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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 |^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 |^TestModel_With TMD.SDB] features a subsystem which simulates the effect of a pendulum damper.


h1. Procedure for Model 2

Model 2 is created using the process which follows:

* *Link properties*: Any spring/[mass|kb:Mass] system may be used to represent the swinging pendulum in 2D. Here, the spring constant is 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. Here, a friction-isolator link element will represent the friction-pendulum device, selected through Define > Section Properties > Link/Support Properties. Translational stiffness along U1, U2, and U3 are defined. The effective and [nonlinear|kb:Nonlinear] stiffness along U1 should be based on the EA/L value of the hangers, where 1.0E6 kN/m is used in the attached file. The effective stiffness properties for U2 and U3 are chosen as Mg/L. In this model, a link is drawn at the top story. Link length is chosen as L = 0.1m, and mass is M = 10 kN-sec{^}2^/m.
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** Length directly relates to the period of the TMD. When length is changed, it must be account for in numerous places, including:
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**# The length of the link element drawn.
**# The effective and actual stiffness (EA/L) of the U1 link property.
**# The effective stiffness (Mg/L) of the U2 and U3 link properties.
**# The sliding radius of the U2 and U3 link properties.
**# The shear location (dj) of the U2 and U3 link properties. This is because the sliding surface is located at the bottom of the link element.
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** Mass M affects how strongly the TMD influences response. Changes to mass must be accounted for in the following sources:
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**# Mass assigned to the lower link [joint|kb:Joint].
**# Weight force (W = Mg) assigned to the lower link joint in the dead [load case|kb:Load case] (DEAD).
**# Effective stiffness (Mg/L) of the U2 and U3 link properties.


* *Setting up the time-history analysis*: Through the Define > Functions > Time History menu, a sine curve is defined with a 0.6 second period, which is the same as the 1{^}st^ Mode of the model without a TMD. Thereafter, a nonlinear-modal [time-history|kb:Time-history analysis] load case is added. 5% [modal|kb:Modal analysis] damping is assumed and 200 output steps are selected, each 1/20{^}th^ the size of the 1{^}st^ time period.

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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.


h1. Attachments

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

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

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