Question C.3.1: Apply the fourth-order Runge-Kutta method to the equations f...

The equations governing rotational motion are

\frac{di(t)}{dt}  =  \frac{1}{L} [V(t) – i(t)(R + R_{coil}) – \frac{∂Λ(i, θ)}{∂θ}ω(t)] ,             (C.19)

\frac{dθ(t)}{dt}   =  ω(t),             (C.20)

and

\frac{dω(t)}{dt}   = \frac{T(i,θ)}{j_{m}}.            (C.21)

Note that there are three dependent variables i(t), θ(t), and ω(t). We need a set of Runge-Kutta coefficients for each variable. We denote these coefficients by (k_{1}, k_{2}, k_{3}, k_{4}), (m_{1}, m_{2}, m_{3}, m_{4}), and (p_{1}, p_{2}, p_{3}, p_{4}), respectively. To implement the Runge-Kutta method, it is convenient to rewrite Eqs. (C.19)—(C.21) as follows:

\frac{di(t)}{dt}  = f(V(t), i(t),θ(t), ω(t)),

\frac{dθ(t)}{dt}  =  g(ω(t)),

and

\frac{dω(t)}{dt}  =  h(i(t),θ(t)),

where

f(V(t), i(t),θ(t), ω(t)) = \frac{1}{L} [V(t) – i(t)(R + R_{coil}) – \frac{∂Λ(i(t),θ(t))}{∂θ}ω(t)] ,

g(ω(t)) = ω(t),

and

h(i(t),θ(t)) = \frac{T(i(t),θ(t))}{j_{m}}.

The Runge-Kutta coefficients are determined from the following relations:

k_{1} =  Δt f(V(n), i(n), θ(n), ω(n))

m_{1} = Δt g(ω(n))

p_{1} = Δt h(i(n), θ(n)),          (C.22)

k_{2} =  Δt f(V(n), i(n) + \frac{k_{1}}{2}, θ(n) + \frac{m_{1}}{2}, ω(n) + \frac{p_{1}}{2})

m_{2} = Δt g(ω(n) +\frac{p_{1}}{2})

p_{2} = Δt h(i(n) +\frac{k_{1}}{2},θ(n) + \frac{m_{1}}{2}),            (C.23)

k_{3} =  Δt f(V(n), i(n) + \frac{k_{2}}{2}, θ(n) + \frac{m_{2}}{2}, ω(n) + \frac{p_{2}}{2})

m_{3} = Δt g(ω(n) +\frac{p_{2}}{2})

p_{3} = Δt h(i(n) + \frac{k_{2}}{2},θ(n) + \frac{m_{2}}{2}),            (C.24)

and

k_{4}  =  Δt  f(V(n), i(n)+k_{3} , θ(n)+m_{3}, ω(n)+p_{3})

m_{4}  =  Δt  g(ω(n)+p_{3})

p_{4}= Δt  h(i(n)+k_{3}, θ(n)+m_{3}).           (C.25)

Once the coefficients have been computed, they are used to determine the subsequent solution values as follows:

i_{n+1} = i_{n} +\frac{1}{6} [k_{1} +2k_{2} +2k_{3}+k_{4}]

θ_{n+1} = θ_{n} +\frac{1}{6} [m_{1} +2m_{2} +2m_{3}+m_{4}]

ω_{n+1} = ω_{n} +\frac{1}{6} [p_{1} +2p_{2} +2p_{3}+p_{4}].      (C.26)

The solution procedure for Eqs. (C.19)—(C.21) is similar to steps 1 through 4 of Euler’s method. Specifically, start with the initial conditions \left[i_0,\theta _0,\omega _0\right]   and use Eqs. (C.22)—(C.25) to determine the coefficients (k_{1}, k_{2}, k_{3}, k_{4}), (m_{1}, m_{2}, m_{3}, m_{4}), and (p_{1}, p_{2}, p_{3}, p_{4}). Once these are known, use Eq. (C.26) to obtain [i_{1},θ_{1}, ω_{1}], which gives the solution at t = Δt. Repeat this process to obtain [i_{2},θ_{2}, ω_{2}],[i_{3},θ_{3}, ω_{3}], . . . ,[i_{n},θ_{n}, ω_{n}] , . . . , until solution values are obtained for the interval (t_{0}, t_{max}).