Energy Correction Factor Collision Force Coefficient of Restitution Damage (Rigid Pole Impact) Damage (Miscellaneous)...

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1 Table of Contents Acceleration / Deceleration Factor... 5 Equivalent... 5 Stop, From or To... 7 Acceleration... 8 Deceleration... 9 Lateral... 0 Lateral Stability... Acceleration / Deceleration Rate... 3 Stop, From or To... 4 Acceleration... 4 Deceleration... 5 Lateral... 6 Aerodynamics... 7 Airborne... Audible... 3 Bicycle... 3 Braking Efficiency Center of Mass Longitudinal Center of Mass Lateral Center of Mass... 4 Trailer, Center of Mass... 4 Collinear Avoidance (Stationary Hazard) Reasonable & Prudent Speed Collinear Impact Safe Following Distance Frontal Sideswipe Rear end Sideswipe Damage Crush... 5 Damage Profile Angular Velocity Rotation Time... 58

2 Energy Correction Factor Collision Force Coefficient of Restitution... 6 Damage (Rigid Pole Impact)... 6 Damage (Miscellaneous) Delta V Momentum Crush Distance... 7 Energy Acceleration / Deceleration, Distance Contained in Motion Dissipation of Energy... 8 Speed Velocity Equivalents Force Grade & Superelevation Gravity... 9 Acceleration of Gravity... 9 Gravitational potential Energy Heavy Truck Impact Equivalent Deceleration Factor; Tractor/Semi Trailer Force & Load During Braking; Tractor/Semi Trailer Equivalent Deceleration Factor; Powered Vehicle/Full trailer Velocity; Weight Shift (articulated) Velocity; Weight Shift (non articulated) Trailer; Center of Mass... 0 Trailer Swing Hydroplane Skip Skid Marks Linear Distance Lateral Distance... 07

3 Braking, Linear Distance Low Speed Impact... Mass... 4 Relating to Force... 4 Effective Mass Coefficient... 5 Inertia, Mass Moment of... 5 Momentum Check... 5 Momentum... 7 In-Line Momentum... 7 Angular Momentum... 8 Motorcycle Impact... 6 Newton s Laws of Motion Oblique Angle Collisions Off Tracking / Low Speed Turn Passing Maneuver (Constant Velocity)... 4 Passing Maneuver (Acceleration)... 4 Pedestrian Impact Power Radius Railroad Crossing Impacts Rollover RPM Speed... 7 Speed Hydroplane Spin Out Tangent Offset Time Perception/Reaction Tires Trigonometry Turn / Swerve Linear Distance, Swerve Lateral Distance, Swerve

4 Critical Speed Scuff Velocity Brake Lag... 0 Kinetic Energy... 0 Skip Skid Marks... 3 Weight Shift... 4 Weight... REFERENCE DATA... 6 Acceleration / Deceleration Factor... 6 Acceleration / Deceleration Rate... 7 Air Bag... 9 Drag Coefficients Lane Change Truck Impact Light Luminous Motorcycle Impact Railroad Crossing Impacts Rollover RPM Speed... 4 Temperatures... 4 Time

5 Acceleration / Deceleration Factor Ratio between the forces required to move an object and the weight of the object, given as a percentage of gravity. Accel/Decelerating. Friction coefficient of an accel/decelerating object, decimal. µ = a / g a = Accel / Decel rate, ft/sec g = Gravitational constant, 3. ft/sec Percentage of Acceptability. Determination of an acceptable friction coefficient range, percent. The percentage acceptability should be within 5%. ( Lµ Sµ ) 00 P a = L µ = Largest friction coefficient during testing, decimal Sµ decimal S µ = Smallest friction coefficient during testing, Equivalent 3. Equivalent friction coefficient of a level surface from a grade, decimal. ( ) ( ( )) ( ) µ e = µ Sin Tan m / Cos Tan m µ = Friction coefficient of grade, decimal m = Grade, decimal (negative value (-) for decline) 4. Equivalent friction coefficient of a grade from a level surface, decimal. ( ( )) ( ) ( ) µ e = Sin Tan m + µ * Cos Tan m µ = Level friction coefficient, decimal m = Grade, decimal (negative value (-) for decline) 5

6 5. Equivalent friction coefficient of a grade from a level surface, decimal. ( ) µ e = µ ± m / ± m µ = Level friction coefficient, decimal m = Grade, decimal {(-) for decline, (+) for incline} 6. Equivalent deceleration factor for a straight line skid on several surfaces, decimal. f e = d f + d f + d f + d f d + d + d + d d d = Distance of each individual surface, ft 4 f f = Deceleration factor of each individual surface, decimal 4 7. Equivalent deceleration factor for a two axle vehicle during a straight line skid, knowing the center of mass location (x Fi and z i ). Center of mass utilized as a decimal fraction of the wheelbase, decimal. f e = ( ) ( ) f x f f F Fi F R z f f i F R f F = Front deceleration factor, decimal f R = Rear deceleration factor, decimal x Fi = Longitudinal center of mass from the front axle, decimal z i = Vertical center of mass height, decimal 8. Adjusted deceleration factor for braking with a grade less than.9%, decimal. f n m = ± µ µ = Level friction coefficient, decimal n = Braking efficiency, decimal m = Grade, decimal {(-) for decline, (+) for incline} 6

7 Stop, From or To 9. Accel/deceleration factor from or to stop, decimal. ( ) f = S / 30dn S = Speed, mi/hr d = Distance, ft n = Braking efficiency, decimal (deceleration only) 0. Accel/deceleration factor from or to a stop over a unit of time, decimal. ( gt ) f =.466S / S = Speed, mi/hr T = Time, sec g = Gravitational constant, 3. ft/sec. Accel/deceleration factor from or to a stop, decimal. ( ) f = V / gdn V = Velocity, ft/sec d = Distance, ft g = Gravitational constant, 3. ft/sec n = Braking efficiency, decimal (deceleration only) Stop, From or To; Time. Accel/deceleration factor from or to a stop over a unit of time, decimal. f = V / ( gt) V = Velocity, ft/sec T = Time, sec g = Gravitational constant, 3. ft/sec 3. Accel/deceleration factor from or to a stop over a unit of time, decimal. ( ) f = d / gt d = Distance, ft T = Time, sec g = Gravitational constant, 3. ft/sec 7

8 4. Accel/deceleration factor from or to a stop over a unit of time, decimal. ( ) f = d / 6. T d = Distance, ft T = Time, sec Acceleration 5. Acceleration factor from one speed to another over a determined distance, decimal. f = Sf So 30d Sf = Speed final, mi/hr So = Speed initial, mi/hr d = Distance, ft 6. Acceleration factor from one velocity to another over a determined distance, decimal. f Vf = Vo gd Vf = Velocity final, ft/sec Vo = Velocity initial, ft/sec d = Distance, ft g = Gravitational constant, 3. ft/sec 7. Acceleration factor from one velocity to another over a unit of time, decimal. f Vf Vo = gt Vf = Velocity final, ft/sec Vo = Velocity initial, ft/sec T = Time, sec g = Gravitational constant, 3. ft/sec 8. Acceleration factor over a determined distance and a unit of time, decimal. f d VoT = T g / Vo = Velocity initial, ft/sec d = Distance, ft T = Time, sec g = Gravitational constant, 3. ft/sec 8

9 9. Acceleration factor over a determined distance and a unit of time, decimal. ( ) ( ) f = d / T g / Vo / Tg / Vo = Velocity initial, ft/sec Deceleration d = Distance, ft T = Time, sec g = Gravitational constant, 3. ft/sec 0. Friction coefficient of a decelerating object knowing the weight and force applied, decimal. µ = F / W F = Force, lb W = Total static weight, lb. Deceleration factor from one speed to another over a determined distance, decimal. f = So Sf 30d So = Speed initial, mi/hr Sf = Speed final, mi/hr d = Distance, ft. Deceleration factor from one velocity to another over a determined distance, decimal. f Vo Vf = gd Vo = Velocity initial, ft/sec Vf = Velocity final, ft/sec d = Distance, ft g = Gravitational constant, 3. ft/sec 3. Deceleration factor from one velocity to another over a unit of time, decimal. f Vo Vf = gt Vo = Velocity initial, ft/sec Vf = Velocity final, ft/sec T = Time, sec g = Gravitational constant, 3. ft/sec 9

10 4. Deceleration factor from one speed to another over a unit of time, decimal. ( So Sf )/(. T ) f = 96 So = Speed initial, mi/hr Sf = Speed final, mi/hr T = Time, sec 5. Deceleration factor over a determined distance and a unit of time, decimal. ( ) ( ) f = Vo / Tg / d / T g / Vo = Velocity initial, ft/sec d = Distance, ft T = Time, sec g = Gravitational constant, 3. ft/sec Lateral 6. Lateral acceleration factor needed to maintain the radius of a level curve at a determined speed, decimal. ( ) f y S / 4. 97r S = Speed, mi/hr r = Radius of roadway, ft 7. Lateral acceleration factor of a vehicle negotiating a level curve at a determined speed with an unknown radius at the center of mass, decimal. f y = S ( r tw) S = Speed, mi/hr r = Radius of yaw mark, ft tw = Track width, ft 0

11 8. Lateral acceleration factor of a vehicle negotiating a banked curve at a determined speed with a known radius at the center of mass, decimal. (( 386) ) ( 386) ( ) f S r e S e r y = /. / / + /. / S = Speed, mi/hr r = Radius traveled by center of mass, ft e = Superelevation of curve, decimal (negative value (-) for decline) 9. Lateral acceleration factor needed to maintain the radius of a level curve at a determined velocity, decimal. f = V y / ( rg) V = Velocity, ft/sec r = Radius of roadway, ft g = Gravitational constant, 3. ft/sec 30. Lateral acceleration factor of a vehicle negotiating a level curve at a determined velocity with an unknown radius at the center of mass, decimal. f y = V ( 0. 5tw) g r V = Velocity, ft/sec r = Radius of yaw mark, ft tw = Track width, ft g = Gravitational constant, 3. ft/sec

12 3. Lateral equivalent deceleration factor for a vehicle sliding in a yaw on different friction surfaces, decimal. f ey ( i + o) ( ) f f tw / = z f + f + tw Weinberg i o tw = Track width, ft f i = Braking coefficient for the surface on which the inner wheels are rolling, decimal f o = Braking coefficient for the surface on which the outer wheels are rolling, decimal z = Vertical center of mass height, ft Lateral Stability 3. Determine a vehicle's lateral stability. The friction coefficient of the roadway must be greater than the value of the solution for the vehicle to rollover, decimal. tw f = y z tw = Track width, in z = Vertical center of mass height, in Rolling Resistance 33. Rolling resistance coefficient for bias or radial tires, decimal. 0.5 b f ( S /00) roll = a + + p p S = Speed, mi/hr p = Tire inflation pressure, lb/in Limpert Radial: a = b = 0.67 Bias Ply: a = b =.0

13 34. Rolling resistance coefficient for radial tires on heavy trucks, decimal. ( ) f f roll = 00004V V = Velocity, ft/sec f = Friction coefficient, decimal University of Michigan.0; smooth concrete.; worn concrete, brick, cold blacktop.5; hot blacktop 35. Rolling resistance coefficient for bias-ply tires on heavy trucks, decimal. ( ) f f roll = V V = Velocity, ft/sec f = Friction coefficient, decimal University of Michigan.0; smooth concrete.; worn concrete, brick, cold blacktop.5; hot blacktop Acceleration / Deceleration Rate Acceleration (positive)/deceleration (negative) is the rate of change of velocity with respect to time. Acceleration/deceleration rate per unit of time, ft/sec. a = fg f = Accel / Decel factor, decimal g = Gravitational constant, 3. ft/sec. Average acceleration/deceleration rate over a unit of time, ft/sec. a =. 466S / T S = Speed constant, mi/hr T = Time, sec 3. Average acceleration/deceleration rate over a unit of time, ft/sec. a V T = / V = Velocity constant, ft/sec T = Time, sec 3

14 Stop, From or To 4. Acceleration/deceleration rate of an object from or to a stop knowing the mass and force applied, ft/sec. a = F / m F = Force, lb m = Mass, lb-sec /ft 5. Acceleration/deceleration rate from or to a stop over a determined distance and a unit of time, ft/sec. a = d / ( 0. 5T ) d = Distance, ft T = Time, sec 6. Acceleration/deceleration rate from or to a stop over a determined distance and a unit of time, ft/sec. a = d / T d = Distance, ft T = Time, sec 7. Acceleration/deceleration rate from or to a stop over a determined distance, ft/sec. a = V / d V = Velocity, ft/sec d = Distance, ft Acceleration 8. Acceleration rate from one velocity to another over a unit of time, ft/sec. Vf a = Vo T Vf = Velocity final, ft/sec Vo = Velocity initial, ft/sec T = Time, sec 4

15 9. Acceleration rate from one velocity to another over a determined distance, ft/sec. Vf a = Vo d Vf = Velocity final, ft/sec Vo = Velocity initial, ft/sec d = Distance, ft 0. Acceleration rate over a determined distance and a unit of time, ft/sec. d Vo T a = Vo = Velocity initial, ft/sec T d = Distance, ft T = Time, sec Deceleration. Deceleration rate from one velocity to another over a unit of time, ft/sec. Vo Vf a = T Vo = Velocity initial, ft/sec Vf = Velocity final, ft/sec T = Time, sec. Deceleration rate from one velocity to another over a determined distance, ft/sec. Vo a = Vf d Vo = Velocity initial, ft/sec Vf = Velocity final, ft/sec d = Distance, ft 5

16 3. Deceleration rate over a determined distance and a unit of time, ft/sec. ( ) a = d VfT / T Vf = Velocity final, ft/sec d = Distance, ft T = Time, sec Lateral 4. Lateral acceleration rate of a vehicle negotiating a level curve at a determined velocity with a known radius at the center of mass, ft/sec. a V r y = / V = Velocity, ft/sec r = Radius traveled by center of mass, ft 5. Lateral acceleration rate of a vehicle negotiating a level curve at a determined velocity with an unknown radius at the center of mass, ft/sec. ( 0 5 ) a V / r. tw V = Velocity, ft/sec y = r = Radius of yaw mark, ft tw = Track width, ft 6. Acceleration factor in the x-direction, decimal. a x Fx max = W = Weight of vehicle, lb W F x max = Maximum tractional force to which a vehicle can Produce 6

17 Aerodynamics. Dynamic pressure of the airflow at a given velocity, lb-ft. P = P + P T Static Dynamic. Dynamic pressure of the airflow at a given velocity, lb-ft. P T = PStatic + 0.5ρV ρ = Mass density of air, lb sec / ft 4 (Eq #3) V = Velocity of air relative to vehicle, ft/sec P = Table Static 3. Air density for any atmospheric condition, absolute units. lb sec / ft 4. Temperatures must be converted to ( P / 9.9) ( 59 /( + T )) ρ = P = Ambient pressure, in (Table ) T = Air temperature, deg (Fahrenheit) 4. Determine a calculated aerodynamic drag force, lb. F A = 0.5ρV C A ρ = Mass density of air, D lb sec / ft 4 (Eq #3) V = Velocity of air relative to vehicle, ft/sec C = Aerodynamic drag coefficient, decimal D (Table a or b) A = Vehicle frontal area, ft 7

18 5. Determine an aerodynamic drag coefficient, decimal. C D ( 0.5 V A) = F / ρ ρ = Mass density of air, A lb sec / ft 4 (Eq #3) V = Velocity of air relative to vehicle, ft/sec F A = Drag Force, lb (Eq #4) A = Vehicle frontal area, ft 6. Horsepower required to move the vehicle against air resistance, hp. H pa = FAV A / 550 F A = Drag Force, lb (Eq #4) V A = Air velocity over the vehicle, ft/sec 7. Total rolling resistance of a vehicle proceeding down road, lb. ( f m) FR = W roll ± W = Total static weight, lb f roll = Rolling drag factor, decimal (Accel/Decel Factor section Eq # 33-36) m = Slope, pct (maximum of 0%) (+ if uphill, - if down hill) 8. Horsepower required to move a vehicle against its rolling resistance, hp. H F V / 550 = pr R R F = Rolling resistance, lb (Eq #7) V = Velocity of vehicle with respect to the road, ft/sec 9. Determine the total horsepower, hp. H + p = H pa H pr pa H = Air resistance, required horsepower, hp (Eq #6) H pr = Rolling resistance, required horsepower, hp (Eq #8) 8

19 0. Determine side forces in a constant wind, lb. F S = 0.5ρVw CS A ρ = Mass density of air, lb sec / ft 4 (Eq #3) V w = Total wind velocity, ft/sec C S = Side force coefficient which is a function of relative wind angle, decimal A = Vehicle frontal area, ft (Not the side area of the vehicle). Yaw moment with side force winds, ft-lb. Y m = 0.5ρV C ym A V = Velocity of vehicle, ft/sec ρ = Mass density of air, lb sec / ft 4 (Eq #3) A = Vehicle frontal area, ft = Wheelbase, ft C = Yaw moment coefficient, decimal ym. Determine the aerodynamic drag force, lb. F AD = 0.005C D AVr C D = Aerodynamic drag coefficient, decimal (Table a or b) V = Relative velocity between vehicle and wind, r ft/sec A = Vehicle frontal area, ft 9

20 Wind Speed Required, Rollover 3. Theoretical wind speed required to cause wheel lift or rollover, mi/hr. ( 0.5Wtw) /( 0. Az) S = 0056 W = Gross weight of vehicle, lb tw = Track width, ft Ravensdale A = area of windward side, ft z = Vertical center of mass height, ft 4. Determine a lateral stability of a vehicle. The friction coefficient of the roadway must be greater than the value of the solution for the vehicle to rollover, decimal. tw f y = tw = Track width, in z z = Vertical center of mass height, in Note: Wind direction must be perpendicular to the vehicle. If the roadway friction is less than the vehicle's calculated overturn friction coefficient (stability), the vehicle will slide rather then rollover. 5. Velocity from transmission measurements incorporating air resistance, ft/sec. V i i n ( Te / R) f C A( ρ / ) W T A roll = T D i = Transmission gear ratio, 00: i A = Axle ratio, 00: n = Mechanical efficiency of drive train, decimal Te = Torque at maximum rpm, ft/lb R = Radius of drive wheel, ft f roll = Rolling resistance coefficient, decimal W = Weight of vehicle, lb C D = Aerodynamic drag coefficient, decimal A = Vehicle frontal area, ft 4 ρ = Mass density of air, lb sec / ft 0

21 Airborne Galileo Galilei (564 64). Initial speed during a fall from a level take-off, mi/hr. S =. 73d / h d = Horizontal distance center of mass traveled from takeoff to landing, ft h = Vertical fall distance, ft. Initial velocity during a fall from a level take-off, ft/sec. V = 4. 0d / h d = Horizontal distance center of mass traveled from takeoff to landing, ft h = Vertical fall distance, ft 3. Speed required to flip at a 45 take-off with a level center of mass landing, mi/hr. S = d d = Horizontal distance center of mass traveled from takeoff to landing, ft 4. Velocity required to flip at a 45 take-off with a level center of mass landing, ft/sec. V = d g / d d = Horizontal distance center of mass traveled from takeoff to landing, ft g = Gravitational constant, 3. ft/sec

22 5. Speed required to vault with a grade less than 6.8, mi/hr. S =. 73d / dm h d = Horizontal distance center of mass traveled from takeoff to landing, ft h = Vertical fall distance, ft (negative value (-) for a lower center of mass landing) m = Grade, maximum 6.8, decimal (negative value (-) for decline) 6. Velocity required to vault with a grade less than 6.8, ft/sec. V = 4.0d / dm h d = Horizontal distance center of mass traveled from takeoff to landing, ft h = Vertical fall distance, ft (negative value (-) for a lower center of mass landing) m = Grade, maximum 6.8, decimal (negative value (-) for decline) 7. Speed required to vault or flip with a 45 take-off, mi/hr. S = 3. 86d / d h d = Horizontal distance center of mass traveled from takeoff to landing, ft h = Vertical distance from the plane of take-off to landing, ft (negative value (-) for a lower center of mass landing)

23 8. Speed required to vault with a grade greater than 6.8, mi/hr.. 73d S = dcosθsinθ hcos θ landing, d = Horizontal distance center of mass traveled from takeoff to landing, ft h = Vertical distance from the plane of take-off to ft (negative value (-) for a lower center of mass landing) θ = Angle of grade, deg 9. Speed required to vault with an angle of take-off exceeding 6.8, mi/hr.. 73d S = Cosθ dtanθ h landing, d = Horizontal distance center of mass traveled from takeoff to landing, ft h = Vertical distance from the plane of take-off to ft (negative value (-) for a lower center of mass landing) θ = Angle of take-off, deg 0. Velocity required to vault with an angle of take-off exceeding 6.8, ft/sec. 4.0d V = Cosθ dtanθ h takelanding, d = Horizontal distance center of mass traveled from off to landing, ft h = Vertical distance from the plane of take-off to ft (negative value (-) for a lower center of mass landing) θ = Angle of take-off, deg 3

24 . Velocity required to fall with a grade less than 6.8, ft/sec. V = d g ( dm h) d = Horizontal distance center of mass traveled from takeoff to landing, ft h = Vertical fall distance, ft (negative value (-) for a lower center of mass landing) g = Gravitational constant, 3. ft/sec m = Grade, maximum 6.8, decimal (negative value (-) for decline). Velocity required to vault or flip with a 45 take-off angle, ft/sec. ( ) V = d g / d h d = Horizontal distance center of mass traveled from takeoff to landing, ft h = Vertical distance from the plane of take-off to landing, ft (negative value (-) for a lower center of mass landing) g = Gravitational constant, 3. ft/sec 3. Velocity required to vault with an angle of take-off exceeding 6.8, ft/sec. V = 4. 0d dcosθsinθ hcos θ d = Horizontal distance center of mass traveled from take-off to landing, ft h = Vertical distance from the plane of take-off to landing, ft (negative value (-) for a lower center of mass landing) θ = Angle of take-off, deg 4

25 4. Velocity required to vault with an angle of take-off exceeding 6.8, ft/sec. ( / ) / ( ( )) V = d g Cos θ dtanθ h d = Horizontal distance center of mass traveled from take- landing, off to landing, ft h = Vertical distance from the plane of take-off to ft (negative value (-) for a lower center of mass landing) θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec 5. Velocity required to vault with an angle of take-off exceeding 6.8, ft/sec. [ ( )] V = d g / Cosθ dsinθ hcosθ d = Horizontal distance center of mass traveled from take- landing, off to landing, ft h = Vertical distance from the plane of take-off to ft (negative value (-) for a lower center of mass landing) θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec 6. Velocity at landing from a fall with a level take-off, ft/sec. VL = gh h = Vertical fall distance, ft g = Gravitational constant, 3. ft/sec 7. Velocity at landing from a vault, ft/sec. V = V + V Lh h f V h = Initial horizontal velocity prior to take-off slope, ft/sec (Eq #8) V f = Final vertical velocity, ft/sec (Eq #0) 5

26 8. Initial horizontal velocity prior to the take-off slope, ft/sec. Vh = VCosθ V = Velocity, ft/sec (Eq #0 thru 5) θ = Angle of take-off, deg 9. Initial vertical velocity during the take-off, ft/sec. Vv = VSin θ V = Velocity, ft/sec (Eq #0 thru 5) θ = Angle of take-off, deg 0. Final vertical velocity, ft/sec. ( ) V f = VSinθ + gh V = Velocity, ft/sec h = Maximum vertical height to landing, ft (Eq #3) θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec. Optimum angle of take-off during a flip or vault to determine minimum required speed, deg. ( ) θ = 0. 5Cos h / d + h d = Horizontal distance center of mass traveled from take- landing, off to landing, ft h = Vertical distance from the plane of take-off to ft (negative value (-) for a lower center of mass landing) 6

27 . Optimum angle of take-off during a flip or vault to determine minimum required speed, deg. Add 90 if the solution is negative. ( ) θ = 0. 5Tan d / h d = Horizontal distance center of mass traveled from take- landing, off to landing, ft h = Vertical distance from the plane of take-off to ft (negative value (-) for a lower center of mass landing) 3. Time of flight to a particular point along the trajectory, sec. point T = d / ( VCosθ ) d = Horizontal distance from take-off to the specific along the trajectory, ft V = Initial take-off velocity, ft/sec θ = Angle of take-off, deg 4. Time of flight, sec. ( v v ) T = V + V h g / g V v = Initial vertical velocity, ft/sec (Eq #9) landing, h = Vertical distance from the plane of take-off to ft (negative value (-) for a lower center of mass landing) g = Gravitational constant, 3. ft/sec 7

28 5. Time of flight, sec. ( ) VSin VSin h g T = θ θ g landing, V = Velocity, ft/sec h = Vertical distance from the plane of take-off to ft (negative value (-) for a lower center of mass landing) θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec 6. Time of flight from take-off to maximum vertical height, sec. Tm = VSin θ / g V = Velocity, ft/sec θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec 7. Maximum vertical height reached above the plane of take-off, ft. hm = TmVSin θ 0. 5gTm V = Velocity, ft/sec T m = Time to maximum vertical height, sec (Eq #6) θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec 8. Maximum vertical height reached above the plane of take-off, ft. ( ) hm = VSinθ / g V = Velocity, ft/sec θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec 8

29 9. Maximum vertical height reached above the plane of take-off, ft. hm = V v / g V v = Initial vertical velocity, ft/sec (Eq #9) g = Gravitational constant, 3. ft/sec 30. Maximum vertical height reached above the plane of take-off, ft. h m = 0.033S Sin θ S = Speed, mi/hr θ = Angle of take-off, deg 3. Vertical distance from maximum height to landing, ft. h = h m h h m = Maximum vertical height above the plane of takeoff, ft (Eq #5 thru 7) h = Vertical distance from the plane of take-off to landing, ft (negative value (-) for a lower center of mass landing) 3. Time from maximum vertical height to landing for a vehicle, which has gone airborne, sec. TL = h / g h = Distance from maximum vertical height to landing, ft (Eq #3) g = Gravitational constant, 3. ft/sec 9

30 33. Horizontal distance from the take-off to a point perpendicular to the maximum vertical height, ft. dm = Tanθ( VCosθ) / g V = Velocity, ft/sec θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec 34. Distance traveled during flight from take-off to the maximum vertical height, ft. d a = V SinθCosθ / g V = Velocity, ft/sec θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec 35. Vertical fall distance from any point along the trajectory measured along a horizontal plane, ft. (negative value (-) for a lower center of mass landing) ( θ) h = Tanθd 0. 5gd / VCos V = Initial take-off velocity, ft/sec d = Horizontal distance from take-off to the specific point along the trajectory, ft θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec 36. Vertical fall distance from the take-off point to the landing, ft. (negative value (-) for a lower center of mass landing) h = VSin θt 0. 5gT V = Initial take-off velocity, ft/sec T = Time of flight, sec (Eq #4, 5) θ = Angle of take-off, deg g = Gravitational constant, 3. ft/sec 30

31 37. Horizontal distance traveled from a level take-off to landing knowing the initial velocity and vertical fall distance, ft. d = V h / g V = Velocity, ft/sec h = Vertical fall distance, ft g = Gravitational constant, 3. ft/sec 38. Horizontal distance traveled from take-off to landing knowing the initial velocity and angle at take-off, ft. d = VCosθ T V = Velocity, ft/sec T = Time of flight, sec (Eq #4, 5) θ = Angle of take-off, deg 39. Maximum flight distance with a 45-degree take-off if initial velocity is known, ft. d = Vo / g Vo = Velocity original, ft/sec g = Gravitational constant, 3. ft/sec Audible Audible Levels Source Distinguishable Insertion loss for vehicle Inside Vehicle: 50 mph ~ Average Windows closed / no radio Horn from Locomotive: 00 ft Interior vehicle ~ Average Interior cab of Truck ~ Average Train Accident Reconstruction p.8 ~ Loumiet dba 9 0 dba above threshold 30 dba 7 dba 96 dba 49CFR dba 85 + dba 3

32 Example Findings A Time Before Impact B Outside dba Level C Insertion dba Loss D Signal dba Inside E Operating dba Level F Signal-to- Noise Ratio 5 Seconds Seconds Seconds Seconds Second Seconds Second Train Accident Reconstruction p.33 ~ Loumiet Bicycle A vehicle consisting of a light frame mounted on two wire-spoked wheels one behind the other and having a seat, handlebars for steering, brakes, and two pedals or a small motor by which it is driven. Acceleration Rate Over distance of 40 feet Speed Men Women Slow 6.54 sec 5.84 sec Fast 3.97 sec 3.95 sec Average 4.90 sec 5.0 sec Acceleration rate 3.3 ft/sec 3.7 ft/sec Acceleration factor Haight Revolutions per Minute Normal Riding Serious Exercising Racing rpm 00 5 rpm rpm 3

33 Lean Angle Average Maximum Lateral Acceleration Factor Maximum =.5 to.30. Calculate the gear inch knowing the number of teeth on the chain wheel and on the freewheel, in. Cn G i = Wdπ C n = Number of teeth on chain wheel, # Fn F n = Number of teeth on freewheel, # W d = Diameter of rear wheel, in π = Pi, Riding velocity knowing gear inch and pedal revolutions per minute, ft/sec. V = G i R p G i = Gear inch (Eq #), in R p = Pedal revolutions per minute, rpm 3. Speed of vehicle striking a bicyclist, ft/sec. NOTE: Adult & Child Pontoon Vehicles and Adult V-Contour Vehicles equation V a d a c a d a d a c a d = a 3 Sturtz Child V-Contour Vehicles dt h V = a Box Vehicles (Forward Projections) dt h V =

34 User Inputs d t Throw Distance h Height of pedestrian s Center of Mass d h - d t a & c Equation Constants Solved in: fps or m/s Miscellaneous Value of µ ranged from 0.4 to 0.7 V-Contour Vehicles - Low pointed front-end vehicles Pontoon Vehicles - Traditional style front end vehicles.. Box Vehicle equation (forward projection) quickly becomes unstable as the throw distance exceeds 50 feet95 meters. Constant values a - Imperial a -Metric c-value d-value Pontoon Vehicles (adult) V-Contour Vehicles (adult) Pontoon Vehicles (child) V-Contour Vehicles (child) N/A Bicycle / ped-com d 34

35 Braking Efficiency Table ; Vehicle Braking Percentage (forward frontal heading) Vehicle Type Passenger Vehicles Only front wheels locked Only rear wheels locked ABS equipped; full braking Motorcycles Free rolling Front/Rear Full Lockup Moderate/heavy front brake application with rear wheel lockup Percentage of Braking (%) ( ) f = µ / + µ / Front Wheel Only Clean, dry surface Rear Wheel Only Clean, dry surface Soft soil, sand Hard soil e Proper Brake Adjustment For The Following Values Apply Straight Trucks Loaded Tractor/Semi Trailer (5 axle) (0% steer, 36% drives, 4% trailer) Doubles (Cab over Engine tractor & twin 8's) 80 Dump Trucks Concrete Mixers (Caution: Limited Testing) Motor Homes (Caution: Limited Testing)

36 Commercial Buses MC-9 (Greyhound style) Transit (including articulated city buses) School Bobtails Cab over Engine Conventional Note: Front axle brakes may slightly increase the braking coefficient: However the coefficient will still fall within the range of CAUTION: If the vehicle is equipped with a brake proportioning valve, the percentage will increase dramatically to 80-85% Bobtails w/ BP- & BP- values 'Anteaters' w/bp- Frontlines (86+) w/wabco 6 Channel anti-lock (Westinghouse Air Brake Company) Percentage of braking applied during a deceleration with a known friction coefficient for the surface and a deceleration factor for the vehicle, pct. ( f ) n = / µ 00 f = Deceleration factor, decimal µ = Friction coefficient, decimal. Percentage of braking applied during a deceleration to a stop, decimal. ( dµ ) n = S / 30 S = Speed, mi/hr d = Distance, ft µ = Friction coefficient, decimal 36

37 3. Percentage of braking applied during a deceleration to a stop, decimal. ( ) n = V / gdµ V = Velocity, ft/sec d = Distance, ft µ = Friction coefficient, decimal g = Gravitational constant, 3. ft/sec Trailer, Equivalent Deceleration Factor 4. Equivalent deceleration factor for a vehicle/trailer combination with no braking of the trailer, decimal. ( ) fe = fwv / WV + WT f = Vehicle deceleration factor, decimal W V = Static weight of vehicle, lb W T = Static weight of trailer, lb 5. Braking force applied to a tire, which is at its frictional limit during a cornering maneuver, lb. F xb = Wµ Sinα µ = Friction coefficient, decimal α = Tire slip angle, deg W = Weight on tire, lb Brake Lag 6. Velocity at commencement of brake activation incorporating the time of brake lag, ft/sec. Utilize for standard, hydraulic fluid transfer brake systems only. Vb = V aTb V = Initial velocity calculated, ft/sec a = Deceleration rate, ft/sec T b = Brake lag time, sec Recommended brake lag time Eubanks / Reed of seconds for standard brake systems. 37

38 7. Distance traveled during brake lag time, ft. d b = VT gf T V = Initial velocity calculated, ft/sec b b f = Deceleration factor, decimal T b = Brake lag time, sec Recommended brake lag time of seconds for standard brake systems. g = Gravitational constant, 3. ft/sec 8. Velocity at commencement of brake activation incorporating distance traveled during brake lag time, ft/sec. Utilize for standard, hydraulic fluid transfer brake systems only. V V gf 0. 6 b = d b b d = Brake lag distance, ft (Eq # 7) V = Initial velocity calculated, ft/sec f = Deceleration factor, decimal g = Gravitational constant, 3. ft/sec 9. Distance traveled at commencement of brake activation incorporating distance traveled during brake lag time, ft. Utilize for standard, hydraulic fluid transfer brake systems only. ( V gf 0.6d )/ fg d b = b d = Brake lag distance, ft (Eq # 7) V = Initial velocity calculated, ft/sec f = Deceleration factor, decimal g = Gravitational constant, 3. ft/sec 38

39 Center of Mass The point in a system of bodies at which the mass of the system may be considered to be concentrated and at which external forces may be considered to be applied. Also called barycenter, centroid. Table ; Rule of Thumb Thumb ger Cars Trucks CM height =.0 in.9 in ±.5 in 6.7 in ± 4.0 in CM height = 40% Roof Hgt 39.5% ±.6 % 38.7% ± 3.5% CM for BobTail snubnose semi tractor is about inches from ground U.S. Federal regulations do not permit CM over 75 inches CM for pedestrian can be estimated in three ways: o at the iliac crest (Spitz) o third lumbar vertebrae (Snyder & Hermance) o 57% of the pedestrian s height (Wood) Table ; Inertial Parameters Thumb ger Cars Trucks I pitch I roll I yaw W = Total static weight, lb ft lb sec 0.99W-49.W-657 ft lb sec 0.8W-50 0.W-35 ft lb sec.03w-06.03w-343 NHTSA Longitudinal Center of Mass. Longitudinal center of mass measured from the front axle, ft. x F WR = W = Wheelbase, ft W R = Static rear axle weight, lb W = Total static weight, lb 39

40 . Longitudinal center of mass measured from the front axle, ft. x F ( W ) = = Wheelbase, ft Fi W Fi = Fraction of weight on front wheels, decimal W W ( ) F / 3. Longitudinal center of mass measured from the front axle as a decimal fraction of the wheelbase, decimal. x Fi = xf / x F = Longitudinal center of mass from the front axle, ft (Eq #) = Wheelbase, ft 4. Longitudinal center of mass measured from the rear axle, ft. x R WF = W = Wheelbase, ft W F = Static front axle weight, lb W = Total static weight, lb 5. Longitudinal center of mass measured from the rear axle, ft. x R ( W ) = = Wheelbase, ft Ri W Ri = Fraction of weight on rear wheels, decimal W W ( ) R / 6. Longitudinal center of mass measured from the rear axle as a decimal fraction of the wheelbase, decimal. x Ri = xr / x R = Longitudinal center of mass from the rear axle, ft (Eq #4, 5) = Wheelbase, f 40

41 Lateral Center of Mass 7. Lateral center of mass measured from the left side, ft. y l Wr tw = tw = Track width, ft W W r = Static right side weight, lb W = Total static weight, lb 8. Lateral center of mass measured from the right side, ft. y r Wtw l = tw = Track width, ft W W l = Static left side weight, lb W = Total static weight, lb Vertical Center of Mass 9. Vertical center of mass height, ft. Rear elevated. ( Wh WF) ( h r) W( h r) z = + r = Wheelbase, ft h = Vertical height rear axle elevated, ft (/3 of wheelbase) r = Radius of drive wheels, ft W h = Front axle weight, rear elevated, lb W F = Static front axle weight, lb W = Total static weight, lb 4

42 0. Vertical center of mass height, ft. Front elevated. ( Wh WR) ( h r) W( h r) z = + r = Wheelbase, ft h = Vertical height front axle elevated, ft (/3 of wheelbase) r = Radius of drive wheels, ft W h = Rear axle weight, front elevated, lb W R = Static rear axle weight, lb W = Total static weight, lb. Vertical center of mass height as a decimal fraction of the wheelbase, decimal. zi = z / z = Vertical center of mass height, ft (Eq #9, 0) = Wheelbase, ft Trailer, Center of Mass. Longitudinal center of mass of combined trailer with load measured from a datum line, ft. x W xl WT x W L + T = L x = Longitudinal distance from the datum line to center of mass of load, ft x = Longitudinal distance from the datum line to T the trailer s center of mass, ft W L = Static weight of load, lb W = Static weight of trailer, lb T W = Total static weight of semi trailer and load, lb 4

43 3. Lateral center of mass of combined trailer with load measured from a datum line, ft. y Wy + Wy W L L T T = L y = Lateral distance from the datum line to center of mass of load, ft y = Lateral distance from the datum line to T the trailer s center of mass, ft W L = Static weight of load, lb W = Static weight of trailer, lb T W = Total static weight of semi trailer and load, lb 4. Vertical center of mass height of combined trailer with load, ft. W zl WT z z = W L + T z L = Vertical center of mass height of load from the ground, ft z T = Vertical center of mass height of trailer, ft W L = Static weight of load, lb W T = Static weight of trailer, lb W = Total static weight of semi trailer and load, lb Collinear Avoidance (Stationary Hazard) Maximum Speed/Velocity. Maximum speed possible in order to stop from a known distance; (hill crest, bend in roadway) when first perception of an obstacle occurs, mi/hr. [ T d f T ] S =.96 f / f = Deceleration factor, decimal d = Total distance to Impact, ft {including P/R distance} T = Perception/Reaction time, sec 43

44 . Maximum velocity possible in order to stop from a known distance; (hill crest, bend in roadway) when first perception of an obstacle occurs, ft/sec. [ T + d ( fg) T ] V = fg / f = Deceleration factor, decimal d = Total distance to Impact, ft {including P/R distance} T = Perception/Reaction time, sec g = Gravitational constant, 3. ft/sec Reasonable & Prudent Speed 3. Reasonable and prudent speed under adverse conditions knowing the speed limit and the friction coefficient for the normal and adverse conditions, mi/hr. R ( S L f a ) f n S = / S L = Posted speed limit, mi/hr f a = Friction coefficient for adverse conditions, decimal f n = Friction coefficient for normal conditions, decimal Original Speed 4. Original speed knowing the total distance to impact, speed at impact, perception/reaction time and deceleration factor, mi/hr. (.96 ft ) + Sf fd So =.96 ft + 30 O S = Speed original, mi/hr f = Deceleration factor, decimal {negative value for deceleration} d = Total distance to Impact, ft {including P/R distance} T = Perception/Reaction time, sec 44

45 Maximum Distance 5. Distance required to perceive/react and stop to avoid a hazard from a known velocity, ft. d = V / fg + VT V = Velocity, ft/sec f = Deceleration factor, decimal T = Perception/Reaction time, sec g = Gravitational constant, 3. ft/sec 6. Total distance required including perception/reaction time to decelerate from one velocity to another, ft. ( Vo Vf )/( fg) d = VoT + f = Deceleration factor, decimal Vo = Velocity original, ft/sec Vf = Velocity final, ft/sec g = Gravitational constant, 3. ft/sec T = Perception/Reaction time, sec Collinear Impact * For equations through 3, vehicles must depart after collision as one unit. Closing Velocity. Closing velocity of a trailing vehicle on the lead vehicle in a collinear collision, ft/sec. ( ) VC = ge D WT + WL / WTWL E D = Total combined crush energy for both vehicles, ft-lb W T = Weight, trailing vehicle, lb W L = Weight, lead vehicle, lb g = Gravitational constant, 3. ft/sec 45

46 . Pre-impact velocity of the closing vehicle (trailing vehicle) in a collinear collision, ft/sec. ( ) VT = VCWL / WT + WL + V V C = Closing velocity of trailing vehicle on the lead vehicle, ft/sec (Eq #) W T = Weight of trailing vehicle, lb W L = Weight of lead vehicle, lb V = Post-impact velocity of both vehicles as one unit, ft/sec 3. Velocity of the lead vehicle knowing the closing velocity and pre-impact velocity of the trailing vehicle, ft/sec. VL = VT VC V T = Pre-impact velocity of trailing vehicle, ft/sec (Eq #) V C = Closing velocity of trailing vehicle on the lead vehicle, ft/sec (Eq #) 4. Closing velocity of two vehicles during a collinear impact, ft/sec. V c = E d W + W g W W ( e ) Wells, Atkinson, Hennessy E d = Total absorbed energy for damage from both vehicles, ft-lb W = Weight of vehicle #, lb W = Weight of vehicle #, lb e = Coefficient of restitution, decimal g = Gravitational constant, 3. ft/sec 46

47 5. Velocity for vehicle #; Inline Collision; Vehicles traveling in same direction, ft/sec. ( ) V = V + W W V V 3 / 4 W = Weight, vehicle #, lb W = Weight, vehicle #, lb V = Pre-impact velocity veh #, ft/sec V 3 = Post-impact velocity veh #, ft/sec V = Post-impact velocity veh #, ft/sec 4 6. Velocity veh # Inline Collision; Elastic (minimal damage), ft/sec. V ( + / ) + ( / ) V W W V W W = 4 W = Weight, vehicle #, lb W = Weight, vehicle #, lb V = Pre-impact velocity veh #, ft/sec V = Post-impact velocity veh #, ft/sec 4 7. Velocity veh #; Inline Collision, utilizing a coefficient of restitution, ft/sec. V V = ( + W / W ) + V ( e W / W ) 4 + e e = Coefficient of Restitution, decimal W = Weight, vehicle #, lb W = Weight, vehicle #, lb V = Pre-impact velocity veh #, ft/sec V = Post-impact velocity veh #, ft/sec 4 8. Velocity for vehicle #3; Inline Collision; Vehicles traveling in same direction, ft/sec. ( ) V = V W W V V 3 / 4 W = Weight, vehicle #, lb W = Weight, vehicle #, lb V = Pre-impact velocity veh #, ft/sec V = Pre-impact velocity veh #, ft/sec V = Post-impact velocity veh #, ft/sec 4 47

48 Coefficient of Restitution 9. Coefficient of restitution, decimal. (Collinear impacts) ( V V ) ( V ) e = 3 4 / V V = Pre-impact velocity veh #, ft/sec V = Pre-impact velocity veh #, ft/sec V 3 = Post-impact velocity veh #, ft/sec V 4 = Post-impact velocity veh #, ft/sec For perfect elastic collision e =. For inelastic collisions e <. If vehicles lodge together after collision, V = V, e = Safe Following Distance 0. Safe following distance between a lead and trailing vehicle prior to a collinear collision, ft. [ ( / )( / / )] ds = d + V TP + TR + V at al V = Initial velocity of vehicles, ft/sec d = Distance between vehicles at points of rest, ft a L = Lead vehicle deceleration rate, ft/sec a T = Trailing vehicle deceleration rate, ft/sec T P = Trailing vehicle perception time, sec T R = Trailing vehicle reaction time, sec 48

49 Frontal Sideswipe. Determine the pre-impact speed for vehicle # for in-line/sideswipe frontal collisions, ft/sec. m + m m + m ( m + m ) ( m )( m V m V ) + ( V + V ) + bev + V = m / / bev m m Limpert bev = Barrier equivalent velocity for vehicle #, ft/sec (Eq #8 Crush Damage section) bev = Barrier equivalent velocity for vehicle #, ft/sec (Eq #8 Crush Damage section) m = Mass of vehicle #, lb-sec /ft m = Mass of vehicle #, lb-sec /ft V 3 = Post-impact velocity veh #, ft/sec V = Post-impact velocity veh #, ft/sec 4. Determine the pre-impact speed for vehicle # for in-line/sideswipe frontal collisions, ft/sec. ( m / m ) V 3 ( m m ) V V = V 4 + / m = Mass of vehicle #, lb-sec /ft m = Mass of vehicle #, lb-sec /ft Limpert V = Pre-impact velocity veh #, ft/sec (Eq #) V 3 = Post-impact velocity veh #, ft/sec V = Post-impact velocity veh #, ft/sec 4 49

50 Rear end Sideswipe 3. Determine the pre-impact speed for vehicle # for in-line/sideswipe rear end collisions, ft/sec. m + m m + m ( m + m ) ( m )( m V + m V ) + ( V V ) + bev + V = m / / bev m m Limpert bev = Barrier equivalent velocity for vehicle #, ft/sec (Eq #8 Crush Damage section) bev = Barrier equivalent velocity for vehicle #, ft/sec (Eq #8 Crush Damage section) m = Mass of vehicle #, lb-sec /ft m = Mass of vehicle #, lb-sec /ft V 3 = Post-impact velocity veh #, ft/sec V = Post-impact velocity veh #, ft/sec 4 4. Determine the pre-impact speed for vehicle # for in-line/sideswipe rear end collisions, ft/sec. ( m / m ) V 3 ( m m ) V V = V 4 + / m = Mass of vehicle #, lb-sec /ft m = Mass of vehicle #, lb-sec /ft Limpert V = Pre-impact velocity veh #, ft/sec (Eq #3) V 3 = Post-impact velocity veh #, ft/sec V = Post-impact velocity veh #, ft/sec 4 Post Impact Speed 5. Post impact speed of Veh # during a collinear collision with vehicle # stationary prior to impact, ft/sec. V [ W /( W + W )]( e) V 4 = Wells, Atkinson, Hennessy W = Weight of vehicle #, lb W = Weight of vehicle #, lb e = Coefficient of restitution, decimal V = Pre-impact speed of Veh #, ft/sec 50

51 6. Post impact speed of Veh # during a collinear collision with vehicle # stationary prior to impact, ft/sec. [( W + ew ) ( W W )] V 3 / + V = Wells, Atkinson, Hennessy Delta V 7. Delta V for the bullet vehicle during a collinear impact, ft/sec. W = Weight of vehicle #, lb W = Weight of vehicle #, lb e = Coefficient of restitution, decimal V = Pre-impact speed of Veh #, ft/sec V = B E gw D ( e) ( + )( ) W W W e E D = Total absorbed energy for damage from both vehicles, ft-lb W = Weight of vehicle #, lb W = Weight of vehicle #, lb e = Coefficient of restitution, decimal g = Gravitational constant, 3. ft/sec 8. Delta V for the target vehicle during a collinear impact, ft/sec. V = T E gw D ( e) ( + )( ) W W W e E D = Total absorbed energy for damage from both vehicles, ft-lb W = Weight of vehicle #, lb W = Weight of vehicle #, lb e = Coefficient of restitution, decimal g = Gravitational constant, 3. ft/sec 5

52 Damage Crush The following variables are used in equations through 0 of this section: A = Stiffness coefficient, lb/in B = Stiffness coefficient, lb/in G = Stiffness coefficient, lb g = Acceleration of gravity, 3. ft/sec (386.4 in/sec ) b o = Intercept (maximum barrier velocity w/o permanent damage), in/sec (4.398 to 0.6 ft/sec or to 3.44 in/sec) b = Slope of the speed versus crush relation, /sec (change in impact speed to the change in crush) E = Energy dissipated due to crush, in-lb L C = Width of crush region (crash vehicle), in L T = Width of crush region (test vehicle), in W T = Total static weight of the test vehicle, lb σ = Angle of attack at impact, deg (angle between the PDOF (ρ) and the damaged side axis) Do not exceed 45 degrees. V imp = Impact velocity of test vehicle, ft/sec Cr ave = Average crush depth of test vehicle, in C through C 6 = Crush measurements, in Centroid of Damage. Centroid of Damage measured from the center of the damage width along the x-axis direction (depth), in. C x = + C + C 3 + C 3 4 ( C + C + C + C + C + C ) + C 5 + C 6 + C C C C C C C C C C 5 6 5

53 . Centroid of Damage measured from the center of the damage width along the y-axis direction (width), in. L 3C 8C 6C3 + 6C4 + 8C5 + 3C y = 30 C + C + C3 + C4 + C5 + C6 The following variables ( g, b o and V imp ) are converted to in/sec prior to their entry into the following equations. This is done by multiplying the variables (ft/sec) by. Use the crush data available by NHTSA Slope of the speed versus crush relation, /sec. Campbell b = V imp Cr b ave o 4. Maximum force per inch of damage width without permanent damage, lb/in. * Campbell A W b b T o = gl T 5. Crush resistance per inch of damage width, lb/in. * Campbell B W b T = gl T 6. Energy dissipated without permanent damage, lb. Campbell G = A / ( B) 53

54 Table ; Stiffness Values; Average Vehicle Type Pasenger Cars Pickup Trucks Vans Value Frontal Crash A = 35 ± 0 % B = 43 ± 0 % Rear-end Crash A = 364 ± 0 % B = 48 ± 0 % Side Crash A = 4 ± 35 % B = 5 ± 35 % Frontal Crash A = 456 ± 0 % B = 90 ± 5 % Rear-end Crash A = 350 ± 0 % B = 5 ± 0 % Side Crash A = 60 ± 0 % B = 45 ± 0 % Frontal Crash A = 380 ± 0 % B = 5 ± 0 % Rear-end Crash A = 300 ± 0 % B = 55 ± 0 % Damage Profile Place the above values for (A, B, G) into one of the following Damage Profile Equations: 7. Two Point Damage Profile (crush energy), in-lb: ( + tan σ ) L [( A/ )( C + C ) + ( B / 6)( C + C C + C ) G] E = C + 8. Four Point Damage Profile (crush energy), in-lb: C + C + C3 + C4 + E = ( + tan σ )( LC / 3 ) ( A/ )( C+ C + C3 + C4) + ( B/ 6) + 3G CC + CC 3 + CC

55 9. Six Point Damage Profile (crush energy), in-lb: E = + L A C + C + C + C + C + C 3 ( tan σ )( C / 5) ( / ) + ( B / 6) Average Crush Depth C + C + C + C + C + C G CC + CC3 + C3C4 + C4C5 + C5C6 0. C / + C + C3 + C4 + C5 + C6 / C ave = Average crush depth for a six point 5 damage profile, in. Barrier Equivalent Velocity The barrier equivalent velocity (bev) ft/sec can then be calculated from the energy ( E) in-lb produced from the above Damage Profile Equations. First, rewrite the variable ( E ) from in-lb to ft-lb by division of. Then place the variable ( E ) ft-lb into the following equation:. Barrier equivalent velocity utilizing energy, ft/sec. bev = Eγ / m E = Collision energy dissipated due to crush, ft-lb (Eq #8 thru 0) m = Mass of vehicle, lb-sec /ft γ = Effective mass coefficient at the center of gravity, decimal (Eq #4). Barrier equivalent velocity utilizing energy, ft/sec. bev = ge / W E = Collision energy dissipated due to crush, ft-lb (Eq #8 thru 0) W = Total static weight, lb g = Gravitational constant, 3. ft/sec 55

56 Delta V 3. Delta V due to crush for vehicle #, ft/sec. ( ) ( ) ( ) V = E + E / m + m / m E = Collision energy dissipated due to crush, vehicle #, ft-lb (Eq #8 thru 0) E = Collision energy dissipated due to crush, vehicle #, ft-lb (Eq #8 thru 0) m = Mass of vehicle #, lb-sec /ft m = Mass of vehicle #, lb-sec /ft 4. Delta V due to crush for vehicle #, ft/sec. ( ) ( ) ( ) V = E + E / m + m / m E = Collision energy dissipated due to crush, vehicle #, ft-lb E = Collision energy dissipated due to crush, vehicle #, ft-lb m = Mass of vehicle #, lb-sec /ft m = Mass of vehicle #, lb-sec /ft 5. Delta V for either vehicle # or, ft/sec. V = P / m P = Impulse, lb-sec (Eq #7) m = Mass, lb-sec /ft 6. Longitudinal component of a Delta V, ft/sec. ( ) V long VCos ρ 80 V = Delta V. Magnitude of the velocity change for the center of gravity, ft/sec (Eq #3, 4, 5) SAE # ρ = Principal direction of force, deg (Eq #4, 5 Momentum section) 56