Modeling Golf Ball Bounce: Experimental Observations and Mathematical Analysis

When considering the bounce of a tennis ball, basketball, or football, researchers typically study the impact of a deformable ball against a rigid surface. However, such assumptions seem rather obsolete in the context of a golf ball that bounces on turf; the ball remains rigid while the turf deforms upon impact. Although scientists have not studied this scenario in much detail, the fundamental science behind golf ball bounce is of interest to many stakeholders — including the legislative bodies of the game, equipment manufacturers, and developers of golf simulators.

The most recent large-scale study of golf ball bounce dates back to Stephen John Haake’s Ph.D. thesis in 1989 [6], which used a two-layer model to account for the turf’s elastic (where the material returns to its original shape after deformation) and plastic (where the material remains permanently deformed) properties. Other approaches have altered the model for the bounce of a rigid ball against a rigid surface [3, 8, 9].

Our goal is to collect data from large-scale experimental campaigns that span a wide range of landing conditions and present a suitable model that matches our data-based observations. The resulting model should be generic enough to apply to a variety of surfaces while simultaneously accounting for the effect of friction and the turf’s elastic and plastic properties.

Experimental Data

Our analytical model is motivated by data from two experimental campaigns in which we recorded 330 slow-motion videos of a golf ball bouncing against artificial turf and 693 videos of a golf ball bouncing against a well-maintained teeing surface [2]. The ball travelled in a plane that was parallel to the camera’s field of view with a spin that was imposed in the plane of flight. From the videos, we extracted the horizontal and vertical velocity components as well as the ball’s spin immediately before and after impact.

For the sake of simplicity, let us assume that the ball is in a point contact with the deformable horizontal surface. This point \(P\) is the lowest point on the ball; Figure 1 depicts the velocity of \(P\) after the bounce, which is a function of the same velocity before the bounce.

Consider the case of \(v_P=0\). If the velocity of the contact point is zero, then the ball must be rolling. This value at liftoff is denoted by a dashed line in Figure 1, which divides the data set at liftoff into two distinct groups. Additionally, the liftoff tangential velocities appear to be shifted away from the zero velocity, thus suggesting liftoff with some slip. We associate this behavior with the effects of friction. How can we model friction in a way that is both physically and mathematically consistent and sound?

Friction and Nonsmooth Differential Equations

Here, we consider two simple cases of the effects of friction. Two bodies are in contact, and the associated friction forces between them can either stick or slip. In one-dimensional motion, sticking is the state between slipping to the left or right. Given the vector of state variables \(\boldsymbol{x}\), we mathematically describe the slipping motion of a body as

\[\dot{\boldsymbol{x}} = \begin{cases}
    F_1(\boldsymbol{x}, t) & \qquad \mbox{if } v_P<0 \quad \mbox{(slipping to the left)} \\
    F_2(\boldsymbol{x}, t) & \qquad \mbox{if } v_P>0 \quad \mbox{(slipping to the right)}. \tag1
    \end{cases}\]

A friction law determines the vector fields \(F_1\) and \(F_2\). In Coulomb’s law of friction, for example, the friction force \(\lambda_F\) is related to the normal forces during slipping by \(\lambda_F = \pm \mu \lambda_N\). If we know \(\lambda_N\) and the coefficient of friction \(\mu\), we can hence determine the behavior of the entire system. However, the dynamics when \(v_P=0\)—that is, during the stick (or in case of our ball, during the roll)—remain unknown. We thus seek to determine the dynamics on a lower-dimensional manifold in a way that assures continuity (but not necessarily smoothness) with the neighboring vector fields \(F_1\) and \(F_2\).

Aleksei Filippov proposed a general framework for discontinuous dynamical systems in 1988 [5]. While various definitions have materialized over the years [4], we choose to define the dynamics along the discontinuity as the unique convex combination of the two vector fields (see Figure 2). Mathematically, this definition is

\[F_S(\boldsymbol{x}, t) = (1-\alpha(\boldsymbol{x})) F_1(\boldsymbol{x}, t) + \alpha(\boldsymbol{x}) F_2(\boldsymbol{x}, t).\tag2\]

Here, \(\alpha(\boldsymbol{x})\) is a scalar function that takes values between \(0\) and \(1\) and is determined such that \(F_S\) agrees with our discontinuity constraint; in the case of \((1)\), we require \(v_P=0\).

Modeling the Turf’s Elastoplastic Properties

We can now propose a generic mathematical model for golf ball bounce. Once again, we model the ball to be rigid and travel in a plane, with only the spin \(\omega\) in the same plane and measured counterclockwise. The coordinates \((x,y)\) denote the position of the center of mass. The elastic and plastic properties behave like nonlinear springs and dampers that are connected in parallel (see Figure 3). The states of these springs and dampers depend on the system’s displacement and the ball’s velocity. At the beginning of impact, the center of mass is located at the origin \((x,y)=(0,0)\).

Next, we must impose some constraints on our system. For simplicity, let us rescale the measurements to use the unit mass and radius of the ball. We can then write the dynamics of the golf ball as

\[\begin{aligned}
    \ddot{x} +  u(x,\dot{x}, y, \dot{y}, \omega) \, \dot{x} + z(x,\dot{x},
    y, \dot{y}, \omega)\,x & = \lambda_F,\\
    \ddot{y} +  d(y, \dot{y}) \, \dot{y} + k( y, \dot{y})\, y & = -g ,\\
    \dot{\omega} & = \frac{5}{2} \lambda_F. 
\end{aligned}\tag3\]

Here, \(\lambda_F\) is the friction force, \(k\) and \(z\) are scalar functions that respectively denote the stiffness of the springs in the normal and horizontal directions, and \(d\) and \(u\) are scalar functions that respectively denote the behavior of the dampers in the normal and horizontal directions. The constant \(\frac{5}{2}\) arises due to a rigid sphere’s moment of inertia.

The idea of varying stiffness and elastic functions \(k\), \(z\), \(d\), and \(u\) mimics and generalizes Haake’s original model, wherein the elastoplastic effects increase with the indentation of the ball [6]. Intuitively, one would expect a golf ball that is fully embedded in the turf to move slower than one that is gliding on the surface due to increased stiffness.

Based on the experimental data, the ball remains in contact with the surface for approximately \(10^{-4}\) seconds. This information enables us to appropriately rescale the functions in the normal directions, allowing for a balance of terms and the introduction of some physics-based constraints on the system in the horizontal direction [1]. 

During the slipping component, we assume Coulomb’s law of friction: \(\lambda_F = \pm \mu \lambda_N = \) \(\mp \mu (d(y, \dot{y}) \, \dot{y} + k( y, \dot{y})y +g)\), where the positive sign corresponds to the left slip (defining vector field \(F_1\)) and the negative sign corresponds to the right slip (defining \(F_2\)). Throughout the roll, we describe the dynamics according to Filippov’s formalism, as in \((2)\).

Next, we explore whether the golf ball can lift off while rolling. To justify this challenge, consider Figure 4. The ball moves along the discontinuity surface \(v_P=0\) and can enter the left slipping field when \(\lambda_F = \mu \lambda_N\) or the right slipping field when \(\lambda_F = - \mu \lambda_N\). Furthermore, \(\lambda_N=0\) (a balance of the normal forces) is the condition for liftoff, i.e., the rolling liftoff is also the point on the discontinuity surface at which the switching curves overlap. The literature refers to this point as the two-fold bifurcation [7]. 

With an appropriate normal form, the point becomes stable and we can invoke the usual tools of linear analysis to investigate its behavior. Under our parsimonious assumptions, the two-fold bifurcation that represents rolling liftoff is a non-attractive saddle point (further details are available in the literature [1, 7]). In terms of the dynamics, only a single trajectory can lead to such liftoff; we therefore observe a rolling liftoff with probability \(0\).

Conclusions

Previous models of golf ball bounce either focused on alterations of simple bounce models or numerical investigations and fits to data. However, such models fail to account for the elastoplastic and friction effects of the turf. Motivated by a large data set, we revisited this challenging problem and presented rigorous analysis of the generic model of elastoplastic bounce. Both the model and data reveal that the dynamics occupy two disparate classes based on whether the ball slips or rolls through the bounce. Furthermore, our model confirms that the introduction of elastoplastic effects in the horizontal direction allows for a spin reversal and prevents the ball from lifting off while rolling — which is precisely what we see in the data.

Stanisław Biber delivered a minisymposium presentation on this research at the 2023 SIAM Conference on Applications of Dynamical Systems (DS23), which took place in Portland, Ore., in May 2023. He received funding to attend DS23 through a SIAM Student Travel Award. To learn more about Student Travel Awards and submit an application, visit the online page.

SIAM Student Travel Awards are made possible in part by the generous support of our community. To make a gift to the Student Travel Fund, visit the SIAM website.

References
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