1、Thinking Outside the Box251Thinking Outside the Box and Over the ElephantMelissa J. Banister Matthew Macauley Micah J. Smukler Harvey Mudd College Claremont, CAAdvisor: Jon JacobsenAbstractWe present a mathematical model of the collapsing of a box structure, which is to be used to protect a stunt mo

2、torcyclist who jumps over an elephant. Reasonable values of the models parameters cause it to predict that we should construct the structure out of fty 6 in 28 in 28 in boxes, stacked ve high, two wide, and ve long. In general, the model predicts that we should use boxes whose height is one-quarter

3、of the harmonic mean of their length and width. We discuss the assumptions, derivation, and limitations of this model.IntroductionA stunt motorcyclist jumps over an elephant; we use cardboard boxes to cushion the landing. Our goal is to determine how to arrange the boxes to protect the motorcyclist.

4、 We determine how many boxes to use, the size of the boxes. the arrangement of the boxes, and any modications to the boxes.In addition, our model must accommodate motorcyclists jumping from dif- ferent heights and on motorcycles of different weights. Our goal is to reduce the impulse at landing, thu

5、s essentially simulating a much lower jump (of which we assume the rider is capable).The UMAP Journal 24 (3) (2003) 251262. 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

6、provided that copies are not made or distributed for prot 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

7、redistribute to lists requires prior permission from COMAP.更多數(shù)學(xué)建模資料請(qǐng)關(guān)注微店店鋪“數(shù)學(xué)建模學(xué)習(xí)交流”/RHO6PSpA252The UMAP Journal24.3 (2003)Figure 1. The landing platform (graphic from JustIn Designs 2001).As the rider breaks through the top layer of boxes, crashing through card- board at a high

8、horizontal speed, it will be difcult to maintain balance. It is too dangerous to rely on the cardboard to stop the horizontal motion of the rider, unless we use such a large pile that keeping it from being visible to the camera would be nearly impossible.We are faced with how to cushion the riders l

9、anding without creating merely a pit of boxes. Imagine jumping from a 10-ft roof. If the jumper lands on a large wooden platform resting on a deep foam pit, the risk for injury is much less; the foam spreads out the jumpers deceleration over a much longer time, simulating a much lower jumping height

10、.Our goal is to create a landing platform for the motorcyclist that behaves much like the wooden platform on the foam pit. We simulate the foam pit by stacks of boxes. Our “platform” is constructed from boxes unfolded into cardboard ats and placed in a layer on top of the “pit” (Figure 1). The idea

11、is that the motorcyclist should never break through this layer of ats but should merely break the boxes underneath it.Safety Considerations Once the motorcycle has landed on the stack of cardboard boxes, its decel- eration should to be as uniform as possible as the structure collapsesthemore uniform

12、 the deceleration, the easier to maintain balance. We want the platform to remain as level and rigid as possible. If it is not level, the rider may lose balance; it it is insufciently rigid, it may bend andcollapse into the pile of boxes.Terminology The ute type of a gauge of cardboard refers to its

13、 corrugated core structure. Three sheets of linerboard compose one sheet of corrugated cardboard;themiddle one is shaped into utes, or waves, by a machine, and then the outer two sheets are glued on either side of it. For example, C-ute corrugated cardboard is the most common form Mall City Containe

14、rs n.d.Thinking Outside the Box253The edge crush test (ECT) value of a gauge of cardboard is the force per unit length that must be applied along the edge before the edge breaks. We make extensive use of the concept of such a value; however, the actual numbers given for gauges of cardboard apply to

15、ideal situations that would not be replicated in the cases that we are considering Boxland Online 1999.The atwise compression test (FCT) value of a gauge of cardboard is the pressure that must be applied to collapse it. It does not directly correlate to the stress placed on boxes in practice and the

16、refore is not used as an industry standard Pug et al. 2000.The bursting strength of a gauge of cardboard is the amount of air pressure required to break a sample. Because our model is concerned mainly with the strength of a box edge, but the bursting strength more accurately measures the strength of

17、 a face, we do not make use of it Boxland Online 1999.The stacking weight of a box is the weight that can be applied uniformly to the top of a box without crushing it. In general, the stacking weight of a box is smaller than the ECT or bursting strength, because it takes into account the structural

18、weaknesses of the particular box. We derive most of our numerical values for box strength from manufacturers specied stacking weights Bankers Box 2003.AssumptionsThe force exerted on every layer beneath the top platform is horizontally uniform. The force that the motorcycle exerts on the top platfor

19、m is concentrated where the wheels touch. Ideally, however, this top platform is perfectly at and rigid, so it distributes the force evenly to all lower layers. We approach this ideal by adding additional ats to the top platform.The stacking weight of a box is proportional to its perimeter and inver

20、sely propor- tional to its height. This assumption is physically reasonable, because weight on a box is supported by the edges and because the material in a taller box on average is farther from the boxs points of stability than in a shorter box. This claim can be veried from data Clean Sweep Supply

21、 2002.Nearly all of the work done to crush a box is used to initially damage its structural integrity. After the structure of a box is damaged, the remaining compression follows much more easily; indeed, we suppose it to be negligible. We denote by d the distance through which this initial work is d

22、one and assume for simplicity that the work is done uniformly throughout d. Through roughexperiments performed in our workroom, we nd that d 0.03 m. We assume that this value is constant but also discuss the effect of making it afunction of the size of box and of the speed of the crushing object.254

23、The UMAP Journal24.3 (2003)Figure 2. Tire before and after landing, acting as a shock absorber.When the motorcycle lands, we ignore the effects of any shock absorbers and assume that the motorcyclist does not shift position to cushion the fall. This is a worst- case scenario. To calculate how much f

24、orce the tires experience per unit area, we consider a standard 19-inch tire of height 90 mm and width 120 mm Kawasaki 2002. It compresses no less than 50 mm (Figure 2). A simple geometry calculation then tells us that the surface area of the tire touching the platform is approximately 3000 mm2. We

25、assume that the force exerted on the motorcycle on landing is uniformly distributed over this area.The pressure required to compress a stack of cardboard ats completely is the sum of the pressures required to compress each individual at.In a uniformly layered stack of boxes, each layer collapses com

26、pletely before the layer beneath it begins to collapse. This is probably an oversimplication; however, it is reasonable to suppose that the motorcycle is falling nearly as fast as the force that it is transmitting.The motorcyclist can easily land a jump 0.25 m high.The ModelCrushing an Individual Bo

27、xFor a cardboard box of height h, width w, and length l, by the assumptions made above, the stacking weight S isS(h, l, w)= k(l + w) ,hwhere k is a constant (with units of force).Once a box is compressed by a small amount, its spine breaks and very little additional force is required to atten it. Th

28、us, most of the force that the box exerts on the bike is done over the distance d h, and we assume that the work is done uniformly over this distance; this work isk(l + w)d.W = (force)(distance) =hThinking Outside the Box255Crushing a Layer of BoxesTo ensure that a layer of boxes collapses uniformly

29、, we build it out of nidentical boxes. The total amount of work required to crush such a layer isWT = n k(l + w)d.hOnce the structure starts to collapse, we want the rider to maintain a roughly constant average deceleration g over each layer. It follows that the layer should do total work m(g + g)h,

30、 soWT = nkd(l + w) = m(g + g )h.(1)hDene A = nlw to be the cross-sectional area of a layer of boxes; rearranging(1) producesAdkh2 lwB =.(2)m(g + g )l + wThe constant B gives a necessary relationship among the dimensions of the box if we wish to maintain constant deceleration throughout the collision

31、.Finally, we would like to minimize the total amount of material, subject to the above constraint. To do so, consider the efciency of a layer with a given composition of boxes to be the ratio of amount of work done to amount of material used. If the motorcyclist peaks at a height h0, we must do work

32、 mgh0 to stop the motorcycle. We minimize the total material needed by maximizing the efciency of each layer.The amount of material in a box is roughly proportional to its surface area, S = 2(hl + lw + wh). Thus the amount of material used by the layer is pro- portional to nS = 2n(hl + lw + wh). It

33、follows that the efciency E of a layer composed of boxes of dimensions h l w isnkd(l + w)kd(l + w)WTE =.nS2nh(hl + lw + wh)2h(hl + lw + hw)We maximize E for each layer, subject to the constraint (2). The calculations are easier if we minimize 1/E. Neglecting constant factors, weminimizehl + wf(h,l,w

34、 )= (hl + hw + lw)subject to the constrainth2 lw l + w= B,where B is the constant dened in (2). However, as long as we are obeying this constraint (that each layer does the same total work), we can writeB, hlw l + wf (h, l, w) = h2 += h2 +h256The UMAP Journal24.3 (2003)and thus f depends only on h.

35、The function f is minimized at Adk.B2=h =33(3)2m(g + g )At this value of h, the constraint reduces to3 lw B=4B.l + wh2This implies that the harmonic mean of l and w should be3 2lw l + w= 24B = 4h.H So, in the optimal situation, the box should be roughly four times as long and wide as it is tall. How

36、ever, there are other considerations. For commercially available boxes, we must choose among some discrete set of realizable box dimensions. That the number n of boxes in a layer must be an integer affects the possible values of many of the parameters on which h is based (most notably A, thecross-se

37、ctional area of the layer). We select the box among the potential candidates which most nearly compensates for this change in parameters.The Entire StructureWe need to determine the gross parameters of the entire structure. We determine the cross-sectional area of the structure by considering how mu

38、ch space the motorcyclist needs to land safely. The motorcycle has length about 1 m; we should leave at least this much space perpendicular to the direction of travel. We need substantially more in the direction of travel, to ensure that the motorcyclist can land safely with a reasonable margin of e

39、rror. So we let the structure be 1 m wide and 3 m long, for a cross-sectional area of 3 m2.How many corrugated cardboard ats should we use? We do not want them all to crease under the weight of the motorcycle and rider, for then the motorcyclist could be thrown off balance. So we must determine how

40、much we can expect the ats to bend as a result of the force exerted by the motorcycle. To calculate the number of ats required, we use the atwise compression test (FCT) data in Pug et al. 2000 for C-ute cardboard. Though our goal is to prevent substantial creasing of the entire layer of ats, we note

41、 that if we have a reasonable number of ats, creasing the bottom at requires completely crushing a substantial area along most of the remaining layers. Less than 20% of this pressure is required to dimple a sheet of cardboard to the point where it may be creased. Since we assume that the pressure re

42、quired to crush the ats scales linearly with the number of ats, we nd the maximum pressure thatThinking Outside the Box257the motorcycle will ever exert on the ats and divide it by the FCT value for a single C-ute cardboard at order to obtain the total number of sheets needed. A brief examination of

43、 a piece of cardboard demonstrates that bending it in the direction parallel to the utes is much easier than bending it in the direction perpendicular to the utes. Hence, it would be risky to orient all ats in the same direction; if force is applied along a strip of the surface, all of the ats may e

44、asily give way and bend in succession. So, it would be wise to alternate the orientation of the ats in the stack. To make sure that we have the full strength in any direction, we use twice the number of ats calculated, alternating the direction of the utes of each at as we build the stack. Then, no

45、matter how the motorcycle is oriented when it lands on the stack, at least the requiredstrength exists in every direction.Finally, we determine the overall height h1 of the structure as follows. The motorcycle accelerates downward at a rate of g from the peak of its ight to thetop of the structure;

46、this distance is h0 h1 . It then decelerates constantly at arate of g until it reaches the ground, over a distance h1 . Since the motorcycle is at rest both at the apex of its ight and when it reaches the ground, the relation(h0 h1 )g = h1 gmust hold. It follows, then, that h0 g g + gh =.(4)1Finding

47、 the Desired DecelerationTo determine g , we need the height H of the jump. We discuss how to build the platform so that the rider does not experience a deceleration greater than for a 0.25-m jump.We assume that most of the cushion of the landing is in the compression of the tires, which compress x

48、= 5 cm. The vertical velocity of the rideron impact is 2gH. We also make the approximation that the motorcycleexperiences constant deceleration after its tires hit the ground. So the ridertravels at an initial speed of2gH and stops after 0.05 m. We determine the acceleration:v2 + v2 = 2ax,0f 22gH +0

49、 = 2a(0.05).Solving yields a = 20gH. That is, if the motorcyclist jumps to a height of 4 m, on landing the ground exerts a force on the motorcycle that feels like 80 times the force of gravity; if the jump were from 0.25 m, this force would be only 5mg. To simulate a 0.25-m fall, we should have the

50、motorcyclist decelerate at a rate of 5g.258The UMAP Journal24.3 (2003)Numerical ResultsWe now return to the question of determining the number of ats needed for the top layer. By Pug et al. 2000, the atwise compression test (FCT) resultfor C-ute board is 1.5 105 Pa. We expect the motorcyclist to exp

51、erience anacceleration of approximately 5g upon landing, distributed over a surface area of 3000 mm2 = 0.003 m2. We assume that the motorcycle has mass 100 kg Kawasaki 2002 and the rider has mass 60 kg. The pressure exerted on the cardboard is(160 kg)(5)(9.8 m/s2)0.003 m2P = 2.61 106 Pa.For the card

52、board at the bottom of the stack of ats to be bent signicantly,enough pressure must be applied to crush most of the cardboard above it. Thusa lower bound on the number of ats is (2.61 106 Pa)/(1.50 105 Pa) =17.4 = 18 ats. To be perfectly safe, we double this gure and cross-hatch theats; that is, we

53、want 36 ats in the top platform, the utes of which alternatein direction.Next, we calculate the total mass of cardboard for these ats. We assumethat the ats are 1 m 3 m. From Gilchrist et al. 1999, we know that the density of C-ute corrugated cardboard is 537 g/m2; we obtain a mass of 1.611 kg perat

54、, or about 60 kg for 36 ats, which is comparable to the weight of a second person. The thickness of a C-ute at is 4.4 mm Mall City Containers n.d.; with 36 ats, the height of the stack is 158.4 mm.We now plug some reasonable values into (3) and get a good approximation of the desired height of the b

55、oxes. Let the stacking weight constant k = 800 N; this is roughly the mean value found in Clean Sweep Supply 2002. These values, along with g = 5g, give an optimal h of roughly(3 m2)(0.05 m)(800 N) 2(220 kg) 9.8+ 5(9.8) m/s2h =3 0 17 m.So the harmonic mean of l and w must be on the order of 4h = 0.6

56、7 m.Converting these values into inches gives h = 6.5 in and a value of roughly26.5 in for the harmonic mean of l and w. The two commercially available box sizes that most closely approximate these values are 6 in 26 in 26 inand 6 in 28 in 28 in Uline Shipping Supplies 2002. Note that we must increa

57、se the cross-sectional area beyond what was calculated in order to keepthe number of boxes per layer an integer. Doing so increases the total value of B = Akd/m(g + g ); thus, we ideally want a somewhat larger box than calculated. Since we cannot increase h (any commercially available box of this rough shape and size has a height of 6 in), we increase l and w. This choice increases the amount by which the cross-sectional area is larger than previously calc