Damped Harmonic Oscillator - Gravity not constant.

[mlewis14]

Hello,

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

 

[BvU]

Hello Matthew,

If you have an ideal spring and ideal measurement equipment, both of which is a greater source of problems than the change in $g$ with height, then you simply replace the $m\ddot x + Kx = 0$ equation by $$ m \ddot x + K(x-x_{\rm eq}) = - GM_{\rm earth} \; m \left ( {1 \over x_{\rm eq}^2 } - { 1 \over x_\strut^2} \right ) $$(note that $x$ is the distance to the Earth center).

 

[mlewis14]

Dear BvU,

Thanks for the reply, but I know how to generate the equation. My attempt to write the equation for a dampened harmonic spring was:

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

With your formatting, the equation looks significantly nicer (note: damping is missing). Also, your assumptions for an ideal spring and an ideal measurement equipment are fine. But, what what I am really interested in is a generic transient solution to the differential equation that you posted but with damping included. I know this can be solved numerically, but I hoping for general solution.

Thanks,

Matthew

 

[BvU]

OK, so you see that what I wrote in first instance is nonsense. On the way home I realized I had forgotten the $x_{\rm eq}$ on the left so I put it in $$m \ddot x + K(x-x_{\rm eq}) = - GM_{\rm earth} \; m \left ( {1 \over x_{\rm eq}^2 } - { 1 \over x_\strut^2} \right )$$I repeat it, because your modification of the equation had 'separation' as if that is another (independent?) variable.

Next step is write an expression in terms of $x - x_{\rm eq}$, write as a power series and see if one of the perturbation approaches in

http://web.physics.ucsb.edu/~fratus/phys103/LN/NLO.pdf

gives an order of magnitude for the distortion.

There is no analytical solution afaik

 

[mlewis14]

Hello BvU,

Thanks again for the quick response. "There is no analytical solution afaik." is unfortunately the same solution I came too. Regarding the use of Perturbation Theory, that was one of the reasons I stated in my original post the assumption that the oscillation occurs within a specific range. In fact, if I produce a few plots using the force equation above, and limit the movement curves (i.e. x(t)) for ranges where F_spring > F_gravity, the curves look like they might have a solution very similar to the normal damped harmonic oscillation:

x(t) = Ae(...t) + Be(...t) (1)

but where the coefficients and other other terms are defined differently. For example, in an overdamped setup, where I initial pull the object 1 towards the center of the object 2. At some separation to the center of the object 2 F_spring will equal F_gravity. If I release object 1 just before this point, the initial acceleration of object 1 will be significantly slower than equation (1) expect in the case of a standard damped oscillator due to the increased force of gravity at close distances. However, I can make equation (1) fit the response curve that I generate using the force equations, but then the coefficient are just made up. This is a hint that the solution to my question will have the form of equation (1)... just need to solve it somehow so that I can define the coefficients using mass, k, and other properties of the system.

Regards,

Matthew

 

[BvU]

However, I can make equation (1) fit the response curve that I generate using the force equations

O ? what curve is that ?

 

[mlewis14]

Hello,

I can simulate a curve using the force equations we have listed above and by setting the mass, K, and other system properties to anything I desire. Using a small program I can set the initial conditions (velocity, acceleration, position), and then using small desecrate steps I can produce the response curve of the entire system over a given length of time. This is fairly accurate if the time steps are small enough. It's also fast and easy and allows me to get a rough idea of how the system behaves.

Regards,

Matthew

 

[sophiecentaur]

A very simple example of an oscillator in which the restoring force is not proportional to displacement is the very simplest one - the Simple Pendulum.
The restoring force is not proportional to the angle of displacement but the sine of the angle. (See this link for a very full treatment)

 

[BvU]

I can simulate a curve using the force equations we have listed above and by setting the mass, K, and other system properties to anything I desire.

I understand that. But I mean: what did you do to get a noticeable deviation from a perfect sine ?

 

[mlewis14]

I understand that. But I mean: what did you do to get a noticeable deviation from a perfect sine ?

Well, if I have an underdamped system I do have non-sinusoidal oscillations. However, in my comment I limited the curve fitting to an overdamped system. In this case, the fit does seem to match the sum of two algorithmic decaying functions. But, since I have no exact way to calculate the exponents, it could be luck that it fits in this case. And even if it is a fit, it will only show the motion of the system if it is overdamped and would not be a general solution. I would be fine though if it was a part of a general solution that is only defined for a certain region of operation. But currently I doubt this as well...