Small angles around the motion of the point don't make much difference. Even if the coordinate axis is the worst it can be, like 45 degrees, there is only a small reduction in peak amplitude.
Mathematically, I can simulate a single DOF mass-spring-damper system with 4% damping and a 7 Hz natural frequency with a plot like this:
You can see that a 7 Hz natural frequency lasts longer than 0.2 seconds compared with a 90 Hz natural frequency. If your model decays much faster than this, then perhaps the model needs further adjustment.