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.

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 object in which one joint is attached to the structure, and the other joint is free.

**Mass**– Mass and weight are then assigned to the free joint.

**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.

# Reference files

For reference, two SAP2000 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.

# Procedure for Model 2

Model 2 is created through the process which follows:

#### Specifying link properties

Any spring-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 linear link will represent the pendulum device by selecting Define > Section Properties > Link/Support Properties, then translational stiffnesses are defined along U1, U2, and U3. The linear stiffness along U1 (axial stiffness) should be based on the EA/L value of the hangers, where 1.0e6 kN/m is used in the attached file. The linear stiffness properties for U2 and U3 are chosen as Mg/L. In Model 2, a link is drawn at the top story. The link length is chosen as L = 0.1m, and mass is M = 10 kN-sec^{2}/m.

**Length**– The 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 can 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, the shear distance from end J can be set to zero for the U2 and U3 degrees of freedom in the linear link property.

**Mass**– The mass M strongly affects how strongly the TMD influences response. Changes to mass must be accounted for in the following locations:

- 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.

- Mass M should be assigned to the free joint (J-end of the link).

**Period**– Generally, the period (T) of the TMD is chosen to closely match the period of the structure that is to be counteracted. The period of the TMD is given by:

- T = √(M/K) = √(M/(Mg/L) = √(L/g)

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.

- T = √(M/K) = √(M/(Mg/L) = √(L/g)

**Damping**– Damping is defined as a linear C coefficient or a nonlinear C value plus an exponent on the velocity term. The damping values should be chosen based on the physical characteristics of the TMD device. This damping affects the TMD itself but is not the primary energy dissipation mechanism for the structure as a whole. For a linear damper one can estimate the fraction of critical damping, ξ, for the TMD as:

- ξ = C / (2 √(K M)) = C / (2 √(M
^{2}g/L)) = C / (2 M √(g/L))

- ξ = C / (2 √(K M)) = C / (2 √(M

#### Setting up the time-history analysis (for this example)

Time-history analysis should be performed using 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 that may be caused by damping in the TMD device. If the damping is small, reasonable results may possibly be obtained using linear modal time-history analysis, and even response-spectrum 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 load case is added. 5% modal damping is assumed and 200 output steps are selected, each 1/20^{th}the size of the 1^{st}time period.

#### Analysis and 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.

# 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)