Estimating vehicle inertia and centre-of-gravity height using the NHTSA light vehicle inertial parameter database
PREFACE: this page is best experienced on desktop
Vehicle dynamic simulation relies on having known information about the vehicle under study. Vehicle mass and inertial properties are required for any meaningful handling simulation. These values are non-trivial to obtain and makes use of highly specialized equipment to measure. Without access to a kinematics and compliance machine or an inertia measurement rig, vehicle dynamic simulation is out-of-reach to most individuals outside of the automotive industry.
From 1992 to 1998, the National Highway Traffic Safety Administration (NHTSA) published a series of papers with a database of measured mass and inertial properties for light passenger vehicles. Given that the underlying architecture of the modern light vehicle has not fundamentally changed, we can use this information to identify trends in the data to better inform an estimation of mass and inertial properties of a given vehicle.
Update 2022/08/07 A new study using NHTSA Light Vehicle Inertial Parameter Database and NCAP Rollover Stability measurements is now available at the link below.
This new study uses a much larger dataset and includes CG height measurements for up to year 2020. It is recommended to view this article if your primary interest is in identifying trends in vehicle centre of gravity height.
Vehicle properties
The mass properties of a four-wheeled vehicle typically refer to its mass and the position of its centre-of-gravity (CG). We will assume that the lateral position of the CG lies along the XZ-plane (ie. the vehicle centre plane). This yields four parameters of interest.
- The vehicle mass, \(m\) in [\(kg\)]
- The longitudinal distance from the CG to the front axle, \(a\) in [\(m\)]
- The longitudinal distance from the CG to the rear axle, \(b\) in [\(m\)]
- The vertical distance from the CG to the ground, \(h\) in [\(m\)]
Of the four parameters, only the vertical distance from the CG height to the ground (simply known as the CG height) is difficult to obtain. The remaining properties can be measured using a set of floor scales. Consequently, we will limit our discussion to focus on estimating the vehicle CG height.
The mass moment of inertia properties of an object can be described by a tensor, \(I\). In matrix form:
\[I = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \\ \end{bmatrix} = \begin{bmatrix} I_{xx} & 0 & I_{xz} \\ 0 & I_{yy} & 0 \\ I_{zx} & 0 & I_{zz} \\ \end{bmatrix}\]The diagonal terms represent the moments of inertia. The off-diagonal terms represent the products of inertia. Because we assume the vehicle is symmetric along the XZ-plane, we can set \(I_{xy} = 0\) and \(I_{yz} = 0\). This leaves us with four inertial parameters of interest.
- The roll inertia, \(I_{xx}\) in [\(kg \cdot m^2\)]
- The pitch inertia, \(I_{yy}\) in [\(kg \cdot m^2\)]
- The yaw inertia, \(I_{zz}\) in [\(kg \cdot m^2\)]
- The roll/yaw product of inertia, \(I_{xz}\) in [\(kg \cdot m^2\)]
All four inertial parameters are non-trivial to obtain and will be in scope of our discussion.
Data treatment
The NHTSA Light Vehicle Inertial Parameter Database contains measurements for a variety of different passenger vehicles in different loading loading conditions. Measurements that have ballast are excluded. If duplicate entry is found for a vehicle, only the first measurement is kept.
The database provides a vehicle type code to describe the chassis style. Unfortunately, no key is provided for type code in the database. The best guess legend key and sample vehicles are shown below.
- 4S - four door sedan - 1987 Ford Tempo, 1991 Honda Accord LX
- 2S - two door sedan - 1998 Chevrolet Metro, 1986 BMW 325i
- SW - station wagon - 1979 Datsun 210, 1986 Buick Century Estate
- PU - pick up truck - 1986 Chevrolet S-10 Pickup, 1998 Toyota Tacoma
- VN - minivan - 1992 Dodge Caravan, 1991 Toyota Previa LE
- MP - “multi-purpose” SUV - 1983 Chevrolet S-10 Blazer, 1998 Toyota 4Runner
- 3H - three door hatchback - 1991 Geo Metro, 1986 Mazda 323
- 5H - five door hatchback - 1983 Dodge Omni, 1983 Toyota Camry
- 2C - two door coupe - 1986 Toyota MR2, 1989 Pontiac Grand Am
Trends
With data from the NHTSA Inertial Parameter Database, we can observe trends with respect to vehicle mass. To begin, we will assess trends in vehicle centre of gravity height.
From this graph, we can see that the CG is likely to be higher in heavier vehicles. Observe the clustering of cars and trucks; trucks and SUVs tend to have a higher CG where the CG height increases faster with vehicle mass than in cars. The table below summarizes the results of the linear regression.
| Dataset | Transfer Function | R-Squared | Count |
|---|---|---|---|
| All Vehicles | y = 0.00018x + 0.34338 | 0.723 | 204 |
| Cars | y = 0.00005x + 0.46983 | 0.405 | 73 |
| Trucks | y = 0.00014x + 0.40760 | 0.631 | 131 |
Next, we will assess the vehicle inertial properties. All four inertial parameters (\(I_{xx}\), \(I_{yy}\), \(I_{zz}\) and \(I_{xz}\)) are shown in the graph below. Note that the roll/yaw product contains fewer data points. This is because only a select number of vehicles have the roll/yaw product measured.
The moments of inertia have good correlation with vehicle mass in consideration of all vehicle types. The relationship starts to diverge with higher vehicle mass, especially between minivans and SUVs. The roll/yaw product will likely be positive for most vehicles with the exception of pickup trucks. There is some correlation for cars but that relationship is weak. The results are summarized in the table below.
| Dataset | Parameter | Transfer Function | R-Squared | Count |
|---|---|---|---|---|
| All Vehicles | Roll, \(I_{xx}\) | y = 0.566x - 276.379 | 0.836 | 204 |
| Cars | Roll, \(I_{xx}\) | y = 0.497x - 181.445 | 0.857 | 73 |
| Trucks | Roll, \(I_{xx}\) | y = 0.609x - 355.532 | 0.760 | 131 |
| All Vehicles | Pitch, \(I_{yy}\) | y = 2.978x - 1697.108 | 0.834 | 204 |
| Cars | Pitch, \(I_{yy}\) | y = 3.079x - 1728.758 | 0.913 | 73 |
| Trucks | Pitch, \(I_{yy}\) | y = 3.182x - 2107.468 | 0.751 | 131 |
| All Vehicles | Yaw, \(I_{zz}\) | y = 2.961x - 1596.431 | 0.845 | 204 |
| Cars | Yaw, \(I_{zz}\) | y = 3.176x - 1754.164 | 0.920 | 73 |
| Trucks | Yaw, \(I_{zz}\) | y = 3.168x - 2021.319 | 0.770 | 131 |
| All Vehicles | Roll/yaw, \(I_{xz}\) | y = 0.049x - 13.996 | 0.078 | 41 |
| Cars | Roll/yaw, \(I_{xz}\) | y = 0.060x - 10.578 | 0.627 | 10 |
| Trucks | Roll/yaw, \(I_{xz}\) | y = 0.078x - 75.000 | 0.098 | 31 |
Static stability factor (SSF)
The static stability factor (SSF) is used as a measure of rollover resistance based on static vehicle properties. It is defined as:
\[SSF = \frac{T}{2h}\]Where \(T\) is the average vehicle track and \(h\) is the centre of gravity height from the ground. A higher static stability factor indicates higher resistance to vehicle rollover. It is convenient to describe the vehicle CG height in terms of its SSF because it is non dimensionalized to the vehicle geometry. The trend in static stability factor is shown in the graph and histogram shown below.
There is a weak negative correlation between SSF and vehicle mass when considering all vehicle types. Upon further inspection, this is because of the clustering of cars and trucks. Trucks and SUVs tend to have a lower SSF whereas cars tend to have a higher SSF.
| Static Stability Factor (SSF) | All Vehicles | Cars | Trucks |
|---|---|---|---|
| Mean | 1.212 | 1.341 | 1.140 |
| Min | 0.964 | 1.220 | 0.964 |
| 25th percentile | 1.110 | 1.310 | 1.079 |
| 50th percentile | 1.220 | 1.342 | 1.128 |
| 75th percentile | 1.316 | 1.378 | 1.211 |
| Max | 1.478 | 1.435 | 1.478 |
| Count | 204 | 73 | 131 |
Dynamic indices
The dynamic index (DI) is a non dimensionalized value used to describe the inertial properties of the vehicle. The yaw dynamic index dates back to the 1930s with the \(k^2\) experiments conducted by Olley. The yaw dynamic index defined as:
\[DI_{yaw} = \frac{k_z^2}{ab}\]Where:
- \(k_z\) is the yaw radius of gyration [\(m\)]
- \(a\) is the longitudinal distance from the CG to the front axle [\(m\)]
- \(b\) is the longitudinal distance from the CG to the rear axle [\(m\)]
The following equation shows the relationship between the yaw radius of gyration and the yaw inertia. Rearranging this relation for \(I_{zz}\) completes the relationship between DI and inertia.
\[k_z^2 = \frac{I_{zz}}{m}\]The same concept can be applied to the pitch inertia to find the pitch dynamic index.
\[DI_{pitch} = \frac{k_y^2}{ab}\]Applying such index for roll is not common. From experimentation, the following equation normalizes the roll inertial measurements. Note that this equation is completely arbitrary and is not based on any literature.
\[DI_{roll} = \frac{k_y}{T}\]Where \(T\) is the average track width. The trends in dynamic indices are shown in the graphs and histograms below.
Notice how all of the dynamic indices are weakly correlated with vehicle mass. These values are better described by its average since they are centred around some value. The results are shown in the tables below.
| Roll Dynamic Index | All Vehicles | Cars | Trucks |
|---|---|---|---|
| Mean | 0.413 | 0.409 | 0.415 |
| Min | 0.367 | 0.378 | 0.367 |
| 25th percentile | 0.399 | 0.399 | 0.399 |
| 50th percentile | 0.409 | 0.408 | 0.412 |
| 75th percentile | 0.423 | 0.415 | 0.427 |
| Max | 0.512 | 0.471 | 0.512 |
| Pitch Dynamic Index | All Vehicles | Cars | Trucks |
|---|---|---|---|
| Mean | 1.004 | 1.042 | 0.982 |
| Min | 0.816 | 0.831 | 0.816 |
| 25th percentile | 0.927 | 0.977 | 0.908 |
| 50th percentile | 0.998 | 1.061 | 0.963 |
| 75th percentile | 1.078 | 1.095 | 1.038 |
| Max | 1.298 | 1.225 | 1.298 |
| Yaw Dynamic Index | All Vehicles | Cars | Trucks |
|---|---|---|---|
| Mean | 1.034 | 1.091 | 1.002 |
| Min | 0.837 | 0.914 | 0.837 |
| 25th percentile | 0.962 | 1.037 | 0.932 |
| 50th percentile | 1.033 | 1.105 | 1.000 |
| 75th percentile | 1.111 | 1.148 | 1.055 |
| Max | 1.242 | 1.242 | 1.190 |
| Metric | All Vehicles | Cars | Trucks |
|---|---|---|---|
| Count | 204 | 73 | 131 |
Discussion
Based on the findings from the 1998 NHTSA Light Vehicle Inertial Parameters Database, vehicles tend to have similar dynamic indices and static stability factors. The average values can be used to estimate the vehicle’s centre of gravity height and inertial properties with data that is easily obtainable. For cars, the average values are:
- Static stability factor: 1.341
- Roll dynamic index: 0.409
- Pitch dynamic index: 1.042
- Yaw dynamic index: 1.091
The roll/yaw product can be estimated using the transfer function with respect to vehicle mass. Note that this is subject to greater uncertainty because of the limited number of measurements and the weak correlation observed in the data.
Final comments
Vehicle dynamic simulation is contingent on having data on your vehicle. Centre of gravity position and inertial properties are some of the most basic pieces of information needed parameterize the car. With public databases and journals, one can identify trends in the industry to make a reasonable guess of a vehicle’s centre of gravity position and inertial properties.
This analysis identified average values for the static stability factor and the dynamic indices. These values are non dimensionalized to the vehicle geometry like track width and weight distribution. Manufacturers typically publish vehicle geometry data, making the calculation simple to do. This enables anyone to obtain reasonable values for the centre of gravity height and inertia without needing expensive measurement equipment.
The Light Vehicle Inertial Parameters Database is available from the NHTSA here.
References
- Heydinger, Gary J., Ronald A. Bixel, W. Riley Garrott, Michael Pyne, J. Gavin Howe, and Dennis A. Guenther. “Measured vehicle inertial parameters-NHTSA’s data through November 1998.” SAE transactions (1999): 2462-2485.
- Walz, M. C. (2005). Trends in the static stability factor of passenger cars, light trucks, and vans (No. HS-809 868).
- Basso, G. L. (1974). Functional derivation of vehicle parameters for dynamic studies (No. LTR-ST. 747).
- Gillespie, T. D. (1992). Fundamentals of vehicle dynamics (Vol. 400). Warrendale, PA: Society of automotive engineers.