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A *tuned-mass damper* (TMD), also known as a pendulum [damper|kb:Damping], 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|kb:Modal analysis], thereby reducing the wind and/or seismic response of those modes.

Within SAP2000 or ETABS, a TMD may be modeled using a spring-mass system with damping. Guidelines for this subsystem are described as follows:

* *Spring* -- Assign spring properties to a linear two-joint [link|kb:Link] object in which one [joint|kb:Joint] is attached to the structure, and the other joint is free.

* *Mass* -- [Mass|kb:Mass] and weight are then assigned to the free joint.

* *Damping* -- Within [SAP2000|sap2000:home], 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|etabs:Home], whether the system is linear or nonlinear, these damping objects are modeled in parallel.


h1. Reference files

For reference, two [SAP2000|sap2000:home] models are attached, each identical except that Model 1 does not use a TMD, whereas Model 2 does. These models, also available in the Attachments section as a zipped file, are described as follows:

* *Model 1* -- [TestModel_Without TMD.SDB |Tuned-mass damper^TestModel_Without TMD.SDB] does not use a TMD, serves as the control system, and is used to determine the frequency of the structure.

* *Model 2* -- [TestModel_With TMD.SDB |Tuned-mass damper^TestModel_With TMD.SDB] features a subsystem which simulates the effect of a pendulum damper.

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!Figure 3.png|align=center,border=0,width=500pxpxpxpx500px!

{center-text}Model 2 - Test model with TMD{center-text}


h1. Procedure

The forgeneral Model 2

Model 2 is created through the process whichprocedure for modeling a tuned-mass damper is given as follows:


h4. 1. Specify link properties

Any spring-mass system may be used to represent the swinging pendulum in 2D. Here, the spring constant is given as Mg/L, where M is [mass|kb: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. HereIn this case, a [linear [link|kb:Link] is willcreated to represent the pendulum device. bySelect selecting Define > Section Properties > Link/Support Properties, then define translational stiffnesses are defined along U1, U2, and U3. The linear stiffness along U1 represents (axial stiffness)properties, and should be based on the EA/L value of the hangers, wherewhich is 1.0e6 kN/m is used in theModel attached file2. The linear stiffness properties forof U2 and U3 are chosen as Mg/L. In Model 2, athe link is drawn at the top story. TheLink link length is chosen as L = 0.1m, and mass is M = 10 kN-sec{^}2^/m.

* *Length* -- ThePendulum 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 canmay be chosen for convenience; it may even be zero. We recommend drawing the link such that the [I-end|kb:Joint] (first joint) attaches to the structure, and the [J-end|kb:Joint] (second joint) is free. In this case, within the linear link property, the shear distance from end J canmay be set to zero for the U2 and U3 degrees of freedom in the linear link propertyDOF.

* *Mass* -- The mass MMass strongly affects how strongly the TMD influences response. Changes to mass must be accounted for in the following locations:
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*#* Mass (M) should be assigned to the free joint (J-end of the link).
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*#* 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|kb:Load pattern].
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*#* Effective stiffness (Mg/L) of the U2 and U3 link properties.

* *Period* -- Generally, the period (T) of the TMD is chosen to closely match the structural period of the structure thatwhich is to be counteracted. Note that, although the mass does not affect the period, 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:

!Figure 1.png|align=center,border=0!

* *Damping* -- Damping is defined as either a linear C coefficient or a nonlinear C value plus an exponent on the velocity term. TheDamping 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, onean canestimate estimateof the fraction of critical damping, (ξ,) for the TMD asis:

!Figure 2.png|align=center,border=0!


h4. 2. Define the time-history analysis (for this example)

[Time-history|kb:Time-history analysis] analysis should be performed using either [nonlinear modal (FNA)|kb:Fast Nonlinear Analysis (FNA)] or [direct-integration|kb:Direct-integration time-history analysis] (linear or nonlinear) time-history [load cases|kb:Load case]. These types of analyses correctly account for the coupling of the modes, thatan effect which may be caused by damping in the TMD device. If the damping is small, it reasonablemight be resultspossible mayto possiblyobtain bereasonable obtainedresults using a linear modal time-history analysis, and possibly even [response-spectrum|kb:Response-spectrum analysis] analysis.

To provide overview for this procedure, the time-history load case of Model 2 is defined as follows:

* * Through the TMD period* -- Create a control model which does not have a TMD device. Run [modal analysis|kb:Modal analysis] and measure the fundamental period of the 1{^}st^ mode. The TMD will be designed to counteract response which results from this mode.

* *Time function* -- Select Define > Functions > Time History, menu,then define a sine curve iswhich defined withhas a 0.6 second period, whichequal isto thethat sameof as the 1{^}st^ Modemode of the control model without a TMD. Thereafter,. In this case, a period of 0.6 seconds is obtained.

* *Load case* -- Add a nonlinear- modal [time-history|kb:Time-history analysis] load case iswhich added.assumes 5% modal [modaldamping|kb:Modal analysisDamping] damping is assumed andand uses 200 output steps are selected, each 1/20{^}th^ the size of the 1{^}st^ time period.


h4. 3.* *Response* -- Run analysis and review output

Analysis may be run and various response measures may be reviewed through Display > Show Plot Functions. As expected, response is found to be reduced for the tuned-mass-damper model is found to be reduced.


h1. Attachments

* *Model 1* -- [TestModel_Without TMD.SDB |Tuned-mass damper^TestModel_Without TMD.SDB] (SDB file)

* *Model 2* -- [TestModel_With TMD.SDB |Tuned-mass damper^TestModel_With TMD.SDB] (SDB file)

* *Zipped File* -- [SAP2000 V14.2.4 models |Tuned-mass damper^SAP2000 V14.2.4 models.zip] (zipped SDB files)