Zhiqiang Et Al. (2011) Vibration Reduction of a Stadium Corridor Using Tuned Mass Dampers

Vibration Reduction of a Stadium Corridor Using Tuned Mass Dampers

Zhiqiang Zhang1,2, Guowei Zhou3, Aiqun Li1,2, Xiaofeng Bill Zhang4

1School of Civil Engineering, Southeast University, 2 Si Pai Lou, Nanjing 210096, China 2Southeast University, Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education,2 Si Pai Lou, Nanjing 210096, China 3Huasen Architectural & Engineering Designing Consultant Ltd., Nanjing 210096, China 4Department of Civil & Environmental Engineering, Temple University, 1947 N. 12th St., Philadelphia, PA 19122; PH (215) 204-3046; FAX (215) 204-4696; email: [email protected] ABSTRACT The dynamic characteristics of the long-span corridor in a stadium are usually very complicated. The vibration may be significant when subjected to heavy walking excitations, which requires vibration control systems, such as dampers, to be implemented. In this paper, a stadium corridor using Tuned Mass Dampers (TMDs) for vibration control was studied. The eigenvalue analysis of the corridor showed that a vertical vibration mode is prominent, and the frequency of this mode is close to human fast-walking frequency, which can cause resonance in the corridor. Simulated walking excitation loads were applied to the finite element model. A variety of walking scenarios, and different TMD frequencies and distributions were investigated in order to optimize the frequencies and layout of the TMDs. The analysis results indicated that TMDs can significantly reduce the vibration response of the corridor due to walking, particularly near the resonance frequency. Though the TMDs were tuned to or near the resonance frequency, when subjected to other walking frequencies, the corridors responses with TMDs also reduced substantially. The sensitivity of TMDs to structural frequency necessitates the field dynamic tests before fine tuning the dampers to optimize the vibration control effects. INTRODUCTION

Contemporary architectural design evolves with radical changes in functionality, aesthetics, and space arrangement, and brings irregular and unique building configurations. It poses challenges to structural engineers in achieving structural

2807Structures Congress 2011 ASCE 2011

Structures Congress 2011

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design to meet the safety and comfort requirements. Shown in Figure 1 is the 3-D model of a stadium, an example of such a challenging structure.

As shown in Figure 1, the bowl is hung at the outer rim by cables to the space trusses at the top. There is no support from the bottom except a few diagonal braces to the giant lattice columns on the left side. The top of the bowl is partially connected to the space trusses. The vertical vibration may be significant because of the low vertical stiffness of the bowl. The ring corridor inside the bowl is supported by vertical posts on the latticed shells of the bowl structure (Figure 2). As a result of the long span, and being supported on the locations where the vertical vibration mode is pronounced, the corridors vibration may be substantial under human walking excitation, causing discomfort of the occupants and affecting the use of the structure. Therefore, it is necessary to control the vibration of the corridor.

DYNAMIC ANALYSIS OF THE STRUCTURE

A 3-D finite element model (Figure 1) was developed in MIDAS in order to study the dynamic behavior of the structure, and to provide a basis for the settings of the damper parameters. The space beam element is used to model concrete and steel beams and columns. The shell element is used to model floor slabs, taking account of in-plane and out-of-plane deformations. The cable element is used for the cables

Figure 1. 3-D model of the stadium

Figure 2. Bowl section and corridor

Corridor



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hanging the bowl. Modal analysis is performed using the Ritz vector method by Wilson and others.

The damping ratio varies among different modes for a steel-concrete composite structure like this. To ensure the accuracy of subsequent response history analysis, the method provided by MIDAS is used to calculate the damping ratios of the structures different modes. The theoretical base is the Stain Energy Weighted Average Method, which averages the strain energy of members of different materials in a mode, weighted by their damping ratios, to obtain the damping ratio of that mode. The damping ratios of the first ten modes are shown in Table 1.

Table 1. Frequencies and damping ratios of the first 10 modes

Mode No. 1 2 3 4 5 6 7 8 9 10 Frequency

(Hz) 2.34 3.21 3.47 4.53 4.89 6.32 7.20 7.38 7.41 7.48

Damping ratio 0.031 0.027 0.036 0.033 0.044 0.036 0.021 0.025 0.047 0.024 The results of the modal analysis reveal that the vertical vibration mode of the

structure is prominent. The second vibration mode turns out to be a vertical mode of the bowl (see Figure 3) and the third mode is a torsion mode. Meanwhile, the corridor shows significant vertical vibration in every mode. Therefore, the vertical vibration is significant for the corridor. The first two modal frequencies of the structure are 2.3Hz and 3.2Hz. Though the first mode is mainly translational, the corridor exhibits certain vertical vibration. The frequencies of 2.3Hz and 3.2Hz are very close to human walking frequencies ranging from 1.8Hz to 2.5Hz, leading to potential resonance in the structure. The structure is adequate in strength. However, the excessive vibration beyond the acceleration tolerance of human comfort may cause psychological panic. Therefore, it is necessary to take measures to damp the vertical vibration of the corridor. The modal periods and vertical participating mass ratios of the top 20 modes are shown in Table 2.

Table 2. Modal properties of first 20 modes

Mode No. 1 2 3 4 5 6 7 8 9 10 Freq. (Hz) 2.34 3.21 3.47 4.53 4.89 6.32 7.20 7.38 7.41 7.48 Vert. parti. mass ratios

(%) 0.14 7.90 0.01 2.00 0.51 0.10 1.53 6.62 3.03 3.70

Mode No. 11 12 13 14 15 16 17 18 19 20 Freq. (Hz) 8.07 8.50 8.83 9.56 9.88 10.22 10.39 10.67 10.86 10.95Vert. parti. mass ratios

(%) 3.86 0.01 2.12 4.31 1.26 0.14 1.16 0.14 0 0.42



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It can be seen from Table 2 that the 2nd, 8th, 9th and 10th modes are primarily vertical vibration modes, of which the vertical participating mass ratio of the 2nd mode is the largest. The vertical vibration modes are distributed. However, almost every mode has some vertical participating mass. Its observed that significant vertical vibration occurs in every mode in the corridor. With regard to higher order modes, their vibration frequencies are far away from that of the walking excitation, and are less likely to cause resonance. Hence, the vibration control is performed mainly on the first two modal frequencies. Figure 3 shows the first two mode shape.

During the response history analysis, sixty modes were included for combing. The damping ratios for the first ten modes are shown in Table 1, while the other modes having a uniform damping ratio of 0.035, 0.04, and 0.05, respectively, by input. The analysis demonstrated that the results of using the above three values are almost the same. This indicates that the damping ratios for the upper fifty modes have little

Mode 1

Mode 2

Figure 3. First two structural vibration modes



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effects on the analysis.

PRINCIPLES AND DESIGN PROCEDURES OF TMD

The main task of controlling the vibration of the corridor under walking excitation is to reduce vertical vibrations. Tuned mass damper (TMD), as a passive control technology, has been successfully used in control of wind-excited vibration in high-rise structures. Numerical studies have shown that TMD can also effectively reduce the vertical vibration of long-span structures.

To maximize the effect of vibration control, the frequency of TMD shall be set near the free vibration frequency of the structure to be controlled. However, the damping effect of TMD is very sensitive to the fluctuations of the free vibration frequency, and there are always discrepancies between the actual free vibration frequencies of the structure and the calculated ones as a result of structural analysis errors and construction variations. Damping effect of TMD will be affected by these factors. More importantly, for the vibration modes lower than the mode being controlled, TMD has a good curbing effect to their response. For the modes higher than the one being controlled, TMD has little damping effect or may even amplify them. Given these characteristics, Clark (1988) proposed the concept of MTMD, which aims at a certain range of frequencies to be controlled, resolving the TMD system of a certain frequency bandwidth into multiple sub-TMD systems to improve the stability of the control system.

With MTMD attached, the dynamic equations of the multi-degree of freedom system under vertical load become:

Where:

MS, CS, and KS denote the mass, damping and the stiffness matrices of the main structure, respectively. Md, Cd, and Kd denote the mass, damping and the stiffness matrices of the TMD system. {xd} stands for the displacement vector of each TMD relative to the main structure, only with the vertical component, while horizontal components being zero. E is the acting position matrix of TMD, where, the jth column vector (with ith element being unity, all others zero) represents that jth TMD is set at the ith node of the structure. f(t) is the time history of the walking load, which was obtained by the standard of IABSE.



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To reduce the vertical vibration of the corridor, the primary vertical modes must be controlled. In this project, the second mode is the primary vertical mode. Considering the sensitivity of TMD systems to the frequencies, the TMD frequencies need to be fine tuned within the range of frequencies being controlled, to prevent the amplification of other walking frequencies and enhance the stability of the structure. The following are the basic steps of the method. 1. Analyze the structures dynamic properties. Obtain the vertical vibration mode of

the structure and the distribution of the corridors vertical vibration. 2. Determine the frequency that needs to be controlled according to the results of

modal analysis. 3. Locate the TMDs. Generally speaking, installing TMDs targeting a vibration

frequency at the peak(s) of the mode shape appears to provide the most damping effect.

4. Add TMDs to the structure, and fine tune the controlling frequency of TMD in a certain spectrum. With the consideration of period change due to the additional masses of TMDs and variations of the walking excitations, determine the most appropriate frequency value(s), to enhance the stability of the system.

COMPARISON OF TMD DESIGNS AND ANALYSIS ON DAMPING

To find out the maximum response locations under the walking load, a group of 30 people were considering walking along the corridor on two floors from Endpoint A to Endpoint B. With two people in a row and the length of the group on the corridor being 14m, that is equivalent to a uniform load of to 0.7kN/m2, when taking the weight of a person as 70kg, referring to Section 2.2.1 in American Institute of Steel Construction (AISC) Steel Design Guide Series 11 Floor Vibrations Due to Human Activity (Murray et al 1997). The acceleration response was calculated every time the group moving forward by three sections. Then the maximum response was found, which occurred at the exterior on the upper and lower levels, i.e., Node 4999 and Node 5015. Therefore, the locations near these two nodes are the major spots to set TMDs.

Figure 4. Sketch of the corridor

A

BA

B

4999

5015



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According to the analysis above, more TMDs are set at the curved segment on the exterior side of the corridor, and fewer near the end points A and B. TMDs are also located on the entrance platforms of the corridor. There are a total of 33 sets of TMD systems over the two corridors. Since the frequencies of walking excitation range from 1.8Hz to 2.5Hz, and the first two natural frequencies of the structure are 2.3Hz and 3.2Hz, the following vibration control schemes are compared: 1) TMDs at 2.5Hz; 2) TMDs at 2.8Hz; 3) TMDs at 3.2Hz. After finding the most effective damping frequency, the TMDs are divided into three groups having different frequencies within the narrow band around that frequency. The damping effects of the above schemes are compared under different loading conditions. The following loading conditions were considered: 1) slow walking, 50 people stretching 20m on the corridor, at the most unfavorable location (i.e., the outermost arc), with a uniform load of 0.7 kN/m2, and exciting frequency of 1.8Hz, 2.0Hz and 2.2Hz, respectively; 2) fast walking, 30 people stretching 14m on the outermost arc, with a uniform load of 0.7 kN/m2, and exciting frequency of 2.5Hz and 3.2Hz, respectively. The walking excitation curves are shown in Figure 5.

The comparisons between the acceleration responses of Node 4999 and Node

5015 under each loading condition in three TMD schemes are shown in Table 3.

Figure 5. Walking excitations

1.8Hz slow walking curve

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2.5Hz fast walking curve

00.20.40.60.81

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The following was observed from the analysis, as shown in Table 3. 1) The second vibration mode of the structure plays a decisive role in the response of the corridor, with the frequency of 3.2Hz. 2) The resonance is significant under the walking frequency of 3.2Hz. 3) The responses are much less under the walking frequencies ranging from 1.8Hz to 2.5Hz than that of 3.2Hz, since the resonance frequency is outside this range. 4) As TMDs tuning frequency increases, the damping effects under different conditions vary. The tuning frequency of 3.2Hz has the best effects on restraining the resonance. But under low excitation frequencies, the damping effects are limited. The tuning frequencies of 2.5Hz and 2.8Hz better restrain vibrations of the lower modes, less effective at 3.2Hz excitation.

Table 3. Acceleration responses of Nodes 4999 and 5015 (mm/s2)

Tuning frequency

Excitation frequency 1.8Hz 2.0Hz 2.2Hz 2.5Hz 3.2Hz Node Node Node Node Node

4999 5015 4999 5015 4999 5015 4999 5015 4999 5015Before TMD 182 142 207 163 200 160 160 135 665 531

After TMD

2.5Hz 168 128 179 138 176 137 132 101 464 323 2.8Hz 169 128 179 135 176 133 132 100 353 251 3.2Hz 175 138 187 145 183 143 143 124 201 183

In order to avoid the resonance, the tuning frequency should be set around 3.2Hz.

Table 3 indicates that TMD with one single frequency cannot behave well on both cases of slow and fast walking. Using the concept of MTMD, the whole system of TMDs is divided into three groups. Considering that 3.2Hz is about the upper bound for walking, the groups have the frequencies ranging from 2.8Hz to 3.2Hz. These groups are mostly laid around the peaks of the second mode shape, at the outer side of the arc. TMD with different frequencies are laid in alternative pattern. Doing so reduces TMDs sensitivity to the excitation frequency, and makes the system to gain damping effects in a wider frequency range. The parameters of the three groups of TMDs are shown in Table 4.

Table 4. TMD system parameters

Number Mass (kg) Tuning frequency (Hz) Period (s) Stiffness (kN/m)TMD1 500 3.0 0.333 177.5 TMD2 500 3.1 0.322 189.5 TMD5 500 3.2 0.313 201.9 With the MTMD systems in place, the response accelerations of Nodes 4999 and

5015 under each operating condition are shown in Table 5.



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Compared with results in Table 3 and Table 5, it can be seen that after grouping TMDs with separate frequencies within a small range, the responses changed little under the resonance, with significant damping effects. It improved damping effects under slow walking.

Figure 6 shows the comparison of acceleration responses of Nodes 4999 and 5015 before and after TMD installation under the excitation of the frequency of 3.2Hz. It can be seen from Figure 6, before installing TMDs, the structures acceleration response is of resonance characteristics. After the installation of TMDs, the resonance phenomenon diminishes. At the resonance frequency of 3.2Hz, the damping effect on the acceleration reaches nearly 65%.

Table 5. Max acceleration responses under MTMD (mm/s2)

Excitation frequency Node Before TMD After TMD

Damping ratios

1.8Hz 4999 182 169 7.14% 5015 142 129 9.15%

2.0Hz 4999 207 181 12.56% 5015 163 141 13.50%

2.2Hz 4999 200 175 12.50% 5015 160 135 15.63%

2.5Hz 4999 160 135 15.63% 5015 135 110 18.52%

3.2Hz 4999 665 235 64.66% 5015 531 212 60.08%

CONCLUSIONS

Based on the damping analysis using MTMD on the corridor in a stadium under walking excitations, the following conclusions can be drawn. 1. When the vertical natural frequency of the structure is close to the walking

frequency, the resonance response is significant. In order to meet the human comfort requirements, the vibration control should be implemented.

2. TMDs need to be deployed in the vicinity of the peaks of the mode shape corresponding to the frequency to be controlled to achieve the most vibration control effects.

3. TMDs perform the best when tuned to the excitation frequencies. They are less effective to other vibration modes. To control more vibration modes, each group of TMDs should be located at the peaks of the corresponding mode shape.

4. Grouping TMDs into different frequencies within the controlling frequency range



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can reduce the amplification of the structures vibration under other modes. 5. Due to TMDs sensitivity to the structural free vibration frequencies, it is

necessary to conduct field measurements of the dynamic properties of the structures before setting the final damper parameters.

REFERENCES Clark, A. J. (1988). Multiple passive tuned mass dampers for reducing earthquake

induced building motion. Proceedings of 9th World Conference on Earthquake Engineering, Tokyo, Japan, Vol V: 779-784.

Comparison of point 4999's acceleration

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Comparison of point 5015's acceleration

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Figure 6. Acceleration responses of Nodes 4999 and 5015 before and after TMD installation



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Li, A. (2007). Structural Vibration Control, China machine press, Beijing, China. Murray, T. M., Allen, D. E., and Ungar, E. E. (1997). Floor Vibrations Due to Hunam

Activity, AISC Steel Design Guide Series 11, American Institute of Steel Construction, Chicago.

Wang, Z. (1997). Vibration Control of Tall Structures, Tongji University Press, Shanghai, China

Ye, J. (1998). Theory on Vibration Control of Latticed Shell Structure With TMD, Post-doctoral research report, Harbin Jianzhu University, Harbin, China.

Ye, Z., Li, A., Ding, Y. (2003). Study on Vibration Energy Dissipation Design of A Pedestrian Bridge, Special Structures, 20(1): 68-70.

Zhang, Z., Li, A., and Xu, Y. (2007). Research on Seismic Vibration Control of Hefei TV Tower with TMD, Earthquake Resistant Engineering and Retrofitting, 29(5): 12-16.



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Documents

Zhiqiang Et Al. (2011) Vibration Reduction of a Stadium Corridor Using Tuned Mass Dampers