[分享]TMD在SAP2000中的建模及解析
A tuned-mass damper (TMD), also known as a pendulum damper, is not really a damper, but rather a pendulum or another gravity-based oscillator which is attached to the structure in such a way that it counteracts the vibration of one or more fundamental modes, thereby reducing the wind and/or seismic response of those modes.
调谐质量阻尼器，又叫钟摆阻尼器，事实上并非真实一个阻尼器，更像是一个附属到主体结构上的钟摆或重力振荡器，它的摆动可以抵消主体结构的基本振动模式，从而减小风或者地震的反应。
Within SAP2000 or ETABS, a TMD may be modeled using a spring-mass system with damping. Guidelines for this subsystem are described as follows:
在SAP2000或者ETABS中，可以用带有阻尼的弹簧-质量系统来模拟，该系统建模原则如下：
Spring – Assign spring properties to a linear two-joint link object in which one joint is attached to the structure, and the other joint is free.
弹簧：将弹簧属性指定给一个两节点线性连接单元，一段节点连接到主体结构上，另一端节点自由。
Mass – Mass and weight are then assigned to the free joint.
质量：质量及重量指定给自由节点。
Damping – Within SAP2000, linear damping is included directly in the linear link property, while nonlinear damping is modeled using a viscous-damping link object in parallel with the linear link. Within ETABS, whether the system is linear or nonlinear, these damping objects are modeled in parallel.
阻尼：在SAP2000中线性阻尼直接包含在线性连接属性中，当需要考虑非线性阻尼时，需要用一个平行的粘滞阻尼连接对象来建模。 在ETABS中，无论是线性还是非线性阻尼，都直接平行建模。
Procedure
The general procedure for modeling a tuned-mass damper is given as follows:
TMD建模的一般过程如下：
1. Specify link properties 定义连接属性
Any spring-mass system may represent the swinging pendulum in 2D. Here, spring constant is given as Mg/L, where M is mass, L is pendulum length, and g is gravity. It is slightly more challenging to model a pendulum which is free to translate in 3D. In this case, a linear link is created to represent the pendulum device. Select Define > Section Properties > Link/Support Properties, then define translational stiffnesses along U1, U2, and U3. The linear stiffness along U1 represents axial properties, and should be based on the EA/L value of the hangers, which is 1.0e6 kN/m in Model 2. The linear stiffness properties of U2 and U3 are chosen as Mg/L. In Model 2, the link is drawn at the top story. Link length is chosen as L = 0.1m, and mass is M = 10 kN-sec2/m.
每个弹簧-质量体系都能代表一个二维的自由单摆。此处弹簧常数为Mg/L，M为质量，L为单摆长度，g是重力加速度。不过要模拟一个三维摆动还是比较有挑战性的。
因为用一个线性连接来表示一个单摆装置，定义-截面属性-连接属性，打开对话框来定义U1、U2、U3方向的刚度，U1代表摆绳或悬挂装置的轴向刚度EA/L，U2和U3方向的刚度用Mg/L来表示，例子中U1为1000000,U2、U3为1000,L为0.1m，M为10吨。
Length – Pendulum length directly affects the period of the TMD. This is accounted for in the spring and mass properties used. However, the drawn length of the link object is arbitrary and may be chosen for convenience; it may even be zero. We recommend drawing the link such that the I-end (first joint) attaches to the structure, and the J-end (second joint) is free. In this case, within the linear link property, the shear distance from end J may be set to zero for the U2 and U3 DOF.
长度：单摆长度影响TMD的周期，这已经在弹簧和质量属性中考虑过，因此模型中绘制连接对象的长度可以任意长度、随意选取，也可以是零。一般绘制连接单元一端节点在主体结构，另一端节点自由。由此，线性连接属性中U2、U3方向J端的剪切距离可以设置为零。
Mass – Mass strongly affects how the TMD influences response. Changes to mass must be accounted for in the following locations:
质量强烈影响到到TMD如何对外部作用的反应，只能在一下位置改变质量。
Mass (M) should be assigned to the free joint (J-end of the link).
Weight (W = Mg) should be assigned to the free joint (J-end of the link) as a joint force load in the gravity direction in any self-weight load pattern.
Effective stiffness (Mg/L) of the U2 and U3 link properties.
质量和重量都必须在两点连接对象的自由节点，重力必须在重力方向施加，可以是任意自重荷载模式。
U2和U3方向的有效刚度为Mg/L。
Period – Generally, the period (T) of the TMD is chosen to closely match the structural period to be counteracted, taken as the first period of vibration of control model. Note that, although the mass does not affect the period of the TMD, it does affect how strongly the TMD affects the rest of the structure, with larger masses typically having a larger effect. The period of the TMD is given by：
Once the period of the TMD is derived, TMD length is calculated from this value and gravity. This length, along with gravity and the arbitrary mass value, are then converted into the effective link stiffness of the U2 and U3 directions, given as Mg/L. This being the case, the length of the TMD in the real structure is not explicitly modeled, but is accounted for in the link property.
TMD的周期一旦得到，可以利用重力加速度计算出TMD的长度L。长度、重力加速度g和任意质量M被转换为U2、U3方向的有效连接刚度Mg/L.因此实际结构中的TMD的长度L不需要真实模拟，她利用连接属性来表达。
Damping – Damping is defined as either a linear C coefficient or a nonlinear C value plus an exponent on the velocity term. Damping values should be chosen based on the physical characteristics of the TMD device. This damping source affects the TMD itself, but it is not the primary energy-dissipation mechanism for the structure as a whole. For a linear damper, an estimate of the fraction of critical damping (ξ) for the TMD is:
阻尼被定义为一个线性系数C或者一个非线性C加上一个速度的指数项，应该基于TMD装置的物理特性来选择阻尼值，不过阻尼并不是整个结构的主要耗散能量机制，所以有时可以默认为零，当然线性阻尼器的临界阻尼比可以按下式计算：
2. Define the time-history analysis 定义时程分析
Time-history analysis should be performed using either nonlinear modal (FNA) or direct-integration (linear or nonlinear) time-history load cases. These types of analyses correctly account for the coupling of the modes, an effect which may be caused by damping in the TMD device. If damping is small, it might be possible to obtain reasonable results using a linear modal time-history analysis, and possibly even response-spectrum analysis.
To provide overview for this procedure, the time-history load case of Model 2 is defined as follows:
TMD period – Create a control model which does not have a TMD device. Run modal analysis and measure the fundamental period of the 1st mode. The TMD will be designed to counteract response which results from this mode. 现运行不含TMD的主体结构的模态分析，得到第一模态的周期，TMD设计为抵消该模态震动，即TMD周期接近或等于此周期
Time function – Select Define > Functions > Time History, then define a sine curve which has a period equal to that of the 1st mode of the control model. In this case, a period of 0.6 seconds is obtained.
Load case – Add a nonlinear modal time-history load case which assumes 5% modal damping and uses 200 output steps, each 1/20th the size of the 1st time period.
Response – Run analysis and review various response measures through Display > Show Plot Functions. As expected, response for the tuned-mass-damper model is found to be reduced.
节点14的位移曲线，不带TMD
节点14的位移曲线，带TMD，有明显减小。
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