I'm investigating how increasing the mass on a pendulum will affect its damping ratio, by comparing angular amplitude at some nth oscillation. There appears to be a clear connection, but after a couple hours of looking for it, I'm still having trouble formulating it mathematically — especially as my angles are too large to approximate.

I calculated the angular amplitude to be the below, where $m$ is mass, $L$ string length to measured point, $t$ time, and $A$, $k$, and $b$ constants.


However, translating this into what a graph of the $nth$ stationary point would look like as mass varied is somewhat beyond me — all experimental results suggest are an $e^{-(1/x)}$-like curve.

I've considered using another equation (the second-order differential one below) to solve the problem. However, this next one once more only models angular amplitude over time — finding its maxima per oscillation over different masses still isn't clear, and any tips would be much appreciated.

$$(d^2 \theta)/dt^2 +(b/m)*(d \theta)/dt +(g/L)*sin \theta =0$$

  • $\begingroup$ Do you have access to plotting aoftware like mathematica or matlab? Ever consider just plotting the nonlinear ode and seeing what happens to get a feeling? $\endgroup$ Commented Oct 9, 2022 at 18:19
  • $\begingroup$ @DrMrstheMonarch I've plotted it (admittedly without specialist software), hence the $e^{-1/x}$ 'look-alike'. But I'd rather have a theory beforehand than just fitting my datapoints to some line of best fit. $\endgroup$
    – Destaq
    Commented Oct 9, 2022 at 18:22
  • $\begingroup$ maybe wiki logarithmic decrement then! $\endgroup$ Commented Oct 9, 2022 at 18:24

1 Answer 1


The second-order nonlinear equation is likely that has no simple analytical solution.

The homogeneous second-order linear equation, $m \ddot x + c \dot x + k x = 0$, that is an approximation of the nonlinear one for small-amplitude oscillations, has solution that can be obtained as a combination of the solutions of form $x(t) = e^{st}$, where the values of $s$ are the solutions of

$m s^2 + c s + k = 0$$\qquad \rightarrow \qquad $ $s_{1,2} = -\dfrac{c}{2m} \mp \sqrt{\left(\dfrac{c}{2m}\right)^2 - \dfrac{k}{m} }$

and if the system ha subcritical damping,

$s_{1,2} = -\dfrac{c}{2m} \mp i \sqrt{\dfrac{k}{m} - \left(\dfrac{c}{2m}\right)^2 } = -\xi \omega_n \mp i \omega_n \sqrt{1 - \xi^2} = -\dfrac{\xi}{\sqrt{1-\xi^2}} \omega \mp i \omega$

in terms of:

  • the natural frequency (of the undamped oscillation): $\omega_n = \sqrt{\frac{k}{m}}$
  • the damping coefficient $\xi = \frac{c}{\sqrt{2m \omega_n}}$

as $x(t) = e^{-\xi \omega_n t} \left( A \cos\left( \omega t\right) + B \sin\left( \omega t \right) \right)$. You need to provide the initial conditions to the differential problem. As an example, if the initial conditions are

$x(0) = x_0, \qquad \dot{x}(0) = 0$,

the solution reads $x(t) = x_0 e^{-\xi \omega_n t} \cos\left( \omega t\right) $, and you can readily observe that this is a damped oscillation with angular velocity $\omega$, whose graph is comprised between the decaying exponentials $\pm e^{-\xi \omega_n t}$.

  • $\begingroup$ Thank you for the detailed answer. However, as my investigation did not use the small-angle approximation, I can't make use of the linear ODE. You shared that there are no 'simple analytical solution[s]' to the non-linear angular amplitude function — is a computational, not-fully-accurate approach thus the only viable one? $\endgroup$
    – Destaq
    Commented Oct 9, 2022 at 18:41
  • $\begingroup$ What do you mean exactly? What is the range of interest of the angle? Is the system not forced? $\endgroup$
    – basics
    Commented Oct 9, 2022 at 19:10
  • $\begingroup$ I mean that the initial angle was beyond any point that could be approximated with small angles — it was released from roughly 55 degrees / 1 radian. Otherwise, the pendulum was simple and not forced — it was simply left to swing for ten different weights, and the final angular amplitude measured after ten oscillations. I'm just looking to mathematically formulate this mass-amplitude relationship. $\endgroup$
    – Destaq
    Commented Oct 9, 2022 at 19:16
  • $\begingroup$ I'm busy right now, so I can't perform calculation on my own, but you could try to evaluate the decrease of mechanical energy as a function of time for the nonlinear equations $\endgroup$
    – basics
    Commented Oct 9, 2022 at 19:35

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