1E10 Lecture in Design Mechanical & Manufacturing Engineering
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Transcript of 1E10 Lecture in Design Mechanical & Manufacturing Engineering
Dr. Gareth J. BennettTrinity College Dublin
Page 1
1E10 Lecture in Design1E10 Lecture in DesignMechanical & Manufacturing EngineeringMechanical & Manufacturing Engineering
“Dynamics for the Mangonel”. Dr. Gareth J. Bennett
Dr. Gareth J. BennettTrinity College Dublin
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ObjectiveObjective
A small model Mangonel
Dr. Gareth J. BennettTrinity College Dublin
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ObjectiveObjective
Can we predict the distance?
Dr. Gareth J. BennettTrinity College Dublin
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ObjectiveObjective
A larger version!
Dr. Gareth J. BennettTrinity College Dublin
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ObjectiveObjective
What are the factors that control the distance? (The dynamics)
Dr. Gareth J. BennettTrinity College Dublin
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ModellingModelling
Bigger means further?Some of the issues related to
scaling up are discussed in Prof. Fitzpatrick’s lecture!
(Reflect on these)
Dr. Gareth J. BennettTrinity College Dublin
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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?
Dr. Gareth J. BennettTrinity College Dublin
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FundamentalsFundamentals
force = mass x acceleration (ma)
work = force x distance (Fs) energy== work power = rate of work
(work/time)
Dr. Gareth J. BennettTrinity College Dublin
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Derived Units
Force (1N=1kgm/s2) Work (1J=1Nm=1kgm2/s2) Energy (J) Power (1W=1J/s)
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
Starting with some basic equations
Speedav=distance/time
Accelerationav=velocity/time
Dr. Gareth J. BennettTrinity College Dublin
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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
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
Example 1: (1-D)Kick a ball straight up. Given
a given initial velocity, how high will it go?
Dr. Gareth J. BennettTrinity College Dublin
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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=?
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
Example 2: (1-D)Drop a rock from a cliff. How
long will it take to hit the ground/sea?
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
Example 2: (1-D)
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
Example 2: (1-D)
Use equation:s=ut+1/2at2
s=1/2at2
t (from stopwatch)
u=0 (at top)
s=?
Dr. Gareth J. BennettTrinity College Dublin
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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 to square of t!
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
Can we use these equations to model the trajectory of the missile?
And hence predict the distance?A 2-D problem!
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
y
x
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
y
x
Discretise the curve
1
2
3 4
s
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
y
x
Not u and v now but
v1, v2, v3, v4, etc…..
1
2
3 4
v1
v2
v3v4
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
y
x
We can decompose vectors (v, s, a) into x, y components
1
2
3 4
s1x
s1s1y
Dr. Gareth J. BennettTrinity College Dublin
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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)
Dr. Gareth J. BennettTrinity College Dublin
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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)
Dr. Gareth J. BennettTrinity College Dublin
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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
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
x2=x1+vx1Δt+1/2ax1Δt2
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
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
y2=y1+vy1Δt+1/2ay1Δt2
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
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
vx2=vx1+ax1Δt
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
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
vy2=vy1+ay1Δt
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
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
Const=0!
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
Dr. Gareth J. BennettTrinity College Dublin
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Dynamics
Const=-g
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
Dr. Gareth J. BennettTrinity College Dublin
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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
Dr. Gareth J. BennettTrinity College Dublin
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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
Dr. Gareth J. BennettTrinity College Dublin
Page 34
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!