Simulations of Pedestrian Impact Collisions with Virtual CRASH 3 and Comparisons with IPTM Staged Tests

Simulations of Pedestrian Impact Collisions with Virtual CRASH 3 and Comparisons with IPTM Staged Tests Tony Becker, ACTAR, Mike Reade, Bob Scurlock, Ph.D., ACTAR Introduction and dummy systems as point-like particles. From the dummy was used rather than the puck. Again we see work-energy theorem [3], we expect a change in kinetic excellent agreement between the analytic solution and In this article, we present results from a series of Virtual energy to be associated with dissipative frictional forces. simulation, with only some slight deviation at high CRASH-based pedestrian impact simulations. We That is: friction values. This is due to the dummy’s body compare the results of these Virtual CRASH pedestrian rotating toward the end of its motion at high speeds. impact simulations to data from pedestrian impact 푓 ̅ collisions staged at the Institute of Police Technology 푊 = ∫ 퐹 ∙ 푑푟̅ = ∆퐾퐸 Figure 3 illustrates the sliding distance and speed versus and Management. 푖 time for the puck and dummy, where each is given an 1 2 1 2 = 푚푣푓 − 푚푣푖 (1) initial speed of 30 mph, and a ground contact 2 2 1 Staged Pedestrian Impact Experiments coefficient-of-friction of 0.6 is used in the simulations. where the mass is displaced from point i to point f along We see very good agreement in the behaviors of the Each year the Institute of Police Technology and some path. In the one-dimensional constant frictional puck and dummy. Indeed, focusing on the velocity Management (IPTM) stages a series of pedestrian force approximation, we have: versus time behavior of the two systems in Figure 4, impact experiments as a part of its Pedestrian and first-order polynomial fits are performed to both sets of Bicycle Crash Investigation courses. These experiments 퐹̅ = −휇푚푔푥̂ data. With a simulated ground contact coefficient-of- offer a unique opportunity for participants to gain friction value of 0.6, the puck’s average deceleration hands-on experience setting up controlled experiments, and rate is within 0.3% of the expected value, and the gathering data and evidence, and performing full dummy’s average deceleration rate is within 0.7% of analyses of the collision events using standard 푑푟̅ = 푑푥̅ the expected value. reconstruction approaches in the accident reconstruction community. These analyses can then where 푥̂ points along the direction of displacement. Airborne Trajectory Simulation compared to direct measurements obtained during the Assuming the mass comes to rest at point f, we can write experiments. an expression relating the total sliding distance to the Virtual CRASH is a fully three-dimensional simulation environment; therefore, collision forces can project pre-slide velocity at point i, 푣푖. This is given by: The Virtual CRASH Simulator objects vertically, causing them to go airborne for some 1 time. Indeed, the user can run simulations specifying an Virtual CRASH 3 is a general-purpose fully three- −휇푚푔∆푥 = − 푚푣2 arbitrary initial velocity vector orientation at the start of 2 푖 dimensional accident reconstruction software package simulation, allowing objects to go airborne independent developed by Virtual CRASH, s.r.o., a company based or more familiarly: of collision events. out of Slovakia. Virtual CRASH is a simulation package that uses rigid-body dynamics to simulate 푣2 For clarity, let the x-direction point along the forward collisions between vehicle objects within its ∆푥 = 푖 (2) direction. Let the z-direction point upward vertically, environment; Virtual CRASH simulates multibody 2푔휇 antiparallel to the gravitation acceleration direction. In collisions in a manner similar to packages such as the following, we will neglect restitution effects, and so MADYMO [1] or Articulated Total Body [2]. The where ∆푥 = 푥푓 − 푥푖. We conducted a series of assume that the projectiles either come to rest upon impact dynamics are also determined by pre-impact experiments in Virtual CRASH, testing the distance landing or slide to rest at some time after initial ground geometry and specification of the coefficients-of - required to stop a mass as a function of pre-slide speed. contact. From basic kinematics, for a point-mass object friction and -restitution, as well as the inertial properties The puck and dummy objects started at ground height with initial height at ground level, launched at a velocity and were given an initial horizontal velocity. of the objects, which are all specified in the user of magnitude, |푣푖̅ |, at an angle 휃 with respect to the x- interface. Virtual CRASH offers a unique, fast, and Rearranging (2), one expects the simulation to yield axis, we expect the z-position as a function of time to visually appealing way to simulate pedestrian impacts. results consistent with the relation: be given by: 2 Sanity Checks 푣푖 = 휇 ∙ (2푔∆푥) (3) 1 푧(푡) = 푣 푡 − 푔푡2 푧,푖 2 2 Simulation of Dissipative Forces Figure 1 illustrates the square of the initial velocity, 푣푖 , 1 = |푣̅ |sin(휃)푡 − 푔푡2 (4) plotted as a function of the quantity (2푔∆푥). In the 푖 2 To better understand how Virtual CRASH performs simulations, the puck was given an initial velocity compared to our expectations from classical physics, we between 10 to 50 mph. The coefficient-of-friction was The total time, 푡푎, in which the object is airborne can be conducted a series of ground slide and projectile motion varied from 0.1 to 1.0 and the total sliding distance was solved for by obtaining the solution to the quadratic: experiments for a cylindrical “puck” and a dummy noted. As indicated by equation (3), the slope of a first- model. The puck was given the same mass as the default order polynomial fit to each set of points should yield 1 푧(푡 ) = |푣̅ |sin(휃)푡 − 푔푡2 = 0 (5) pedestrian model. the coefficient-of-friction used in the corresponding 푎 푖 푎 2 푎 simulations. Excellent agreement is observed between First, we evaluated the distances required for the puck the analytic solution and the simulation as indicated by which yields: and dummy systems to slide to a stop via frictional the slopes of the fits, which serve as estimates of . forces as they traveled along level flat terrain. In our Figure 2 illustrates the corresponding results for the treatment below, we mathematically model the puck same experiments, where the Virtual CRASH default 1 See video online at: https://youtu.be/htIwFYLG_W8 2|푣̅ |sin(휃) Therefore, from (9), (10), and (11) we have: 훥푣 = −(1 + 휀) ∙ 푣 (22) 푡 = 푖 (6) 푛 푅푒푙푛,푖 푎 푔 ∆푣푡 = 휇 · ∆푣푛 (12) Figure 5 shows the total airborne time for simulated 30 훥푣푡 = −휇(1 + 휀) ∙ 푣푅푒푙푛,푖 (23) mph launches of the puck and dummy systems as a The sign of the impulse ratio is determined by the function of sin(휃).2 First-order polynomial fits to these relative velocity vector component tangent to the Let us assume the ground is a flat level surface such 푛̂ = data points yield slopes that estimate the quantity contact surface at the moment of contact. The impulse 푧̂ and 푡̂ = 푥̂ . Let object 1 be an object undergoing ratio is typically associated with the inter-object surface 2|푣푖̅ |/g. Excellent agreement is observed between our projectile motion, and let object 2 be the infinitely analytic solution and simulation to the 1/100% level for contact coefficient-of-friction, whose average behavior massive ground plane. In the no-restitution limit, we is often referred to as the drag factor. In this paper, we both systems. have at time 푡푎 + ∆푡: use the terms interchangeably since we neglect any time-dependent behavior of 휇. Since the object undergoes uniform motion along the Δ푣푧 = |푣푧(푡푎)| (24) horizontal direction (neglecting air resistance), we can solve for the total horizontal airborne distance by: The coefficient-of-restitution is given by the ratio of and normal components of the final to initial relative velocities at the point-of-contact. Δ푣푥 = −휇 ∙ |푣푧(푡푎)| (25) 퐷푎 = 푣푥,푖푡푎 = |푣푖̅ |cos(휃)푡푎 2 푣푅푒푙푛,푓 2|푣̅푖| sin(휃) ∙ cos(휃) 휀 = − (13) where the collision pulse width, ∆푡, can be taken as = (7) 푣 푔 푅푒푙푛,푖 vanishingly small. Figure 7 illustrates the relation between the simulated horizontal changes-in-velocity Equation (7), of course, is the famous range equation where the relative velocity is defined as the difference from and vertical changes-in-velocity for the puck and from classical physics. between the velocity vectors of the two interacting dummy systems at the moment just after ground impact, objects at the point-of-contact: after the downward vertical velocity has been arrested. Figure 6 illustrates the total airborne distance as a The plots in this figure were created using 30 mph 푣̅ = 푣̅ − 푣̅ (14) function of the quantity (푠in(휃) ∙ cos(휃)) for 30 mph 푅푒푙 1 2 launch speeds at increasing launch angles between 5 degrees (lowest Δ푣푧) to 85 degrees (largest Δ푣푧) from launches of the puck and dummy systems. First-order The normal and tangent projections of this are given by: polynomial fits to the data serve to estimate the quantity horizontal. There are a few interesting features of note (2|푣̅ |2/푔). Again, excellent agreement between the in this figure. First, from equation (25), we expect a 푖 푣 = 푣̅ ∙ 푛̂ (15) simulations and the analytic solution is observed to 푅푒푙푛 푅푒푙 first-order polynomial fit to yield a slope equal to the and better than a fraction of a percent. ground contact coefficient-of-friction used in the simulations. This is indeed the case to within 0.4% for 푣 = 푣̅ ∙ 푡̂ (16) Ground Contact 푅푒푙푡 푅푒푙 the puck system and 6% for the dummy model.

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