The equation of motion for the system is
where m is the mass, c the
damping, k the stiffness, and u the
displacement.
Rayleigh damping defines the damping as ,
where
is the mass damping factor and
is the stiffness damping factor.
Assuming a solution of the form ,
we have
where
is the undamped frequency of vibration (25.118 rad/sec for the parameters of
this example). Critical damping occurs when the value of c
causes the discriminant of this equation to be zero, so
We define the damping ratio, ,
as the ratio of damping to critical damping:
The relationships in this equation are often used as a basis for choosing
and .
The equation defining
can be rewritten
We choose the damping in this case to be less than critical, so
1
and the system can vibrate. The initial conditions are
1 and
0, so the dynamic part of the motion is
where
is the damped frequency of the system.
The amplitudes of this oscillatory equation before and after one period of
vibration, ,
have the ratio
so the logarithmic decrement over n cycles of response
is
Table 1
shows the values of
calculated from
Abaqus
for the various test cases examined, together with their corresponding exact
solution. A sample time history from which the logarithmic decrements are
calculated is shown in
Figure 2.
All the
Abaqus
runs use fixed time increments of .01 seconds. The integrator used in the modal
method is exact, so the results of that analysis are exact. The integrator used
in the direct integration method is not exact; however, since the period of the
system is 0.25 seconds, the time increment chosen gives 25 increments per
cycle, so those results are also quite accurate.