Question C.1.1: Apply Euler’s method to the equations of motion (5.86) for a...
Apply Euler’s method to the equations of motion (5.86) for an electromechanical device.
We reduce Eq. (5.86) to a set of algebraic equations using Euler’s method in which derivatives are approximated by the forward difference quotient (C.7). This results in the following system of equations:
Eq. (5.86): Linear Motion
\frac{di(t)}{dt} = \frac{1}{L} [V_{s}(t) – i(t)(R + R_{coil}) – \frac{∂Λ(i, x)}{∂x}v(t)]\frac{dv(t)}{dt} = \frac{1}{m}F(i, x)
\frac{dx(t)}{dt} = v(t)
i(t + Δt) = i(t) + \frac{Δt}{L} [V(t) – i(t)(R + R_{coil}) – \frac{∂Λ(i, x)}{∂x}x(t)]
v(t + Δt) = v(t) + \frac{Δt}{m} F(i, x)
x(t + Δt) = x(t) + Δt v(t). (C.11)
Equations (C.11) can be written using index notation,
i_{n+1} = i_{n}\frac{Δt}{L} [V_{n} – i_{n}(R + R_{coil}) – \frac{∂Φ(i_{n}, x_{n})}{∂x}v_{n}]
v_{n+1} = v_{n} + \frac{Δt}{m}F(i_{n}, x_{n})
x_{n+1} = x_{n} + Δt v_{n} . (C.12)
The solution procedure for Eq. (C.12) is similar to the preceding steps 1 through 4. Specifically, start with the initial conditions [i_{0}, v_{0}, x_{0}] and use Eq. (C.12) to obtain [i_{1}, v_{1}, x_{1}] , which gives the solution at t = Δt. Repeat this process to obtain [i_{2}, v_{2}, x_{2}] , [i_{3}, v_{3}, x_{3}] , . . . ,[i_{n}, v_{n}, x_{n}] , . . . , until solution values are obtained for the entire interval (t_{0}, t_{max}).