Multi-Model Verification of a Pushrod-Actuated Double-Wishbone Suspension for Formula Racing Applications

High-performance racing cars are not merely fast; they are machines of extreme precision, where the connection between the tire and the road is the single most critical factor for speed and safety. At the heart of this connection lies the suspension system, a complex assembly of arms, pivots, and springs designed to keep the tire pressed firmly against the track surface despite bumps, turns, and rapid acceleration. In the world of Formula racing, engineers often use a specific design known as a double-wishbone suspension, which uses two triangular arms to control the wheel's movement. To save space and lower the car's center of gravity, these systems frequently employ a pushrod mechanism, where a vertical rod transfers the wheel's motion inward to a spring and shock absorber mounted on the car's frame. However, this clever arrangement introduces a subtle but powerful geometric trick: the force felt by the wheel is not the same as the force felt by the spring. The relationship between the two depends on the precise angles and lengths of the connecting parts, a ratio that changes as the wheel moves up and down. If an engineer misunderstands this relationship, they might select a spring that is too soft or too stiff, leading to a car that handles poorly or loses grip when it matters most.

A team of researchers from Izmir Kâtip Çelebi University set out to untangle this complexity by building a rigorous digital verification framework for such a suspension. Their work was not about inventing a new type of car or a new physical component, but rather about ensuring that the different computer models used to design these vehicles all agreed with one another. In modern engineering, a suspension is often analyzed in several different ways: by calculating the geometry of the moving parts, by simulating the forces on the shock absorber, and by testing the entire system in a dynamic model that predicts how the car will bounce over a bump. The challenge is that these different models often speak different "languages" regarding units, coordinates, and assumptions. The researchers created a workflow that linked these separate models together, ensuring that the physical variables—such as wheel travel, spring compression, and the resulting stiffness at the tire—remained consistent across every stage of the analysis. They focused specifically on a pushrod-actuated double-wishbone setup, tracing the path of force from the wheel, through the pushrod and a rotating lever called a rocker, and finally into the spring and damper unit.

The core of their investigation revealed a counterintuitive truth about how these racing suspensions behave. The team calculated the stiffness of the springs used in the front and rear of the vehicle. The rear spring was physically stiffer, with a rate of 56.517 newtons per millimeter, while the front spring was softer, at 39.181 newtons per millimeter. In a simple system where the spring connects directly to the wheel, one would expect the rear of the car to be significantly stiffer than the front. However, because of the geometry of the pushrod and rocker mechanism, the story changed completely. The researchers measured the "motion ratio," which describes how much the spring compresses for every millimeter the wheel moves. The front suspension had a motion ratio of 0.8967, meaning the spring compressed almost as much as the wheel moved. The rear suspension had a much lower ratio of 0.6700, meaning the spring compressed significantly less than the wheel moved. When these ratios were applied to the spring stiffness, the result was a reversal of expectations: the front wheel ended up with a vertical stiffness of 31.504 newtons per millimeter, while the rear wheel was softer at 25.371 newtons per millimeter. Despite the front spring being physically softer, the front of the car was actually 24.2% stiffer at the wheel because of the mechanical advantage provided by the rocker geometry.

This finding highlighted a significant risk in the design process. The researchers demonstrated that if an engineer were to ignore the motion ratio and simply assume the spring stiffness was the same as the wheel stiffness, the error would be substantial. For the front of the car, such a mistake would lead to an overestimation of the stiffness by 11.5%. For the rear, the error would be even more dramatic, inflating the estimated stiffness by 49.3%. This discrepancy is critical because the stiffness of the suspension directly influences the frequency at which the car bounces, a key factor in how the tires maintain contact with the road. By treating the geometry as a minor detail, a designer could fundamentally misjudge the dynamic behavior of the vehicle. The study confirmed that the geometry-dependent transformation of stiffness is not a small correction but a dominant factor that dictates whether the front or rear of the car is dynamically stiffer.

Beyond the stiffness calculations, the researchers also examined how the suspension affected the steering of the wheels. As the wheel moves up and down over a bump, the geometry of the suspension arms can cause the wheel to turn slightly inward or outward, a phenomenon known as bump steer. The team analyzed the steering response of their model and found that over a vertical travel range of 30 millimeters, the wheel angle changed by approximately 3.30 degrees. For a racing car traveling at high speeds, this is a large and potentially dangerous amount of unintended steering input, as it alters the angle at which the tire meets the road and can upset the car's balance. The study identified the specific connection points of the steering rods as the primary area needing refinement. The researchers suggested that future design efforts should focus on adjusting these hardpoints to minimize this unwanted steering variation, ensuring that the driver's input remains the only factor controlling the car's direction.

To ensure their results were trustworthy, the team subjected their dynamic models to a strict numerical verification process. They simulated the car's response to a standard road bump, represented as a smooth, raised-cosine pulse 30 millimeters high, and compared the results of two different mathematical methods. One method used a highly precise, adaptive calculation, while the other used a simpler, fixed-step approach. The two methods agreed with each other to within 0.011 percent for the peak acceleration and travel metrics. This level of agreement confirmed that the computer models were solving the equations correctly and that the results were not artifacts of the calculation method. The simulations showed that the stiffer front setup produced a peak vertical acceleration of 1.145 times the force of gravity, while the softer rear setup produced 1.002 times gravity, further illustrating how the geometric differences translated into distinct physical behaviors.

The study concluded by emphasizing the importance of this verification-oriented approach. The researchers made it clear that their work was a numerical exercise, a way to ensure the internal logic of the design models was sound before any physical car was built. They did not claim to have replaced physical testing with computer simulations, noting that real-world validation would still require measured data from actual springs, tires, and track telemetry. However, by establishing a consistent framework that linked the geometry, the forces, and the dynamic response, they provided a reliable baseline for future development. Their work serves as a reminder that in the high-stakes world of Formula racing, the most subtle geometric details can have the most profound effects on performance, and that understanding the true relationship between a spring and a wheel is essential for building a car that can truly compete.