《结构防灾技术》课程教学课件（讲稿）第三篇 结构抗风 Wind-resistant structure 第十一章 结构顺风向随机风振响应 Alongwind random vibration response of structure

第11章结构顺风向随机风振响应Chapter 11 Alongwind random vibration response ofstructure11.1结构的风振响应的运动方程ll.1 the eguation of motion of structure under fluctuating wind作用在结构上的顺风向风压可以分解为：平均风、脉动风。平均风作用下的平均响应x(t)采用静力计算方法确定。Downwind wind pressureacting on the structure can bedivided into mean wind and fluctuating wind.The meanresponse under mean wind action, x(t) is calculated by staticmethod.脉动风作用下的脉动响应方差采用随机振动的方法确定。The fluctuating response variance oxis determinedbyrandomvibrationmethod结构中总的响应Totalstructuralresponse：x=x+gox式中g为峰值因子（peakgustfactor）

第11章 结构顺风向随机风振响应 Chapter 11 Alongwind random vibration response of structure 11.1 结构的风振响应的运动方程 11.1 the equation of motion of structure under fluctuating wind

11.1结构的风振响应的运动方程11.1 the eguation of motion of structure under fluctuating windThe equation of motion of structure under wind loads:[M ](x(0) +[C] (x(t) +[K] /x(t) = F(t,/x(t),/x(0),(x(t)■方程右边的荷载包括了作用于结构的脉动风荷载以及由结构自身运动而产生的自激力。The right of the equation includes thefluctuating wind load on the structure and the self-excited forceproduced by the structural motion.尽管结构体系被假定为线性，由于自激力的影响，方程本身仍然为非线性的，对该方程解析求解的困难就在于方程右端项的不确定性。Although the structural system is considered to be linear,because of the influence of self-excited force, the equationitself is non-linear. The difficulty to solve the equation is theuncertainness of the right of equation

 The equation of motion of structure under wind loads:  方程右边的荷载包括了作用于结构的脉动风荷载以及由结构自身运动 而产生的自激力。The right of the equation includes the fluctuating wind load on the structure and the self-excited force produced by the structural motion.  尽管结构体系被假定为线性，由于自激力的影响，方程本身仍然为非 线性的，对该方程解析求解的困难就在于方程右端项的不确定性。 Although the structural system is considered to be linear, because of the influence of self-excited force, the equation itself is non-linear. The difficulty to solve the equation is the uncertainness of the right of equation 11.1 结构的风振响应的运动方程 11.1 the equation of motion of structure under fluctuating wind

11.1结构的风振响应的运动方程11.1 the equation of motion of structure under fluctuating wind研究表明：自激力项可以通过气动阻尼的方式来考虑，通常的做法在结构阻尼比上叠加气动阻尼比，因此结构的运动方程可写为：Studies shows that: self-excited force may be considered byway of aerodynamic damping. generally the structuraldamping ratio superimposed on the aerodynamic dampingratio, so the structure of the eguations of motion:[M ] (x(t) +[C] (x(t) +[K ](x(t)) = (p(t))在时域内求解时，方程右端为脉动风荷载时程；在频域内求解时，方程右端为脉动风荷载的率谱密度函数。When solving the equation in the time domain, the right ofthe equation is the time history of wind load. When solving itin the frequency domain, the right of the equation is thespectral density function of fluctuating wind loads

研究表明：自激力项可以通过气动阷尼的方式来考虑，通常的做法： 在结构阷尼比上叠加气动阷尼比，因此结构的运动方程可写为： Studies shows that: self-excited force may be considered by way of aerodynamic damping. generally the structural damping ratio superimposed on the aerodynamic damping ratio, so the structure of the equations of motion: 在时域内求解时，方程右端为脉动风荷载时程；在频域内求解时， 方程右端为脉动风荷载的率谱密度函数。 When solving the equation in the time domain, the right of the equation is the time history of wind load. When solving it in the frequency domain, the right of the equation is the spectral density function of fluctuating wind loads. 11.1 结构的风振响应的运动方程 11.1 the equation of motion of structure under fluctuating wind

11.1结构的风振响应的运动方程11.1 the equation of motion of structure under fluctuating wind[M /(x(t)) +[cl(x(t)) +[K] (x(t)) = (p(t))时域求解：时程分析、基于振型分解的时程分析Solution in Time domain : Time-history analysis, time -history analysis based on modal decompositionP(t)x(t)频域求解：多自由度随机振动分析、基于振型分解的频域分析Solution in frequency domain: random vibration analysisofMDOFsystem,andfrequencydomainanalysisbasedonmodaldecompositionSp(n)Sx(n)

 时域求解：时程分析、基于振型分解的时程分析 Solution in Time domain : Time-history analysis, time - history analysis based on modal decomposition  频域求解：多自由度随机振动分析、基于振型分解的频域分析 Solution in frequency domain: random vibration analysis of MDOF system, and frequency domain analysis based on modal decomposition 11.1 结构的风振响应的运动方程 11.1 the equation of motion of structure under fluctuating wind

11.2结构的风振响应的时程分析11.2 time-history analysis of the equation of motion时程分析Timehistoryanalysis■也称直接积分法，实质数值计算方法It is also known as direct integral method, actually annumerical methods按假定的加速度变化规律时程分析法可分为therearedifferent time history method based on the assumptionof acceleration variation:linearaccelerationmethodoWilson-0methodoNewmark-βmethodRang-kutamethod

时程分析 Time history analysis 也称直接积分法，实质数值计算方法 It is also known as direct integral method, actually an numerical methods 按假定的加速度变化规律时程分析法可分为 there are different time history method based on the assumption of acceleration variation:  linear acceleration method Wilson-θ method Newmark-β method Rang-kuta method 11.2 结构的风振响应的时程分析 11.2 time-history analysis of the equation of motion

第11章结构顺风向随机风振响应Chapter 11 Alongwind random vibration response ofstructure11.3顺风向随机风振响应的频域分析11.3 frequency domainanalysis of vibration response ofstructure underdownwindfluctuating wind，多自由度随机振动分析：[S (o)]=[H(o)][ S,(o) [H (α)荷载功率谱矩阵Sp,(o)Sp,(o)... Sp(o)2Sp. (o)S pa,(0)Sp (o)...222[s,(o)]:·.Sp(o)S.(o)Sp (o)·

11.3 顺风向随机风振响应的频域分析 11.3 frequency domain analysis of vibration response of structure under downwind fluctuating wind 第11章 结构顺风向随机风振响应 Chapter 11 Alongwind random vibration response of structure

第11章结构顺风向随机风振响应Chapter 11 Alongwind random vibration response ofstructure11.3顺风向随机风振响应的频域分析11.3 freguency domain analysisof vibration response ofstructure under downwind fluctuating wind频率响应函数矩阵[H,(α)H,(o)H12(0)H2(0)H2()H2n()[H(α)] =：::H(o)Hr2(0)Hm(o)

11.3 顺风向随机风振响应的频域分析 11.3 frequency domain analysis of vibration response of structure under downwind fluctuating wind 第11章 结构顺风向随机风振响应 Chapter 11 Alongwind random vibration response of structure

11.3顺风向随机风振响应的频域分析11.3 frequency domainanalysis of vibrationresponseofstructure under downwind fluctuating wind■基于振型分解的频域方法Frequencydomainmethodbased on modal decomposition:（1）采用振型分解，将多自由运动方程转换为n个单自由度运动方程；By using modal decomposition, decouple the MDOFequation of motion of MDOF system into n number ofequations of motion of SDOF system（2）采用频域法对单个自由度运动方程进行求解；Solve SDOF equation of motion using frequency domainmethod（3）采用叠加原理得到原结构的解superpose the solutions of SDOF system to obtain thesolution of original structure

 基于振型分解的频域方法 Frequency domain method based on modal decomposition: （1）采用振型分解，将多自由运动方程转换为n个单自由度运动 方程； By using modal decomposition，decouple the MDOF equation of motion of MDOF system into n number of equations of motion of SDOF system （2）采用频域法对单个自由度运动方程进行求解； Solve SDOF equation of motion using frequency domain method （3）采用叠加原理得到原结构的解。 superpose the solutions of SDOF system to obtain the solution of original structure. 11.3 顺风向随机风振响应的频域分析 11.3 frequency domain analysis of vibration response of structure under downwind fluctuating wind

11.3顺风向随机风振响应的频域分析11.3 frequency domain analysis of vibration response ofstructure under downwind fluctuating wind模态向量振型分解[x(t) =2(g,q,(t)=[0](g(t)[M x(t) +[c](x(t) +[K](x(t)) = (p(t))?j=1[M [@(g(t)) +[c[l(g(t)) +[][@(q(t) =(p(t))[Φ]' [m[](q(t)] +[]"[c[](q(t)) +[]" [][](q(t)) =[](p(t))[]' [M ][] =[]][Φ]' [k][Φ] =[α] = diag(α,)[]"[c][Φ] =diag(25,o,)

11.3 顺风向随机风振响应的频域分析 11.3 frequency domain analysis of vibration response of structure under downwind fluctuating wind

11.3顺风向随机风振响应的频域分析11.3 frequency domainanalysis of vibration response ofstructure underdownwindfluctuating winddiag(1)(g(t)) + diag(25 ,o,)(g(t)) + diag(o))(q(t)) =[] ( p(t))α?[25,01A0?25,0(q(t))g(t)/g(t)-125,0nw"=[0] (p(t)g,(t)+25,0,q,(t) +oq,;(t) ={Φ, (p(t) = f, (t)模态力

11.3 顺风向随机风振响应的频域分析 11.3 frequency domain analysis of vibration response of structure under downwind fluctuating wind