*KEYWORD  
*comment
An x-force is ramped up and then released.  Observe the x-displacement oscillation to determine the 
effect of damping.

Two cases look at stiffness damping (beta) and mass damping (alpha) separately.   

f from eigenvalue analysis is 42.5 Hz.

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case1:  5% mass damping desired.  Alpha = 0.05 * (4*PI*f) = 26.7

From resulting time history, log decrement* is 0.318.    0.318/(2*PI) = 0.0505.    Checks out!
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case2:  6.67% stiffness damping desired.   Beta = 0.0667 / (PI * f) = 0.0005

Note the negative value given to COEF!   This means beta = abs(COEF).    See the description of COEF in the User's Manual.

From resulting time history, log decrement* is 0.419.   0.419/(2*PI) = 0.0667.   Checks out.
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Ref: Dynamics of Structures by Clough and Penzien.
Regarding "Free-Vibration Decay",
Damping ratio can be determined from the ratio of two displacement amplitudes measured at an interval of m cycles.
Damping ratio = ln(dn/d(n+m))/(2*pi*m*(omega/omega_damped))
For damping ratios less than 0.2, the change in frequency omega due to damping can be neglected with little (<2 percent) error.

"log decrement" mentioned above is  ln(dn/d(n+m)), where dn is the amplitude of displacement at cycle n and 
  d(n+m) is the amplitude of displacement at cycle(n+m).
__________________________________________________________________________
*case_begin_1
*damping_global
,26.7
*case_end_1
*case_begin_2
*DAMPING_PART_STIFFNESS
$#     pid      coef
1,-.0005
*case_end_2
*TITLE
$# title
Test Model (42.5 Hz)
*CONTROL_IMPLICIT_GENERAL
         1  0.0005
*CONTROL_IMPLICIT_DYNAMICS
         1
*CONTROL_ENERGY
$#    hgen      rwen    slnten     rylen
         2         2         2         2
*CONTROL_TERMINATION
$#  endtim    endcyc     dtmin    endeng    endmas
  4.000000         0     0.000     0.000     0.000
*DATABASE_BINARY_D3PLOT
$#      dt      lcdt      beam     npltc    psetid
  0.000500         0         0         0         0
$#   ioopt
         0
*PART
$# title
Steel
$#     pid     secid       mid     eosid      hgid      grav    adpopt      tmid
         1         1         1         0         0         0         0         0
*SECTION_BEAM
$#   secid    elform      shrf   qr/irid       cst     scoor       nsm
         1         1  0.833300         5         0     0.000     0.000
$#     ts1       ts2       tt1       tt2     nsloc     ntloc
  1.000000  1.000000  1.000000  1.000000     0.000     0.000
*MAT_ELASTIC
$#     mid        ro         e        pr        da        db  not used
         1  0.015200 4.1760E+6  0.300000     0.000     0.000         0
*LOAD_NODE_POINT
$#     nid       dof      lcid        sf       cid        m1        m2        m3
         2         1         1  2.000000         0         0         0         0
*DEFINE_CURVE
$#    lcid      sidr       sfa       sfo      offa      offo    dattyp
         1         0     0.000     0.000     0.000     0.000         0
$#                a1                  o1
               0.000               0.000
           2.0000000           1.0000000
           2.0100000               0.000
          99.0000000               0.000
*ELEMENT_BEAM
$#   eid     pid      n1      n2      n3     rt1     rr1     rt2     rr2   local
       1       1       1       2       3       0       0       0       0       0
*ELEMENT_MASS
$#   eid     nid            mass     pid
       2       2       1.9200000       1
*NODE
$#   nid               x               y               z      tc      rc
       1           0.000           0.000           0.000       7       7
       2           0.000           0.000       2.0000000  
       3       1.0000000           0.000           0.000 
*END