**Blog Post | Motorcycle T-bone Impacts**

Last update: June 17, 2017

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Virtual CRASH can be used to reconstruct motorcycle impact cases quickly and easily. In this blog post we will use Virtual CRASH to reconstruct “Case Study 1” presented in “Linear and Rotational Motion Analysis in Traffic Crash Reconstruction” by Keifer, Conte, and Reckamp [1], which can be purchased at the Institute of Police Technology and Management book store. We will solve this case both analytically and by using Virtual CRASH simulations.

In this case, the driver of a 1991 Buick Century sedan was in the process of making a left turn, driving across two northbound lanes as she was making her way to the median. The driver of the Buick stopped, with the rear of the Buick still occupying a portion of the inside northbound lane. The Buick was then struck by a 1997 Kawasaki Vulcan 750, which was headed north in the inside through lane. As a result of the impact, the Buick rotated counterclockwise, as viewed from above, with an angular displacement of 160 degrees.

The Buick was struck near the driver side rear wheel (see figures below). Damage the Buick’s C pillar was noted. The speed limit for the northbound lanes was 60 mph. The primary task, in this case, is to estimate the motorcycle’s pre-impact speed.

In order to solve for the pre-impact speed of the motorcycle, the authors used a rigid body dynamics based analysis, treating the motorcycle as a point-mass, but taking into account the torque caused by the impulse exchange, whose lever-arm is estimated from crush damage. Starting from our single-body collision model treatment in Appendix 1, we will re-derive the relation used to estimate the motorcycle’s pre-impact speed. We first start by writing the expression for the post-impact relative velocity (separation velocity), \(\bar{v}^{P}_{Rel,f}\), for at the point of contact for the two vehicles. This is given by equation (33) of Appendix 1 of the Virtual CRASH User's Guide.

$$\bar{v}^{P}_{Rel,f} = \bar{v}^{P}_{1,f} - \bar{v}^{P}_{2,f} \tag{1} $$

$$\bar{v}^{P}_{Rel,f} = \bar{v}^{P}_{Rel,i} + {\bar{J}_{1} \over \bar{m}} + \left({\bar{r}_{1} \times \bar{J}_{1} \times \bar{r}_{1} \over I_{1}} + {\bar{r}_{2} \times \bar{J}_{1} \times \bar{r}_{2} \over I_{2}}\right) \tag{2} $$

where \(\bar{m}\) is the system reduced mass.

Here we label the Buick vehicle 1 and the motorcycle as vehicle 2.

Following the model used in reference [1], we will treat the motorcycle as a point mass (zero lever-arm approximation). Therefore we have:

$$ \bar{r}_{2} = 0$$

The lever-arm between the Buick's CG and the point of impact is given by:

$$ \bar{r}_{1} = -D \hat{y}$$

where the vehicle 1 lever-arm is anti-parallel with the \(\hat{y}\) axis.

The impulse on Buick is given by:

$$ \bar{J}_{1} = m_{1}\Delta v_{1} \hat{x} \tag{3} $$

where the impulse vectors are aligned with the \(\hat{x}\) axis. The triple vector product (equation 30 of Appendix 1) is given by:

$$ \bar{r}_{1} \times \bar{J}_{1} \times \bar{r}_{1} = \bar{J}_{1} (|\bar{r}_1|^{2}) - \bar{r}_{1}(\bar{r}_{1} \cdot \bar{J}_{1}) \tag{4} $$

$$ \bar{r}_{1} \cdot \bar{J}_{1} = (m_{1} \Delta v_{1} D) \left( \hat{x} \cdot \hat{y} \right) =0 \tag{5} $$

$$ \bar{r}_{1} \times \bar{J}_{1} \times \bar{r}_{1} = m_{1} \Delta {v}_{1} D^{2} \hat{x} \tag{6} $$

The authors of [1] model of the vehicles as coming to a common velocity at the point of contact, with no restitution. Thus, we have:

$$\bar{v}^{P}_{Rel,f} = 0 = \bar{v}^{P}_{Rel,i} +{m_{1} \Delta v_{1} \over \bar{m}} \hat{x} + \left({m_{1} \Delta v_{1} D^{2} \over I_{1}} \right) \hat{x} \tag{7} $$

The Buick was initially at rest, therefore, the pre-impact closing velocity is given by:

$$\bar{v}^{P}_{Rel,i}=\bar{v}^{P}_{1,i} - \bar{v}^{P}_{2,i}=- \bar{v}^{P}_{2,i} \tag{8}$$

Substituting (8) into (7), we can write an expression for the motorcycle's initial velocity:

$$ \bar{v}^{P}_{2,i} = \left( {1 \over \bar{m}} + { D^{2} \over I_{1}} \right) m_{1} \Delta v_{1}\hat{x} \tag{9}$$

So, we now have an expression for the motorcycle's initial velocity in terms of the Buick's change-in-velocity. We will obtain an estimate for the change-in-velocity by using the Buick's post-impact angular displacement. First, let's write an expression for the post-impact angular velocity (yaw rate) for the Buick. This is given by equation (27) in Appendix 1:

$$ \bar{\omega}_{1,f} = \bar{\omega}_{1,i} + {\bar{r}_{1} \times \Delta \bar{p}_{1} \over I_{1}} = {m_{1} \Delta v_{1} D \over I_{1} } \hat{z} \tag{10}$$

Solving for the Buick's change-in-velocity in terms of its final angular velocity:

$$ \Delta v_{1} = {I_{1} \over m_{1} D} \cdot \omega_{1,f} \tag{11} $$

The authors of [1] model the post-impact rotational kinetic energy as being solely dissipated by the effective torque caused by friction at the ground-tire contact patch:

$$ {1 \over 2} I_{1} \omega_{1,f}^{2} \approx \int_{0}^{\Delta\theta} d\theta \cdot W_{1} f_{r}{L_{WB} \over 2} = {W_{1} f_{r} L_{WB} \Delta\theta \over 2} \tag{12} $$

where \(f_{r}\) is the "coefficient of rotational friction", \(L_{WB}\) is the wheelbase of the Buick, and \(\Delta\theta\) is the Buick's post-impact angular displacement. Note, the authors of [1] obtain values for \(f_{r}\) based on an EDSMAC simulation study, whereby various tbone impacts were staged, and the resulting angular displacements and final yaw rates where analyzed (see reference [2]). These were converted into "rotational friction factors" by using the effective torque model above. These friction factors are then multiplied by the ground contact coefficient of friction to obtain the coefficient of rotational friction used in the effective torque model (see Chapter 4 of reference [1]). Because EDSMAC, with a 2D collision model, is a different simulation model than Virtual CRASH (a fully 3D collision and 3D vehicle dynamics simulator), we can expect to see slightly different behavior between the two during post-impact motion. We should therefore, anticipate differences between full Virtual CRASH simulated results and the analytic results presented here. In addition, and more importantly, the impact configuration of the subject t-bone impact does not exactly match the simulation conditions of reference [2] which are used to derive the rotational friction factors, as the subject motorcycle strikes the Buick with a 70 degree difference in heading compared to a 90 degree difference in the reference [2] dataset. Further below, we illustrate this initial heading difference mismatch as the cause for the largest differences between the analytic solution and the Virtual CRASH simulation.

Solving for the post-impact angular velocity needed to produce this angular displacement is given by:

$$ \omega_{1,f} = \sqrt{ {W_{1} f_{r} L_{WB} \Delta\theta \over I_{1}}} \tag{13} $$

Substituting this expression into equation (11), we have:

$$ \Delta v_{1} = {I_{1} \over m_{1} D} \cdot \sqrt{ {W_{1} f_{r} L_{WB} \Delta\theta \over I_{1} } }= {\sqrt{I_{1} W_{1} f_{r} L_{WB} \Delta\theta } \over m_{1} D } \tag{14} $$

Finally, using (14) in (9), we have our expression for the motorcycle's pre-impact velocity:

$$ v_{2,i} = \left({m_{1} \over \bar{m}} + {m_{1} D^2 \over I_{1}}\right) \cdot {\sqrt{I_{1} W_{1} f_{r} L_{WB} \Delta\theta} \over m_{1} D} \tag{15}$$

which can be written as:

$$v_{2,i} = \left({1 \over D m_{1}} + {1 \over D m_{2}} + {D \over I_{1}}\right) \cdot \sqrt{I_{1} W_{1} f_{r} L_{WB} \Delta\theta} \tag{16} $$

This is equivalent to Equation 2-73 in [1].

The authors of [1] give the following values for the vehicles:

$$W_{1} = 2,954 ~ lbs$$

$$W_{2}=743 ~ lbs$$

$$L_{WB}=8.75 ~ ft$$

$$ f_{r} = 0.43$$

$$I_{1} = 1,823.4 ~ lbs \cdot ft \cdot s^{2}$$

$$D = 5.14~ft $$

Roadway friction is reported to be 0.6.

Plugging these values into equation (16), we have:

$$v_{2,i} = \left({1 \over (5.14~ft)(2,954~lbs/g)} + {1 \over (5.14~ft)(743~lbs/g)} + {5.14~ft \over 1,823.4~lbs \cdot ft \cdot s^{2}}\right) \times $$

$$\sqrt{ (1,823.4 ~ lbs \cdot ft \cdot s^{2})(2,954 ~ lbs) (0.43) (8.75 ~ ft)\left (160~deg \times {\pi \over 180~deg}\right)} $$

which simplifies to:

$$v_{2,i} = \left(0.013~slug^{-1}ft^{-1}\right)\times\left(7.521 \times 10^{3} slug \cdot {ft^2 \over s}\right) = 100.5~{ft\over s} = 68.6~mph $$

It is therefore estimated that the motorcycle's pre-impact speed was 68.6 mph.

**Post-Impact Velocity Scan**

As has been demonstrated in our various RICSAC test reconstructions, we can quickly obtain first estimates for the post-impact velocity of our Buick by taking advantage of Virtual CRASH's trajectory model. With only the Buick in place, simply scan across possible final velocities and yaw rates that land the Buick in the needed post-impact position. In this particular case, since we do not have tight constraints on the exact final position of the Buick, but only its total post-impact travel distance and final orientation, we must take particular care not to treat the final velocity and yaw rate as uncorrelated.

A circle is drawn, centered about the initial CG position of the Buick, and extending 21 ft to indicate the target rest distance from impact. The authors of [1] only provide the total travel distance, but not the specific post-impact trajectory of the Buick. Our ultimate goal is to impact the Buick and find a solution that lands the Buick within the circle at final rest. Because the Buick will tend to roll backward after its rotation has stopped, we will apply braking only after rotation stops. This is in keeping with the approach taken by reference [1] in that the rotation factors were obtained based on EDSMAC simulation runs without vehicle braking enabled.

Recall, the final velocity, in this case, is simply equal to the Buick's \(\Delta v\). We can take into account the correlation between \(\Delta v\) and \(\omega_{f}\) by starting with equation (11) above. Simply solve for \(\omega_{1,f}\):

$$\omega_{1,f} = \left(m_{1} D \over I_{1}\right) \cdot \Delta v_{1} $$

Plugging in the values for \(m_{1}\), \(D\), and \(I_{1}\) we have:

$$\omega_{1,f}[rad/s] = \left(0.378\right) \cdot \Delta v_{1}[mph] $$

where \(\omega_{1,f}\) is given in units of rad/s and \(\Delta v_{1}\) is in units of mph.

The scan is shown below:

With this analysis, we arrive at a \(\Delta v_{1} \) value of 10.4 mph. With an estimate for \(\Delta v_{1}\), we can now simply plug this into equation (9) to obtain our estimate for \(v^{P}_{2,i}\). This is given by:

$$ \bar{v}^{P}_{2,i} = \left( {1 \over 18.441~slugs} + { (5.14~ft)^{2} \over (1,823.4 ~ lbs \cdot ft \cdot s^{2})} \right) (91.533~slugs) (10.4~mph) \hat{x} $$

or

$$ \bar{v}^{P}_{2,i} =65.4~mph$$

which is within 4.6% of the analytic solution above which relies on EDSMAC simulation results.

We can understand the 4.6% difference between our simulation and the analytic solution as having mostly to do with the origin of the rotational friction factors, which were derived in reference [2] for perpendular impacts. Indeed, rearranging our system such that our Buick's heading is 90 degrees to the sphere (see below), we obtain a post-impact speed \(\Delta v_{1}\) for the Buick of 10.85 mph with \(\omega_{1,f}\) = 4.101 rad/s. This gives an implied value for \(\bar{v}^{P}_{2,i} =68.3~mph\). This is within 0.5% of the analytic solution, thus demonstrating the import differences between the subject case and the simulation conditions used in reference [2]. This also demonstrates that the trajectory models of EDSMAC and Virtual CRASH are in good agreement for this post-impact vehicle motion.

**Point mass simulation**

We can make an immediate connection between the above effective point mass model and a Virtual CRASH simulation by simulating a rigid body sphere impacting a Buick Century. Below we show our simulation setup in Virtual CRASH 4. Since the Buick Century vehicle model is not contained in the vehicle database, our Buick Century was obtained from 3dcadbrowser.com. The Buick's geometrical and inertial properties are obtained from Expert Autostats. The yaw moment of inertia is set to \(I_{1}\) above, and the pitch and roll moments of inertia are linearly rescaled to account for the increased vehicle weight \(W_{1}\) from the curb weight. A 1 ft radius sphere is created and converted into a rigid body using Create > Physics > Make Rigid Body From Selection. The sphere's weight is set to \(W_{2}\). The sphere's moments of inertia are \(10^{7} lbs \cdot ft \cdot s^{2}\) in order to ensure that any torques acting on the sphere are essentially ignored, thus giving us a point mass like approximation.

In order to further approximate the point mass calculation, the sphere's initial height is set to the Buick's CG height. The sphere's contact model is set to Kudlich-Slibar (recall "default-auto" is the default contact model for objects converted into rigid bodies), and a user contact ees object is created, whose height is specified at the Buick's CG height. The sphere's initial position is adjusted such that its volume overlaps with the Buick's volume. The depth of penetration parameter is set to 0 seconds to ensure that the impulses are exchanged immediately at the start of the simulation so that the sphere does not begin falling due to gravity prior to impact. The impulse centroid is positioned such that the moment arm is effectively 5.14 ft from the Buick's CG. The coefficient of restitution is set to 0 (see below).

The video below shows the workflow to obtain a solution for this scenario:

The initial velocity required to produce the required rotational displacement is 65.6 mph. This is within 4.3% of the analytic solution above. The Buick's post-impact velocity is 10.402 mph, and its final yaw rate is 3.939 rad/s, giving a ratio of 0.379 which is in excellent agreement with our expectation from above.

Again, we can understand the 4.3% difference between our simulation and the analytic solution as having mostly to do with the origin of the rotational friction factors. Once again, rearranging our system such that our Buick's heading is 90 degrees to the sphere, we obtain a pre-impact speed for the sphere of 68.5 mph. This is within 0.2% of the analytic solution.

**Full Simulation**

Next, we conduct a full simulation of the motorcycle with rider system. We use the 1994 Moto Guzzi California as an exemplar for the Kawasaki (saddle bags are deleted).

Kawasaki specifications can be obtained from services such as Lightpoint Data (see below).

The motorcycle's weight is set to the total weight given in [1] of 523 lbs. The multibody model is set into the rider pose and given the specified weight of 220 lbs. The auto-ees restitution is adjusted to 0 and depth of penetration is left to the default value of 30 msec. Finally, with our rider and motorcycle system aligned with the x-axis, we scan pre-impact velocities until we obtain the correct angular displacement for the Buick. Again, braking is engaged on the Buick once it has undergone is maximum angular displacement. With a pre-impact speed of 69.2 mph, we satisfy our post-impact constraints for the Buick. Our final result is 0.87% different from the analytic solution. Note we focus here only on the Buick's post-impact motion, as the authors of [1] did not provide enough information to attempt to match final conditions for the rider and motorcycle.

Note a full 3D simulation accounts for effects not considered by a 2D model. For example, the rider is not rigidly attached to the motorcycle, and so takes about 40 msec to make contact with the motorcycle and Buick after first contact. There is also some sensitivity to the motorcycle's CG height (1.25 ft is the default for all motorcycles in the database), as well as to the automatically positioned impulse centroid z location (1.209 ft). In particular, the auto-positioned impulse centroid z location adds a z-component to the lever arm, and thus contributes a roll torque that is not accounted for in the analytic solution. This same effect is explored in the vCRASH Academy video: Pre-Enlistment Fitness Challenge | An Inline Collision Check. Making adjustments to the impulse centroid height and the motorcycle's CG height up to the Buick's CG height causes variations in the needed motorcycle pre-impact speed from 67 mph to 70.9 mph, or -2.3 to 3.4% of the analytic solution value. Thus, we have good agreement between the two approaches despite differences between a full 3D simulation and an effective 2D point mass model, and differences in the subject accident configuration and the configuration used in reference [2] to obtain the rotational friction values. A video of the workflow for this case can be found below.

**References**

[1] "Linear and Rotational Motion Analysis in Traffic Crash Reconstruction", O. Kiefer, R. Conte, and B. Reckamp, IPTM, ISBN: 1-884566-70-7.

[2] SAE 2005-01-1203, "A Parametric Study of Frictional Resistance to Vehicular Rotation Resulting from a Motor Vehicle Impact", Keifer et al, SAE 2005-01-1203.

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