The study, published open access in the journal Automotive and Engine Technology by Thomas Gerstorfer, Martin Schabauer, and Cornelia Lex, addresses a long-standing tension in vehicle dynamics. Independent wheel suspensions, such as double wishbone or multi-link layouts, allow each wheel to move with a degree of freedom that solid axles do not possess. Describing their kinematics with a single roll center requires assumptions that do not always hold, particularly when the suspension geometry generates vertical forces as a byproduct of lateral tire forces. These so-called jacking forces can literally lift or lower the vehicle body during cornering, changing the ride height and altering the handling balance in ways that a classic roll angle calculation struggles to capture.

The core of the new approach is the principle of virtual work, a classical tool of analytical mechanics. Rather than choosing an arbitrary point around which the body rolls, the researchers model the suspension as a constrained mechanical system in which the geometry itself dictates how lateral tire forces are transmitted to the body. When a tire generates a lateral force, the links and joints of the suspension guide that force along specific paths, and some components of those paths point vertically. By computing the virtual work done by these force components through small virtual displacements of the mechanism, the model captures the geometric load transfer directly, without ever needing to define a roll center height. The result is a description of both roll and heave motion that emerges naturally from the suspension’s constraints.

What makes this modeling strategy particularly attractive is its efficiency. A complete multi-body simulation of a vehicle, with all its joints, bushings, dampers, and compliance elements, can be computationally expensive, requiring detailed parameterization and significant processing time. Such models are indispensable for final validation, but they are cumbersome during early design exploration, when engineers need to sweep through dozens of suspension geometry variants quickly. The Austrian team’s model sits in a sweet spot: it is simple enough to run rapidly, yet it retains physical effects that cruder analytical methods ignore. The authors emphasize that this transient single-axle model, while far lighter than a full vehicle multi-body model, still accounts for phenomena that matter in real driving.

Among those phenomena is aerodynamic downforce. Modern passenger cars and especially performance vehicles generate significant vertical aerodynamic loads at speed, which press the body toward the road and alter the forces flowing through each suspension link. The new framework incorporates downforce explicitly, allowing engineers to see how it interacts with the suspension geometry during cornering. The model also accounts for camber effects, specifically camber thrust, the lateral force generated when a tire leans relative to the road surface. Because independent suspensions change camber as the body rolls and the wheels travel, these effects feed back into the load paths and the resulting jacking forces, and the proposed model handles them within the same virtual work framework.

Another subtle but important capability concerns the left and right sides of the axle. During cornering, the outer tire typically carries more load and generates more lateral force than the inner tire, so the two sides of the axle are rarely symmetric in their force contribution. Classic roll center methods often implicitly assume symmetric behavior, masking the asymmetries that influence how the body actually moves. The new model takes unequal lateral tire forces into account, computing the jacking forces they produce on each side independently. This means characteristics such as ride height changes, including the tendency of some suspensions to squat or lift under sustained cornering, can be determined directly from the model outputs rather than estimated through empirical corrections.

To establish credibility, the researchers compared their approach against two established benchmarks found in the literature. The first was an existing model that does consider the kinematics of independent suspension systems in more detail, providing a like-for-like test of whether the new formulation reproduces known behavior. The second was the classic roll angle calculation, the textbook method that relies on roll center positions and represents the traditional baseline every vehicle dynamics engineer learns. The comparisons served to position the new model within the existing landscape: more faithful than the simplified classic method, yet more efficient than the detailed kinematic alternative.

The decisive test, however, came from validation against a multi-body simulation, the de facto gold standard for suspension and vehicle dynamics analysis. The authors report that the proposed model was able to predict the roll and heave motions of independent wheel suspension systems in an efficient and physically plausible manner, matching the reference simulation without requiring the computational overhead of a full multi-body environment. For engineers iterating on suspension hardpoints, kinematic properties, or force paths, this combination of accuracy and speed is precisely what makes a modeling approach useful in practice rather than merely elegant on paper.

The broader significance of the work lies in its willingness to question a foundational concept. The roll center has been criticized for decades precisely because its definition becomes ambiguous for independent suspensions, and different definitions can yield different predictions for the same mechanism. By sidestepping the concept entirely and grounding the description of load transfer in the actual constraints of the suspension, the Graz researchers offer a formulation that is easier to defend physically and more transparent to interpret. Effects that had to be patched in with corrections under the roll center framework, such as jacking, camber thrust, and aerodynamic loading, arise organically from the virtual work formulation.

For the automotive industry, the implications are practical. Simulation tools built on such models could accelerate the early stages of chassis development, letting teams evaluate ride height behavior, roll characteristics, and geometric load transfer across wide design spaces before committing to expensive detailed simulations or physical prototypes. The work also has educational value, giving students and practitioners a clearer picture of how suspension geometry converts tire forces into body motion. As vehicles grow heavier with batteries and more reliant on aerodynamics and sophisticated chassis control, the demands on suspension modeling will only increase. An approach that captures the essential physics with minimal computational cost, and that has been validated against multi-body simulation, offers a timely contribution to that effort. The full article is available open access under a Creative Commons license, allowing engineers and researchers worldwide to examine the formulation and apply it to their own suspension systems.

Subject of Research: A virtual work-based modeling approach for roll and heave motions of independent wheel suspension systems without roll centers

Article Title: An effective modeling approach for roll and heave motions for independent wheel suspension systems

Article References: Gerstorfer, T., Schabauer, M., & Lex, C. (2026). An effective modeling approach for roll and heave motions for independent wheel suspension systems. Automotive and Engine Technology. https://doi.org/10.1007/s41104-026-00180-2

Image Credits: AI Generated

DOI: 10.1007/s41104-026-00180-2

Keywords: vehicle dynamics, suspension kinematics, independent wheel suspension, roll center, jacking forces, principle of virtual work, roll motion, heave motion, camber thrust, aerodynamic downforce, multi-body simulation, automotive engineering

The research, led by Pascal Ambrosoli with co-authors Carl Sandberg, Werner Krantz, Jens Neubeck and Andreas Wagner, addresses a central problem in modern vehicle development. As automakers shift toward systems engineering approaches built around the well-known V-model, the earliest phases of a program require target values for overall vehicle characteristics to be fixed at the top of the V, long before hardware exists. Those targets must then be cascaded step by step down to subsystem level, so that suspension kinematics, tire selection, steering tuning and electronic controls can be designed in a decoupled way while still guaranteeing the behavior of the finished car. For this cascade to work, the top-level characteristics must be measurable in a precise sense: they must be physically grounded quantities that a virtual vehicle model can actually compute.

The team focuses on lateral dynamics, the domain that governs cornering behavior and, crucially, driving safety. Within this domain they define a family of objective metrics covering both controllability and stability. These include the steering wheel gradient, which describes how much steering angle is needed per unit of lateral acceleration in the linear range from zero to four meters per second squared; the steady-state and dynamic lateral acceleration peaks, which mark how much cornering power the vehicle can generate; the yaw acceleration peak, which captures how quickly the car can be rotated into a corner; and stability and controllability measures evaluated both in straight-ahead driving and near the limit area, defined as eighty percent of the steady-state lateral acceleration peak. Side-slip angles at the center of gravity and at the rear axle, along with slip angles at the front and rear axles, form the physical backbone of these quantities.

Why does the nonlinear part of lateral dynamics deserve special attention? Because that is where accidents happen. In the linear regime, a car responds predictably: double the steering input and the lateral response roughly doubles. But as tires approach the limit of their grip, their force potential saturates and rolls off, and the vehicle’s behavior becomes strongly nonlinear. Small changes in design or tuning can produce disproportionate shifts in how the car behaves when a driver demands maximum cornering. A chassis development process that only characterizes the benign, linear middle of the handling envelope leaves exactly the region that matters most for safety unmeasured, and therefore unmanaged.

The conventional way to extract such metrics is time-domain simulation: excite a full vehicle model with steering inputs, sweep the maneuvers, and post-process the resulting trajectories. The problem is computational cost. A target-oriented development process built on the solution space method is inherently iterative. Design variables are varied, targets are checked, the design is adjusted, and the cycle repeats, often thousands of times across many subsystems. If every evaluation of a stability or controllability metric requires a lengthy transient simulation, the iteration grinds to a halt. The Stuttgart team’s answer is to stop simulating time altogether for these evaluations and instead compute the metrics from partial equilibria of the vehicle’s equations of motion.

At the heart of the methodology lies the Newton-Raphson method, a classical numerical root-finding technique that iteratively drives the residuals of a nonlinear system to zero. Instead of integrating the vehicle model forward in time, the researchers formulate the steady-state and dynamic partial equilibrium conditions of the lateral dynamics and solve them directly. The Jacobian matrix of the model states, which contains the partial derivatives of the state derivatives with respect to the states themselves, guides each Newton step toward the equilibrium point. Inputs such as the steering wheel angle are varied systematically, and at each operating point the equilibrium states, including lateral velocities at the rear axle, side-slip angles and lateral tire forces at the front and rear axles, are recovered analytically rather than simulated. Tire relaxation length, which describes the lag with which a tire builds up lateral force, is retained in the formulation so that dynamic partial equilibria remain physically meaningful.

This equilibrium-based approach makes the defined characteristics computable automatically and efficiently, which is precisely what the iterative solution space method demands. Within a target cascading workflow, each subsystem design proposal can be checked against the top-level handling targets in seconds rather than hours, and the robustness of a design, meaning how much its metrics degrade as parameters vary, becomes visible early. The decoupling that target cascading promises, in which a suspension engineer and a tire engineer can work independently yet converge on a coherent vehicle, only functions if the shared top-level metrics are cheap enough to evaluate on every iteration. The Newton-Raphson methodology supplies that missing computational ingredient.

To demonstrate that the new metrics are not merely convenient but meaningful, the authors compare them against full time-domain simulations of the vehicle. The comparison shows that characteristics derived from steady-state and dynamic partial equilibria capture the relevant features of the vehicle’s lateral behavior, including the location of the lateral acceleration peak and the yaw acceleration response near that peak. The metrics thus serve as faithful proxies for quantities that would otherwise require expensive transient maneuvers, while remaining stable and well-defined across the design space, a property that time-domain peak values, which can jump discontinuously as designs change, often lack. The work also connects to established tools of the vehicle dynamics trade, including the Milliken moment method, whose diagram conventions for mapping stability and controllability the new characteristics are designed to complement, and alternative formulations such as the multibody system transfer matrix method.

The implications reach beyond the specific metrics. Battery electric vehicles, with their heavy floor-mounted packs, low centers of gravity and high curb masses, are reshaping the chassis design space, and development cycles are compressing as competition intensifies. A characteristics-based, V-model-oriented process in which handling targets are defined, monitored and tracked from the first concept sketch to production validation offers a way to keep safety-critical qualities under control amid that turbulence. Because the metrics are physically based, they can be carried consistently from early idealized models through to detailed multibody simulations and, ultimately, to test-track confirmation, giving program managers a single quantitative thread through the entire development process.

The study, received in March 2026 and accepted in August 2026, was published open access on 18 September 2026 with funding enabled by Projekt DEAL. Its authors report no conflict of interest. For an industry in which a single physical prototype iteration can cost millions and months, the ability to determine controllability and stability metrics automatically, efficiently and reliably in the virtual domain marks a practical step toward chassis development that is simultaneously faster, more systematic and more safety-aware. The nonlinear edge of the handling envelope, long the blind spot of early-phase engineering, is now a quantity that can be targeted, cascaded and verified before rubber ever meets asphalt.

Subject of Research: Efficient determination of vehicle lateral controllability and stability metrics for target-oriented virtual chassis development

Article Title: Determination of vehicle controllability and stability metrics in a target-oriented virtual chassis development process

Article References: Ambrosoli, P., Sandberg, C., Krantz, W., Neubeck, J., & Wagner, A. (2026). Determination of vehicle controllability and stability metrics in a target-oriented virtual chassis development process. Automotive and Engine Technology, 11(1), Article 15. https://doi.org/10.1007/s41104-026-00179-9

Image Credits: AI Generated

DOI: 10.1007/s41104-026-00179-9

Keywords: vehicle dynamics, chassis development, lateral dynamics, Newton-Raphson method, solution space method, V-model, driving safety, steering wheel gradient, yaw acceleration, virtual vehicle development, automotive engineering, target cascading