- #1
logan233
- 11
- 0
Homework Statement
Generate an open loop u(t) and simulate. Plot x(t) and y(t)
[itex]\dot{x}[/itex] = Vcos(θ)
[itex]\dot{y}[/itex] = Vsin(θ)
[itex]\dot{θ}[/itex] = u
I am given initial values. All are 0 except for [itex]\dot{x}[/itex](0) = V.
Homework Equations
Laplace Transform Tables
The Attempt at a Solution
I think I know how to find the transfer functions [itex]\frac{X}{U}[/itex] and [itex]\frac{Y}{U}[/itex], which is what I'm assuming the problem statement is asking for. I first found [itex]\frac{θ}{U}[/itex] which is equal to [itex]\frac{1}{s}[/itex]. I am now trying to find the transfer functions [itex]\frac{X}{θ}[/itex] and [itex]\frac{Y}{θ}[/itex] so that I can multiply it by the transfer function θ/U. My problem is finding the Laplace transform of the given [itex]\dot{x}[/itex] and [itex]\dot{y}[/itex] equations. I know that the Laplace transform of cos(at) and sin(bt) are s/(s^2 + a^2) and b/(s^2 + b^2) respectively, however I am not sure of these transforms when the input is inside of the cosine and sine functions. My original thinking was the transfer functions X/θ and Y/θ would simply be given as V*s/(s^2 + 1) and V/(s^2+1) respectively but because the cosine and sine arguments aren't simply at and bt I am being thrown off.
SUMMARY:
I am trying to find the Laplace transforms of [itex]\dot{x}[/itex] = Vcos(θ) and [itex]\dot{y}[/itex] = Vsin(θ) when [itex]\dot{θ}[/itex] = u ([itex]\frac{θ}{U}[/itex] = [itex]\frac{1}{s}[/itex])