A Time-Independent model 233 A Time-Independent model of Box Safety for Stunt motorcyclists van corwin Sheel Ganatra Nikita rozenblyum Harvard University Cambridge, ma Advisor: Clifford h. Taubes Abstract We develop a knowledge of the workings of corrugated fiberboard and create an extensive time-independent model of motorcycle collision with one box, our Single-Box Model. We identify important factors in box-to-box and frictional interactions, as well as several extensions of the Single-Box Model. Taking into account such effects as cracking, buckling, and buckling under other boxes, we use the energy-dependent Dual-Impact Model to show that the pyramid"configuration of large 90-cm cubic boxes-a configuration of boxes in which every box is resting equally upon four others-is optimal for absorption of the most energy while maintaining a reasonable deceleration. We show how variations in height and weight affect the model and calculate a bound on the number of boxes needed General assumptions The temperature and weather are assumed to be"ideal conditions"-they do not affect the strength of the box The wind is negligible, because the combined weight of the motorcycle and the person is sufficiently large The ground on which the boxes are arranged is a rigid flat surface that can take any level of force All boxes are cubic, which makes for the greatest strength [Urbanik 19971 The UMAP Journal 24(3)(2003)233-250. 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
A Time-Independent Model 233 A Time-Independent Model of Box Safety for Stunt Motorcyclists Ivan Corwin Sheel Ganatra Nikita Rozenblyum Harvard University Cambridge, MA Advisor: Clifford H. Taubes Abstract We develop a knowledge of the workings of corrugated fiberboard and create an extensive time-independent model of motorcycle collision with one box, our Single-Box Model. We identify important factors in box-to-box and frictional interactions, as well as several extensions of the Single-Box Model. Taking into account such effects as cracking, buckling, and buckling under other boxes, we use the energy-dependent Dual-Impact Model to show that the “pyramid” configuration of large 90-cm cubic boxes—a configuration of boxes in which every box is resting equally upon four others—is optimal for absorption of the most energy while maintaining a reasonable deceleration. We show how variations in height and weight affect the model and calculate a bound on the number of boxes needed. General Assumptions • The temperature and weather are assumed to be “ideal conditions”—they do not affect the strength of the box. • The wind is negligible, because the combined weight of the motorcycle and the person is sufficiently large. • The ground on which the boxes are arranged is a rigid flat surface that can take any level of force. • All boxes are cubic, which makes for the greatest strength [Urbanik 1997]. The UMAP Journal 24 (3) (2003) 233–250. 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
234 The UMAP Journal 24.3 (2003) Table 1 Variables Variable Definition V(t) Velocity as a function of time(vo is velocity of impact At Energy change due to the top of the box kg-cm2/52 Energy change due to the buckling of the box g-cm ANET Total energy including gravity for top ACD-EDGE Energy absorbed by CD springs in modelled buckling kg-cm-/s AMD- EDGE Energy absorbed by MD springs in modelled buckling kg-cm/s NET Total depression of the top △ r Change in a distance am DOWn Component of edge's depression in the x-direction r(t) ownward displacement of the top of the box( F Final depression before top failure oL Offset from top center. Motorcycle and stuntman combined mass PcD Tensile strength in the cross-machine direction PMd Tensile strength in the machine direction kg/(s-cm) Maximum strength as measured with the Edge Crush Test Maximum strength as measured with the Mullen Test Force in the cross-machine direction kg-cm/st FCDMAx Maximum force in the cross-machine direction kg-cm Force in the machine direction FmDMAx Maximum force in the machine direction kg-cm/s FUP Net dampening force the box exerts on the motorcycle FUPMAX Maximum dampening force the box exerts on the motorcycle kg-cm/s Force box exerts on the frame kg-cm/s Maximum force the box exerts on the frame before yielding g-cm/s2 FNI Total force k Depression at which MD tensile strength is exceeded aCD Depression at which CD tensile strength is exceeded ECt Depression at which a buckle occurs Depression at which a puncture occurs Velocity in the r-direction cm/s Velocity in the y-direction cm/s Initial vele Initial velocity in the y-direction cm/s Vfr Final velocity in the x-direction cm/s Final velocity in the y-direction Mass of boxes displaced Energy in the r-direction kg-cm2/s2 Energy in the y-direction kg-cm/s Energy in the z-direction m2/s2 Distance
234 The UMAP Journal 24.3 (2003) Table 1. Variables. Variable Definition Units l Box edge length cm V (t) Velocity as a function of time (v0 is velocity of impact) cm/s AT Energy change due to the top of the box kg-cm2/s2 AB Energy change due to the buckling of the box kg-cm2/s2 ANET Total energy including gravity for top kg-cm2/s2 ACD-EDGE Energy absorbed by CD springs in modelled buckling kg-cm2/s2 AMD-EDGE Energy absorbed by MD springs in modelled buckling kg-cm2/s2 xNET Total depression of the top cm ∆x Change in a distance cm xDOWN Component of edge’s depression in the z-direction cm x(t) Downward displacement of the top of the box (x0 = 0) cm xF Final depression before top failure. cm δL Offset from top center. cm M Motorcycle and stuntman combined mass kg PCD Tensile strength in the cross-machine direction kg/(s2cm) PMD Tensile strength in the machine direction kg/(s2cm) PECT Maximum strength as measured with the Edge Crush Test kg/s2 PML Maximum strength as measured with the Mullen Test kg/(s2cm) FCD Force in the cross-machine direction kg-cm/s2 FCDMAX Maximum force in the cross-machine direction kg-cm/s2 FMD Force in the machine direction kg-cm/s2 FMDMAX Maximum force in the machine direction kg-cm/s2 FUP Net dampening force the box exerts on the motorcycle kg-cm/s2 FUPMAX Maximum dampening force the box exerts on the motorcycle kg-cm/s2 FECT Force box exerts on the frame kg-cm/s2 FECTMAX Maximum force the box exerts on the frame before yielding kg-cm/s2 FNET Total force kg-cm/s2 xMD Depression at which MD tensile strength is exceeded cm xCD Depression at which CD tensile strength is exceeded cm xECT Depression at which a buckle occurs cm xML Depression at which a puncture occurs cm Vx Velocity in the x-direction cm/s Vy Velocity in the y-direction cm/s Vix Initial velocity in the x-direction cm/s Viy Initial velocity in the y-direction cm/s Vfx Final velocity in the x-direction cm/s Vfy Final velocity in the y-direction cm/s MB Mass of boxes displaced kg t Time s Ax Energy in the x-direction kg-cm2/s2 Ay Energy in the y-direction kg-cm2/s2 Az Energy in the z-direction kg-cm2/s2 d Distance cm
A Time-Independent model 235 Table 2 Constants Variable Definition Value EcD Young s modulus in the cross-machine direction 80000kg/(2-cm) E Sum of Emd and ecD 3800kg/(2-cm) Tire rect length in the machine direction 7cm front, 10cm back PcD Tensile strength in the cross-Machine direction 2000kg/(s2-cm) PMD Tensile strength in the machine direction 2500kg/(s2-cm) PECT Max strength as measured with the edge crush test 10000kg/ PML Max strength as measured with the Mullen test 25000kg/(s2-cm) Gravitational Constant 980cm/s2 Coefficient of kinetic friction Cardboard thickness peed variation 200 cm/s Angle variation away from y-axis leaving the ramp Mass of rider and motorcycle vi Initial velocity leaving ramp 1500cm/s 6 np angle of elevation Definitions and Key terms Buckling is the process by which a stiff plane develops a crack due to a stress exceeding the yield stres Compressive strength is the maximum force per unit area that a material can withstand, under compression, prior to yielding Corrugation is the style found in cardboard of sinusoidal waves of liner paper sandwiched between inner and outer papers. We use boxes with the most common corrugation, C-flute corrugation(see below for definition of a flute Cracking is a resulting state when the tensile force exceeds the tensile strength Cross-machine direction is the direction perpendicular to the sinusoidal wave of the corrugation Depression is when, due to a force, a section of a side or edge moves down wards ECT is the acronym for a common test of strength, the edge crush test . Fiberboard is the formal name for cardboard Flute is a single wavelength of a sinusoidal wave between the inner and outer portion papers that extends throughout the length of the cardboard A C-flute has a height of 0. 35 cm and there are 138 flutes per meter[Packaging glossary n d I
A Time-Independent Model 235 Table 2. Constants. Variable Definition Value EMD Young’s modulus in the machine direction 3000000 kg/(s2-cm) ECD Young’s modulus in the cross-machine direction 800000 kg/(s2-cm) E Sum of EMD and ECD 3800000 kg/(s2-cm) LMD Tire rect. length in the machine direction 7cm front, 10cm back LCD Tire rect. length in the cross-machine direction 10 cm PCD Tensile strength in the cross-Machine direction 2000 kg/(s2-cm) PMD Tensile strength in the machine direction 2500 kg/(s2-cm) PECT Max. strength as measured with the edge crush test 10000 kg/s2 PML Max. strength as measured with the Mullen test 25000 kg/(s2-cm) g Gravitational Constant 980 cm/s2 µ Coefficient of kinetic friction 0.4 w Cardboard thickness 0.5 cm δv Speed variation 200 cm/s δφ Angle variation away from y-axis leaving the ramp π/36 M Mass of rider and motorcycle 300 kg vi Initial velocity leaving ramp 1500 cm/s θ Ramp angle of elevation π/6 Definitions and Key Terms • Buckling is the process by which a stiff plane develops a crack due to a stress exceeding the yield stress. • Compressive strength is the maximum force per unit area that a material can withstand, under compression, prior to yielding. • Corrugation is the style found in cardboard of sinusoidal waves of liner paper sandwiched between inner and outer papers. We use boxes with the most common corrugation, C-flute corrugation (see below for definition of a flute). • Cracking is a resulting state when the tensile force exceeds the tensile strength. • Cross-machine direction is the direction perpendicular to the sinusoidal wave of the corrugation. • Depression is when, due to a force, a section of a side or edge moves downwards. • ECT is the acronym for a common test of strength, the edge crush test. • Fiberboard is the formal name for cardboard. • Flute is a single wavelength of a sinusoidal wave between the inner and outer portion papers that extends throughout the length of the cardboard. A C-flute has a height of 0.35 cm and there are 138 flutes per meter [Packaging glossary n.d.]
236 The UMAP Journal 24.3(2003) Machine direction is parallel to the direction of the sinusoidal waves Motorcycle We use the 1999 BMw R1200c model motorcycle for structural information; it has a 7. 0-cm front wheel and a 10-cm back wheel, with radii 46 and 38 cm. It is 3 m long and has 17 cm ground clearance. The weight is 220 kg(dry)and 257 kg(fueled)[Motorcycle specs n d I Mullen test is a common measurement of the maximum force that a piece of cardboard can stand before bursting or puncturing Puncturing is when a force causes an area to burst through the cardboard surface Pyramid configuration is a configuration of boxes in which each box is rest ing equally on four others Strain is the dimensionless ratio of elongation to entire length. Stress is the force per unit area to which a material is subject Tensile strength is the maximum force per unit area which a material can withstand while under tension prior to yielding Youngs modulus of elasticity is the value of stress divided by strain and relates to the ability of a material to stretch Developing the Single-Box Model Expectations Given sufficiently small in area, the surface plane either punctures or cracks before the frame bu Given sufficiently large impact area the frame should buckle and no punc turing occurs. During buckling, corners are more resistive to crushing than edges Preliminary Assumptions The force is exerted at or around the center of the top The top of the box faces upward. This assumption allows us to use ECT (Edge Crush Test)results. This orientation also ensures that the flutes along the side are oriented perpendicular to the ground so they serve as columns
236 The UMAP Journal 24.3 (2003) • Machine direction is parallel to the direction of the sinusoidal waves. • Motorcycle We use the 1999 BMW R1200C model motorcycle for structural information; it has a 7.0-cm front wheel and a 10-cm back wheel, with radii 46 and 38 cm. It is 3 m long and has 17 cm ground clearance. The weight is 220 kg (dry) and 257 kg (fueled) [Motorcycle specs n.d.]. • Mullen test is a common measurement of the maximum force that a piece of cardboard can stand before bursting or puncturing. • Puncturing is when a force causes an area to burst through the cardboard surface. • Pyramid configuration is a configuration of boxes in which each box is resting equally on four others. • Strain is the dimensionless ratio of elongation to entire length. • Stress is the force per unit area to which a material is subject. • Tensile strength is the maximum force per unit area which a material can withstand while under tension prior to yielding. • Young’s modulus of elasticity is the value of stress divided by strain and relates to the ability of a material to stretch. Developing the Single-Box Model Expectations • Given sufficiently small impact area, the surface plane either punctures or cracks before the frame buckles. • Given sufficiently large impact area, the frame should buckle and no puncturing occurs. • During buckling, corners are more resistive to crushing than edges. Preliminary Assumptions • The force is exerted at or around the center of the top. • The top of the box faces upward. This assumption allows us to use ECT (Edge Crush Test) results. This orientation also ensures that the flutes along the side are oriented perpendicular to the ground, so they serve as columns
A Time-Independent model The Conceptual Single-Box Model During the puncturing or cracking of the top of the box The frame stays rigid The cardboard can be modeled as two springs with spring constants equal to the length of board times its modulus of elasticity [Urbanik 1999](Figure 1) Cross-Machine Direction (CD) Machine Direction (MD) Figure 1. Our model for a tire hitting the top of one box. We treat the portions of the box directly vertical or horizontal from the edges of the motorcycle tire(the rectangle in the middle)as ideal springs, neglecting the effect of the rest of the box. The surface of the motorcycle tire that strikes the box is approximated as a rectangle with dimensions LMD(the wheel's width) and Lcd( the length of the wheel in contact with the cardboard surface). We neglect the spin and tread of the tire The part of the spring beneath the tire does not undergo any tension. In re- ality, this is not the case; but with this assumption, cracking and puncturing occur along the edges of the tire. The force still comes from the rigid frame, and the springs have the same constant; therefore, we believe this assump tion affects only the position of the cracking and puncturing, not when it occurs or how much energy is dissipated There is no torque on the box during this first process. This assumption can be made since the force is at the center of the top The top of the cardboard box can fail in several ways If the resistive upwards force from the top, FUP, exceeds PML. LMD. LCD (the Mullen maximum allowable pressure over this area), then puncturing occurs
A Time-Independent Model 237 The Conceptual Single-Box Model During the puncturing or cracking of the top of the box: • The frame stays rigid. • The cardboard can be modeled as two springs with spring constants equal to the length of board times its modulus of elasticity [Urbanik 1999] (Figure 1). L L Machine Direction (MD) Cross-Machine Direction (CD) l MD CD EMD E CD Figure 1. Our model for a tire hitting the top of one box. We treat the portions of the box directly vertical or horizontal from the edges of the motorcycle tire (the rectangle in the middle) as ideal springs, neglecting the effect of the rest of the box. • The surface of the motorcycle tire that strikes the box is approximated as a rectangle with dimensions LMD (the wheel’s width) and LCD (the length of the wheel in contact with the cardboard surface). We neglect the spin and tread of the tire. • The part of the spring beneath the tire does not undergo any tension. In reality, this is not the case; but with this assumption, cracking and puncturing occur along the edges of the tire. The force still comes from the rigid frame, and the springs have the same constant; therefore, we believe this assumption affects only the position of the cracking and puncturing, not when it occurs or how much energy is dissipated. • There is no torque on the box during this first process. This assumption can be made since the force is at the center of the top. The top of the cardboard box can fail in several ways: • If the resistive upwards force from the top, FUP, exceeds PML · LMD · LCD (the Mullen maximum allowable pressure over this area), then puncturing occurs