pendulum and its equations etc.

[PG1995]

Hi

Could you please help me with these queries? In the Q3 I'm referring to this post. Thanks a lot.

Regards
PG

 

[steveB]

Q1, basically you have it correct. The first equation is an equation of motion, while the others are just force and torque equations.

Q2, is an example where you are being too literal. Sometimes you do that. Despite the fact that it would be preferable that people use perfect precision in language, usually they don't and you often need to read between the lines. Here, obviously the coefficients are constants, so the fact that they don't depend on the dependent variable is not a problem for linearity. In a sense a "constant" function can be considered to be a function of time or any independent variable.

Q3, shows that there can be different viewpoints and approaches to linearization. Most of the time, these methods produce equivalent results. Generally, the formal approach to linearization is used in the more complicated cases where you find it difficult to linearize by common sense or intuition or by inspection.

Q4, that answer comes from a proper nonlinear analysis. Someone once figured out how to derive this equations, and so the solution has been known for a long time. The details are not important for you right now with regards to this topic. However, someday you might want to go through the derivations to satisfy your curiosity.

 

[PG1995]

Thank you very much, Steve.

Q3, shows that there can be different viewpoints and approaches to linearization. Most of the time, these methods produce equivalent results. Generally, the formal approach to linearization is used in the more complicated cases where you find it difficult to linearize by common sense or intuition or by inspection.

So, in Q3 I was correct in saying that there exist different approaches to linearization. Specifically, in Q3 I was discussing two different linearlization approaches. In one linearlization is done using the fact that for small θ(t), sin(θ(t))≈θ(t), and in the other higher order terms of Taylor expansion are discarded. But as you indicate the end result of both approaches could be the same. Let me elaborate on this for clarity. For example, if linearization is performed on f1, f2, etc. after creating a non-linear state space equations using the equation 4.8 (which is non-linear differential equation) then the end result could be the same if we had instead used linearized differential equation 4.9 for state space equations.

By the way, which one of the two you consider to be a formal approach to linearization? Kindly let me know. Thanks.

Regards
PG

 

[steveB]

Thank you very much, Steve.



So, in Q3 I was correct in saying that there exist different approaches to linearization. Specifically, in Q3 I was discussing two different linearlization approaches. In one linearlization is done using the fact that for small θ(t), sin(θ(t))≈θ(t), and in the other higher order terms of Taylor expansion are discarded. But as you indicate the end result of both approaches could be the same. Let me elaborate on this for clarity. For example, if linearization is performed on f1, f2, etc. after creating a non-linear state space equations using the equation 4.8 (which is non-linear differential equation) then the end result could be the same if we had instead used linearized differential equation 4.9 for state space equations.

By the way, which one of the two you consider to be a formal approach to linearization? Kindly let me know. Thanks.

Regards
PG

This description seems correct to me. I consider the generation of state space matrices A,B,C and D as Jacobian matrices calculated from partial derivatives to be the formal approach. I would recommend that you actually derive the linearized state space system for a pendulum using the formal approach.

 

[MrAl]

Hello again,


I decided to test the highlighted solution just for the fun of it. Here's what came of it.

For the following parameters:
g=9.8 meters/sec^2, L=1 meter, ang(0)=0 rads, ang'(0)=0.3 rads/second

where L is the length of the 'arm', ang(0) is the angle when the mass passes through x=0, and ang'(0) is the rate when the mass passes that same point. The equation for the pendulum was reduced to a set of two non linear ODE's and then calculated using a numerical routine.

The results were:

analytic max angle: 0.0958681928991643
max angle : 0.0958681928991863
analytic period: 2.0082434419777684
actual period: 2.0082436000490524
Last derivative: -3.705899254203889e-08
run time: 71.3 seconds

where
'analytic max angle' is the precise angle calculated using conservation of mechanical energy,
'max angle' is the angle calculated using a Runge-Kutta Fehlberg fourth/fifth numerical ODE solution routine,
'analytic period' is the precise period calculated using the highlighted formula up to N=22 (which causes high accuracy convergence),
'actual period' is the actual period tested,
'Last derivative' is the last derivative at the farthest swing angle to one side (would be 0.0 in an analytical calculation),
'run time' is simply the time it took to run the algorithm to completion.

Looking over the results we can see they do agree within a reasonable degree, especially the max angle calculations where the numerical calculation came out very close to the theoretical calculation. The actual period varies a bit more because the algorithm works by using a small increment and it can only be made just so small or it takes too long to complete, so the period accuracy does suffer, but it's close enough anyway and tests show that as the increment is reduced the actual period matches the analytic period closer and closer. The Last derivative would be zero if we did an exact calculation because the pendulum comes to a complete stop at the end of one swing to one side.
It took over a minute to calculate because i used such a small step time at an attempt to get decent accuracy