1E10 Lecture in Design Mechanical & Manufacturing Engineering

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Presentation transcript:

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

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

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

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

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

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

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

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

Derived Units Force (1N=1kgm/s2) Work (1J=1Nm=1kgm2/s2) Energy (J) Power (1W=1J/s) Dr. Gareth J. Bennett Trinity College Dublin

Dynamics Starting with some basic equations Speedav=distance/time Accelerationav=velocity/time Dr. Gareth J. Bennett Trinity College Dublin

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Dynamics x2=x1+vx1Δt+1/2ax1Δt2 Dr. Gareth J. Bennett Trinity College Dublin

Dynamics y2=y1+vy1Δt+1/2ay1Δt2 Dr. Gareth J. Bennett Trinity College Dublin

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

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

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

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

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

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

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