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**1E10 Lecture in Design Mechanical & Manufacturing Engineering**

“Dynamics for the Mangonel”. Dr. Gareth J. Bennett Dr. Gareth J. Bennett Trinity College Dublin

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**Objective A small model Mangonel Dr. Gareth J. Bennett**

Trinity College Dublin

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**Can we predict the distance?**

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

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**Objective A larger version! Dr. Gareth J. Bennett**

Trinity College Dublin

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

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

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**Modelling Bigger means further?**

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

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**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? Dr. Gareth J. Bennett Trinity College Dublin

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**Fundamentals force = mass x acceleration (ma)**

work = force x distance (Fs) energy== work power = rate of work (work/time) Dr. Gareth J. Bennett Trinity 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. Bennett Trinity College Dublin

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**Dynamics Starting with some basic equations Speedav=distance/time**

Accelerationav=velocity/time Dr. Gareth J. Bennett Trinity College Dublin

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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 Dr. Gareth J. Bennett Trinity 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. Bennett Trinity College Dublin

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**Dynamics v=0 (at top) Example 1: (1-D) Use equation: v2=u2+2as s=u2/2g**

a=-g s=? u Dr. Gareth J. Bennett Trinity 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. Bennett Trinity College Dublin

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

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**Dynamics u=0 (at top) Example 2: (1-D) Use equation: s=ut+1/2at2**

s=1/2at2 s=? t (from stopwatch) Dr. Gareth J. Bennett Trinity College Dublin

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**Dynamics Example 2: (1-D) s=1/2at2 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! t (from stopwatch) u=0 (at top) s=? Dr. Gareth J. Bennett Trinity 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. Bennett Trinity College Dublin

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

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**Dynamics s Discretise the curve 4 3 2 y 1 x Dr. Gareth J. Bennett**

Trinity College Dublin

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**Dynamics v4 v3 v2 v1 Not u and v now but v1, v2, v3, v4, etc….. 4 3 2**

x v1 Not u and v now but v1, v2, v3, v4, etc….. Dr. Gareth J. Bennett Trinity College Dublin

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

Dynamics 4 3 y 2 s1y s1 1 x s1x We can decompose vectors (v, s, a) into x, y components Dr. Gareth J. Bennett Trinity 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. Bennett Trinity College Dublin

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**Use Excel to study trajectory of missile**

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

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Dynamics t2=t1+Δt Dr. Gareth J. Bennett Trinity College Dublin

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**Dynamics x2=x1+vx1Δt+1/2ax1Δt2 Dr. Gareth J. Bennett**

Trinity College Dublin

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**Dynamics y2=y1+vy1Δt+1/2ay1Δt2 Dr. Gareth J. Bennett**

Trinity College Dublin

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Dynamics vx2=vx1+ax1Δt Dr. Gareth J. Bennett Trinity College Dublin

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Dynamics vy2=vy1+ay1Δt Dr. Gareth J. Bennett Trinity College Dublin

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

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

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**Dynamics Copy formula down Dr. Gareth J. Bennett**

Trinity College Dublin

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**Dynamics Plot x versus y using chart wizard Dr. Gareth J. Bennett**

Trinity College Dublin

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**Mangonel Dynamics Design Tool using Excel**

Assignment 1 Mangonel Dynamics Design Tool using Excel Work in groups and/or individually in computer rooms today and during week to Create excel spreadsheet as demonstrated Plot x versus y Study effect of changing velocity Study effect of changing angle An assignment will be set based on this work. Assignment to be submitted individually – no copying! Dr. Gareth J. Bennett Trinity College Dublin

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