Problem 1: A. Find the linearized equation for z = f(x) = 0.4x3 about x- = 2, z- = 3.2.
B. Use Matlab and plot both the nonlinear function and its linear version to confirm.
Problem 2: Linearize the nonlinear equation z =x/y. in the region defined by 90 ≤ x ≤ 110 and 45 ≤ y ≤ 55.
Problem 3: The mechanical system shown below has a nonlinear spring. The relationship between the force exerted by the spring and the change of the spring's length measured from the spring's relaxed position has been approximated mathematically by the following nonlinear equation:
FNLS = 2.4√x
Obtain the linearized mathematical model of the system that approximates the system dynamics in the vicinity of the normal operating point determined by the average value of the input force, F = 0.1N.
Problem 4: Figure shows a simple pendulum system in which a cord is wrapped around a fixed cylinder. The motion of the system that results is described by the differential equation
(l+Rθ)θ·· + gsinθ + Rθ·2 = 0.
where
I = length of the cord in the vertical (down) position,
R = radius of the cylinder.
(a) Write the State-variable equation for this system.
(b) Linearize the equation around the point θ = 0, and show that for small values of θ, the system equation reduces to an equation for a simple pendulum-that is, θ·· + (g/l) = 0.