In summary, I need the damping ratio and damped frequency of a system excluding air resistance, as the system will be operating in a vacuum. The system can be modeled as a mass-spring-damper, and the results I've got from an experiment conducted in air give me a damping ratio and frequency with air resistance. I want to know what the damping ratio and damped frequency of the system will be when it's operating in a vacuum. The system is like a long flexible beam. I've attached a diagram below showing the experiment setup before the string is cut and measurements taken.

I've run an experiment on a system (attached a known weight to the system using string and then cut the string, recorded it with a camera, and then used Tracker software) to find the damping ratio and damped frequency of the system. I've noticed in my analysis that as the oscillations reduce in amplitude, the damping ratio appears to decrease, and the frequency increase. This is most likely due to air resistance (which using the drag equation is related to velocity squared), so as the amplitude decreases, the velocity decreases and so does the drag and therefore the system is less damped.

Is there any way to mathematically calculate the damping ratio and frequency of the system excluding air resistance (the system will be in a vacuum environment).

So far I've taken the standard mass spring damper equation: $$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0$$ If you then add air resistance, $$\frac{1}{2}\rho C_dAv^2$$ You get this, $$m\frac{d^2x}{dt^2} + c\frac{dx}{dt}+\frac{1}{2}\rho C_dA\frac{dx}{dt}^2+ kx = 0$$

You can then say $$c' = (c + \frac{1}{2}\rho C_dA\frac{dx}{dt})$$ and the new damping part of the equation is $$c'\frac{dx}{dt}$$ which simplifies the equation to: $$m\frac{d^2x}{dt^2} + c'\frac{dx}{dt} + kx = 0$$

It gets a little more complicated, because my system is effectively a beam, with length R, with the weight attached at the end. Therefore velocity isn't constant along the length of the beam. So if you say v = wr, to convert to angular velocity as that is constant along the beam, then integrate the drag equation along the length of the beam, you can find the total force across the whole beam.

I can then solve the equation fine using a general Soln for the 2nd Order DE, but I can't figure out a way to get an equation excluding drag to find c.

I then tried changing the equation to angular, so using moment of inertia instead of mass, dtheta/dt instead of dx/dt, exc, so that I could then use laplace transform on the equation but that was a bit of dead end.

I have values for the moment of inertia, I, the frequency, w, and damping coefficient including air resistance, c', damping ratio of the system including air resistance, lambda, and the initial displacement, x, and the spring stiffness of the system, k.

Any guidance or help would be very appreciated.

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  • $\begingroup$ You might want to re look the term $\frac{1}{2}\rho C_dA (\frac{dx}{dt})^2$ if $\rho,\ C_d,\ A$ are constants. A damping force acts against the velocity direction. by using the square term $(\frac{dx}{dt})^2$, the force is always the same sign and need not have the opposite sign to velocity. $\endgroup$
    – AJN
    Aug 19, 2022 at 8:33
  • $\begingroup$ 1 Shouldn't you also have to account for the friction in the pulley system and the drag of the weight ? 2 Would it be possible to repeat the experiment in vacuum or partial vacuum ? 3 Are, $m,\ k,\ x,\ dx/dt,\ d^2x/dt^2$ well known ? $\endgroup$
    – AJN
    Aug 19, 2022 at 8:38

1 Answer 1


Most real world ODEs don't have an analytical solution, and ones involving v^2 drag are among them. You have to do a numerical solution.


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