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Exercise 1: Parachute jump
Figure 1 shows a skydiver jumping from a height h
0.
Figure 1: Parachute jump
After the jump, the skydiver falls towards the earth with increasing speed.
The fall speed increases until the weight (P = mg) and the drag F
L
are in equilibrium.
At the height h1 the skydiver opens the parachute, which causes the drag to increase
considerably and the fall speed to decrease as desired.
Definitions and numerical values:
h Parachutist height relative to the ground
h
0
= 3000 m Initial height of the jump
h
1
= 1500 m Opening height of the parachute
As = 0.5 m
2
Area covered by the parachutist in the air
A
FS
= 30 m
2
Area covered by the parachute in the air
m = 85 kg Mass of the parachutist
c
w
= 1.3 Drag coefficient (same value for parachutist and parachute)
ρ= 1.2 kg/m
3
Density of air (considered constant)
g = 9.81m/s
2
Gravitational acceleration
FL Drag
1.1 Equation of Motion
To determine the equation of motion, the masses and forces involved are
represented in a pseudo-isolated system (see Figure 1). The weight P = mg is oriented
downward, the drag F
L
(air resistance) is oriented upward and by applying
d'Alembert's principle, the force F = m*h ̈
that must be applied to the skydiver for the system to remain in equilibrium is
oriented downward (thus against the positive direction). The height h is the
coordinate used here to describe the motion.
At the equilibrium of forces according to D'Alembert's principle we have:
m
= F
L
– m*g (1)
with the drag F
L
= C
w
*A*
*v
2
(2)
which is proportional to the square of the speed v = ḣ
Equation (1) is a differential equation that can be solved using several methods
including the analytical method and the numerical method with Simulink. First, the
analytical method will be used to solve this differential equation, then the numerical
method with Simulink will be used, this method also allows the simulation and
visualization of the parachutist's displacement. At the end you will get an overview of
Simulink's performance.
1.2 Analytical method
the constant speed of the falling parachutist in case of a closed parachute is first
calculated.
in case of constant speed, the acceleration is zero (
.
If
then from (1) we obtain :
m*g = C
w
*A*
*v
2
(3)
If we isolate v from (3) we obtain:
V = ḣ = -
= -
= - 46,2 m/s = - 166,5 km/h (4)
The sign ''-'' before the value of v in (4) is due to the fact that the height h decreases
during the fall (
is negative).
With
equation (1) can also be written as follows :
+ g – a*v
2
= 0 (5)
with a =
=
=
=
(6)
From (5) we obtain :
g – a*v
2
= -
(7)
= -
(8)
= - 1 (9)
By separating the variables (here the speed v and the time t) we obtain:
= - dt (10)
By integrating the two members of the equality, we obtain:
=
(11)
The integral
is an integral of type
(12)
With A = - a = -
A < 0, B = 0, C = g.
We then have 4AC - B
2
< 0, so
= I =
(13)
We deduce that
=
=
(14)
From (11) we obtain :
=
=
-
=
-
=
-
=
-
=
because
-=
= -
t = -
t =
t =
=
=
(15)
From (15) we can determine v as follows:
Since 4AC - B
2
< 0 and then |x| = |(-1)x|, so =
=
=
ln(
)
=
ln(
)
=
(ln
)
=
( 0
)
=
=
=
=
= (
)
=
=
(16)
Equation (16) gives the expression of the speed of the falling parachutist as a function
of time, starting from the moment of the jump until an instant t and this is verifiable:
At the time of the jump (t = 0), we have:
= 0 m/s
At an instant t = , we have:
= -
= -
(17)
The same expression as in (4) is found.
At the end of the equation (1) solving using the analytical method we can clearly see
that this method is rather laborious. Let's move on now to the numerical method
with Simulink.
1.3 Numerical method with Simulink
1.3.1 Simulation of parachute jump
For the purpose of realizing the Simulink model of the parachute jump,
in equation
(1) must be isolated:
m
= F
L
– m*g
=
=
=
=
(18)
The right side of the equality (18) is expanded with the function block "Fcn" and is
then connected to the first integrator block "Integrator1". This integrator block is
initialized with
.
Figure 2: Simulink model of the parachute jump (parachute_jump.mdl)
The Integrator2 block with the output h is initialized with
. The Switch,
with control input h, changes the surface from
to
when the parachutist
reaches the height h = 1500 m. With the function block Fcn1, the simulation stops
when the parachutist reaches the height h = 0 (when he touches the ground).
Figure 3 shows the speed of the falling parachutist.
Figure 3: Visualization of the speed of the falling parachutist as a function of time
The parachutist reaches, with the parachute closed, the constant speed calculated in
(4) after about 25 s. At t = 36 s, the height h of the parachutist is 1500 m (see figure 4)
and the Switch passes to the large surface of the parachutist (the parachute is open)
and from then on the speed of the falling parachutist decreases considerably and
reaches a constant speed of about -6 m/s.
Figure 4: Visualization of the height of the falling parachutist as a function of time
The acceleration of the falling parachutist (see figure 5) at t = 36 s is not realistic
(extremely high). This is due to the fact that the opening of the parachute was done
with a simple switch (sudden change from 0.5 m
2
to 30 m
2
in Fcn). This can be
improved by connecting the output of the switch with a well-initialized transfer
function of a first order system.
Figure 5: Visualization of the acceleration of the falling parachutist as a function of
time
