page 1 dr. gareth j. bennett trinity college dublin 1e10 lecture in design mechanical &...

Dr. Gareth J. BennettTrinity College Dublin

1E10 Lecture in Design1E10 Lecture in DesignMechanical & Manufacturing EngineeringMechanical & Manufacturing Engineering

“Dynamics for the Mangonel”. Dr. Gareth J. Bennett


ObjectiveObjective

A small model Mangonel


ObjectiveObjective

Can we predict the distance?


ObjectiveObjective

A larger version!


ObjectiveObjective

What are the factors that control the distance? (The dynamics)


ModellingModelling

Bigger means further?Some of the issues related to

scaling up are discussed in Prof. Fitzpatrick’s lecture!

(Reflect on these)


ModellingModelling

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?


FundamentalsFundamentals

force = mass x acceleration (ma)

work = force x distance (Fs) energy== work power = rate of work

(work/time)


Derived Units

Force (1N=1kgm/s2) Work (1J=1Nm=1kgm2/s2) Energy (J) Power (1W=1J/s)


Dynamics

Starting with some basic equations

Speedav=distance/time

Accelerationav=velocity/time


Dynamics

Can derive equations for linear motion (for constant acceleration)

v=u+ats=ut+1/2at2

v2=u2+2as

u=initial velocity

v=final velocity

t=time duration

a=acceleration

s=distance travelled


Dynamics

Example 1: (1-D)Kick a ball straight up. Given

a given initial velocity, how high will it go?


Dynamics

Example 1: (1-D)

Use equation:v2=u2+2as

s=u2/2g

a=-g

u

v=0 (at top)

s=?


Dynamics

Example 2: (1-D)Drop a rock from a cliff. How

long will it take to hit the ground/sea?


Dynamics

Example 2: (1-D)


Dynamics

Example 2: (1-D)

Use equation:s=ut+1/2at2

s=1/2at2

t (from stopwatch)

u=0 (at top)

s=?


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 to square of t!


Dynamics

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

And hence predict the distance?A 2-D problem!


Dynamics

y

x


Dynamics

y

x

Discretise the curve

1

2

3 4

s


Dynamics

y

x

Not u and v now but

v1, v2, v3, v4, etc…..

1

2

3 4

v1

v2

v3v4


Dynamics

y

x

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

1

2

3 4

s1x

s1s1y


Dynamics

v=u+at becomes:

•vx2=vx1+ax1Δt

•vy2=vy1+ay1Δt

s=ut+1/2at2 becomes:

•Δsx=vx1Δt+1/2ax1Δt2

•Δsy=vy1Δt+1/2ay1Δt2

Acceleration is constant (for no drag of lift – we’ll return to this point later)

ax=0!

ay=-g

t2-t1= Δt (keep time interval constant)


Dynamics – Assignment1

Use Excel to study trajectory of missile

Position t x y vx vy ax ayVel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01theta (degrees) 30.00theta (radians) 0.52

Input Data

Initial Conditions

vx=Vel*cos(theta)

vy=Vel*sin(theta)

Position t x y vx vy ax ayVel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01theta (degrees) 30.00theta (radians) 0.52

Input Data

Initial Conditions

vx=Vel*cos(theta)

vy=Vel*sin(theta)


Dynamics

t2=t1+

Δt

Position t x y vx vy ax ayVel 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 -9.81theta (degrees) 30.00theta (radians) 0.52

Input Data


Dynamics

x2=x1+vx1Δt+1/2ax1Δt2


Input Data


Dynamics

y2=y1+vy1Δt+1/2ay1Δt2


Input Data


Dynamics

vx2=vx1+ax1Δt


Input Data


Dynamics

vy2=vy1+ay1Δt


Input Data


Dynamics

Const=0!


Input Data


Dynamics

Const=-g


Input Data


Dynamics

Copy formula down

Position t x y vx vy ax ayVel 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 -9.81theta (degrees) 30.00 3.00 0.02 0.17 0.10 8.66 4.80 0.00 -9.81theta (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 -9.816.00 0.05 0.43 0.24 8.66 4.51 0.00 -9.817.00 0.06 0.52 0.28 8.66 4.41 0.00 -9.818.00 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 -9.8111.00 0.10 0.87 0.45 8.66 4.02 0.00 -9.8112.00 0.11 0.95 0.49 8.66 3.92 0.00 -9.8113.00 0.12 1.04 0.53 8.66 3.82 0.00 -9.8114.00 0.13 1.13 0.57 8.66 3.72 0.00 -9.8115.00 0.14 1.21 0.60 8.66 3.62 0.00 -9.8116.00 0.15 1.30 0.64 8.66 3.53 0.00 -9.8117.00 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 -9.8119.00 0.18 1.56 0.74 8.66 3.23 0.00 -9.81

Input Data


Dynamics

Plot x versus y using chart wizard

Position t x y vx vy ax ayVel 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 -9.81theta (degrees) 30.00 3.00 30.01 0.17 0.10 8.66 4.80 0.00 -9.81theta (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 -9.816.00 30.53 0.43 0.24 8.66 4.51 0.00 -9.817.00 30.53 0.52 0.28 8.66 4.41 0.00 -9.818.00 30.53 0.61 0.33 8.66 4.31 0.00 -9.819.00 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 -9.8111.00 30.53 0.87 0.45 8.66 4.02 0.00 -9.8112.00 30.53 0.95 0.49 8.66 3.92 0.00 -9.8113.00 30.53 1.04 0.53 8.66 3.82 0.00 -9.8114.00 30.53 1.13 0.57 8.66 3.72 0.00 -9.8115.00 30.53 1.21 0.60 8.66 3.62 0.00 -9.8116.00 30.53 1.30 0.64 8.66 3.53 0.00 -9.8117.00 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 -9.8119.00 30.53 1.56 0.74 8.66 3.23 0.00 -9.8120.00 30.53 1.65 0.77 8.66 3.13 0.00 -9.8121.00 30.53 1.73 0.80 8.66 3.04 0.00 -9.8122.00 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


Assignment 1

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!

page 1 dr. gareth j. bennett trinity college dublin 1e10 lecture in design mechanical &...

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