Dynamics 1e10 1no Drag -Sks

7/25/2019 Dynamics 1e10 1no Drag -Sks

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Page 1

UTA010 (ED

-

II)

Dynamics for the Catapult(No Drag)

Sandeep K SharmaMED, Thapar University, Patiala


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Page 2

Modelling

For a given size, can we maximise thedistance?

What are the key parameters thatcontrol the distance?

Can we formulate a model that will helpus design our Catapult?



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Page 3

Objective of Modelling

create the simplest possible projectilemotion model using standard kinematicformulas and variables.

-initial height

- initial speed

-

initial angle- time step


Animation of themodel for ease ofunderstanding


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Page 4

Fundamentals

force = mass x acceleration (ma)

work = force x distance (Fs)

energy== workpower = rate of work (work/time)



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Page 5

Derived Units

Force (1N=1kgm/s2)

Work (1J=1Nm=1kgm2/s2)

Energy (J)Power (1W=1J/s)



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Page 6

Dynamics


Uniformly accelerated body moving along axis X


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Dynamics

Speed av=distance/time

Accelerationav=velocity/time



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Dynamics

Can derive equations for linear motion(for constant acceleration)

v = u + at

s = ut + 1/2at2

v2 = u2 + 2as

u=initial velocity

v=final velocity

t=time duration

a=acceleration

s=distance travelled



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Dynamics

Example 1: (1-D)

Kick a ball straight up. Given a giveninitial velocity, how high will it go?

DISTANCE.. ????



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Dynamics

Example 1: (1-D)

Use equation:v2=u2+2as

s=u2/2g

a=-g

u

v=0 (at top)

s=?



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Dynamics

Example 2: (1-D)

Drop a rock from a cliff of

height s. How long will ittake to hit the ground/sea?

Time.???

s = ut + 1/2at2


s=1/2at2


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Dynamics

Example 2: (1-D)

Reverse Case

Use equation:

s=ut+1/2at2

s=1/2at2

t (from stopwatch)

u=0 (at top)

s=?



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Dynamics

Example 2: (1-D)

s=1/2at2

t (from stopwatch)

u=0 (at top)

s=?

Example Result: t=3s =>s=44m

However!

t=2.5s =>s=31m

t=3.5s =>s=60m

Sensitive to error: proportional tosquare of t!



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Dynamics


0

10

20

30

40

50

60

70

80

90

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

Time

S


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Page 15

Dynamics

Can we use these equations to modelthe trajectory of the missile?

And hence predict the distance?

A 2-D problem!



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Page 16

Dynamics (2-D)

y

x



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Page 17

Dynamics (2-D)

y

x

Discretise the curve

1

2

3 4

s



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Page 18

Dynamics (2-D)

y

x

Not u and v now but

v1, v2, v3, v4, etc..

1

2

3 4

v1

v2

v3v4



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Page 19

Dynamics (2-D)

y

x

We can decompose vectors (v, s, a)into x, y components

1

2

3 4

s1x

s1s1y



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Page 20

Dynamics (2-D)

v=u+at becomes:

vx2=vx1+ax1t

vy2=vy1+ay1t

s=ut+1/2at2 becomes:

sx=vx1t+1/2ax1t2

sy=vy1t+1/2ay1t2

Acceleration is constant

(for no drag of lift wellreturn to this point later)

ax=0!

ay=-g

t2-t1= t(keep time interval constantthroughout the flight)



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Page 21

Dynamics (2-D)

s2=s1+vavt

s2=s1+ (v1+(at)/2)t

s2=s1+v1t +(at2)/2

s=s2-s1s= v1 t +(at2)/2

x2=x1+v1xt +(axt2)/2

y2=y1+v1yt +(ayt2

)/2


For constant acceleration-

a=(v2-v1)/(t2-t1)

v2=v1+at

s2=s1+vavt

where vav=(v1+v2)/2

vav=(v1+v1+at)/2

vav=v1 +(at)/2


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Page 22

Dynamics (2-D)

s2=s1+vavt

s2=s1+ (v1+(at)/2)t

s2=s1+v1t +(at2)/2

s=s2-s1s= v1 t +(at2)/2

x2=x1+v1xt +(axt2)/2

y2=y1+v1yt +(ayt2

)/2


sx=vx1t+1/2ax1t2

sy=vy1t+1/2ay1t2

For constant acceleration-

a=(v2-v1)/(t2-t1)

v2=v1+at

s2=s1+vavt

where vav=(v1+v2)/2

vav=(v1+v1+at)/2

vav=v1 +(at)/2


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Page 23

Dynamics Assignment1

Use Excel to study trajectory of missile

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

theta (degrees) 30.00

theta (radians) 0.52

Input Data

Initial Conditions

vx=Vel*cos(theta)

vy=Vel*sin(theta)


Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81

delt t 0.01



Input Data

Initial Conditions

vx=Vel*cos(theta)

vy=Vel*sin(theta)



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Page 24

Dynamics

t2=t1+t


Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81



Input Data



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Page 25

Dynamics

x2=x1+vx1t+1/2ax1t2


Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81



Input Data



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Page 26

Dynamics

y2=y1+vy1t+1/2ay1t2


Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81



Input Data



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Page 27

Dynamics

vx2=vx1+ax1t


Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81



Input Data



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Page 28

Dynamics

vy2=vy1+ay1t


Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81



Input Data



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Page 29

Dynamics

Const=0!


Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81



Input Data



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Page 30

Dynamics

Const=-g


Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81



Input Data



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Dynamics

Copy formuladown


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 -9.819.00 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 -9.8118.00 0.17 1.47 0.71 8.66 3.33 0.00 -9.81

19.00 0.18 1.56 0.74 8.66 3.23 0.00 -9.81

Input Data



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Dynamics

Plot x versus yusing chartwizard


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 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 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 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 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 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 30.53 1.21 0.60 8.66 3.62 0.00 -9.81

16.00 30.53 1.30 0.64 8.66 3.53 0.00 -9.81

17.00 30.53 1.39 0.67 8.66 3.43 0.00 -9.8118.00 30.53 1.47 0.71 8.66 3.33 0.00 -9.81

19.00 30.53 1.56 0.74 8.66 3.23 0.00 -9.81

20.00 30.53 1.65 0.77 8.66 3.13 0.00 -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

Input Data

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 2.00 4.00 6.00 8.00 10.00



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Assignment 1

Catapult Dynamics Design Tool using Excel

Individual tutorial exercise

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 submittedindividually no copying!

Sandeep K Sharmah i i i l

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Dynamics 1e10 1no Drag -Sks