The key equation for SHM is: a=–ω^2xa=–ω2x
Where the angular frequency, ω=2πfω=2πf aa is the acceleration,
and xx is the displacement.
**As you can see the frequency is not related to the amplitude. ** Actually ωω is a constant for this equation. It is the proportionality constant for aprop-xa∝−x. Thus frequency is dependent only on the dimensions of the oscillator (e.g. mass of bob and length of string for pendula or force constant of spring and mass for mass-spring systems). This is the important point here, frequency is independent of amplitude only for a given oscillator. All SHM oscillators do not have the same frequency!
The equation, a=–ω^2xa=–ω2x, can be proven mathematically but it is quite a long derivation. I will add it below when I have time.
In practical situations the presence of friction and drag do cause the frequency to change if amplitude changes, especially at large amplitudes.
