Engin e e r in g & Tech

te of Dr. Vishal Gupta nol

itu Assistant Professor ogy
st MED-TIET- Patiala –
In P
ar

at
iala
Thap

Contact Details
Office BC102
9729002917
vishal.gupta@thapar.edu
 Overview of the course
Mechanical Engineering Department
Course coordinator Course Co-coordinator
Dr. Vishal Gupta Dr. Gautam Setia
Assistant Professor Assistant Professor
Course Co-coordinator
Dr. Bikramjit Sharma
Associate Professor
Electronics and Communication Engineering Department
Course Coordinator
Dr. Poonam Verma
Assistant Professor
Electronics and Communication Engineering Department
Dr. Vishal Gupta, Asst. Prof. MED, TIET, Patiala
1
ENGINEERING DESIGN PROJECT-I
UTA016

Lecture - 2

2
 Instructional objective

➢Study of the dynamics for the Mangonal

➢ No DRAG Condition (Part 1)

➢ With DRAG

3
4
5
No DRAG Condition

6
 Objective

A Smaller version!
A larger version!
7
Challenges in Modelling

8
 Challenges in Modelling
• For a given “size”, can we maximise the
distance?
• What are the key parameters that control the
distance?
• Can we formulate a model that will help us
design our Mangonel?

Horizontal Distance of travel = D

Launch Velocity = v
Angle of launch = θ
Acceleration due to gravity = g 9
10
 Equation of Motion
Speed=distance/time u=initial velocity
v=final velocity
Acceleration=velocity/time
t=time duration
a=acceleration
v=u+at s=ut+1/2at2 v2=u2+2as
s=distance travelled

11
One Dimensional Motion
Example 1: (1-D) v=0 (at top)
Use equation:

v2=u2+2as a=-g
s=?

s=u2/2g

12
 Two Dimensional Motion : Dynamics v=u+at s=ut+1/2at2
• Can we use the three equations to model v2=u2+2as
the trajectory of the missile?
• Can we predict the distance?

13
Dynamics

14
Dynamics

3 4

y 2

s
1 x

Discretise the curve

15
Dynamics

3 4

y 2 v3 v4
v2

1 x
v1

Not u and v now but

v1, v2, v3, v4, etc…..

16
Dynamics
3 4

y 2

t2-t1= Δt (keep time interval constant)

s1 s1y
1 x
s1x We can decompose vectors (v, s, a) into
x, y components

v=u+at becomes: S = ut+1/2at2 becomes:

•Vx2 = vx1+ax1Δt •Δsx= vx1Δt+1/2ax1Δt2
•Vy2 = vy1+ay1Δt •Δsy= vy1Δt+1/2ay1Δt2
17
 Dynamics

Acceleration is constant (for no drag

of lift – we’ll return to this point later)
ax=0!
ay=-g

v=u+at becomes: s=ut+1/2at2 becomes:

•vx2=vx1+ax1Δt •Δsx=vx1Δt+1/2ax1Δt2 t2-t1= Δt (keep time interval constant)
•Δsy=vy1Δt+1/2ay1Δt2
•vy2=vy1+ay1Δt

18
Dynamics – Assignment1 Use Excel to study trajectory of missile

19
Conti…

Input Data Position t x y vx vy ax ay

Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52

t2=t1+Δt

20
 Conti…

Input Data Position t x y vx vy ax ay

Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52

x2=x1+vx1Δt+1/2ax1Δt2

21
 Conti…

Input Data Position t x y vx vy ax ay

Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52

y2=y1+vy1Δt+1/2ay1Δt2

22
Conti…

Input Data Position t x y vx vy ax ay

Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52

vx2=vx1+ax1Δt

23
 Conti…

Input Data Position t x y vx vy ax ay

Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52

vy2=vy1+ay1Δt

24
Conti…

Input Data Position t x y vx vy ax ay

Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52

Const=0!

25
 Conti…

Input Data Position t x y vx vy ax ay

Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52

Const=-g

26
Conti…

Input Data Position t x y vx vy ax ay

Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00 3.00 0.02 0.17 0.10 8.66 4.80 0.00 -9.81
theta (radians) 0.52 4.00 0.03 0.26 0.15 8.66 4.70 0.00 -9.81
5.00 0.04 0.35 0.19 8.66 4.61 0.00 -9.81
6.00 0.05 0.43 0.24 8.66 4.51 0.00 -9.81
7.00 0.06 0.52 0.28 8.66 4.41 0.00 -9.81
8.00 0.07 0.61 0.33 8.66 4.31 0.00 -9.81
9.00 0.08 0.69 0.37 8.66 4.21 0.00 -9.81
10.00 0.09 0.78 0.41 8.66 4.11 0.00 -9.81
11.00 0.10 0.87 0.45 8.66 4.02 0.00 -9.81
12.00 0.11 0.95 0.49 8.66 3.92 0.00 -9.81
13.00 0.12 1.04 0.53 8.66 3.82 0.00 -9.81
14.00 0.13 1.13 0.57 8.66 3.72 0.00 -9.81
15.00 0.14 1.21 0.60 8.66 3.62 0.00 -9.81
16.00 0.15 1.30 0.64 8.66 3.53 0.00 -9.81
17.00 0.16 1.39 0.67 8.66 3.43 0.00 -9.81
18.00 0.17 1.47 0.71 8.66 3.33 0.00 -9.81
Copy formula 19.00 0.18 1.56 0.74 8.66 3.23 0.00 -9.81
down 27
 Input Data Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00 3.00 30.01 0.17 0.10 8.66 4.80 0.00 -9.81
theta (radians) 0.52 4.00 30.53 0.26 0.15 8.66 4.70 0.00 -9.81
5.00 1.40
30.53 0.35 0.19 8.66 4.61 0.00 -9.81
6.00 30.53 0.43 0.24 8.66 4.51 0.00 -9.81
7.00 1.20
30.53 0.52 0.28 8.66 4.41 0.00 -9.81
8.00 30.53 0.61 0.33 8.66 4.31 0.00 -9.81
9.00 1.00
30.53 0.69 0.37 8.66 4.21 0.00 -9.81
10.00 30.53 0.78 0.41 8.66 4.11 0.00 -9.81
11.00 0.80
30.53 0.87 0.45 8.66 4.02 0.00 -9.81
12.00 30.53 0.95 0.49 8.66 3.92 0.00 -9.81
13.00 0.60
30.53 1.04 0.53 8.66 3.82 0.00 -9.81
14.00 30.53 1.13 0.57 8.66 3.72 0.00 -9.81
15.00 0.40
30.53 1.21 0.60 8.66 3.62 0.00 -9.81
Plot x versus y using chart 16.00 30.53 1.30 0.64 8.66 3.53 0.00 -9.81
17.00 0.20
30.53 1.39 0.67 8.66 3.43 0.00 -9.81
wizard 18.00 30.53 1.47 0.71 8.66 3.33 0.00 -9.81
19.00 30.53
0.00 1.56 0.74 8.66 3.23 0.00 -9.81
20.00 30.53
0.00 1.65
2.00 0.77
4.00 8.66
6.00 3.13
8.00 0.00
10.00 -9.81
21.00 30.53 1.73 0.80 8.66 3.04 0.00 -9.81
22.00 30.53 1.82 0.83 8.66 2.94 0.00 -9.81

28
Assignment1
Mangonel Dynamics Design Tool using Excel
Work in groups and/or individually in computer rooms today and during week to

1.Create excel spreadsheet as demonstrated

2.Plot x versus y
3.Study effect of changing velocity
4.Study effect of changing angle
An assignment will be set based on this work. Assignment to be submitted individually
– no copying!

29
With DRAG Condition

30
 Drag force
A drag force is the resistance force caused by the motion of a body through a fluid, such as
water or air. A drag force acts opposite to the direction of the oncoming flow velocity

31
 A A
Section line
33
Golf ball
34
Drag on a Smooth ball

Drag on a ball with dimple 35

Assignment 2 - Add Drag!
The “projectile” which our Mangonel will fire will be a squash-ball which is spherical See
fig 1.

a. b.
Fig 1. Prototype of our Mangonel!

36
Factors that control drag
Drag reduces the distance travelled by the ball before it hits the ground.
• What are the factors that control drag? and
• How can you reduce it?
If you can reduce drag then your object will travel further!

37
Empirical formula of drag

The equation that expresses the drag force (Fd) experienced by an object moving in air is:

ρ =Density of air
Fd=1/2 ρ A V2 Cd
A =Projected area of the body
V =Velocity of the body
Cd =Drag Coefficient

38
Empirical formula of drag

39
Assignment 2 - Add Drag!

Fd=1/2 ρ A V2 Cd

So, we see the drag force increases with size (A) and velocity (V2).

It makes sense to reduce the size but what about the velocity?

If we reduce the velocity the drag force reduces but the missile wont go as far!

40
Concept of darg

41
42
Acceleration due to drag
Newton’s 2nd Law Fd=1/2 ρ A V2 Cd=mad ad= 1/(2m) Cd ρ A V2

The magnitude of the acceleration is thus expressed ad = 1/(2m) Cd ρ A (vx2+vy2)

Thus Constant (k)

axd = ad Cos(ß) ayd = ad Sin (ß)

axd = k (vx2+vy2) Cos(ß) ayd = k (vx2+vy2) Sin(ß)

ax=0-k (vx2+vy2) Cos(ß) and ay=-9.81-k (vx2+vy2) Sin(ß)

where
ß=tan-1(vy/vx) 43
Assignment 2 - Add Drag!
Input Data Position t x y vx vy beta [rads] cos(beta) sin(beta) ax ay
Change to see
Vel 15.00 impact!!!! 1.00 0.00 0.00 0.00 7.51 12.99 1.04667 0.50046 0.86576 -0.85919 -11.29633
delt t 0.01 2.00 0.01 0.08 0.13 7.50 12.87 1.04337 0.50331 0.86411 -0.85237 -11.27338
Change to see
theta (degrees) 60.00 impact!!!! 3.00 0.02 0.15 0.26 7.49 12.76 1.04003 0.50619 0.86242 -0.84559 -11.25067
theta (radians) 1.05 4.00 0.03 0.22 0.38 7.48 12.65 1.03666 0.50910 0.86071 -0.83886 -11.22820
5.00 0.04 0.30 0.51 7.47 12.54 1.03323 0.51204 0.85896 -0.83217 -11.20597
6.00 0.05 0.37 0.64 7.46 12.42 1.02977 0.51502 0.85718 -0.82552 -11.18397
7.00 0.06 0.45 0.76 7.46 12.31 1.02626 0.51802 0.85537 -0.81892 -11.16220
8.00 0.07 0.52 0.88 7.45 12.20 1.02270 0.52106 0.85352 -0.81236 -11.14067
9.00 0.08 0.60 1.00 7.44 12.09 1.01910 0.52413 0.85164 -0.80584 -11.11937
Drag Data 10.00 0.09 0.67 1.12 7.43 11.98 1.01545 0.52723 0.84972 -0.79936 -11.09829
The value at
atmospheric
rho 1.20 conditions 11.00 0.10 0.75 1.24 7.42 11.87 1.01176 0.53037 0.84777 -0.79292 -11.07744
This is a typical
value, however
try and change
Cd 0.40 it! 12.00 0.11 0.82 1.36 7.42 11.76 1.00802 0.53354 0.84577 -0.78653 -11.05682
Change to see
m 0.050 impact!!!! 13.00 0.12 0.89 1.48 7.41 11.65 1.00422 0.53674 0.84375 -0.78018 -11.03642
Change to see
D 0.045 impact!!!! 14.00 0.13 0.97 1.59 7.40 11.54 1.00038 0.53998 0.84168 -0.77387 -11.01624
Area 0.0016 15.00 0.14 1.04 1.71 7.39 11.42 0.99649 0.54325 0.83957 -0.76760 -10.99628
Constant, K 0.01 16.00 0.15 1.12 1.82 7.38 11.31 0.99254 0.54656 0.83742 -0.76137 -10.97654

Calculate ß, for each step, depending on the velocity 44

Assignment 2 - Add Drag!
Input Data Position t x y vx vy beta [rads] cos(beta) sin(beta) ax ay
Change to see
Vel 15.00 impact!!!! 1.00 0.00 0.00 0.00 7.51 12.99 1.04667 0.50046 0.86576 -0.85919 -11.29633
delt t 0.01 2.00 0.01 0.08 0.13 7.50 12.87 1.04337 0.50331 0.86411 -0.85237 -11.27338
Change to see
theta (degrees) 60.00 impact!!!! 3.00 0.02 0.15 0.26 7.49 12.76 1.04003 0.50619 0.86242 -0.84559 -11.25067
theta (radians) 1.05 4.00 0.03 0.22 0.38 7.48 12.65 1.03666 0.50910 0.86071 -0.83886 -11.22820
5.00 0.04 0.30 0.51 7.47 12.54 1.03323 0.51204 0.85896 -0.83217 -11.20597
6.00 0.05 0.37 0.64 7.46 12.42 1.02977 0.51502 0.85718 -0.82552 -11.18397
7.00 0.06 0.45 0.76 7.46 12.31 1.02626 0.51802 0.85537 -0.81892 -11.16220
8.00 0.07 0.52 0.88 7.45 12.20 1.02270 0.52106 0.85352 -0.81236 -11.14067
9.00 0.08 0.60 1.00 7.44 12.09 1.01910 0.52413 0.85164 -0.80584 -11.11937
Drag Data 10.00 0.09 0.67 1.12 7.43 11.98 1.01545 0.52723 0.84972 -0.79936 -11.09829
The value at
atmospheric
rho 1.20 conditions 11.00 0.10 0.75 1.24 7.42 11.87 1.01176 0.53037 0.84777 -0.79292 -11.07744
This is a typical
value, however
try and change
Cd 0.40 it! 12.00 0.11 0.82 1.36 7.42 11.76 1.00802 0.53354 0.84577 -0.78653 -11.05682
Change to see
m 0.050 impact!!!! 13.00 0.12 0.89 1.48 7.41 11.65 1.00422 0.53674 0.84375 -0.78018 -11.03642
Change to see
D 0.045 impact!!!! 14.00 0.13 0.97 1.59 7.40 11.54 1.00038 0.53998 0.84168 -0.77387 -11.01624
Area 0.0016 15.00 0.14 1.04 1.71 7.39 11.42 0.99649 0.54325 0.83957 -0.76760 -10.99628
Constant, K 0.01 16.00 0.15 1.12 1.82 7.38 11.31 0.99254 0.54656 0.83742 -0.76137 -10.97654

Modify the accelerations for each step depending on ß and on the velocities
45
Assignment 2 - Add Drag!

Superimpose the “no drag”

and “with drag” plots!

Two sheets!

46
 Assignment 2 - Add Drag!

Adjust Cd, mass and the diameter

the see effect as well as the initial
velocity and launch angle!

47
Assignment 2 - Add Drag!

48
Assignment 2 - Add Drag!

49
Assignment 2 - Add Drag!

What causes drag? In figure 3, although much smaller, the

cylinder experiences the same drag force
The drag force is due to pressure as the much larger but more aero-
losses caused by recirculation of flow. dynamic airfoil. This is due to the
Simply put; eddies and vortices which relatively greater amount of “turbulence”
are caused by abrupt changes in in the wake.
geometry.

Fig 3.
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