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I have a question regarding Damped Harmonic Motion and I was wondering if anyone out there could help me out? Under normal conditions, gravity will not have an affect on a damped spring oscillator that goes up and down. Gravity will just change the offset, and the normal force equation can be written as follows:

mass*accel + damp*velocity + k_spring*displacement = 0.

From this we can get the standard solution which has the form:

x(t) = Ae

^{(...t)}+ Be

^{(...t)}

This is all pretty straight forward. However my question is, what happens when gravity is not constant over the entire range of the oscillation (e.g. Very Very large oscillations relative to size, or very accurate measurement..)? Then we need to compensate for it. To do this, we need to include the force equation for gravity which would then produce a total force equation like:

mass*accel + damp*velocity + k_spring*displacement + G*m1*m2 / separation^2 = 0.

To simplify things, we can assume that the object has a stable position that it will return to under standard stimulus conditions (e.g. The distance separating the object never becomes so small that the gravitational force overwhelms everything else). We can then simplify this equation slightly as "displacement + separation = constant". However, the oscillations are big enough that we cannot use a linear approximation for the change in gravity.

Anyway, this is then my main question: is there a general solution to this, and what would the oscillations look like?

Thanks,

Matthew