CODEWITHSUNDEEP
pythonodetaylor-series
damu_d
damu_d

Reputation: 53

Taylor series for pendulum in python

I have a question from school to simulate pendulum motion based on taylor series expansion.

Angular frequency d 2 θ d t 2 = − m G R I sin ⁡ ( θ )

I am quite new to python. I know now how to simulate using Euler's method.

  for n in range(N_t):

  u[n+1] = u[n] + dt*v[n]

  v[n+1] = v[n] + dt*(m*g*r/I)*sin(u[n])

How can I simulate it using taylor expansion? Should I run it the code only as below?

f′′(x0) = 2a2

Upvotes: 0

Views: 370

Answers (2)

Lutz Lehmann
Lutz Lehmann

Reputation: 25992

In the equation u''=f(u) you get higher order derivatives by deriving this equation by applying chain and product rule and substituting back the ODE for all second derivatives of u. The values of u and u' are taken from the current state vector.

u''' = f'(u)u'
u^{(4)} = f''(u)*u'^2 + f'(u)*u'' 
        = f''(u)*u'^2 + f'(u)*f(u)
u^{(5)} = f'''(u)*u'^3 + 3f''(u)*u'*f(u) + f'(u)^2*u'

There is also a systematic way using Taylor series arithmetics of automatic/algorithmic differentiation.

(2022/06/29) To solve x''=-sin(x) -- the constant factors become trivial by rescaling the time -- via Taylor series anywhere, reformulate this as system

x'=y
y'=-v
u=cos(x)  ==> u' = -v*y
v=sin(x)  ==> v' =  u*y

the last two via the trigonometric derivatives and the chain rule. Comparing coefficients left and right results in coupled incremental formulas for the coefficients of the Taylor series of all variables, with x(t0)=x0, y(t0)=y0, u(t0)=cos(x0), v(t0)=sin(x0).

Upvotes: 1

Mochan
Mochan

Reputation: 1369

I assume you meant this from your code,

enter image description here

I also assume that u := θ and v := θ'.

So, the Taylor expansion of sin(x) is

enter image description here

and your equation is now

enter image description here

So, you can calculate u and v from the above equation. Or, I don't know if your teacher wanted you to calculate the integrals first and then use Taylor series.

Upvotes: 0

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