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Trampoline effect


Introduction

A golf ball when struck by a clubhead behaves like an anelastic medium. This means that the deformation depends both on the magnitude and the time rate of change of the applied load. The resulting hysteresis in the response to an applied load represents its anelastic behavior.

This means that not all of the kinetic energy can be stored as strain energy during the compression and not all of this strain energy can be recovered when the golf ball regains its shape. Hence unlike elastic collisions some fraction of the energy is lost. However deformations are reversible for both.

The modern titanium drivers have a very thin walled club face starting to act as a spring with a response time fast enough to have a noticeable effect on the dynamics of impact. Instead of one there are now two mass spring systems interacting at impact.

To conveniently analyze this problem we will use simple equivalent physical models to represent both clubhead and ball, assuming non-linear Hertzian behavior for ball and linear behavior for clubface.

Our goal is not to be exact, this requires experimental work for validation, but uniquely to show the mechanism behind the trampoline effect. Hence to go beyond the popular explanation invoked such as 'being like
jumping into a swimming pool off a springboard compared to a concrete platform'.

[Graphics:HTMLFiles/trampoline_effect_2_1.gif]

The trampoline model is shown in Fig1. The clubhead is modeled having two masses, clubhead mass m0,  clubface mass m1,& and connected by a spring and a dashpot. The ball is modeled as a mass m2 and a Hertzian dashpot and spring. The derived governing DV equations are given below.

m0 x0''[t] =       k1 (x1[t] - x0[t]) + c1 (x1 '[t] -  ...  x1[t]) Abs[x2[t] - x1[t]] ^a   - c2 (x2 '[t] - x1 '[t]) Abs[x2[t] - x1[t]] ^b

[Graphics:HTMLFiles/trampoline_effect_2_3.gif]


To have a reference for the above model we will use another model for the clubhead with zero compliance for the clubface, Fig2. The value of the mass m1 is taken equal to the sum of masses m0 and m1. Comparing the two models allows more readily to show effect of the trampoline mechanism. The derived governing DV equations are shown below.

m1 x1''[t]    =       k2 (x2[t] - x0[t]) Abs ... Abs[z2[t] - x1[t]] ^a   - c2 (x2 '[t] - x0 '[t]) Abs[x2[t] - x1[t]] ^b

[Graphics:HTMLFiles/trampoline_effect_2_5.gif]


The spring stiffness and damping are shown in Figs 3 and 4, (red→ club, blue→ ball). The numerical values for k2 and c2 are determined from the known impact duration Δt and the known coefficient of restitution (COR), e. The damping coefficient c1 is chosen to account for some light damping for the clubface and k1 is determined from optimizing the trampoline effect.

We will below compare systematically the trampoline model with the reference model - left side reference model, right side trampoline model. This makes it very simple to see clearly the effect of the compliance of the clubface. The first vertical dashed line indicates maximum compression, the second vertical dashed line indicates when ball and clubface separate, hence the end of collision.

[Graphics:HTMLFiles/trampoline_effect_2_6.gif]


Figs 5a and 5b show the force, in the two models, acting on the clubface. The peak force is reduced from 13300 to 8400 Newton. In Fig5b and for all subsequent figures on the right side it is to be noted that the impact duration increases noticeably for the trampoline model.

[Graphics:HTMLFiles/trampoline_effect_2_7.gif]


Figs 6a and 6b show the comparative force levels, in the two models, acting on the ball. The peak force is reduced from 13300 to 7800 Newton.

[Graphics:HTMLFiles/trampoline_effect_2_8.gif]


As for the spring forces the damping forces equally reduce substantially in magnitude in the trampoline model. Since the damping represents the energy dissipation it is clear that the trampoline model has less energy loss associated with its operation.

[Graphics:HTMLFiles/trampoline_effect_2_9.gif]


Figs 8a and 8b1,2 show the damping forces in a different way, as a function of compression/deflection, instead of time. Note that the area under the curves represents the energy lost during collision. It is immediately evident that the losses for the trampoline model are substantially less - the sum of the surfaces under the curves in Figs 8b1 and 8b2 is less than the surface under the curve in Fig 8a, for the reference model.

[Graphics:HTMLFiles/trampoline_effect_2_10.gif]


Comparing the compression curves of ball and club face for both models show clearly two very essential features of the trampoline effect. There is both a substantial reduction of the deformation of the ball - compare the blue curves - and a substantial increase in the duration of the collision. The additional mass spring system of the clubhead increases the overall response time. This spreads out the energy / momentum over more time causing lower force levels and hence less deformation and subsequently less energy loss.

[Graphics:HTMLFiles/trampoline_effect_2_11.gif]


The potential energy is stored in the form of strain energy in ball and clubface. The trampoline model shows the potential energy almost equally shared between ball and clubface.

[Graphics:HTMLFiles/trampoline_effect_2_12.gif]


There is quite an improvement for the coefficient of restitution, COR or e, in the trampoline model. For our reference input clubhead speed of 100 mph (44.7 m/s) this means an increase for the ball separation speed of 7.3 mph and hence a potential increase in carry of about 22 yards.    

[Graphics:HTMLFiles/trampoline_effect_2_13.gif]


Figs 12a and b show clearly the transfer of kinetic energy from clubhead to ball. The dashed line represents the kinetic energy of the clubhead. The dotted lines represent the total kinetic energy, the sum of the kinetic energy of 'clubhead' (red), clubface (green) and ball.

[Graphics:HTMLFiles/trampoline_effect_2_14.gif]

Figs 13a and b show the energy loss during the collision for both models. The kinetic input energy, for both models, is 200 Joule. The reference model shows an energy loss of 11.36 Joule ( 5.7 %) and the trampoline model has a energy loss of 5.45 Joule (2.7 %).




Conclusions

Using a collison is an effective way to obtain speed multiplication, approaching 1.5 for a driver. The thin walled flexible driver clubface is a means to further increase the coefficient of restitution of impact between ball and clubhead and hence further increasing the attainable speed multplication factor.

The additional springiness of the thin walled clubface softens the blow of impact by spreading the momentum/energy out over a larger time interval. This means that the ball compresses less, experiences less force, and as a consequence dissipates less energy.

The workload, in the trampoline model, is shared almost equally between ball and clubhead. The ball dissipates less energy, the clubface spring exhibits very little loss and hence the combined operation of clubhead/face and ball constitutes now a more energy efficient operation.

The energy percentage dissipated in a golf ball is decreasing with less force/deformation. Hence there is both an absolute and a percentage gain in the efficiency of the conversion process, occurring in the ball, from kinetic energy to strain energy, and back to kinetic energy during the collision.

Notice that the trampoline effect increases substantially the impact duration, from 0.00045 sec to 0.00071 sec, an increase of 58 %.   I like to remind that our reference model has a clubface with zero compliance which means an infinitely rigid clubface.

mandrin