Page 314 - Applied Numerical Methods Using MATLAB

PROBLEMS 303
where the values of the mass, the viscous friction coefficient, and the spring
constant are given as M = 1 kg, B = 0.1Ns/m, and K = 0.1 N/m, respec-
tively. The input to this system is the movement u(t) of the wheel part
causing the movement y(t) of the body as the output of the system and is
approximated to a triangular wave of height 1 m, duration 1 s, and period
2 s as depicted in Fig. P6.4b. After converting this equation into a state
equation as

x (t) 0 1 x 1 (t) 0

1 = +

x (t) −K/M −B/M x 2 (t) (B/M)u (t) + (K/M)u(t)

2
(P6.4.2)

x 1 (0) 0
with =
x 2 (0) 0
we can use such routines as ode_Ham(), ode45(), ... to solve this state
equation and use some graphic functions to draw not only the graphs of
y(t) and u(t), but also the animated simulation diagram. You can run
the above MATLAB program “do_MBK.m” to see the results. Does the
suspension system made of a spring and a damper as depicted in Fig.
P6.4a absorb effectively the shock caused by the rolling wheel so that the
amplitude of vehicle body oscillation is less than 1/5 times that of wheel
oscillation?
(cf) If one is interested in graphic visualization with MATLAB, he/she can refer to
[N-1].
6.5 A Nonlinear Differential Equation for an Orbit of a Satellite

Consider the problem of an orbit of a satellite, whose position and velocity
are obtained as the solution of the following state equation:

x (t) = x 3 (t)
1

x (t) = x 4 (t)
2
2
2

x (t) =−GM E x 1 (t)/(x (t) + x (t)) 3/2 (P6.5.1)
3 1 2
2
2
x (t) =−GM E x 2 (t)/(x (t) + x (t)) 3/2

4 1 2
2
where G = 6.672 × 10 −11 Nm /kg 2 is the gravitational constant, and
M E = 5.97 × 10 24 kg is the mass of the earth. Note that (x 1 ,x 2 )and (x 3 ,x 4 )
denote the position and velocity, respectively, of the satellite on the plane
having the earth at its origin. This state equation is defined in the M-file
‘df_sat.m’below.
(a) Supplement the following program “nm6p05.m” which uses the three
routines ode_RK4(), ode45(),and ode23() to find the paths of the
satellite with the following initial positions/velocities for one day.