Chapter 4 — Calculation Method¶
Basis for Analysis¶
EDSMAC4 is a dynamic analysis of a crash sequence involving one or more vehicles. The program uses a tire model to calculate forces acting on the vehicle at the tire-road interface. If contact between vehicles is detected, collision forces are also computed. Newton's laws of motion are solved at discrete, user-entered time steps for each vehicle. The resulting accelerations are integrated to update the position and velocity at the end of each time step.
The mathematical model for EDSMAC4 was extended from EDC's EDSMAC model, which, in turn, was developed from the SMAC model originally developed and validated at Calspan [1-5, 11-13]. Extensions provided by EDC are explained in references 6, 7, 22 and 23.
To use EDSMAC4 successfully, a good understanding of vehicle dynamics is extremely useful and is recommended. However, the following discussion is limited to the general analytical approach. The discussion describes the solution procedure while the vehicles are in contact during the collision phase and while they are separated (before and after impact) during the trajectory phases. For a detailed treatise on the subject of vehicle dynamics and simulation programs, the user is referred to the references listed at the end of this manual (Chapter 7) as well as those listed in the User's Manual, Appendix VI.
Equations of Motion¶
EDSMAC4 is a 3-degree-of-freedom analysis. Vehicle motion is calculated in the earth-fixed X and Y directions and rotation about the vertical Z axis. The vector velocity (amplitude and direction) is resolved in the vehicle-fixed coordinate system by resolving its components along the vehicle's forward (u-velocity) and lateral (v-velocity) directions. For more information, refer to SAE J670e — Vehicle Dynamics Terminology.
The required equations of motion, applicable to each vehicle, are:
where:
- \(m\) = mass
- \(I_z\) = yaw moment of inertia
- \(u\) = forward velocity
- \(v\) = lateral velocity
- \(r\) = angular yaw velocity about the z axis
- \(\Sigma F_x\) = summation of external forces in the vehicle-fixed x-direction
- \(\Sigma F_y\) = summation of external forces in the vehicle-fixed y-direction
- \(\Sigma M_z\) = summation of external moments about the vehicle-fixed z axis
EDSMAC4 solves these equations of motion at discrete, user-entered time intervals (see Simulation Controls, Timesteps), then uses numerical integration to advance to the next time step, and the calculations are repeated.
There are no degrees of freedom (and hence, no equations of motion) for vertical translation nor pitch or roll rotation. Thus, roll, pitch and vertical accelerations are ignored.
In HVE, at each time step, the vehicle's roll, pitch and CG elevation are computed using HVE's surface-information search [24]. This approach allows the vehicle to sense when it is on a grade, and the equations of motion account for the grade. Surface slopes up to about 15 degrees are allowed (a message will be issued if this angle is exceeded). Thus, vehicle motion on hills, super-elevations, road crowns and even sloped shoulders may be modeled using EDSMAC4.
(updated: the current version also includes an optional dynamic steer degree of freedom — when enabled via the Steer DOF calculation option, wheel steer angles are computed from the torques acting on the steering system (tire friction, steering stops, gyroscopic precession, steering column friction) instead of being read directly from the driver steer tables; see EDSMAC4 Calculation Options.)
Collision Phase¶
The EDSMAC4 collision model is described in detail in reference 22.
During the collision phase, each vehicle's crush perimeter is described by a set of radial RHO vectors extending from the CG to the vehicle exterior, spaced at the user-entered Vector Spacing angle (default 2 degrees). When the perimeters of two vehicles overlap, the collision calculation iteratively adjusts the lengths of corresponding RHO vectors, in steps of the Vector Adjustment Increment, until force equilibrium between the two vehicles is achieved within the Vector Force Tolerance. Collision forces are computed from the A and B stiffness coefficients of the crushed surfaces; friction forces tangent to the crush surface are computed using the inter-vehicle friction coefficient (set per vehicle pair in the Vehicle Mesh dialog), reduced linearly below the Minimum Velocity for Friction. Restitution is applied using the parametric model with coefficients \(C_0\), \(C_1\) and \(C_2\). (See references 21, 21a and 22 for the full derivation.)
The crush-force relationship¶
Along each RHO vector the crush force per unit width — the pressure, in the sense used by the collision model — follows the vehicle's A and B stiffness coefficients for the surface engaged:
where \(C\) is the crush depth along that vector and \(C_0\) is a small initial-crush threshold of 0.5 in.
NOTE: The second branch matters more than its size suggests. The A coefficient is the force needed to begin producing damage, so the unmodified relationship would jump from zero to A the instant contact occurred. Ramping linearly up to A over the first half inch removes that step, which would otherwise appear as a spike in the collision force at first contact.
Crush is taken along the vector's deformation components. Where the vector engages a single face it is the deformation normal to that face; in the corner region, where a vector engages both a side and an end, the two are blended so that the crush varies smoothly around the corner rather than switching abruptly from one face to the other.
Resolving the collision force onto each vehicle¶
Once a vector pair is in equilibrium, the force carried along it is resolved into each vehicle's own frame. The normal force acts along the crush surface, and a friction force acts tangentially to it, so in the struck vehicle's frame
where \(F_n\) is the normal force, \(\psi_s\) the orientation of the crush surface and \(\mu_t\) the effective inter-vehicle friction coefficient (the entered value, reduced in proportion to sliding speed below the Minimum Velocity for Friction). The moment about that vehicle's CG follows from the mid-point of the vector pair:
The force on the other vehicle is the same force rotated through the relative heading of the two vehicles and reversed, so that Newton's third law holds exactly at every vector pair rather than only in the sum. Its moment is then taken about its own CG using its own vector mid-point — which is why two vehicles in the same collision generally see different moments from the same force.
Restitution¶
At the end of each timestep every crushed vector is allowed to rebound toward its undamaged length. The restitution is a fraction computed from that vector's crush depth,
using the three restitution coefficients from the calculation options, and the restored length is a blend between the crushed and original lengths:
so \(e = 0\) leaves the vector fully crushed and \(e = 1\) restores it completely. Restitution is applied only while the crush depth is below the vertex of the parabola, \(C_1/2C_2\); beyond that depth the vector does not rebound.
Reaching force equilibrium¶
The forces along a pair of corresponding RHO vectors must be equal and opposite. They are made so by iteration. At each pair, the two pressures are compared, and while
— where \(\lambda\) is the Vector Force Tolerance — both vectors are shortened and the pressures recomputed. The iteration stops as soon as the difference falls within the tolerance.
The amount by which the vectors are shortened depends on how well matched the two vehicles are:
- Comparable stiffnesses — a single Vector Adjustment Increment, \(\Delta\rho\), per pass.
- Stiffnesses differing by more than a factor of two — a single increment would converge far too slowly, so the program estimates how many increments are needed from the combined stiffness of the pair and jumps most of the way in one step:
- Either vehicle still below the initial-crush threshold — the increment is reduced to 5% of \(\Delta\rho\), because the pressure is changing rapidly along the ramp described above and a full increment would overshoot. The same estimate is used where the A coefficients differ by more than a factor of two.
Up to 3200 adjustments are allowed at a vector pair. Exceeding that is a fatal error, and it means the tolerance is too tight for the stiffnesses and increment in use — see the guidance under Chapter 6.
Trajectory Phase¶
All of the external forces applied to the vehicle which direct its motion during the pre- and post-impact phases are applied at the tires.
NOTE: The tire forces continue to exist during the collision phase.
Accurate computation of the tire forces is essential. Unfortunately, pneumatic tires do not behave in an easy-to-calculate manner. Rather, these properties are functions of several variables, and are extremely non-linear.
In order to calculate tire forces, EDSMAC4 uses the Fiala tire model [10]. This model has seen use in many popular simulations, and is used in EDSVS, EDVTS, and EDCRASH. The user supplies the tire cornering stiffness, \(C_\alpha\), and tire/road slide friction data. The friction may be velocity dependent. A friction circle is used to limit longitudinal and lateral tire forces during combined braking and steering.
EDSMAC4 allows the vehicle to accelerate, brake and steer. The attempted acceleration, braking and steering are supplied by the user in tabular form using the Event Editor. It is important to understand these driver controls result in attempted forces; the Fiala tire model determines if these forces are sustainable at the tire-road interface and accounts for the condition if the available force is exceeded.
EDSMAC4 models quasi-static longitudinal and lateral load transfers. This is accomplished by applying the current inertial longitudinal and lateral forces at the vehicle's CG elevation, thus producing a pitch and/or roll moment. Lateral load transfer is apportioned to the front and rear axles using the vehicle's roll couple distribution. Roll couple distribution is primarily a function of the front-to-rear lateral stiffness ratio of the front and rear suspensions.
The transfers are computed as follows. Longitudinal transfer is taken about the wheelbase and is equal and opposite front to rear:
where \(h\) is the CG height, \(L\) the wheelbase and \(a_x = \dot u - v r\) the vehicle-fixed longitudinal acceleration. Lateral transfer is formed at each axle using that axle's share of the roll couple:
with \(\gamma\) the roll couple distribution at that axle — the entered value at the front, its complement at the rear — and \(T\) that axle's track width. The road pitch and roll terms hold a vehicle statically correct on a grade or a superelevated surface.
The axle loads are then split between the two wheels, and divided again where an axle is a tandem:
NOTE: A towed unit's lateral transfer is computed differently, and does not use roll couple distribution. A trailer's roll moment is applied through the hitch rather than through its own suspension geometry, so its lateral transfer is formed from the connection force acting at the hitch height together with the inertial term taken about that height. Roll couple distribution entered for a trailer therefore has no effect.
NOTE: The tire forces used in the transfer are those of the previous timestep. A load transfer is a consequence of the tire forces, and those forces in turn depend on the transferred load, so the circularity is broken by lagging one of them by a step.
(updated: earlier editions described the load transfer in prose without giving the relationships, and did not mention that towed units are treated differently.)
EDSMAC4's vehicle model allows the user to study vehicles with tandem axles and dual tires.
Tire force calculation¶
At each wheel, the friction coefficient is the tire's slide friction scaled by the friction multiplier of the surface beneath it — which the hydroplaning model further reduces when one is selected — and then adjusted for speed:
where \(s\) is the wheel's total slip speed and \(C_\mu\) the speed reduction factor. A zero speed reduction factor leaves the friction speed-independent. The force available at the tire-road interface is then
The attempted longitudinal force is the sum of the throttle and brake table entries at the current time; for the Percent Available Friction table method those entries are fractions and are multiplied by \(F_{Avail}\). The longitudinal force developed, \(F_c\), is the attempted force limited to what the tire can supply. At forward speeds below 2 in/sec the braking force is scaled by \(\left|u\right|/2\), so that a nearly stopped wheel does not develop full braking force; drive force is not reduced at low speed.
Whatever longitudinal force is used is unavailable laterally. The remaining lateral capacity follows the friction circle:
The Fiala model is applied to that remaining capacity through the non-dimensional sideslip parameter
where \(\delta_{dual}\) is 1 at a dual-tire wheel location and 0 otherwise, giving
The two components are then resolved through the wheel's steer angle, \(\delta\), into vehicle-fixed components:
(updated: earlier editions described the tire model in prose only. Note that the dual-tire factor is applied to the cornering stiffness rather than to the available force, which differs from EDSVS and EDVTS, where the factor multiplies the force; and that the friction circle is built on the slide friction rather than a peak friction value, so EDSMAC4 has no distinct peak-versus-slide behavior.)
(updated: when the Hydroplane Model calculation option is set to NASA, the available tire friction is further modified at each tire travelling over a water polygon. The NASA model predicts the hydroplaning speed from the tire's inflation pressure (\(182.16\sqrt{P}\) in/sec, \(P\) in lb/in²); above that speed the tire's friction switches to the water polygon's friction value. The extra drag of displacing water is not modeled. See EDSMAC4 Calculation Options.)
Impact and Separation Times¶
The impact and separation times reported in the Accident History bound the collision phase. They are found by testing two conditions at every timestep.
The first condition is geometric: the original (undamaged) perimeter of one vehicle must overlap the other, which is tested by checking whether any corner of either vehicle lies inside the other. The second condition is selected by the Accident History Basis calculation option:
- Impact Force (default) — inter-vehicle collision force is acting at one or more points on the two vehicles.
- Acceleration — the total acceleration of either vehicle exceeds the Threshold value (default 1 g).
The impact time is the first time at which both conditions hold. Note that both are required: with the Acceleration basis, the collision phase still cannot begin before the perimeters overlap. At this time the integration timestep changes to the Vehicle Collision Integration Timestep.
The impact conditions stored for the Accident History — time, position, heading, total speed, forward and lateral components and yaw rate for both vehicles — are latched at that first detection and are never updated by later contact between the same pair. In a multiple-impact event the impact row therefore always describes the first contact between that pair. Impact and separation are tracked separately for each pair of vehicles, so a three-vehicle event produces one impact and one separation entry for each colliding pair.
The end of the collision phase uses the reverse test:
- Impact Force — no collision force is acting between the two vehicles.
- Acceleration — the total acceleration of both vehicles is below the Threshold.
The separation condition must be satisfied on consecutive timesteps before separation is declared: one timestep with the Impact Force basis, six with the Acceleration basis. Separation is then declared at the next time falling on a Vehicle Separation Integration Timestep boundary, at which point the timestep changes and the separation conditions are recorded. The reported separation time consequently lags the true end of contact slightly — by about one timestep with the Impact Force basis, and by up to six collision timesteps plus one separation timestep with the Acceleration basis.
NOTE: The acceleration used by the Acceleration basis is the vehicle's total acceleration, which includes tire forces. If the Threshold is set low enough that hard braking or cornering alone exceeds it, separation can never be declared and the event will remain in the collision phase — and at the collision timestep — for the rest of the run. The 1 g default is above what tires normally produce, but a reduced threshold on a high-friction surface can reach this condition.
Repeated impact and separation¶
Nothing restricts the collision phase to a single occurrence. If the vehicle perimeters are still overlapping when contact resumes after separation has been declared, the collision phase simply restarts: the integration timestep returns to the Vehicle Collision Integration Timestep and the separation test begins again. With the Impact Force basis, where one force-free timestep is enough to declare separation, a disengagement in which contact flickers on and off can cycle through this several times within a few tens of milliseconds.
Only the first and the last of these appear in the Accident History. The impact conditions are recorded once, at first contact, and are never updated; the separation conditions are re-recorded at every separation, so the values shown are those of the final one. The report therefore spans the event as a whole — first contact to last separation — and gives no indication of intermittent contact in between.
Collision Severity Results: PDOF, Delta-V and Peak Acceleration¶
The PDOF, delta-V and peak acceleration reported in the Damage Data report are computed by one of two entirely separate methods, selected by the Damage Data Format calculation option. The two methods do not produce identical numbers; the Traditional format is retained for comparison with results from earlier releases.
In both cases the PDOF is reported in the conventional CDC sense (see reference 8) — a 12 o'clock PDOF is a force directed from front to rear, so the reported angle is 180 degrees opposite the direction in which the force acts on the vehicle. Angles are measured in the vehicle-fixed frame with zero forward and positive toward the right side, and are reported in the range ±180 degrees.
Collision Data format (default)¶
Each row of the Collision Summary table is one collision pulse — one period of sustained contact between that vehicle and one collision partner. For each pulse the report gives the pulse number, the vehicle or environment struck, the start and end times and the duration, the peak acceleration, the peak collision force, the delta-V and the PDOF, followed by the CDC, damage width and offset, maximum crush and the crush profile measurements.
How a pulse is delimited¶
The rule that decides where one pulse ends and the next begins is separate from the one that fixes the impact and separation times in the Accident History, and it is far less sensitive. The two reports can therefore describe the same contact differently.
Start. A pulse opens on the first timestep at which any point on the vehicle carries collision force from that partner. There is no force threshold — any non-zero contact force opens the pulse.
Continuation. On every timestep that contact persists, a provisional closing deadline is set 0.025 seconds ahead. The pulse is closed only once contact has been absent continuously until that deadline passes. Because the deadline is re-set on each contact timestep it is a rolling one: the pulse survives until a full 0.025 seconds have elapsed with no contact at all, however many times contact has come and gone before that. If contact resumes before the deadline, the vehicle rejoins the pulse already in progress against that partner instead of opening a new one.
The intent is that the intermittent contact typical of two vehicles disengaging is reported as one collision rather than a string of small ones. The consequence is that any contact separated from the previous one by less than 0.025 seconds is absorbed into it: its impulse contributes to that pulse's PDOF, its acceleration to the pulse's delta-V, and the vertices it touches to the pulse's CDC, with nothing in the report to show that a distinct contact occurred.
NOTE: This matters when the later contact is severe. A secondary strike reaching a third of the peak collision force, occurring twenty milliseconds after contact was lost, is merged into the preceding pulse and appears nowhere as an event of its own. If the trajectory suggests a separate secondary contact, examine the collision force in the Variable Output, Kinetics group rather than relying on the pulse count.
A gap longer than 0.025 seconds does close the pulse, and the next contact opens a new one, producing a second row.
End. The time in the End column is not the time the pulse was closed. It is the last timestep at which the vehicle had not been declared separated from that partner, using the Accident History criterion described above; Length is simply End minus Start. Because the collision phase may be entered more than once, the End time follows the last contact episode rather than the first separation. The pulse's own closure — 0.025 seconds after the final contact — governs only whether a later contact starts a new row, and is never reported.
Limits. Up to ten pulses are tracked for each vehicle; an eleventh ends the run with a message. A pulse still open when the run ends is counted as complete and a diagnostic message is issued.
PDOF. The inter-vehicle collision forces acting on the vehicle are accumulated over the pulse, giving the total collision impulse in vehicle-fixed coordinates. The PDOF is the direction of that impulse:
where \(J_x\) and \(J_y\) are the forward and lateral components of the accumulated collision impulse. A zenith angle, the elevation of the impulse out of the horizontal plane, is also stored, but since EDSMAC4 computes no vertical collision force it is always zero. The clock direction used for the first two characters of the CDC is this same angle rounded to the nearest hour. Both are frozen once separation is declared.
NOTE: The PDOF is derived from the collision force alone. Tire forces do not affect it.
Delta-V. The vehicle's acceleration components are integrated through the pulse using the trapezoidal rule, accumulating a velocity-change vector in vehicle-fixed coordinates. The reported delta-V is the largest magnitude that vector reaches at any point during the pulse:
Two properties of this calculation are worth understanding:
- The acceleration integrated is the vehicle's total acceleration, not the collision acceleration alone. Tire forces acting during contact therefore contribute to the reported delta-V. In a severe impact the contribution is negligible — a fraction of a g against tens of g — but in a low-severity sideswipe with heavy braking it is not.
- The components are integrated in the vehicle-fixed frame while the vehicle is yawing. For the short durations typical of a collision the yaw change is small and the result is very close to the change in the earth-fixed velocity vector, but a long, rotating sideswipe will accumulate a delta-V that differs from the vector difference between the velocities at impact and separation.
Peak acceleration is the largest total acceleration reached at any timestep on which collision force was acting; peak force is the largest total collision force, and the time at which it occurred is stored with it. As with delta-V, the peak acceleration includes the tire contribution.
Traditional format¶
The Traditional format reproduces the original SMAC reporting, which works from the acceleration history rather than from the contact force. It is available only for events with more than one vehicle.
Acceleration peaks. A peak is recorded each time the vehicle's total acceleration rises above the Threshold value and falls back below it. The peak magnitude, its forward and lateral components and its time are stored. Up to ten peaks are kept, after which a message is issued; the peaks are then sorted into descending order of magnitude.
Delta-V of a peak is the integral of the acceleration magnitude over the peak, again by the trapezoidal rule:
Because this is a scalar integral rather than a vector one, a peak whose direction rotates produces a larger delta-V than the corresponding change in velocity. To limit this, the direction of the acceleration is tracked in sectors, and the contribution of an acceleration that has changed sector is suppressed.
PDOF of a peak is the direction of the acceleration at the instant of the peak, converted to the conventional sense as described above.
Matching peaks to damage. Each damage range found on the crush profile is matched to the acceleration peak whose clock direction is closest to the mid-point of that range. If no peak lies within 60 degrees of the mid-point, the largest peak is used instead and a message is issued — a common and harmless outcome for sideswipes and secondary impacts.
Total delta-V of a damage range is the sum of the delta-Vs of all acceleration peaks whose clock direction lies within one hour either side of the matched peak's clock direction. Delta-V arising from separate impacts in the same general direction is therefore combined into a single figure for that damage range. If only one peak was found, all delta-V is assigned to it.
NOTE: The Threshold value is used by the Traditional format regardless of the Accident History Basis. The dialog disables the Threshold field when the basis is Impact Force, but the stored value — 1 g by default — still sets the minimum acceleration at which an acceleration peak is recognized and delta-V is accumulated.
Assumptions¶
In order to provide a useful analysis without becoming burdensome and overly complex, EDSMAC4 makes several simplifying assumptions. If the user is to use EDSMAC4 properly, it is important these assumptions and their consequences be understood. In some cases, data which violate these assumptions will cause a fatal error, along with a message indicating the reason for the error. In other cases, the error is not with the data but with the use of the program under conditions which violate the assumptions inherent to the computations. EDSMAC4 will issue results which may not be valid for the circumstances of the accident. Before using EDSMAC4, be sure your accident is within the scope of EDSMAC4's original design.
A thorough list of assumptions and how to properly use the program can be found in the References [1-7, 11-13, 15]. Some of the major program assumptions are defined below.
Surface Slope Up To 15 Degrees¶
The HVE physical environment allows EDSMAC4 to consider the general effects of roll and pitch, up to about 15 degrees (the limit is the result of the small angle assumption). Negotiating slopes within this range is generally well modeled. However, traveling off an embankment is completely beyond the scope of EDSMAC4.
No Rollover Allowed¶
EDSMAC4 is a 3-degree-of-freedom analysis and does not consider roll or pitch degrees of freedom. If a vehicle rolls over, the equations of motion that apply during the trajectory phases no longer apply, since forces are no longer applied at the tires but are applied between the vehicle exterior and road. EDSMAC4 makes no provision for this condition.
All External Forces Applied at Tires¶
Other than during the collision, all forces which affect vehicle motion are applied at the tire/road interface. Aerodynamic effects are ignored. This assumption may become a factor at excessive speeds. In addition, forces caused by contact between the undercarriage and ground (sometimes resulting from impact damage) are not considered.
Homogeneous Vehicle Exterior¶
The collision phase analysis assumes there is a linear force vs crush relationship for each vehicle surface (front, sides or rear). The stiffness is assumed to be uniform for the entire surface. This means, for example, that the side stiffness is the same at the front fender, wheels, bumpers, doors and quarterpanel.
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