Putt too slow and you will leave the ball short - putt too fast and it will skip the hole. What are the max and min values for a successful putt? Is there an optimum speed for the ball to easily fall into the centre of the hole? This blog shows how SOLIDWORKS Motion simulation allows you to explore this type of dynamic problem using real-world physics.
Whilst watching the US Masters golf championship a couple of weekends ago and admiring the skill of the players on those treacherous greens I noticed how some players "rammed the ball home" whilst others "tickled it" to cause it to just drop into the cup. Having frequently suffered the frustration of either leaving the ball short or seeing it leap the hole, I started wondering what range of ball speeds would allow the ball to drop. Would it be quite restrictive or would there be a reasonable tolerance?
To test this I set up a quick
Motion study using SOLIDWORKS Premium. This uses real world physics and is a lot easier than trying to do the applied maths! I set up a flat part to act as the putting green. I could use a nicely sculpted shape like a real green but I don't need to do that for my test. I used a flat extrude with a cut to create the cup with a dia of 4 and 1/4 ins. I then modelled a regulation golf ball - 1.68 inch dia and 1.62 oz - complete with a dimpled surface texture to make it look realistic.
In an assy I mated the front plane of the ball to a vertical plane through the hole. This meant that the ball would always be targetted at the centre of the hole and eliminates any sideways variability (oh that I could do that when I play!). I positioned the ball on the surface of the green by eye and 2 metres away.
Having switched on Motion, I created a study and added an 'Initial Velocity' to the golf ball. I then applied 'Gravity' and a contact condition between the ball and the green.
I experimented with various speeds to get the ball to drop. It worked well but I was not happy. I know that the ball is subject to a number of forces that will make it slow down.
Firstly there is aerodynamic drag. This is small on a ball at low speed but in Motion it is easy to include. I added an 'Action only' force to the ball and used an 'Expression' to define the magnitude. This included feedback from a Motion result that I had previously created. This is a great tool as results from one time point in the study can be used to influence the next time point. The equation I used is a well known one below that calculates the drag force from the square of the velocity, air density, frontal area and the coefficient of drag. Here is what it looks like in Motion ....
What about the coefficient of drag I hear you cry? Where did that come from? Well that is where SOLIDWORKS Flow simulation is invaluable. A quick Flow project revealed a Cd of 0.47.
The second force to take into account is friction to account for any sliding of the ball on the grass. This was easy to do as I could just add it to the contacts. I guessed a value for the coefficient of friction to be 0.1. A simple test could confirm this.
The third force is rolling resistance. This would be caused in real life by the effort required to compress the grass as the ball rolls over it and any elastic deformation of the ball and the green. There are academic papers on this but I wanted to keep it simple. I estimated that the balls at Augusta would roll on a 5 degree incline. It will vary from green to green and according to a host of other things but 5 degrees sounds sensible. My applied maths tells me that the downward component of the weight would be {m x g x sin A} where A is the angle of slope and {m x g} is the mass x gravity. I created this as a force in Motion like the drag force but, on calculating, I found the ball rolled, stopped and rolled back! I realised that this was correct as the force was always on. To fix this I set up a conditional 'IF' statement that meant that the force was zero if the ball was rolling forward but zero if it were stationery or negative. This is shown below ...
With these factors set up I ran the study and played with the 'Initial Velocity' and watched as the ball was left short, dropped or skipped the hole.
Now I could adjust this manually for ages to get to my goal - the max and min velocities; but that would be tedious. Why not run a SolidWorks Motion 'Design Study' and let SOLIDWORKS do the work for me? This is also easy. I created a Motion plot to measure vertical distance. This gives me a means of sensing if the ball has dropped or stayed above ground - i.e. my pass / fail condition. I then set up a Motion Design Study 'Parameter' to allow me to vary the 'Initial Velocity'. I then allowed this parameter to vary from small values to high ones and ran the Design Study to see which speeds were successful. This initially gave me a table of results as below.
You can see that the 'scenarios' 2 - 6 were successful by both the table and the graphs. In scenario 1 the ball stopped just short of the hole and in scenario 7 it jumped. Between these extremes it drops!
I then set up further design studies that fine tuned the initial results and gave me a range of successful ball speeds: 2.03 m/s to 3.25 m/s. Any speed slower or faster than this range will result in the ball not dropping.
Watch the video.
Sadly for me this knowledge will probably not overcome my inability to get the ball from tee to green but it might just give me an insight into how hard I can putt. I will let you know if my handicap drops :)
By Andy Fulcher
Technical Manager
Solid Solutions Management Ltd