Our general equation for position in SHM is:
x(t) = xm cos(wt + f)
Let's take this equation apart to analyze each piece separately.
A key characteristic of simple harmonic motion is that the oscillation frequency is independent of the amplitude. Two equal masses connected to identical springs and released from rest at different points have the same angular frequency.
For a spring-mass system, changing the frequency means changing the ratio k/m, because w = (k/m)½. If we kept the k's the same but doubled one mass, for instance, its frequency would be reduced by a factor of the square root of two.
If all that is different is the phase, the graphs of x(t) are identical except that one is shifted relative to the other by the phase angle.