Troubleshooting Nonlinear Pendulum Movement with Air Friction: Tips and Analysis

[Michael Nelo]

The Problem
So, I know the basic, the basic differential equation of movement in terms of the angle it is formed, which has the form: 0 = g⋅sin[α(t)] + α''(t)⋅l. However, I decided to consider the friction force of the air, which is always contrary to the movement, and proportional to the tangential velocity, the expresion I get is this one:

So I thought maybe I could use some of the methods from homogeneous linean differential equations, but it doesn't (or at least I think it doesn't) fulfill the requirements, because the variable that isn't derivated is in a Sin().
What I have tried
I used the usual Consideration of Forces, using an X and Y plane supportive towards l⋅Cos(θ) and l⋅Sin(θ) respectively as can be seen from the picture:

So doing the respective analysis I found said weird differential equation, here's the full process of it:

Then, I went and used the Conservation of Energy, to try and get a expression of θ'(t) to be a function of θ(t), so here's what I got:

Doing this process:

And while yes, I achieved to reduce the weird differential equation to one that doesn't involve θ'(t), the expresion just gets weird, it isn't as easily solved as the would-be equation without the friction force of the air.
What I wish to know, and What I'm going to do
I wish to know if the concepts and analytical expressions I've used are correct (for example, is the velocity used in the kinetic energy the tangential one? If no, then must I take it to be the norm of the vector →v, which includes normal and tangential velocities? If so, would I be better off just including the vectorial components?), as well as any kind of tip I can get to solve the differential equation. For this problem as well, I'm going to try to use parametric equations as well, taking advantage that the pendular movement depicts a trayectory of a circumference, I could use the canonic equation of a circumference, then derivate it regarding to time, and see what I can get.

 

[Orodruin]

In general, please type out your efforts rather than submitting images. It is extremely difficult to see what you are writing. Also, you are just writing down a bunch of equations on your papers rather than explaining what you are actually doing.

So I thought maybe I could use some of the methods from homogeneous linean differential equations, but it doesn't (or at least I think it doesn't) fulfill the requirements, because the variable that isn't derivated is in a Sin().

Indeed, your differential equation is non-linear. You can obtain a solution close to the equilibrium point by linearising it around it.

Then, I went and used the Conservation of Energy, to try and get a expression of θ'(t) to be a function of θ(t), so here's what I got:

You cannot use conservation of energy when you have a dissipative force. Energy will dissipate out of the system.

for example, is the velocity used in the kinetic energy the tangential one?

There is no radial movement, so yes.

 

[Michael Nelo]

In general, please type out your efforts rather than submitting images. It is extremely difficult to see what you are writing. Also, you are just writing down a bunch of equations on your papers rather than explaining what you are actually doing.Indeed, your differential equation is non-linear. You can obtain a solution close to the equilibrium point by linearising it around it.You cannot use conservation of energy when you have a dissipative force. Energy will dissipate out of the system.There is no radial movement, so yes.

Thanks, I actually thought the photos were pretty unorganized and thought that writing what I was doing, on the thread, would give more order, glad to see it doesn't, also, I completely forgot I had to consider the Energy of the friction force of the air, so thank you very much for the reply

 

[Ray Vickson]

The Problem
So, I know the basic, the basic differential equation of movement in terms of the angle it is formed, which has the form: 0 = g⋅sin[α(t)] + α''(t)⋅l. However, I decided to consider the friction force of the air, which is always contrary to the movement, and proportional to the tangential velocity, the expresion I get is this one: View attachment 221188
So I thought maybe I could use some of the methods from homogeneous linean differential equations, but it doesn't (or at least I think it doesn't) fulfill the requirements, because the variable that isn't derivated is in a Sin().
What I have tried
I used the usual Consideration of Forces, using an X and Y plane supportive towards l⋅Cos(θ) and l⋅Sin(θ) respectively as can be seen from the picture:
View attachment 221192
So doing the respective analysis I found said weird differential equation, here's the full process of it:
View attachment 221193
Then, I went and used the Conservation of Energy, to try and get a expression of θ'(t) to be a function of θ(t), so here's what I got: View attachment 221195
Doing this process:
View attachment 221194
And while yes, I achieved to reduce the weird differential equation to one that doesn't involve θ'(t), the expresion just gets weird, it isn't as easily solved as the would-be equation without the friction force of the air.
What I wish to know, and What I'm going to do
I wish to know if the concepts and analytical expressions I've used are correct (for example, is the velocity used in the kinetic energy the tangential one? If no, then must I take it to be the norm of the vector →v, which includes normal and tangential velocities? If so, would I be better off just including the vectorial components?), as well as any kind of tip I can get to solve the differential equation. For this problem as well, I'm going to try to use parametric equations as well, taking advantage that the pendular movement depicts a trayectory of a circumference, I could use the canonic equation of a circumference, then derivate it regarding to time, and see what I can get.

I cannot read your images and am unwilling to even try; typed work is the standard in this forum. However, I will offer an answer of sorts: if you keep the full $\sin \theta$ in your DE (instead of the small-angle approximation $\sin \theta \approx \theta$) then your DE is nonlinear, and must either be solved numerically (when input numbers are given) or else solved using the so-called elliptic functions. These functions are "non-elementary", but have been known for more than 150 years; they are widely tabulated and are easily dealt with on most decent computer algebra systems.

See, eg., https://en.wikipedia.org/wiki/Pendulum_(mathematics)

 

[Michael Nelo]

I cannot read your images and am unwilling to even try; typed work is the standard in this forum. However, I will offer an answer of sorts: if you keep the full $\sin \theta$ in your DE (instead of the small-angle approximation $\sin \theta \approx \theta$) then your DE is nonlinear, and must either be solved numerically (when input numbers are given) or else solved using the so-called elliptic functions. These functions are "non-elementary", but have been known for more than 150 years; they are widely tabulated and are easily dealt with on most decent computer algebra systems.

See, eg., https://en.wikipedia.org/wiki/Pendulum_(mathematics)

Thanks for the answer, sorry for the messy images, I will use other methods to express the process.