Page 308 - Applied Numerical Methods Using MATLAB

PROBLEMS 297
(b) Lorenz Equation: Turbulent Flow and Chaos
Consider a nonlinear state equation.

x (t) = σ(x 2 (t) − x 1 (t)) σ = 10
1

x (t) = (1 + λ − x 3 (t))x 1 (t) − x 2 (t) with λ = 20 ∼ 100 (P6.3.2)
2

x (t) = x 1 (t)x 2 (t) − γx 3 (t) γ = 2
3
Compose a program to solve this equation with λ = 20 and 100 for the
time interval [0,10] and plot x 3 (t) versus x 1 (t). Let the initial condition
be [x 1 (0)x 2 (0)x 3 (0)] = [−8 −16 80].
(c) Chemical Reactor
Consider a nonlinear state equation describing the concentrations of
two reactants and one product in the chemical process.

x (t) = a(u 1 − x 1 (t)) − bx 1 (t)x 2 (t) a = 5
1

x (t) = a(u 2 − x 2 (t)) − bx 1 (t)x 2 (t) with b = 2 (P6.3.3)
2

x (t) =−ax 3 (t) + bx 1 (t)x 2 (t) u 1 = 3,u 2 = 5
3
Compose a program to solve this equation for the time interval
[0, 1] and plot x 1 (t), x 2 (t),and x 3 (t). Let the initial condition be
[x 1 (0)x 2 (0)x 3 (0)] = [1 2 3].
(d) Cantilever Beam: A Differential Equation w.r.t a Spatial Variable
Consider a nonlinear state equation describing the vertical deflection of
a beam due to its own weight
2
2
d y dy
L L 2
JE = ρg 1 + x x − + (P6.3.4)
dx 2 dx 2 2
2
2
3
where JE = 2000 kg · m /s , ρ = 10 kg/m, g = 9.8m/s , L = 2m.
Write a program to solve this equation for the interval [0,L] and plot
y(t). Let the initial condition be [y(0)y (0)] = [0 0]. Note that the

physical meaning of the independent variable for which we usually use
the symbol ‘t’ in writing the differential function is not a time, but
the x-coordinate of the cantilever beam along the horizontal axis in
this problem.
(e) Phase-Locked Loop (PLL)
Consider a nonlinear state equation describing the behavior of a PLL
circuit depicted in Fig. P6.3.1.

a = 1500
au(t) cos(x 2 (t)) − x 1 (t)

x (t) = with τ = 0.002 (P6.3.5a)
1
τ
u(t) = sin(ω o t)

x (t) = x 1 (t) + ω c
2
y(t) = x 1 (t) + ω c (P6.3.5b)