Local maxs and mins occur where the derivative of the function is equal to zero. So:

x(t) = q*sin(Gt-d)*e^(-at)

x'(t) = q*cos(Gt-d)*G*e^(-at) + q*sin(Gt-d)*(-a)*e^(-at)

We wish to find at what times this is zero.

0 = q*cos(Gt-d)*G*e^(-at) + q*sin(Gt-d)*(-a)*e^(-at)

Dividing out the common (non-zero) factors gives us:

0 = G*cos(Gt-d) - a*sin(Gt-d)

or

sin(Gt-d)/cos(Gt-d) = G/a

tan(Gt-d) = G/a

Now, given that tan(Gt-d) = G/a any t' such that Gt' - d = Gt - d + (pi) will produce a local max. (The zeros of the tangent function occur with a period of (pi).) This means that the most general t will have a period of (pi)/G.

-Dan