You Too Can Be james Bond 263 You too Can Be james bond Deng X 8 Xu W Zhang Zhenyu Southeast university Advisor: Chen enhui Abstract We divide the jump into three phases: flying through the air, punching through the stack, and landing on the ground. We construct four models to minimize the number and the cost of boxes In the Ideal Mechanical model, we consider the boxes'force on the motorcycle and stunt person as constant. In the Realistic Mechanical model, we focus on how the boxes support the motorcycle and stunt person, which includes three phases: elastic deformation, plastic deformation, and and crush-down deforma- tion. However, in the Ideal Air Box model, the internal air pressure of each box an' t be ignored. As a matter of fact, the boxes are unsealed, so we amend the Ideal Air Box model to develop a Realistic Air Box model. We discuss the strengths and weaknesses of each model We define a metric U, which is a function of the cost and the number of boxe By mathematical programming, we calculate the size and the number of the boxes In normal conditions, we that assume the safe speed is 5.42 m/s. Fora total weight of stunt person and motorcycle of 187 kg, we need 196 boxes of size 0.7m x07m x 0.5 m. We analyze the accuracy and sensitivity of the result to such factors as the total weight, the contact area, and the velocity. We also offer some important uggestions on how to pile up the boxes and how to change the shape of the Assumptions and Analysis about boxes and the pile of boxes All the boxes are the same size. The ratio of length to width has little effect on the compression strength of a cardboard box; so to simplify the problem, we assume a square cross section The UMAP Journal 24(3)(2003)263-280. Copyright 2003 by COMAP, Inc. Allrights reserved Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial dvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
You Too Can Be James Bond 263 You Too Can Be James Bond Deng Xiaowei Xu Wei Zhang Zhenyu Southeast University Nanjing, China Advisor: Chen Enshui Abstract We divide the jump into three phases: flying through the air, punching through the stack, and landing on the ground. We construct four models to minimize the number and the cost of boxes. In the Ideal Mechanical model, we consider the boxes’ force on the motorcycle and stunt person as constant. In the Realistic Mechanical model, we focus on how the boxes support the motorcycle and stunt person, which includes three phases: elastic deformation, plastic deformation, and and crush-down deformation. However, in the Ideal Air Box model, the internal air pressure of each box can’t be ignored. As a matter of fact, the boxes are unsealed, so we amend the Ideal Air Box model to develop a Realistic Air Box model. We discuss the strengths and weaknesses of each model. We define a metric U, which is a function of the cost and the number of boxes. By mathematical programming, we calculate the size and the number of the boxes. In normal conditions, we that assume the safe speed is 5.42 m/s. For a total weight of stunt person and motorcycle of 187 kg, we need 196 boxes of size 0.7 m × 0.7 m × 0.5 m. We analyze the accuracy and sensitivity of the result to such factors as the total weight, the contact area, and the velocity. We also offer some important suggestions on how to pile up the boxes and how to change the shape of the boxes. Assumptions and Analysis About Boxes and the Pile of Boxes • All the boxes are the same size. The ratio of length to width has little effect on the compression strength of a cardboard box; so to simplify the problem, we assume a square cross section. The UMAP Journal 24 (3) (2003) 263–280. c Copyright 2003 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
264 The UMAP Journal 24.3 (2003) Table 1 Variables, parameters, and physical constant Notation Description Units h height of box zA perimeter around top of bo total surface area of box pressure in box(cylinder) PPkmm pressure in box at time t when it collapses pressure in box at time t, in atmospheres atm rate of air leaking from box m3/s volume of box(cylinder) volume of box in time interval i Pile length of the pile m width of the pile height of the pile L number of layers of boxes in the pile total number of boxes ost of boxes S upper surface area of pile Jump Elephant average height of the elephant Hm naximum height of the jump 4 m fraction of Hmax that a person can reach gle of the ramp launch speed /s safe speed at which to hit the ground mass of motor plus stunt person Kellicut formula F compressive strength of the box P comprehensive annular compressive strength of the pape corrugation constant circumference of the top surface of the box J box shape coefficient Fo maximum supporting force from the box F buffering force of the box N constant concerning the properties of paper Other distance that the cylinder is compressed linder displacement when box collapses ompression distance at which box collapses dt interval of time W )s, k1, k2 quantities related to cost 入h,入 weight factors T, functions to be optimized Constants acceleration due to gravity, at Earths surface standard atmospheric ressure at Earth's surface Pa
264 The UMAP Journal 24.3 (2003) Table 1. Variables, parameters, and physical constants. Notation Description Units Box h height of box m r length of side of box m Z perimeter around top of box m A0 total surface area of box m2 P pressure in box (cylinder) Pa Pt pressure in box at time t when it collapses Pa k pressure in box at time t, in atmospheres atm σ rate of air leaking from box m3/s σi rate of air leaking from box in time interval i m3/s V volume of box (cylinder) m3 Vi volume of box in time interval i m3 Pile Lpile length of the pile m Wpile width of the pile m Hpile height of the pile m L number of layers of boxes in the pile Num total number of boxes Cost cost of boxes S upper surface area of pile m2 Jump Helephant average height of the elephant m Hmax maximum height of the jump 4 m ν fraction of Hmax that a person can reach H0 height of the ramp m θ angle of the ramp v0 launch speed m/s vsafe safe speed at which to hit the ground m/s M mass of motor plus stunt person kg Kellicut formula P compressive strength of the box Px comprehensive annular compressive strength of the paper dx2 corrugation constant Z circumference of the top surface of the box m J box shape coefficient F0 maximum supporting force from the box N F buffering force of the box N b constant concerning the properties of paper Other x, z distance that the cylinder is compressed m xt cylinder displacement when box collapses m zm compression distance at which box collapses m dt interval of time s W work done J Ds, k1, k2 quantities related to cost λh, λc weight factors T, U functions to be optimized Constants g acceleration due to gravity, at Earth’s surface m/s2 P0 standard atmospheric pressure at Earth’s surface Pa
You Too Can Be james Bond 265 After the box has been crushed down to some extent, we ignore the support- ing force that it can still supply Considering the practical production and transport limitations, the size of the box should not be too large The boxes are piled together in the shape of a rectangular solid When the motorcycle impacts one layer, it has little effect on the next layer The layer below is considered to be rigid flat(its displacement is ignored) We ignore the weight of the boxes; they are much lighter than the person plus motorcycle About the Stunt Person and the motorcycle We ignore the resistance of the air to the horizontal velocity of the person and motorcycle. The friction is so little that it is negligible The stunt person has received professional training, is skilled, and is equipped nything allowable for The average weight of a stunt person is 70 kg We choose a certain type of motorcycle(e.g, Yamaha 2003 TT-R225), which weighs 259 Ib [Yamaha Motor Corp. 2003 About the elephant The elephant keeps calm during the jump We adopt the classic value of 3. 5 m for the height of the elephant [PBS Online n.d. about the weather The weather is fine for filming and jumping, including appropriate tem- perature and humidity. On a gusty day, the wind might make the person lose balance in the air about the Camera The most attractive moment is when the person is over the elephant and maximum height Hmax. We have to make sure that the boxes do not appear on camera, namely, we need Pile vHmax, where the coefficient v is best determined empirically In our model, we set v=0625
You Too Can Be James Bond 265 • After the box has been crushed down to some extent, we ignore the supporting force that it can still supply. • Considering the practical production and transport limitations, the size of the box should not be too large. • The boxes are piled together in the shape of a rectangular solid. • When the motorcycle impacts one layer, it has little effect on the next layer. The layer below is considered to be rigid flat (its displacement is ignored). • We ignore the weight of the boxes; they are much lighter than the person plus motorcycle. About the Stunt Person and the Motorcycle • We ignore the resistance of the air to the horizontal velocity of the person and motorcycle. The friction is so little that it is negligible. • The stunt person has received professional training, is skilled, and is equipped with anything allowable for protection. • The average weight of a stunt person is 70 kg. • We choose a certain type of motorcycle (e.g., Yamaha 2003 TT-R225), which weighs 259 lb [Yamaha Motor Corp. 2003]. About the Elephant • The elephant keeps calm during the jump. • We adopt the classic value of 3.5 m for the height of the elephant [PBS Online n.d.]. About the Weather The weather is fine for filming and jumping, including appropriate temperature and humidity. On a gusty day, the wind might make the person lose balance in the air. About the Camera The most attractive moment is when the person is over the elephant and at maximum height Hmax. We have to make sure that the boxes do not appear on camera, namely, we need Hpile ≤ νHmax, where the coefficient ν is best determined empirically. In our model, we set ν = 0.625
266 The UMAP Journal 24.3(2003) About the ramp for Jumping The ramp for jumping is a slope at angle e of length Slope, as determined by the height or horizontal distance for landin The Development of models When the stunt person begins to contact the cardboard boxes or the ground, he or she may suffer great shock. To absorb the momentum, the contact time must be extended We divide the whole process into three independent phases, that is 1. flying through the air, 2. punching through the stack, and 3. landing on the ground find out the maximum height in phase 1, the greatest speed of hitting the ground in phase 3, and how the person plus motorcycle interact with the boxes is based on the results of phases 1 and 3. Phases 1 and 3 are simple and we solve them first Flying through the air The stunt person leaves the ramp with initial speed vo at angle g to the horizontal at height Ho(Figure 1) Initial Person a The Height of Elepl A Pile of Cardboard Boxe gure 1. The jum Q In the air, the stunt person on the motorcycle is affected only by the constant eleration of gravity. Based on Newtons Second Law, we have (t)=(vo cos 8)t, t)=(vo sin 0)t -igt
266 The UMAP Journal 24.3 (2003) About the Ramp for Jumping The ramp for jumping is a slope at angle θ of length Lslope, as determined by the jump height or horizontal distance for landing. The Development of Models When the stunt person begins to contact the cardboard boxes or the ground, he or she may suffer great shock. To absorb the momentum, the contact time must be extended. We divide the whole process into three independent phases, that is: 1. flying through the air, 2. punching through the stack, and 3. landing on the ground. We find out the maximum height in phase 1, the greatest speed of hitting the ground in phase 3, and how the person plus motorcycle interact with the boxes is based on the results of phases 1 and 3. Phases 1 and 3 are simple and we solve them first. Flying through the Air The stunt person leaves the ramp with initial speed v0 at angle θ to the horizontal at height H0 (Figure 1). Figure 1. The jump. In the air, the stunt person on the motorcycle is affected only by the constant acceleration of gravity. Based on Newton’s Second Law, we have x(t)=(v0 cos θ)t, y(t)=(v0 sin θ)t − 1 2 gt2
You Too Can Be james Bond 267 where a(t)and y(t)are the horizontal and vertical displacements from the launch point after t seconds. The launch speed vo and the maximum height h g(Hmax-Ho) For an elephant of height 3.5 m, we take Hmax =4 m. For Ho=0.5 m and 0=30, we get vo=v2.9.8.3. 5/0.5=16.6 m/s With a 2 m-high box-pile the stunt person hits the pile with vertical speed 6.3 m/s and horizontal speed 14.3 m/s; the distance between the landing point and elephant is D=9.2m Would the landing be safe? To simplify the problem, we ignore the complex process when the person begins to touch the ground. We consider that there is a critical safe speed usafe If the speed hitting the ground is less than or equal to that speed the person would not be injured. The safe speed is related to the ground surface(hard, grassplot, mud, etc )and materials used(paper, rubber etc. ) Our simulation uses a typical valu 5.42m/ Is the Pile area Large Enough? The height Hile of the pile of boxes is related to the maximum height Hmax that the stunt person reaches and also to the vertical speed of hitting the boxes The greater Hmax, the greater Pile is required, with Pile- Lh, where L is the number of the layers of boxes and h is the height of a single box Would Lpile equal to the length of the person be enough? The answer is no When accelerating on the ramp, the stunt person cant make the initial jump speed exactly what we calculate. We think that 3-5 times the length of the person is needed The stunt person does not leave the ramp aligned exactly along the central axis and does not keep the motorcycle exactly along that axis after hitting the boxes. That there may be some horizontal movement means that Pile should be 24 times the length of the person n our simulation we let Lpile=6 m, Pile=4 m Boxes: How to Cushion the person Ideal mechanical model is destroying the eneral assumptions, we suppose that while the stunt person boxes of the current layer, boxes in lower layers are seldom affected and ke
You Too Can Be James Bond 267 where x(t) and y(t) are the horizontal and vertical displacements from the launch point after t seconds. The launch speed v0 and the maximum height Hmax are related by v0 sin θ = 2g(Hmax − H0). For an elephant of height 3.5 m, we take Hmax = 4 m. For H0 = 0.5 m and θ = 30◦, we get v0 = √2 · 9.8 · 3.5/0.5 = 16.6 m/s. With a 2 m-high box-pile, the stunt person hits the pile with vertical speed 6.3 m/s and horizontal speed 14.3 m/s; the distance between the landing point and elephant is D = 9.2 m. Would the Landing Be Safe? To simplify the problem, we ignore the complex process when the person begins to touch the ground. We consider that there is a critical safe speed vsafe. If the speed hitting the ground is less than or equal to that speed, the person would not be injured. The safe speed is related to the ground surface (hard, grassplot, mud, etc.) and materials used (paper, rubber etc.). Our simulation uses a typical value, vsafe = 5.42 m/s. Is the Pile Area Large Enough? The height Hpile of the pile of boxes is related to the maximum height Hmax that the stunt person reaches and also to the vertical speed of hitting the boxes. The greater Hmax, the greater Hpile is required, with Hpile = Lh, where L is the number of the layers of boxes and h is the height of a single box. Would Lpile equal to the length of the person be enough? The answer is no. When accelerating on the ramp, the stunt person can’t make the initial jump speed exactly what we calculate. We think that 3–5 times the length of the person is needed. The stunt person does not leave the ramp aligned exactly alomg the central axis and does not keep the motorcycle exactly along that axis after hitting the boxes. That there may be some horizontal movement means that Wpile should be 2–4 times the length of the person. In our simulation, we let Lpile = 6 m, Wpile = 4 m. Boxes: How to Cushion the Person Ideal Mechanical Model Based on our general assumptions, we suppose that while the stunt person is destroying the boxes of the current layer, boxes in lower layers are seldom affected and keep still