.. _URDDamping: Universal Rate-Dependent (URD) Damping ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This command is used to construct a universal rate-dependent damping model. Two ways to define URD damping are provided. **Approach 1** Only the target loss factor distribution is used. The command will calculate the best approximation using the least-squares approach. More details are provided in `reference [1] `_. .. function:: damping URD $dampingTag $n $freq1 $lossFactor1 … $freqn $lossFactorn <-tol $Tlrnc> <-activateTime $Ta> <-deactivateTime $Td> <-fact $tsTagScaleFactorVsTime> <-print> .. csv-table:: :header: "Argument", "Type", "Description" :widths: 10, 10, 40 $dampingTag, |integer|, integer tag identifying damping $n, |integer|, dimension of the frequency–loss factor pairs $fj, |float|, frequency of the **j** th frequency–loss factor pair (in units of T^−1) $lossFactorj, |float|, loss factor of the **j** th frequency–loss factor pair $Tlrnc, |float|, tolerance criteria (relative error) used when calculating the best approximation (optional: default is 0.05) $Ta, |float|, time when the damping is activated $Td, |float|, time when the damping is deactivated $tsTagScaleFactorVsTime, |integer|, time series tag identifying the scale factor of the damping versus time -print, |string|, argument to print the calculated optimal parameters for the URD damping .. admonition:: NOTE The loss factor can be set to twice the viscous damping ratio. For example, if a viscous damping ratio of 0.05 is to be used, then the loss factor would be 2 * 0.05 = 0.1. According to `reference [1] `_ , the following commands are equivalent. .. code-block:: tcl damping Uniform 1 0.05 1.0 100.0 damping URD 1 2 1.0 0.10 100.0 0.10 **Approach 2** Optimal parameters for URD damping are used. These parameters may be obtained using Matlab and other software to achieve the best approximation. More details are provided in `reference [1] `_. .. function:: damping URDbeta $dampingTag $nc $fc_1 $beta_1 … $fc_nc $beta_nc <-activateTime $Ta> <-deactivateTime $Td> <-fact $tsTagScaleFactorVsTime> .. csv-table:: :header: "Argument", "Type", "Description" :widths: 10, 10, 40 $dampingTag, |integer|, integer tag identifying damping $nc, |integer|, dimension of vector **beta_opt** $fc_j, |float|, the **j** th cutoff frequency (in units of T^−1) $beta_j, |float|, **j** th term of the adjustment factor **beta_opt** $Ta, |float|, time when the damping is activated $Td, |float|, time when the damping is deactivated $tsTagScaleFactorVsTime, |integer|, time series tag identifying the scale factor of the damping versus time .. admonition:: NOTE The loss factor at frequency `f` is: .. figure:: URDbetaDamping.png :align: center :width: 300px :figclass: align-center .. admonition:: Example The following shows several examples to construct universal rate-dependent damping models with different target damping-frequency relation. .. code-block:: tcl damping URD 1 2 1.0 0.10 100.0 0.10 -tol 0.01 damping URDbeta 2 9 1.0 0.1101762430 1.7782794100 -0.0783418510 3.1622776602 0.0986952340 5.6234132519 -0.0505663180 10.0000000000 0.0834771114 17.7827941004 -0.0505663180 31.6227766017 0.0986952340 56.2341325190 -0.0783418510 100.0000000000 0.1101762430 damping URD 3 2 1.0 0.10 100.0 0.04 damping URD 4 2 1.0 0.04 100.0 0.10 damping URD 5 3 1.0 0.04 10.0 0.10 100.0 0.04 damping URD 6 5 1.0 0.04 5.0 0.10 10.0 0.06 50.0 0.10 100.0 0.04 damping URD 7 7 0.1 0.04 1.0 0.10 5.0 0.10 10.0 0.06 50.0 0.10 100 0.10 500.0 0.04 damping URD 8 6 0.1 0.02 1.0 0.04 5.0 0.04 10.0 0.10 100.0 0.10 500 0.02 -tol 0.1 .. figure:: URDdamping.jpg :align: center :width: 600px :figclass: align-center URD Damping The following is an example for an SDOF system. .. literalinclude:: URDDamping.tcl :language: tcl **Code Developed by**: `Yuan Tian `_ (University of Science and Technology Beijing), Yuli Huang and `Xinzheng Lu `_ (Tsinghua University). **References** .. [1] Tian Y, Fei Y, Huang Y, Lu X. 2022. `A universal rate-dependent damping model for arbitrary damping-frequency distribution `_. `Engineering Structures`, 255: 113894. `http://dx.doi.org/10.1016/j.engstruct.2022.113894 `_ (Available examples at: `http://pan.ustb.edu.cn/l/NHfFBB `_ )