Finding the sweet-spot of a cricket bat using a mathematical approach - Rogers, Langton (My University, 2016)
The ideal hitting location on a cricket bat, the ‘sweet-spot’, is taken to be defined in two parts: 1) the Location of Impact on a cricket bat that transfers the maximum amount of energy into the batted ball and 2) the Location of Impact that transfers the least amount of energy to the batsman’s hands post-impact with the ball; minimizing the unpleasant stinging sensation felt by the batsman in his hands. An analysis of di↵erent hitting locations on a cricket bat is presented with the cricket bat modelled as a one dimensional beam which is approximated by the Euler-Lagrange Beam Equation. The beam is assumed to have uniform density and constant flexural rigidity. These assumptions allow for the Euler-Lagrange Beam Equation to be simplified considerably and hence solved numerically. The solution is presented via both a Central Time, Central Space finite di↵erence scheme and a Crank-Nicolson scheme. Further, the simplified Euler-Lagrange Beam Equation is solved analytically using a Separation of Variables approach. Boundary conditions, initial conditions and the framework of various collision scenarios between the bat and ball are structured in such a way that the model approximates a batsman playing a defensive cricket shot in the first two collision scenarios and an aggressive shot in the third collision scenario. The first collision scenario models a point-like, impulsive, perpendicular collision between the bat and ball. A circular Hertzian pressure distribution is used to model an elastic, perpendicular collision between the bat and ball in the second collision scenario, and an elliptical Hertzian pressure distribution does similarly for an elastic, oblique collision in the third collision scenario. The pressure distributions are converted into initial velocity distributions through the use of the Lagrange Field Equation. The numerical solution via the Crank-Nicolson scheme and the analytical solution via the Separation of Variables approach are analysed. For di↵erent Locations of Impact along the length on a cricket bat, a post-impact analysis of the displacement of points along the bat and the strain energy in the bat is conducted. Further, through the use of a Fourier Transform, a post-impact frequency analysis of the signals travelling in the cricket bat is performed. Combining the results of these analyses and the two-part definition of a ‘sweet-spot’ allows for the conclusion to be drawn that a Location of Impact as close as possible to the fixed-end of the cricket bat (a point just below the handle of the bat) results in minimum amount of energy transferred to the hands of the batsman. This minimizes the ‘stinging’ sensation felt by the batsman in his hands and satisfies the second part of the definition of a sweet-spot. Due to the heavy emphasis of the frequency analysis in this study, the conclusion is drawn that bat manufacturers should consider the vibrational properties of bats more thoroughly in bat manufacturing. Further, it is concluded that the solutions from the numerical Crank-Nicolson scheme and the analytical Separation of Variables approach are in close agreement.
University Of The Witwatersrand Department Of Computational And Applied Mathematics Masters’ Dissertation 2015
Keywords: Cricket bats; Lagrange equations