Impact Analysis of a Golf Ball

The outcome of a golf stroke is to a large extent influenced by the characteristics of the golf ball, including both its aerodynamics and mechanical properties. The latter, for example, determine how efficiently energy is transferred form the club head to the ball during the swing. The transfer of energy is largely governed by the stiffness of the ball’s core material, or in golf nomenclature, the “compression” of the ball. How well energy is transferred during the impact plays a key role in how far the ball will fly. The “backspin” is another key aspect of the golf stoke influenced by the mechanical properties of the ball. A high spin rate increases the lift, lengthening the carry distance of the shot. Also, increasing the backspin makes the ball stop more rapidly when landing. The stiffness of the cover and core materials plays an important role for the friction and spinning effects. A ball with a thin cover and a soft core will deform more when it comes in contact with the club face, thus generating more friction and a higher spin rate.

This example studies the mechanical impact of a club on a golf ball. The contact between the two parts is modeled using a viscous penalty formulation in order to stabilize this dynamic event. To properly model large deformations, the golf ball is defined using a hyperelastic material model. Additionally, the core material is viscoelastic, which causes dissipation of the elastic energy transfered by the impact. Results are compared to typical golf metrics such as ball speed and spin rate.

Model Definition

The model studies how momentum and energy are transferred form the club to the golf ball during the swing. The simulation focuses on the impact between the two objects, and it only looks at a time period of 2 ms.

Figure 1 shows the geometry of the club and golf ball. The dimensions of the club represent a typical 7 iron with a loft equal to 34°. The club head is approximately 9 cm wide, 6 cm high at the toe, and 3.5 cm high close to the shaft. The shaft is only partially included to simplify the geometry. It is assumed that the club has a dynamic loft of 22° at impact. The dynamic loft is the angle of the club face from the vertical plane at impact.

The the ball’s diameter is 42.67 mm in accordance with the rules of golf (Ref. 1). A three piece design is used: an inner core of 34.67 mm in diameter, a 3 mm thick outer core (or mantle), and a 1 mm thick cover. The outer surface is also covered with 362 dimples.

Figure 1: Model geometry

Material Models And Properties

The golf club is assumed to be made of steel, with a density of 7850 kg/m3, Young’s modulus of 200 GPa, and Poisson’s ration of 0.3.

According to Ref. 1, the golf ball must be constructed entirely form elastomeric materials, that is, viscoelastic materials with a low stiffness. All parts of the ball are thus described using a Neo-Hookean hyperelastic material model. The cover is assumed to be compressible, while both inner and outer core are considered nearly incompressible and modeled using a mixed formulation. The Hartmann–Neff volumetric strain energy density is used with a prescribed bulk modulus of 1 GPa in order to penalize volumetric deformations. The two parts of the core are furthermore considered to be viscoelastic, and thus also include a large strain Generalized Maxwell model with three branches.

Mechanical properties of the materials used in golf balls are propriety and hard to find in the literature. Hence, the material properties summarized in Table 1 are only realistic estimates. The density is, for example, taken so that the total mass of the ball is equal to 45.93 g in accordance with Ref. 1. The other material properties have been tuned to give realistic results.

Table 1: Material properties of the Golf Ball.
Property	Symbol	Unit	Cover	Mantle	Inner core
Density	ρ	kg/m3	1145	1145	1145
Lamé parameters	λ	MPa	1500	N/A	N/A
	μ	MPa	250	6	12
Energy factors	βv1	1	N/A	0.26	0.26
	βv2	1	N/A	0.19	0.19
	βv3	1	N/A	0.22	0.22
Relaxation times	τv1	μs	N/A	10	10
	τv2	μs	N/A	100	100
	τv3	μs	N/A	1000	1000

Constraints and Contact Conditions

The golf club impacts the ball with a velocity equal to 90 mph in a slightly downward direction, defined by an attack angle of -4.3° from the xy-plane. The end of the shaft is constrained to this velocity throughout the simulation. Both the velocity and the attack angle are rough estimates for a professional male golf player when using a 7 iron.

The interaction between the club and the golf ball is modeled using a contact condition. Dynamic effects are important since this is a time-dependent simulation of an impact event, and linear momentum and energy must be consistent during the contact interaction. To ensure this and to stabilize the contact condition, a viscous penalty formulation is added for the pressure contact. Friction is also important to properly capture the interaction between the club and the ball; since it is the effect of friction that causes the golf ball to spin after it is hit by the club head. A Coulomb friction model is added with a friction coefficient equal to 0.15.

Results and Discussion

Table 2 presents some common metrics used in golf to measure the performance of the player and the equipment. Here, the club speed and attack angle are input data to the simulation. The remaining metrics are computed by the COMSOL Multiphysics model. The launch angle is the angle of the ball relative to the ground after impact, and the smash factor is the ball speed divided by the club speed. Given the input parameters, all computed metrics are realistic and close to typical values reported for an average professional male golf player.

Table 2: Metrics of the Club and Golf ball.
Club speed (mph)	Attack angle (deg)	Launch angle (deg)	Ball speed (mph)	Smash factor	Spin rate (rpm)
90	-4.3	17.4	116.1	1.29	6113

Figure 2 shows four snapshots of the simulation before, during, and after the impact. One can clearly see the large deformation of the golf ball when hit by the club head, increasing the contact area between the ball and the club face. This is even more visible when looking at the cut-through image in Figure 3, that also shows the strain distribution inside the ball. Figure 3 shows the minimum principal strain which is in the order of -20% in large parts of the ball.

It is interesting to look at the velocity of the club and ball during the simulation in order to study the kinematics of the problem in detail. Figure 4 shows the velocity magnitude and how momentum is transferred form the club head to the ball during the impact. Due to the flexibility of the shaft, the velocity of the club head decreases slightly during and after the impact, even though the velocity of the shaft is prescribed to a constant value.

Figure 5 shows the variation of the total elastic and kinetic energy of the golf ball during the simulation. The duration of the impact is clearly visible as a peak in both the elastic and kinetic energy content. An interesting observation is how effectively the viscoelastic properties of the core damp out the elastic energy after the impact. In contrast, the kinetic energy reaches a constant value once the ball departures from the club face.

Figure 2: Snapshots of the club and golf ball during the simulation. From top to bottom, the timings are 0 ms, 0.15 ms, 0.30 ms and 0.45 ms.

Figure 3: Deformation of the golf ball and distribution of the third principal (compressive) strain in the interior part of the ball at 0.3 ms.

Figure 4: Average magnitude of the velocity in the club head and the golf ball.

Figure 5: Total elastic and kinetic energy in the golf ball.

Notes About the COMSOL Implementation

The error estimates used by the automatic time-step control will force time steps small enough to resolve the wave propagation within the solid, which is not of primary interest in this analysis. The automatic time-step control will often lead to unnecessarily small time steps, and longer computational time. Therefore, it is often a good practice to use a manual time-step control when the focus is in the kinematics of the contact phenomenon.

The spin rate of the golf ball is determined by adding an node to the model. This node makes it possible to compute average rotation for a set of points in a least-squares sense for rotations that are arbitrarily large.

Reference

1. The Equipment Rules, 1st edition, The R&A and USGA, 2019.

Nonlinear_Structural_Materials_Module/Viscoelasticity/golf_ball_impact

Modeling Instructions

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New

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Model Wizard

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Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
loft	34[deg]	0.59341 rad	Club head loft
attackAngle	-4.3[deg]	-0.075049 rad	Attack angle
dynLoft	22[deg]	0.38397 rad	Dynamic loft
vel	90[mph]	40.234 m/s	Club head speed
velx	cos(attackAngle)*vel	40.12 m/s	Club head velocity, x component
velz	sin(attackAngle)*vel	-3.0167 m/s	Club head velocity, z component

Geometry 1

Import 1 (imp1)

In the toolbar, click

In the window for , locate the section.

Click .

Browse to the model’s Application Libraries folder and double-click the file golf_ball_impact.mphbin.

Click .

Rotate the golf club to the correct dynamic loft at impact.

Rotate 1 (rot1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

From the list, choose .

In the text field, type loft-dynLoft+attackAngle.

Form Union (fin)

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Create selections to be used in the physics and plot creation.

Definitions

Club Head

In the window, expand the node.

Right-click and choose .

In the window for , type Club Head in the text field.

Select Domains 1 and 2 only.

Club Shaft

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In the window for , type Club Shaft in the text field.

Select Domains 5, 6, 8–12, and 14 only.

Club Head and Shaft, Boundary

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In the window for , type Club Head and Shaft, Boundary in the text field.

Locate the section. Under , click

In the dialog box, in the list, choose and .

Click .

Club Ferrule

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In the window for , type Club Ferrule in the text field.

Select Domains 3, 4, 7, and 13 only.

Club Ferrule, Boundary

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In the window for , type Club Ferrule, Boundary in the text field.

Locate the section. Under , click

In the dialog box, select in the list.

Click .

Club

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In the window for , type Club in the text field.

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In the dialog box, in the list, choose , , and .

Click .

Golf Ball

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In the window for , type Golf Ball in the text field.

Click the

button in the toolbar.

Select Domains 15–38 only.

Cover

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In the window for , type Cover in the text field.

Select Domains 15–18, 27, 28, 33, and 38 only.

Mantle

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In the window for , type Mantle in the text field.

Select Domains 19–22, 29, 30, 34, and 37 only.

Inner Core

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In the window for , type Inner Core in the text field.

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Under , click

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Core

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In the window for , type Core in the text field.

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Club Face

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In the text field, type Club Face.

Select Boundary 5 only.

Grooves

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In the window for , type Grooves in the text field.

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Select Boundaries 8–32, 34–48, 50–54, 56–59, 63, 65–69, 76–79, 85, 94–97, and 103 only.

Polished Steel

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In the window for , type Polished Steel in the text field.

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Click .

Contact Pair 1 (p1)

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toggle button.

Select Boundaries 136, 137, 139, and 141 only.

Integration 1 (intop1)

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Average 1 (aveop1)

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Average 2 (aveop2)

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Solid Mechanics (solid)

Hyperelastic Material 1

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Hyperelastic Material 2

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In the κ text field, type 1[GPa].

Viscoelasticity 1

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and choose .

In the window for , locate the section.

Click

In the table, enter the following settings:


Branch	Energy factor (1)	Relaxation time (s)
1	0.26	1e-5
2	0.19	1e-4
3	0.22	1e-3

Initial Values 2

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and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the ∂u/∂t vector as


velx	X
0	Y
velz	Z

Prescribed Displacement 1

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Select the check box.

In the u 0 x text field, type velx*t.

In the u 0 z text field, type velz*t.

Contact 1

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Under , click

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In the τn text field, type .025[ms].

Friction 1

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and choose .

In the window for , locate the section.

In the μ text field, type 0.15.

Add an node to compute typical golf metrics.

Average Rotation 1

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Select Points 158, 161, 164, 166, 169, and 172 only.

In the window for , locate the section.

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Center of Rotation: Point 1

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Select Point 165 only.

Materials

Club Head

In the window, under right-click and choose .

In the window for , type Club Head in the text field.

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	200e9	Pa	Basic
Poisson’s ratio	nu	0.3	1	Basic
Density	rho	7850	kg/m³	Basic

Set material appearance data for use in plot creation.

Click to expand the section. From the list, choose .

Cover

Right-click and choose .

In the window for , type Cover in the text field.

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Lamé parameter λ	lambLame	1.5e9	N/m²	Lamé parameters
Lamé parameter μ	muLame	0.2e9	N/m²	Lamé parameters
Density	rho	1145	kg/m³	Basic