Advanced search
Publication Cover
Vehicle System Dynamics
International Journal of Vehicle Mechanics and Mobility
Latest Articles
Submit an article Journal homepage
0
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Asymmetric vehicle performance assessment using GG diagrams

D. J. N. Limebeera School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg, South AfricaCorrespondence[email protected] [email protected]
,
M. Massarob Department of Industrial Engineering, University of Padova, Padova, Italy
&
A. Dollarc General Motors Charlotte Technical Center, Concord, NC, USA
Received 05 Apr 2024, Accepted 06 Jul 2024, Published online: 16 Jul 2024

Figures & data

Figure 1. Relationship between intrinsic and polar co-ordinate descriptions of a logarithmic spiral. A generic vehicle path is given by C with a generic point P on it. The origin of the polar coordinate system is O, with r and φ the polar co-ordinates of P. The instantaneous centre of rotation of C at P is C. The instantaneous radius of curvature is C, with α the angle between C and the horizontal x-axis. The angles ξ and ν define the orientation of a tangent to C at P with respect to r and the x-axis respectively.

Figure 1. Relationship between intrinsic and polar co-ordinate descriptions of a logarithmic spiral. A generic vehicle path is given by C with a generic point P on it. The origin of the polar coordinate system is O, with r and φ the polar co-ordinates of P. The instantaneous centre of rotation of C at P is C. The instantaneous radius of curvature is C, with α the angle between C and the horizontal x-axis. The angles ξ and ν define the orientation of a tangent to C at P with respect to r and the x-axis respectively.

Figure 2. An exemplar test course comprising a constant radius of turn at fixed speed section OA; an accelerating straight section AB; a section BC that describes braking into a tightening turn, and a section CD requiring acceleration out of a turn that ‘opens up’.

Figure 2. An exemplar test course comprising a constant radius of turn at fixed speed section OA; an accelerating straight section AB; a section BC that describes braking into a tightening turn, and a section CD requiring acceleration out of a turn that ‘opens up’.

Table 1. Drivability metric.

Figure 3. Abstract GG diagram with its periphery described in terms of polar coordinates.

Figure 3. Abstract GG diagram with its periphery described in terms of polar coordinates.

Figure 4. Kinematics of a single-track car model showing its basic geometric parameters [Citation3].

Figure 4. Kinematics of a single-track car model showing its basic geometric parameters [Citation3].

Figure 5. Single-track car model operating at different speeds on inclined planar road surfaces. The left-hand figure shows the vehicle GGV diagram on a horizontal road surface at 70 m/s (red), 50 m/s (black) and 30 m/s (blue). The central figure shows the vehicle operating at 50 m/s on an inclined road surface with a 25∘ inclination angle (red), a level road surface (black), and a 25∘ declination angle (blue). The right-hand figure shows that vehicle operating at 50 m/s on a cambered road surface with a −25∘ camber angle (blue), a level road surface (black), and a 25∘ camber angle (red).

Figure 5. Single-track car model operating at different speeds on inclined planar road surfaces. The left-hand figure shows the vehicle GGV diagram on a horizontal road surface at 70 m/s (red), 50 m/s (black) and 30 m/s (blue). The central figure shows the vehicle operating at 50 m/s on an inclined road surface with a 25∘ inclination angle (red), a level road surface (black), and a 25∘ declination angle (blue). The right-hand figure shows that vehicle operating at 50 m/s on a cambered road surface with a −25∘ camber angle (blue), a level road surface (black), and a 25∘ camber angle (red).

Figure 6. Single-track car model operating at 50 m/s on a horizontal road surface. The left-figure shows the front (red) and rear (blue) tyre cornering stiffnesses and the vehicle understeer coefficient (black). The right-hand figure shows the two-state linear model eigenvalues evaluated on the periphery of the GG diagram; the real parts are shown red and the imaginary parts blue.

Figure 6. Single-track car model operating at 50 m/s on a horizontal road surface. The left-figure shows the front (red) and rear (blue) tyre cornering stiffnesses and the vehicle understeer coefficient (black). The right-hand figure shows the two-state linear model eigenvalues evaluated on the periphery of the GG diagram; the real parts are shown red and the imaginary parts blue.

Figure 7. Stability eigenvalues on the periphery of the GG diagram at 50 m/s on a planar road surface. The real parts of the linearised model eigenvalues are shown in red, while the imaginary parts are illustrated in blue. The left-hand diagram corresponds to a road cambered at −25∘, the central plot is for a level road surface, while the right-hand diagram is for a road cambered at 25∘.

Figure 7. Stability eigenvalues on the periphery of the GG diagram at 50 m/s on a planar road surface. The real parts of the linearised model eigenvalues are shown in red, while the imaginary parts are illustrated in blue. The left-hand diagram corresponds to a road cambered at −25∘, the central plot is for a level road surface, while the right-hand diagram is for a road cambered at 25∘.

Figure 8. Steady-state gain of the transfer function from the steering angle to the yaw rate for the vehicle operating at 50 m/s on a horizontal road surface.

Figure 8. Steady-state gain of the transfer function from the steering angle to the yaw rate for the vehicle operating at 50 m/s on a horizontal road surface.

Figure 9. GGV diagrams for a Gen-7 NASCAR operating on planar road surfaces of variable lateral road camber; the left-hand figure is for −25∘ of camber, the centre figure is for a horizontal road surface and the right-hand figure is for 25∘ of lateral road camber. The black plots correspond to the car travelling at 80 m/s; the red curves correspond to 70 m/s; the blue curves correspond to 60 m/s; the magenta curves are for 50 m/s and the green curves are for 40 m/s.

Figure 9. GGV diagrams for a Gen-7 NASCAR operating on planar road surfaces of variable lateral road camber; the left-hand figure is for −25∘ of camber, the centre figure is for a horizontal road surface and the right-hand figure is for 25∘ of lateral road camber. The black plots correspond to the car travelling at 80 m/s; the red curves correspond to 70 m/s; the blue curves correspond to 60 m/s; the magenta curves are for 50 m/s and the green curves are for 40 m/s.

Figure 10. Normal and lateral tyre loads as a function of the GG diagram sweep angle 0≤α≤2π. The lateral acceleration is Rcos⁡α and the camber angle ϕ. All the car parameters are assumed normalised so that g = 1, M = 1, and R is in G's. All lengths are given as fractions of the vehicle length.

Figure 10. Normal and lateral tyre loads as a function of the GG diagram sweep angle 0≤α≤2π. The lateral acceleration is Rcos⁡α and the camber angle ϕ. All the car parameters are assumed normalised so that g = 1, M = 1, and R is in G's. All lengths are given as fractions of the vehicle length.

Table 2. Idealised and normalised tyre loading parameters.

Figure 11. Total lateral tyre loads as a function of the sweep and camber angles, respectively α and ϕ. All the car parameters are assumed normalised so that g = 1, M = 1, and R is in G's. The black curve corresponds to ϕ=0∘. The red curve is for ϕ=20∘, while the blue curve represents the ϕ=−20∘ case.

Figure 11. Total lateral tyre loads as a function of the sweep and camber angles, respectively α and ϕ. All the car parameters are assumed normalised so that g = 1, M = 1, and R is in G's. The black curve corresponds to ϕ=0∘. The red curve is for ϕ=20∘, while the blue curve represents the ϕ=−20∘ case.

Figure 12. Idealised left- and right-hand tyre normal loads for varying road camber; the left-hand plot is for the left-hand tyre. The black curves are for a level road. The red curve represents a road camber of 15∘, while the blue curve is for a road camber of −15∘.

Figure 12. Idealised left- and right-hand tyre normal loads for varying road camber; the left-hand plot is for the left-hand tyre. The black curves are for a level road. The red curve represents a road camber of 15∘, while the blue curve is for a road camber of −15∘.

Figure 13. Tyre normal loads for a Gen-7 NASCAR travelling at 50 m/s. The black plot corresponds to the car operating on a horizontal road surface. The red plot corresponds to the car operating on a road surface with 20∘ of camber. The blue plot corresponds to the car operating on a road surface with −20∘ of camber.

Figure 13. Tyre normal loads for a Gen-7 NASCAR travelling at 50 m/s. The black plot corresponds to the car operating on a horizontal road surface. The red plot corresponds to the car operating on a road surface with 20∘ of camber. The blue plot corresponds to the car operating on a road surface with −20∘ of camber.

Figure 14. GGV diagrams for a Gen-7 NASCAR operating on a conical road surface with a camber angle of −20∘. The black plots corresponds to the car travelling at 80 m/s; the red curves corresponds to 70 m/s; the blue curves corresponds to 60 m/s; the magenta curves are for 50 m/s and the green curves are for 40 m/s.

Figure 14. GGV diagrams for a Gen-7 NASCAR operating on a conical road surface with a camber angle of −20∘. The black plots corresponds to the car travelling at 80 m/s; the red curves corresponds to 70 m/s; the blue curves corresponds to 60 m/s; the magenta curves are for 50 m/s and the green curves are for 40 m/s.

Figure 15. Measured normal tyre loads captured on an instrumented Gen-6 NASCAR on the Darlington Raceway.

Figure 15. Measured normal tyre loads captured on an instrumented Gen-6 NASCAR on the Darlington Raceway.

Figure 16. Measured lateral and longitudinal accelerations captured on an instrumented Gen-6 NASCAR on the Darlington Raceway.

Figure 16. Measured lateral and longitudinal accelerations captured on an instrumented Gen-6 NASCAR on the Darlington Raceway.

Figure 17. Eigenvalue plots for a Gen-7 NASCAR operating at 50 m/s on a conical road surface. The real parts of the linearised model eigenvalues are shown in red, while the imaginary parts are illustrated in blue. The black dot-dash curve is the maximum achievable acceleration as the GG diagram is traversed. The left-hand figure is for −20∘ of road camber, the central figure is for a horizontal road surface and the right-hand figure is for 20∘ of camber.

Figure 17. Eigenvalue plots for a Gen-7 NASCAR operating at 50 m/s on a conical road surface. The real parts of the linearised model eigenvalues are shown in red, while the imaginary parts are illustrated in blue. The black dot-dash curve is the maximum achievable acceleration as the GG diagram is traversed. The left-hand figure is for −20∘ of road camber, the central figure is for a horizontal road surface and the right-hand figure is for 20∘ of camber.

Figure 18. Steady-state steer angle gains for a Gen-7 NASCAR operating at 50 m/s on a horizontal planar road surface. The left-hand figure is the yaw-rate gain, while the right-hand figure is the side-slip angle gain.

Figure 18. Steady-state steer angle gains for a Gen-7 NASCAR operating at 50 m/s on a horizontal planar road surface. The left-hand figure is the yaw-rate gain, while the right-hand figure is the side-slip angle gain.

Figure 19. Brake balance for a Gen-7 NASCAR travelling at 50 m/s. The left-hand figure is for a level road surface. The right-hand figure is for a curved road surface with −20∘ of camber. The black plot corresponds to the car operating with 67% of the braking torque applied to the front axle. The red plot corresponds to the car operating with 80% of the braking torque applied to the front axle. The blue plot corresponds to the car operating with 90% of the braking torque applied to the front axle.

Figure 19. Brake balance for a Gen-7 NASCAR travelling at 50 m/s. The left-hand figure is for a level road surface. The right-hand figure is for a curved road surface with −20∘ of camber. The black plot corresponds to the car operating with 67% of the braking torque applied to the front axle. The red plot corresponds to the car operating with 80% of the braking torque applied to the front axle. The blue plot corresponds to the car operating with 90% of the braking torque applied to the front axle.

Figure 20. Stability due to variable brake balance for a Gen-7 NASCAR travelling at 50 m/s on a level road surface. The real parts of the linearised model eigenvalues are shown in red, while the imaginary parts are illustrated in blue. The black dot-dash curve is the maximum achievable acceleration as the GG diagram is traversed. The left-hand figure corresponds to a nominal brake balance of 67% on the front wheels. The central figure corresponds to a nominal brake balance of 80% on the front wheels. The right-hand figure corresponds to a nominal brake balance of 90% on the front wheels.

Figure 20. Stability due to variable brake balance for a Gen-7 NASCAR travelling at 50 m/s on a level road surface. The real parts of the linearised model eigenvalues are shown in red, while the imaginary parts are illustrated in blue. The black dot-dash curve is the maximum achievable acceleration as the GG diagram is traversed. The left-hand figure corresponds to a nominal brake balance of 67% on the front wheels. The central figure corresponds to a nominal brake balance of 80% on the front wheels. The right-hand figure corresponds to a nominal brake balance of 90% on the front wheels.

Table B1. Single-track car model parameters.

Table B2. Single-track car model parameters.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.