應用風速功率譜密度函數於調諧質量阻尼器最佳化設計與減振效能之探討

Application of Wind Speed Spectra to the Optimal Design of Tuned Mass Damper and the Performance under Excitation

周琮堯(Cong-Yao Chou)

指導教授 ： 呂良正

國立臺灣大學/工學院/土木工程學研究所/碩士(2019年)

https://doi.org/10.6342/NTU202000686

摘要

隨著都市的發展和科技的進步，許多國家在近幾十年內開始興建超高層建築，相繼角逐世界最高建築。在台灣以及其他地震頻繁的國家，例如:日本、中國…等，以往結構物在設計載重時只需考慮自重帶來的垂直向載重和地震力帶來的橫向載重即可，但隨著建築高度攀升，根據風工程領域學者們的研究，高度愈高，風速愈大，且伴隨而來的風壓力和施加於建築物上的風力亦隨之增加，此時的風力就有可能超過地震力成為主控的設計載重，此為安全性層面的考量。除了安全層面以外，建築本身若為住宅用途尚須考量舒適性或使用性，由於超高層建築多使用高強度建築材料，使其不需加大梁柱尺寸即可抵抗巨量的垂直載重，然而此特性造成結構物的側向勁度過於軟弱，在風力的作用下，頂層居室的側向振動加速度有可能使人體感到不舒適，進而影響居住者的舒適性。綜合以上兩點對於超高層建築既有特性的描述，在設計階段需更加了解風力對其帶來的影響。
由於超高層建築會有頂層側向加速度過大的現象，而過大的加速度則會使人體感到不舒適，因此，許多超高層建築裝設調諧質量阻尼器(Tuned Mass Damper, TMD)來抑制這樣的情形，如台北101大樓、日本大阪的Crystal Tower、美國紐約的Citicorp Center和加拿大的CN Tower等。TMD是一種結構被動控制裝置，裝設完成後不需額外施力即可開始運作並減緩建築物的振動。在調諧質量阻尼器的研究領域，學者們大多假設風力為白噪音型式，此作法可將其功率譜密度假設為定值，便於進行TMD最佳化設計，但實際上風力的功率譜密度函數並非定值，本文採用von Karman (1948)提出的風速功率譜密度函數取代白噪音型式的風力進行TMD的最佳化設計，並將兩種外力作用下的最佳化參數解與折減率做比較，以探討不同外力形式下的最佳化設計變數解對減振效果的影響。
除了探討實際風力譜密度函數與白噪音之間的不同以外，本文進一步討論TMD質量比與減振效果的關係。首先，以Paola(1998)提出的數值模擬流程製造出多筆符合風場特徵的風速歷時，再將風速轉換成風力，接著將風力加載於不同的剪力構架模型上並進行TMD最佳化問題的求解，最後透過動力分析得到裝設TMD前後的位移、速度與加速度歷時。針對舒適性問題，建築物耐風設計規範及解說(以下簡稱規範)4.3節規定頂層居室的振動加速度峰值不得超過5 gal(cm/s^2)，若超過此上限值，則必須依規範4.5節之內容採用可減緩振動加速度的裝置來降低頂層居室的最大加速度，而TMD即為其中一種裝置。設計TMD時所需的頻率比和阻尼比可經由演算法求解最佳化問題得知，而質量比則需考量減振效果與結構物的負重能力。本研究以多筆風力歷時進行時間域的動力分析，將結果進行統計，並提出針對舒適性指標的超越機率，最後得到使各建築物模型的頂層居室加速度峰值降至法規上限值內所需的最低質量比，此為臨界質量比。期望未來工程師與建築師在TMD初步設計階段時，除了考慮風洞實驗的結果之外，也能參考本文臨界質量比的概念進行TMD設計，確保減振效果的同時也避免過度設計的情形。

關鍵字

調諧質量阻尼器最佳化設計 ； 高斯白噪音 ； 風力歷時 ； von Karman風速功率譜密度函數 ； 加速度峰值 ； 舒適性超越機率 ； 臨界質量比

並列摘要

With the development of the city and the progress of technology, many countries compete with others via constructing high-rise buildings to gain the reputation of the tallest building in the world in recent decades. When engineers in Taiwan and other countries where earthquakes are frequent, for instance, Japan, China etc. decided the design load of structures in the past, they usually only needed to take the vertical load from self-weight and horizontal load from earthquake into consideration. However, as the buildings get higher and higher, according to the researches in wind engineering field, the higher the vertical height, the faster the wind speed, and the wind pressure and wind force acting on the buildings become stronger as well. Under this circumstance, the wind force will probably exceed the earthquake force and become the dominant design load. In addition to the safety level, if the buildings are for residential use, engineers also need to consider the accessibility and comfortability. Due to the fact that most high-rise buildings are constructed by high strength construction materials, engineers do not need to scale the size of beams and columns to resist the massive vertical load. Nevertheless, this character makes the lateral stiffness of structures weaker. Under the influence of wind force, the lateral acceleration of the top residential floor has a high probability to make people uncomfortable and then produce great negative effects on residents’ comfortability. As a result of the abovementioned shortcomings of the characters of high-rise buildings, engineers should have more comprehensions of the effects of wind force acting on buildings during the design phase.
Since the high-rise buildings have the phenomenon of the excessive lateral acceleration on their top floor which makes human uncomfortable, many of them are installed tuned mass damper (TMD) to suppress this phenomenon, such as Taipei 101, Osaka Crystal Tower, New York Citicorp Center, Toronto CN Tower etc. TMD is a sort of passive structural control device. After its being installed, it is not in need of additional for its operation and mitigating the vibration response of structures through energy dissipation. In the field of tuned mass damper, vast majority of scholars assume that the wind force is a type of white noise that regards the power spectral density as a constant value for the TMD optimization design. The power spectral density of wind, however, is not a constant value. This thesis substitutes the power spectral density of wind speed proposed by von Karman (1948) for the white noise wind force to investigate the TMD optimization design. Comparing the optimal parameters and reduction indexes from the power spectral density of wind speed suggested by von Karman with those from white noise wind force, this thesis attempts to discuss the effect of optimal parameters on effectiveness of mitigating vibration from different force types.
On top of the discussion of the difference between the power spectral density of real wind and white noise, the relationship between mass ratio of TMD and mitigation effectiveness will be further explored in the thesis. Firstly, multiple wind speed histories that satisfy the characters of wind field will be made by way of digital simulation proposed by Paola (1998), and then they will be transformed into wind force histories. Then, the wind force histories will be loaded on shear frame models and solve TMD optimization problems. Finally, the histories of displacement, velocity, and acceleration before and after installation of TMD will be obtained through dynamic analysis. For comfortability problem, section 4.3 of “Wind Resistance Design Specifications and Commentary of Buildings” (hereinafter referred to as “Specification”) regulates a rule that the peak lateral acceleration on the top floor caused by vibration must be smaller than 5 gal(cm/s^2). If the peak lateral acceleration on the top floor exceeds this limit value, engineers should abide the section 4.5 of Specification to install some devices that can reduce it and TMD is one of them. The frequency ratio and damping ratio required to design the TMD can be solved by algorithm solving optimization problems. Furthermore, the mass ratio of TMD is required to consider both effectiveness of reducing vibrations and load capacity of structures. By analyzing the results from dynamic analysis of buildings loading wind force histories, gathering statistics and proposing the probability of exceedance for comfortability index, this thesis will come up with minimum mass ratios of TMD that can make the peak lateral accelerations on the top floor lower than the upper limit value from Specification, i.e., the critical mass ratio of TMD. Hopefully, this thesis can provide engineers and architects, apart from exclusively relying on the results from wind tunnel test when designing TMDs, with another way of estimating rough values for the mass ratio of TMDs in the future. Therefore, the effectiveness of vibration reduction can be achieved, in the meantime over-design can be avoided as well.

並列關鍵字

Tuned Mass Damper Optimization Design ； White Gaussian Noise ； Wind Force History ； von Karman Power Spectral Density of Wind Speed ； Peak Acceleration ； Probability of Exceedance for Comfortability ； Critical Mass Ratio

參考文獻

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Di Paola, M. (1998). Digital simulation of wind field velocity. Journal of Wind Engineering and Industrial Aerodynamics, 74, 91-109.

Frahm, H. (1911). U.S. Patent No. 989,958. Washington, DC: U.S. Patent and Trademark Office.

Kawaguchi, A., Teramura, A., & Omote, Y. (1992). Time history response of a tall building with a tuned mass damper under wind force. Journal of wind engineering and industrial aerodynamics, 43(1-3), 1949-1960.