Final answer:
The student is looking to find the equilibrium input δ¯ that corresponds to an equilibrium output θ¯ = π/4 rad for a given nonlinear, time-invariant ODE describing a dynamic system. The equation simplifies at equilibrium to 5sin(π/4) = 3tan(δ¯) + 0.5 √δ¯ and must be solved for δ¯ using numerical methods due to its transcendental nature.
Step-by-step explanation:
The student is asking for the equilibrium or trim input δ¯ for the nonlinear, time-invariant ordinary differential equation (ODE) at the equilibrium or trim output θ¯ = π/4 rad. Since the system is at equilibrium, the derivatives with respect to time will be zero. Therefore, the ODE simplifies to the equation 5sin(θ) = 3tan(δ¯) + 0.5 √δ¯. Substituting θ¯ = π/4 into the equation yields 5sin(π/4) = 3tan(δ¯) + 0.5 √δ¯, which simplifies to 5/√2 = 3tan(δ¯) + 0.5 √δ¯.
This equation must be solved for δ¯, but it requires numerical methods or iterative techniques since it is a transcendental equation that cannot be solved algebraically. This equilibrium analysis is an essential part of studying the dynamic behavior of the system described by the ODE and finding the corresponding input that causes the system to remain at a desired output without any change over time.