I need the parametric equations for a simple pendulum

[Stargazer19385]

This is for a personal engineering project. I need the parametric equations y(t) and x(t) for a very simple pendulum. Assume no friction, no forcing, no variation in gravity, a point mass, and the tether angle is significantly less than 30 degrees. It has been a while since I did differential equations, so please link me to some algebraic parametric solutions to this most simple pendulum.

Thank you.

 

[Simon Bridge]

Its pretty straight forward starting from polar coordinates and converting - but the DE is non-linear for large angles.

Online resources vary with you eduation level and need:
http://mathforum.org/mathimages/index.php/Simple_Harmonic_Motion
http://www.codee.org/library/reviews/summaries/gudermann-and-the-simple-pendulum

 

[paisiello2]

x=L*sinθ
y=L*cosθ
θ=θ_maxsin(√(g/L)t)

 

[Stargazer19385]

Thank you for the responses! That is what I needed.

When I woke up this morning, not having checked here again yet, I was going to tackle this with some algebra. I was going to assume that one of them, say x(t), being an oscillating function, was likely a sine function, and that knowing the point weight is stuck on a circular path, I could use basic trig and algebra to find y(t). I had heard that oscillators make a sine function, but needed to convince myself some way that it was indeed a simple sine function, and not one with another function inside the sine function. The above answers look very believable and will tell me what I need to know.
Thank you again.

x=L*sinθ
y=L*cosθ
θ=θmaxsin(√(g/L)t)

Oh, maybe x(t) is a more complicated sine function with respect to time, and it is just x(theta) and y(theta) that are simple sine functions. I need to graph this out and get a better look. It is sad that I don't know the graph of sin(sin(t)) off hand, but I'm rusty.

 

[Stargazer19385]

Its pretty straight forward starting from polar coordinates and converting - but the DE is non-linear for large angles.

Online resources vary with you eduation level and need:

http://mathforum.org/mathimages/index.php/Simple_Harmonic_Motion

Your above link has a derivation I can follow, which shows that x(theta) and y(theta) are simple sine functions. I do not see a x(t), but it is believable that theta(t) is a sine function, so the second respondent's equations for x(t) and y(t) look very believable. That is for small theta, which is close enough for what I'm working with.

http://www.codee.org/library/reviews/summaries/gudermann-and-the-simple-pendulum

Your other link is for very large theta, giving solutions for it. Not what I needed, but interesting to know they have tackled that too.

Thanks again.

I still plan to graph x(t) = sin(sin(t)) in an Open Office spreadsheet to get a better grasp on the shape of the graph.

 

[Stargazer19385]

Sin(sin(z)) looks like sin(z) but with a more leveled off peak maxing at 0.82 instead of 1.0, at least for z under 180 degrees. Under 40 degrees, the two look extremely similar. Sin can probably replace sin(sin) in my calculations at lower values.

 

[Stargazer19385]

Hmmm...
I don't believe that spreadsheet. At 90 degrees, sin(90) = 1, and sin(1) = a bit over zero. Yet it shows it at 0.82.


The spreadsheet uses radians. I thought I corrected for them by using b1 = sin(sin(3.14*a1/180)). I now realize I should have done b1 = sin(180*(sin(3.14*a1/180))/3.14).

 

[Stargazer19385]

I seem to be making errors converting between radians and degrees. Maybe I should just do the x-axis in radians to start with, to keep it simple. I just graphed my above equation, and the result looked very high frequency. Not what I expected.

 

[Stargazer19385]

I just repeated the graphs, this time with the a column going from 3.1, 3.0, etc, to -1.6. Column B is sin(sin(a1)), and column c is sin(a1). I graphed it, and I got the same exact graph as the original above, but different scale.

I just don't understand it. Sin(90 degrees) = 1. Sin(1) = ... Oh. In radians, it is probably about 0.82.Well, I'm glad I'm working out these problems now, since I'm going back to school soon. I'm just amazed at the simple mistakes I'm making.So y= sin(sin(x)) looks very similar to y = sin(x) for all x, but is flatter and a bit lower in the peaks. Nice to know.

 

[Orodruin]

Do note that the solution given in post #3 is only valid for small θ_max. As was mentioned already in post #2, the full equations of motion are non-linear and generally more difficult to solve. Since θ_max is small you can approximate

x = L sin θ ≈ Lθ
y = - L cos θ ≈ - L + L θ²/2

inserting θ = θ_max sin(√(g/L)t) into this will give you a good approximation of the solution for θ_max << 1.

Also note that what you are plotting has θ_max = 1, which is not a small amplitude.

 

[Simon Bridge]

If you have access to open office you probably have access to GNU/Octave ... which is a better tool than a spreadsheet, if you plan to do more of this sort of thing.

note: in general, the small angle oscillations will be:
$$x(t)=x_{max}\sin(\omega t + \delta)$$ ... from that you should be able to figure out y(t) through basic geometry.
$$\omega = \frac{2\pi}{T}$$ ... where T is the period of the horizontal oscillations.

This also means that ##\theta (t) = \omega t + \delta## ... because the thing inside a trig function has to be an angle.

Your ##\sin t## is the special case where ##\omega = 2\pi## radiens per second, and ##\delta=0##radiens.

In the spreadsheet - be careful to use radiens and not degrees for angles.

In the end, what you use will depend on what you need the calculation for.
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