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Page 1 Dr. Gareth J. Bennett Trinity College Dublin 1E10 Lecture in Design Mechanical & Manufacturing Engineering “Dynamics for the Mangonel”. Dr. Gareth J. Bennett

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Page 2 Dr. Gareth J. Bennett Trinity College Dublin Objective A small model Mangonel

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Page 3 Dr. Gareth J. Bennett Trinity College Dublin Objective Can we predict the distance?

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Page 4 Dr. Gareth J. Bennett Trinity College Dublin Objective A larger version!

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Page 5 Dr. Gareth J. Bennett Trinity College Dublin Objective What are the factors that control the distance? (The dynamics)

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Page 6 Dr. Gareth J. Bennett Trinity College Dublin Modelling Bigger means further? Some of the issues related to scaling up are discussed in Prof. Fitzpatrick’s lecture! (Reflect on these)

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Page 7 Dr. Gareth J. Bennett Trinity College Dublin 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?

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Page 8 Dr. Gareth J. Bennett Trinity College Dublin Fundamentals force = mass x acceleration (ma) work = force x distance (Fs) energy== work power = rate of work (work/time)

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Page 9 Dr. Gareth J. Bennett Trinity College Dublin Derived Units Force (1N=1kgm/s 2 ) Work (1J=1Nm=1kgm 2 /s 2 ) Energy (J) Power (1W=1J/s)

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Page 10 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Starting with some basic equations Speed av =distance/time Acceleration av =velocity/time

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Page 11 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Can derive equations for linear motion (for constant acceleration) v=u+at s=ut+1/2at 2 v 2 =u 2 +2as u=initial velocity v=final velocity t=time duration a=acceleration s=distance travelled

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Page 12 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Example 1: (1-D) Kick a ball straight up. Given a given initial velocity, how high will it go?

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Page 13 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Example 1: (1-D) Use equation: v 2 =u 2 +2as s=u 2 /2g a=-g u v=0 (at top) s=?

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Page 14 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Example 2: (1-D) Drop a rock from a cliff. How long will it take to hit the ground/sea?

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Page 15 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Example 2: (1-D)

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Page 16 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Example 2: (1-D) Use equation: s=ut+1/2at 2 s=1/2at 2 t (from stopwatch) u=0 (at top) s=?

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Page 17 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Example 2: (1-D) s=1/2at 2 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!

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Page 18 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Can we use these equations to model the trajectory of the missile? And hence predict the distance? A 2-D problem!

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Page 19 Dr. Gareth J. Bennett Trinity College Dublin Dynamics y x

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Page 20 Dr. Gareth J. Bennett Trinity College Dublin Dynamics y x Discretise the curve 1 2 3 4 s

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Page 21 Dr. Gareth J. Bennett Trinity College Dublin Dynamics y x Not u and v now but v 1, v 2, v 3, v 4, etc….. 1 2 3 4 v1v1 v2v2 v3v3 v4v4

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Page 22 Dr. Gareth J. Bennett Trinity College Dublin Dynamics y x We can decompose vectors (v, s, a) into x, y components 1 2 3 4 s 1x s1s1 s 1y

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Page 23 Dr. Gareth J. Bennett Trinity College Dublin Dynamics v=u+at becomes: v x2 =v x1 +a x1 Δt v y2 =v y1 +a y1 Δt s=ut+1/2at 2 becomes: Δs x =v x1 Δt+1/2a x1 Δt 2 Δs y =v y1 Δt+1/2a y1 Δt 2 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)

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Page 24 Dr. Gareth J. Bennett Trinity College Dublin Dynamics – Assignment1 Use Excel to study trajectory of missile

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Page 25 Dr. Gareth J. Bennett Trinity College Dublin Dynamics t2=t1+ Δ t

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Page 26 Dr. Gareth J. Bennett Trinity College Dublin Dynamics x 2 =x 1 +v x1 Δ t+1/2a x1 Δ t 2

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Page 27 Dr. Gareth J. Bennett Trinity College Dublin Dynamics y 2 =y 1 +v y1 Δ t+1/2a y1 Δ t 2

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Page 28 Dr. Gareth J. Bennett Trinity College Dublin Dynamics v x2 =v x1 +a x1 Δ t

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Page 29 Dr. Gareth J. Bennett Trinity College Dublin Dynamics v y2 =v y1 +a y1 Δ t

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Page 30 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Const=0!

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Page 31 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Const=-g

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Page 32 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Copy formula down

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Page 33 Dr. Gareth J. Bennett Trinity College Dublin Dynamics Plot x versus y using chart wizard

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Page 34 Dr. Gareth J. Bennett Trinity College Dublin 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!

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