Question

Figure 1: Futuristic rendition of a maglev.

Maglevs work by levitating over the tracks and thus eliminating the need for moving parts. There are two sets of magnets in a Maglev system: one levitates the train above the track and the other pulls the train forward.
In this project we focus on the control of levitation by focusing on a simplified model that describes the vertical position of an iron ball (abstracting the train) within a magnetic field created by a single electromagnet. The relevant equations are:
x₁ = x₂
x₂ = 1/2m​λ²−g
​λ =−R/ c ​(1−x1​)λ+u​
where x₁​ denotes the balls height, x₂​ the balls velocity, m is the balls mass, g is gravitys constant, λ is the magnetic flux linkage in the electromagnet, R is the resistance of the electromagnets coil, c is a positive constant modeling the electromagnets geometry and construction, and u is the voltage applied at the coils terminals constituting the input to this system. The values of the parameters are:
m=30000,g=9.8,R=15,c=5
The objective is stabilize the ball's height at 0.01 m. Find an equilibrium pair consistent with this objective.

Answer 1

To find an equilibrium pair consistent with the objective of stabilizing the ball's height at 0.01m, we need to find values of x₁ and x₂ that satisfy the given equations. By substituting the given parameter values and simplifying the equation, we can solve for x₂ and λ.

Step-by-step explanation:

To find an equilibrium pair consistent with the objective of stabilizing the ball's height at 0.01m, we need to find values of x₁ and x₂ that satisfy the given equations. Let's start by substituting the values of the parameters into the equations:

x₁ = x₂

x₂ = 1/2mλ² - gλ = -R/c(1 - x₁)λ + u

Substituting the given parameter values:

x₁ = x₂

x₂ = 1/(2*30000*λ²) - 9.8*λ = -15/5(1 - x₁)λ + u

Simplifying the equation further:

2x₂λ² + 9.8λ - 0.03 = -3(1 - x₁)λ + 3u

Since we want to stabilize the ball's height at 0.01m, x₁ = 0.01. Substituting this value:

2x₂λ² + 9.8λ - 0.03 = -3(1 - 0.01)λ + 3u

2x₂λ² + 9.8λ - 0.03 = -3(0.99)λ + 3u

Simplifying further:

2x₂λ² + 9.8λ - 0.03 = -2.97λ + 3u

Now we have an equation with two variables, x₂ and λ. We can solve this equation to find the values that satisfy it.