# References describing how the initial angular displacement of a pendulum affects its damping ratio?

I'm writing a research paper exploring how damping ratio of a simple pendulum relates on its initial angular displacement.

In order to validate the findings of my paper, I am required to include a comparison to the findings of past research papers, or to a theoretical model derived from an equation connecting initial angular displacement to damping ratios.

Despite best efforts I was unable to find research papers that have explored this research question (e.g the vast majority of them explore how the thickness of string affects the damping ratio), in order to validate my findings. However, if research has been done on this before, I would much appreciate it if someone could direct me to the paper.

Ideally, the reference would contain some equation (possibly a differential equation even of some sort), that connected the initial angular displacement to damping ratios. Alternatively, if there was some equation connecting the relationship between the relationship between initial velocity, to damping ratios, that would be very useful as well.

Thank you!

For clarification, my results currently indicate a positive relationship between the angle of initial release and the damping ratio (which can be modelled either linearly or using a quadratic function)

• Voting to reopen. Obviously not a "do my homework for me" question. Commented Mar 29 at 11:39
• I'm afraid there's a good reason why there is no such research (or hard to find). Damping is inherent property of materials and hence I would assume that it does not depend on any initial conditions of pendulum such as initial displacement or initial speed. Unless there are some non-linear effects. For example, we may give such high initial speed that due to centrifugal force string can change it's length and/or stiffness, which would affect period/damping. But it can be quite challenging to invoke such second-order effects. Hence, no substantial research on this, but it's worth to try. Commented Mar 29 at 12:59
• Hi to clarify, I assumed that as the initial angular displacement increased, the velocity of the pendulum would also increase, leading to a higher air resistance which in turn would lead to a higher damping ratio. Is this assumption incorrect? I used the logarithmic decrement formula and found that there is an upwards correlation between initial angular displacement and the damping ratio Commented Mar 29 at 13:21
• How you define a damping ratio ? Commented Mar 29 at 14:52
• I am using the logarithmic decrement formula to approximate the damping ratio Commented Mar 30 at 3:18