Transcript Dynamics_1E10_GB_2_prot

Assignment 2 - Add Drag!
Mangonel Dynamics Design Tool using Excel
Up until now we have assumed constant acceleration in the x and y
directions (ax=0, ay=-g)
However, large objects that are not very aero-dynamic (not pointy!)
experience a resistance to movement through a fluid such as air. This
resistance is called “drag”. A parachute is an example of how drag
can be used to ones benefit to reduce the falling speed of a person.
The force downwards on the person can be expressed as F=-mg (mass
of the person multiplied by the downwards acceleration). This force,
when a parachute is employed, is counteracted somewhat by the
upward force of drag which serves to reduce the velocity of the
falling person.
Dr. Gareth J. Bennett
Trinity College Dublin
Page 1
Assignment 2 - Add Drag!
The “missile” which our Mangonel will fire will be something like a
squash-ball which is spherical See fig 1.
a.
b.
Fig 1. Prototype of our Mangonel!
Dr. Gareth J. Bennett
Trinity College Dublin
Page 2
Assignment 2 - Add Drag!
This squash ball will experience drag as it is projected through
the air. The questions you need to ask yourselves are; What
are the factors that control drag and how can you reduce it?
If you can reduce drag then your missile will travel further!
Dr. Gareth J. Bennett
Trinity College Dublin
Page 3
Assignment 2 - Add Drag!
You may be familiar to images such as that shown in fig 2 (a). “Flow
visualisation” allows a streamline around a body such as a car to be
determined. Fig 2 (b) shows how cars have become more aero-dynamic
and hence more fuel efficient over time.
a.
b.
Fig 2. Car aero-dynamics
Dr. Gareth J. Bennett
Trinity College Dublin
Page 4
Assignment 2 - Add Drag!
The equation that expresses
the drag force (Fd) is:
Fd=1/2 ρ A V2 Cd
ρ =Density of air
A =Projected area
of the body
V =Velocity of the
body
Cd =Drag Coefficient
Dr. Gareth J. Bennett
Trinity College Dublin
Page 5
Assignment 2 - Add Drag!
So, we see the drag force
increases with size and
velocity.
It makes sense to reduce
the size but what about the
velocity?
Fd=1/2 ρ A V2 Cd
If we reduce the velocity
the drag force reduces but
the missile wont go as far!
Dr. Gareth J. Bennett
Trinity College Dublin
Page 6
Assignment 2 - Add Drag!
Cd: The Drag Coefficient
Typically, the approach in such
situations is, for a fixed velocity,
to reduce the drag coefficient,
where the drag coefficient is the
parameter which characterises how
aero-dynamic the body is.
Fd=1/2 ρ A V2 Cd
In fig. 2b, we see that it is Cd
that is plotted on the y-axis.
Dr. Gareth J. Bennett
Trinity College Dublin
Page 7
Assignment 2 - Add Drag!
What causes drag?
The drag force is due to
pressure losses caused by
recirculation of flow. Simply put;
eddies and vortices which are
caused by abrupt changes in
geometry.
In figure 3, although much
smaller, the cylinder experiences
the same drag force as the much
larger but more aero-dynamic
airfoil. This is due to the
relatively greater amount of
“turbulence” in the wake.
Fig 3.
Dr. Gareth J. Bennett
Trinity College Dublin
Page 8
Assignment 2 - Add Drag!
Acceleration due to drag
For our assignment, in order to calculate the new trajectory, we
need to calculate the acceleration due to drag.
By performing a mass balance (Newton’s 2nd Law), we can write
Fd=1/2 ρ A
Dr. Gareth J. Bennett
Trinity College Dublin
V2
Cd=mad
m= mass of the body
ad=acceleration due to drag
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Assignment 2 - Add Drag!
Acceleration due to drag:
From this we have
ad= 1/(2m) Cd ρ A V2
The magnitude of the acceleration is thus expressed
ad = 1/(2m) Cd ρ A (vx2+vy2)
Constant (k)
Dr. Gareth J. Bennett
Trinity College Dublin
Page 10
Assignment 2 - Add Drag!
Acceleration due to drag:
Thus;
ayd = k (vx2+vy2) Sin(ß)
axd = k (vx2+vy2) Cos(ß)
Therefore;
ax=0-k (vx2+vy2) Cos(ß)
and
ay=-9.81-k (vx2+vy2) Sin(ß)
where
ß=tan-1(vy/vx)
Dr. Gareth J. Bennett
Trinity College Dublin
Page 11
Assignment 2 - Add Drag!
Acceleration due to drag:
We see now that the acceleration depends on the changing velocity
and so is no longer a constant but depends on the velocity at each
step!!!! Don’t confuse the angle ß which changes at each step depending
on the velocity vectors, with the initial launch angle θ which remains
fixed once chosen.
Try and implement the equations in yellow into the spreadsheet to see
the effect of drag!
Dr. Gareth J. Bennett
Trinity College Dublin
Page 12
Assignment 2 - Add Drag!
Input Data
Position
Change to see
15.00 impact!!!!
0.01
Change to see
60.00 impact!!!!
1.05
Vel
delt t
theta (degrees)
theta (radians)
Drag Data
rho
1.20
Cd
0.40
m
0.050
D
Area
Constant, K
0.045
0.0016
0.01
The value at
atmospheric
conditions
This is a typical
value, however
try and change
it!
Change to see
impact!!!!
Change to see
impact!!!!
t
x
y
vx
vy
1.00 0.00 0.00 0.00 7.51 12.99
2.00 0.01 0.08 0.13 7.50 12.87
beta [rads] cos(beta) sin(beta)
ax
ay
1.04667
1.04337
0.50046
0.50331
0.86576
0.86411
-0.85919
-0.85237
-11.29633
-11.27338
12.76
12.65
12.54
12.42
12.31
12.20
12.09
11.98
1.04003
1.03666
1.03323
1.02977
1.02626
1.02270
1.01910
1.01545
0.50619
0.50910
0.51204
0.51502
0.51802
0.52106
0.52413
0.52723
0.86242
0.86071
0.85896
0.85718
0.85537
0.85352
0.85164
0.84972
-0.84559
-0.83886
-0.83217
-0.82552
-0.81892
-0.81236
-0.80584
-0.79936
-11.25067
-11.22820
-11.20597
-11.18397
-11.16220
-11.14067
-11.11937
-11.09829
11.00 0.10 0.75 1.24 7.42 11.87
1.01176
0.53037
0.84777
-0.79292
-11.07744
12.00 0.11 0.82 1.36 7.42 11.76
1.00802
0.53354
0.84577
-0.78653
-11.05682
13.00 0.12 0.89 1.48 7.41 11.65
1.00422
0.53674
0.84375
-0.78018
-11.03642
14.00 0.13 0.97 1.59 7.40 11.54
15.00 0.14 1.04 1.71 7.39 11.42
16.00 0.15 1.12 1.82 7.38 11.31
1.00038
0.99649
0.99254
0.53998
0.54325
0.54656
0.84168
0.83957
0.83742
-0.77387
-0.76760
-0.76137
-11.01624
-10.99628
-10.97654
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.15
0.22
0.30
0.37
0.45
0.52
0.60
0.67
0.26
0.38
0.51
0.64
0.76
0.88
1.00
1.12
7.49
7.48
7.47
7.46
7.46
7.45
7.44
7.43
Calculate ß, for each step,
depending on the velocity
Dr. Gareth J. Bennett
Trinity College Dublin
Page 13
Assignment 2 - Add Drag!
Input Data
Position
Change to see
15.00 impact!!!!
0.01
Change to see
60.00 impact!!!!
1.05
Vel
delt t
theta (degrees)
theta (radians)
Drag Data
rho
1.20
Cd
0.40
m
0.050
D
Area
Constant, K
0.045
0.0016
0.01
The value at
atmospheric
conditions
This is a typical
value, however
try and change
it!
Change to see
impact!!!!
Change to see
impact!!!!
t
x
y
vx
vy
1.00 0.00 0.00 0.00 7.51 12.99
2.00 0.01 0.08 0.13 7.50 12.87
beta [rads] cos(beta) sin(beta)
ax
ay
1.04667
1.04337
0.50046
0.50331
0.86576
0.86411
-0.85919
-0.85237
-11.29633
-11.27338
12.76
12.65
12.54
12.42
12.31
12.20
12.09
11.98
1.04003
1.03666
1.03323
1.02977
1.02626
1.02270
1.01910
1.01545
0.50619
0.50910
0.51204
0.51502
0.51802
0.52106
0.52413
0.52723
0.86242
0.86071
0.85896
0.85718
0.85537
0.85352
0.85164
0.84972
-0.84559
-0.83886
-0.83217
-0.82552
-0.81892
-0.81236
-0.80584
-0.79936
-11.25067
-11.22820
-11.20597
-11.18397
-11.16220
-11.14067
-11.11937
-11.09829
11.00 0.10 0.75 1.24 7.42 11.87
1.01176
0.53037
0.84777
-0.79292
-11.07744
12.00 0.11 0.82 1.36 7.42 11.76
1.00802
0.53354
0.84577
-0.78653
-11.05682
13.00 0.12 0.89 1.48 7.41 11.65
1.00422
0.53674
0.84375
-0.78018
-11.03642
14.00 0.13 0.97 1.59 7.40 11.54
15.00 0.14 1.04 1.71 7.39 11.42
16.00 0.15 1.12 1.82 7.38 11.31
1.00038
0.99649
0.99254
0.53998
0.54325
0.54656
0.84168
0.83957
0.83742
-0.77387
-0.76760
-0.76137
-11.01624
-10.99628
-10.97654
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.15
0.22
0.30
0.37
0.45
0.52
0.60
0.67
0.26
0.38
0.51
0.64
0.76
0.88
1.00
1.12
7.49
7.48
7.47
7.46
7.46
7.45
7.44
7.43
Modify the accelerations for
each step depending on ß and
on the velocities
Dr. Gareth J. Bennett
Trinity College Dublin
Page 14
Assignment 2 - Add Drag!
Try and
superimpose the
“no drag” and
“with drag” plots!
Two sheets!
Dr. Gareth J. Bennett
Trinity College Dublin
Page 15
Assignment 2 - Add Drag!
Adjust Cd, mass
and the diameter
the see effect as
well as the initial
velocity and launch
angle!
Dr. Gareth J. Bennett
Trinity College Dublin
Page 16
Assignment 2 - Add Drag!
One more thing! When you square velocity,
the result will be always positive. However,
if the drag is always going to be acting
against the direction of velocity we need
to be aware of the “sign” of the velocity.
How can we keep that information in the
Excel spread sheet equations?
Good luck!
Dr. Gareth J. Bennett
Trinity College Dublin
Page 17