Mandrin
You state that your mathematical model demonstrates that the hands have to slow down prior to impact irrespective of the shoulder torque pattern. However, the Pingman machine's central arm and peripheral hinge point (equivalent to the hands) doesn't slow down prior to impact and neither does Tiger Woods' hands. JayMessner used a video taken at 1,000 frames/second and demonstrated near-constant velocity of Tiger's hand to 0.03 seconds prior to impact. Even Jay concedes that he may not be sure of his estimations in the last 0.03 seconds because it is guesswork when trying to estimate where the butt end of the club is situated when the hands rotate clockwise near impact, and the "apparent" reversal of the butt end movement at impact could affect the precision of his estimations. Therefore, it is possible that Tiger's hands do not slow prior to impact.Thirdly, Nesbit claims that the hands do not slow down prior to impact. So, which is more accurate - your mathematical model or the three contrary opinions/facts.
I find the assumptions in your mathematical model problematic.
Problem issue number 1:
You are describing a double pendulum swing action model, equivalent to the PingMan machine, where the shoulder torque is the torque force operating at the central hinge point. There is only ONE arm in this model - the central arm of the PingMan machine. There is one free pivoting peripheral hinge joint where the club attaches to the end (hands) of the central arm. In "real" peformance, the PingMan machine can maintain a constant velocity through the downswing, while your mathematical model predicts that the central arm must slow down - irrespective of the torque force pattern at the central hinge point (shoulder). Why does your mathematical model predict arm/hand slowing approximately 0.1 seconds prior to impact when the PingMan machine's central arm , and Tiger Woods hands (equivalent to the end of his left arm) doesn't slow down at that 0.1 second point in time? Your theoretical/mathematical system's arm is losing energy in the late downswing. Why? If you answer that it relates to the transfer of energy from the center to the periphery (from the shoulder to the arm to the hands), then I cannot understand that type of answer. The PingMan machine's central torque generator is capable of behaving in a way that allows the central arm to have constant velocity, with no loss of energy. Now, it could well be that the motorised device that generates torque at the central hinge point can apply additional torque force to maintain a constant arm velocity. Well - so, can a human being! I think that Tiger Woods maintains constant velocity of his left arm and hands to beyond 0.1 seconds prior to impact - presumably by applying the "correct" amount of shoulder torque (really dowswing pivot force torque and mid-upper torso turning force torque) to maintain a constant hand velocity.
Problem issue number 2:
You state that "from the starting position the wrist angle is maintained till the position shown in figure 0b, where the innner segment makes an angle of 60 degrees with the negative axis" - and you apply that "apriori" assumption to all three shoulder torque patterns? On what basis can you pre-specify that the wrist angle is maintained in the early-mid downswing when it is not NECESSARILY concordant with reality. The PingMan machine has a free-hinge joint throughout the entire downswing - all the way from the start of the downswing, and it doesn't only have a free hinge joint later in the downswing (when the central arm reaches 60 degrees from the negative Y axis) The same applies to a good golfer who has relaxed wrists - the wrists are relaxed/passive and function as a free hinge joint throughout the entire downswing. By pre-specifying a "fixed" release point (when the central arm is 60 degrees from the negative Y axis) you are "cooking the books" in the sense that you force the point of release to occur at the SAME point for all three shoulder torque patterns. That doesn't make sense! If a beginner golfer has a shoulder torque pattern (really a downswing arm force pattern) that accelerates very fast from the top and then decelerates later, then that downswing arm-force pattern is often associated with premature release (club throwaway). Your model ARBITRARILY excludes the possibility of premature club release, which means that your mathematical model has an inherent/inbuilt bias.
Problem issue number 3:
Your simplistic model doesn't take into account the fact that the COG of the club is close to the peripheral end of the club, and that "fact" will result in the club acquiring angular momentum during the early-mid downswing as the force moving the club only acts at the grip end (peripheral hinge point). I presume that degree of angular momentum developed in the early-mid downswing, and subsequently the exact TIMING of release, may depend on the behaviour of the linear force acting at the peripheral hinge point - whether it is constant, accelerating or decelerating. Your simplistic mathematical model does not deal with that complex, but highly relevant, issue.
Jeff.
