I think you've been mislead visually.

2 Sin[(2 Pi/4) t +0] + (10/((2 Pi/4)^2 - ((-.0001 + 2 Pi/4)^2))) Sin[(2 Pi/4) t + 0]

2 Sin[(2 Pi/4) t + 0] - (10/(2 Pi/4)) t Cos[(2 Pi/4) t + 0]

try plotting those two, (A==2, w0==2Pi/4, w1 very near w0)

the reason you won't see the same plot? has to do automatic plotting range (that w0 and w1 never approach each other in your plot, and mm adjusts the plot to run them together). as a general rule you should graph them separately and insure nothing strange happens when they combine, because all of strange adjustments can occur.

i'm open to the issue i may be wrong, please correct me