vectors of velocity and displacement of these 21 masses relative to ground, $M$, $C$, and $K$ are the mass, damping, and stiffness matrices, and $F_w$ denotes the vector of the wind forces exciting all $n$ = 20 structural masses, but not the TMDI mass, as the TMDI is installed inside of the building (pendulum mass with index 21 is hung from roof mass with index 20)

$$F_w=f_w\hspace{0.17em}{\left[\begin{array}{cccccc}1& 1& \cdots & 1& 1& 0\end{array}\right]}_{1\times (n+1)}^T$$

(13)

where $f_w$ is the time-based broadband or harmonic excitation force.

The structures of the stiffness matrix $K$ and damping matrix $C$ do not depend on the structural mass to which the inerter is grounded, but $K$ and $C$ slightly depend on the inertance ratio $\beta$ and the inertance $b$ via the tuning of the stiffness coefficient $k_2$ and viscous damping coefficient $c_2$ in (4)–(7) of the TMDI. Therefore, $K$ and $C$ take the well-known form of a homogenous shear frame with stiffness coefficient $k_2$ and viscous coefficient $c_2$ of the TMDI and TMD, respectively.

$$K\hspace{0.17em}={\left[\begin{array}{cccccc}2\hspace{0.17em}k& -k& 0& & & \\ -k& 2\hspace{0.17em}k& -k& 0& & \\ & \ddots & \ddots & \ddots & \ddots & \\ & & -k& 2\hspace{0.17em}k& -k& 0\\ & & 0& -k& (k+k_2)& -k_2\\ & & 0& 0& -k_2& k_2\end{array}\right]}_{(n+1)\times (n+1)}$$

(14)

$$C\hspace{0.17em}={\left[\begin{array}{cccccc}2\hspace{0.17em}c& -c& 0& & & \\ -c& 2\hspace{0.17em}c& -c& 0& & \\ & \ddots & \ddots & \ddots & \ddots & \\ & & -c& 2\hspace{0.17em}c& -c& 0\\ & & 0& -c& (c+c_2)& -c_2\\ & & 0& 0& -c_2& c_2\end{array}\right]}_{(n+1)\times (n+1)}$$

(15)

where $c$ denotes the structural viscous damping coefficient producing ${\zeta }_1$ = 1.5% and $k$ is the structural stiffness coefficient leading to $f_1$ = 0.14 Hz of the structure without TMDI.

In contrast to $C$ and $K$, the mass matrix $M$ strongly depends on the inerter configuration because the inerter force depends on the difference of the absolute acceleration of the TMDI pendulum mass and absolute acceleration of the structural mass to which the inerter is grounded. To make this absolutely clear, the structure of $M$ is given for the following three cases:

• The inerter is grounded to mass $gr$ = 19, which is the most realistic case, as the TMDI pendulum mass is hung from mass 20 and, therefore, the inerter can easily be connected to mass 19.
• The inerter is grounded to mass $gr$ = 18; in a real building, this configuration would require a hole in the floor of level 19 in order to be able to connect the inerter to the floor of level 18.
• The inerter is grounded to mass $gr$ = 1, which is the lowest connection point on the structure; this inerter configuration is not realistic, as it would require an extremely long inerter connection between the pendulum mass and the first level of the building.

Other inerter configurations (grounded to mass $gr$ = 17, 16, …, 2) are not depicted because the structure of $M$ is clear from the three selected inter-configurations given in (16)–(18).

$$M\hspace{0.17em}={\left[\begin{array}{cccccc}m& 0& & & & \\ 0& \ddots & \ddots & & & \\ & \ddots & m& 0& 0& 0\\ & & 0& (m+b)& 0& -b\\ & & 0& 0& m& 0\\ & & 0& -b& 0& (m_2+b)\end{array}\right]}_{(n+1)\times (n+1)}$$

(16)

$$M\hspace{0.17em}={\left[\begin{array}{cccccc}m& 0& & & & \\ 0& \ddots & \ddots & & & \\ & \ddots & (m+b)& 0& 0& -b\\ & & 0& m& 0& 0\\ & & 0& 0& m& 0\\ & & -b& 0& 0& (m_2+b)\end{array}\right]}_{(n+1)\times (n+1)}$$

(17)

$$M\hspace{0.17em}={\left[\begin{array}{cccccc}(m+b)& 0& & & & -b\\ 0& \ddots & \ddots & & & \\ & \ddots & m& 0& 0& 0\\ & & 0& m& 0& 0\\ & & 0& 0& m& 0\\ -b& & 0& 0& 0& (m_2+b)\end{array}\right]}_{(n+1)\times (n+1)}$$

(18)

From the mass matrices (16)–(18), it is observed that the inerter does not only augment the pendulum mass $m_2$ by its virtual mass $b$, but it also introduces the virtual mass $b$ in the rows and columns with an index according to the structural mass to which the inerter is grounded.

2.5. Equations of Motion with TMDI Grounded to the Earth

If the inerter is grounded to the earth, then the inerter force is in proportion to the absolute acceleration of the pendulum mass only because—in absence of earthquakes—the acceleration of the earth is zero.

$$f_I\left(inerter\hspace{0.17em}grounded\hspace{0.17em}to\hspace{0.17em}earth\right)=-b\hspace{0.17em}{\ddot{x}}_{21}$$

(19)

This change in the inerter equation only impacts the mass matrix; the other matrices of (12) remain unchanged. For the inerter being grounded to the earth, the pendulum mass $m_2$ is augmented by the virtual mass $b$ without introducing the virtual mass $b$ at other locations in the mass matrix.

$$M\hspace{0.17em}={\left[\begin{array}{cccccc}m& 0& & & & \\ 0& \ddots & \ddots & & & \\ & \ddots & m& 0& 0& 0\\ & & 0& m& 0& 0\\ & & 0& 0& m& 0\\ & & 0& 0& 0& (m_2+b)\end{array}\right]}_{(n+1)\times (n+1)}$$

(20)

Then, the building with the TMDI behaves exactly the same as a building with the classical TMD with pendulum mass $m_2+b$.

2.6. Equations of Motion with TMD

The equations of motion of the building with the TMD are given by Equations (12)–(16) with $b$ = 0 and with TMD parameters $c_2$ and $k_2$ according to (8)–(11).

3. Simulation Parameters

3.1. Solver

The simulations of the equation of motion in (12) were performed in the time domain in Matlab^®/Simulink^®, adopting the solver ode45 with variable step size, upper bound 0.001 s, and maximum relative tolerance 0.001.

3.2. Broadband Excitation

The broadband excitation force $f_w$, which excites all structural masses, is modeled based on a white noise signal that is (Figure 2):

• High-pass filtered at the corner frequency 0.005 Hz (third-order Butterworth filter) to remove any static excitation force as the static deflection of the building due to the static wind pressure is not of interest for this study, and
• Low-pass filtered at the corner frequency 99 Hz (second-order Butterworth filter) to suppress higher-frequency excitation forces but still ensure white noise excitation of all 20 modes of the building model.

From all simulated structural displacements and accelerations and from the relative motion $x_d=x_{20}-x_{21}$ of the TMDI/TMD, the according spectra are determined by the power spectrum density estimate (PSD). The simulations demonstrate that the structural response of the top mass is greatest. Thus, the following transfer functions $TF$ are of interest for the study under consideration:

$$TF\left(x_{20}/f_w\right)=\frac{PSD\left(x_{20}\right)}{PSD\left(f_w\right)}$$

(21)

$$\hspace{0.17em}TF\left({\ddot{x}}_{20}/f_w\right)=\frac{PSD\left({\ddot{x}}_{20}\right)}{PSD\left(f_w\right)}$$

(22)

$$TF\left(x_d/f_w\right)=\frac{PSD\left(x_d\right)}{PSD\left(f_w\right)}$$

(23)

3.3. Harmonic Excitation

For the simulations with harmonic excitation, the excitation force $f_w$ is a harmonic function that excites all structural masses at excitation frequencies $f{r}_w$, which are defined in relation to the eigenfrequency $f_i$ of mode $i$ ($i$ = 1, 2, 3) to be excited.

$$f{r}_w=\left[0.8:0.01:1.20\right]\hspace{0.17em}\hspace{0.17em}{f}_i$$

(24)

The simulations were run for 800 s at each excitation frequency to obtain steady-state vibrations from which the steady-state amplitudes of the greatest structural displacement $x_{20}$, the greatest structural acceleration ${\ddot{x}}_{20}$, and the relative motion $x_d$ of the TMDI/TMD were determined.

3.4. Considered Modes

The vibration reduction performance of the TMDI, which was designed for mode 1, was computed for the excited modes 1, 2, and 3 to assess the performance of the TMDI not only for the mode for which it was designed, but also for higher-order modes that are excited by the wind load as well.

4. Results and Discussion

The results due to broadband and harmonic excitations are depicted in Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7. All figures show the performance of the TMDI with $\mu$ = 1% and $\beta$ = 1% as a function of the structural mass with number $gr$ to which the inerter is grounded; the corresponding performance curves are the colored lines with a one-point line thickness. The case where the inerter is grounded to the earth is depicted by the thick red performance curve. The benchmark of the building with the classical TMD with $\mu$ = 1% is shown by the thick black performance curve. The observation of the results demonstrates that:

• The TMDI being tuned to mode 1 improves the structural displacement response only if the inerter is grounded to a structural mass within approximately the first third of the building height, i.e., to one of the masses 1, 2, …, 7 (Figure 4a and Figure 6a); the same observation applies to the structural acceleration response (Figure 4b and Figure 6b).
• For the most realistic case, where the inerter is grounded to the floor just below the pendulum mass, i.e., $gr$ = 19, the maximum structural displacement due to the TMDI is approximately 47% greater than for the TMD (Figure 4a and Figure 6a) and the maximum acceleration is approximately 41% greater (Figure 4b and Figure 6b).
• The fact that both the TMDI and TMD are designed for minimum structural displacement response is the reason for the slightly greater right peak of the structural acceleration response because of the term ${(2\hspace{0.17em}\pi \hspace{0.17em}f{r}_w)}^2$ (Figure 4b and Figure 6b). Note that the left peak of the structural displacement response is greater than the right peak although the TMD is designed for minimum structural displacement because ${\zeta }_1$ is not zero, but 1.5%, while the optimum design of TMDs according to [3] assumes that ${\zeta }_1$ = 0.
• The performance curves resulting from of the TMDI with inerter grounded to the upper half of the building ($gr$ ≥ 11) show the characteristics of a structure with over-damped TMD, whereby the pendulum mass is clamped to the structure and therefore the structure with TMDI behaves similarly to a single DOF. This observation agrees with the reduced relative motion of the TMDI (Figure 4c and Figure 6c).
• The structural displacement responses of modes 2 and 3 are almost the same for the TMDI and TMD (Figure 3, Figure 5 and Figure 7). The precise analysis of the results shown in Figure 7a,b reveals that the TMDI improves the vibration reduction of mode 2 by approximately 8% and of mode 3 by approximately 1.7% compared to the TMD. Note that these small improvements may also be influenced by numerical damping, whereby such detailed analysis is questionable.
• The performance of the TMDI with $\mu$ = 1% and $\beta$ = 1% with inerter grounded to earth (red thick line) is exactly equal to the performance of the classical TMD with $\mu$ = 2% because of the fact that the inertance $b$ augments the pendulum mass by 100% ($m_2+b$, see equation 20) but does not appear at other locations in the mass matrix (see equation 20).

5. Conclusions

The vibrations of a building model with 20 lumped masses and with TMDI were computed for different ground levels of an inerter. The numerical results were compared to those resulting from the classical TMD with same pendulum mass as that of the TMDI. The results demonstrate that the TMDI leads to a worse structural performance when the inerter is grounded to a structural mass within the upper half of the building. If the inerter is grounded to a structural mass within the first third of the building height or to the earth, then the TMDI leads to reduced structural vibrations. The interpretation of this observation is that the inerter may enhance the structural vibration reduction if:

• The absolute acceleration of the inerter ground is zero, which is the case when the inerter is grounded to the earth; then, the full inertance is added to the pendulum mass without introducing the inertance term in the equations of motion of the structural masses, or
• The absolute acceleration of the inerter ground is small relative to the acceleration of the pendulum mass, which is the case when the inerter is grounded to a structural mass within the first third of the tall building; then, not all but a certain amount of the inertance is added to the pendulum mass.

Consequently, for real tall building damping, the TMDI cannot improve the vibration reduction because it is hardly feasible, aesthetic, or economic to connect the inerter, which is located in the TMD room, to a structural mass within the first third of the building height. However, if the inerter can be grounded to the earth in an economic and aesthetic way and if the inerter force can be produced in a frictionless manner, then the concept of the TMDI augments the TMDI mass and, therefore, improves the vibration reduction.