The ability of players to swing an implement with optimum speed and accuracy is a key element of performance in many sports. While players have some control over the material performance of the implement through selection decisions based on perceived quality and cost, they have greater control on the selection of physical dimensions, albeit ones often constrained by rules applied by the sport’s governing body. The implement’s material densities and dimensions combine to determine its mass moment of inertia (MOI), which is the inertial property that describes its resistance to angular acceleration. There are three MOIs that act about the principle axes (x, y, z) through the implement centre of mass (Fig. 1a). These MOIs were described for tennis rackets by Brody [1], later defined by Taraborelli et al. [2] as Transverse (x), Polar (y), and Lateral (z), and are shown in Fig. 1b for cricket bats. Application to performance of cricket bats has been described by Eftaxiopoulou [3] and in cricket batting by Headrick et al. [4]. In cricket, the Transverse MOI about the handle is of most interest as this has the biggest effect on the swing of the bat in its predominant mode of use (Fig. 1b).
There is published research involving implement swing in baseball, golf, and tennis, with investigations on swing speed, MOI, and player performance [5,6,7]. Brody [1] was the first to describe the relevance of tennis racket MOI to players and demonstrated a method of measurement for Transverse (x) and polar (y) moments. For baseball, Fleisig et al. [8] showed that bat swing speed decreased as MOI increased. Cross & Nathan [9] went on to show that the ‘intrinsic power’ of a tennis racquet, golf club and baseball bat correlates well with MOI but poorly with mass. Intrinsic power is a colloquial term coined by Cross & Nathan to describe the implement’s apparent coefficient of restitution, which determines the ball rebound speed. More recently, Schorah et al. [10] showed through a meta-analysis across several sports that swing acceleration correlates better with MOI than velocity, in accordance with the governing physics.
The cricket bat provides an interesting case study due to the wide range of inertial properties apparent in consumer products. The quality and performance of the bat materials aside, a functional bat is one that the player can manoeuvre at a range of speeds to make controlled strikes in defence or in attack for run scoring [11]. In a study on the factors affecting cricket batting performance, Peploe et al. [12] concluded that to maximise post-impact ball speed players should focus on striking it with the highest possible bat speed. The bat should also have an effective mass that enables maximum batted ball speed through the momentum transfer at impact. Cross & Nathan [9] described the ‘intrinsic power’ of an implement being dependent on the effective mass at the point of impact. Therefore, finding a bat MOI that satisfies a player’s specific requirements for bat speed and effective mass is an essential driver of performance.
MOI is not used in cricket as a parameter for bat selection, and there are two main limiting factors; (1) MOI is not provided by the manufacturers, and (2) there are no facilities at the point of purchase to measure and assess swing performance. Instead, current practice uses the knowledge of the bat’s mass in conjunction with a subjective assessment of how the bat feels when swung, which is often referred to as ‘pick-up’ (e.g. heavy, light, balanced, toe-heavy). Similar to other sports implements, MOI is problematic in its practical measurement by cricket bat makers and players due to the complex three-dimensional geometry of the bat. Measurement currently requires the use of specialist equipment to implement a simple or bifilar pendulum test as described for tennis rackets by Brody [1], for baseball bats by Koenig [13], for cricket by Eftaxiopoulou [3] and the ASTM [14]. The process of measurement using this method is also time-consuming in the context of production, which exerts a constraint on adoption as common practice.
An alternative to measuring MOI is to make a one-dimensional beam model of the cricket bat that closely matches more easily measured properties, such as mass, and centre of mass (CoM) location. The beam model has been demonstrated as a good estimator of MOI for tennis rackets, firstly by Cross [15], then refined by Goodwill [16], and recently validated for over 400 tennis rackets by Taraborrelli et al. [2]. Cross demonstrated the effectiveness of a simple two-section beam model, where the handle and frame were each represented by a one-dimensional beam of identical length but different mass. The limitation of uniform length and one beam to represent the frame was addressed by Goodwill who developed a one-dimensional five-section beam model that better represented the mass distribution of the frame. This five-section beam model produced identical mass and CoM location against seven test rackets, and MOI values within 2%. Allen et al. [17], working from Goodwill, tested unequal two-section and five-section models on 100 rackets. Model MOI values produced a root mean square error (RMSE) of 0.0015 kg m2 for both models (3% of a typical racket MOI). Thus indicating that the unequal two-section beam model is comparable to a five-section beam model, yet simpler to implement. Taraborelli et al. [2] further demonstrated the effectiveness of the unequal two-section model by testing over 400 diverse rackets, showing model values with RMSE of 0.0013 kg m2.
It is proposed that predicting MOI with beam models can apply in cricket bats. If the model is more accurate than player sensitivity to MOI changes, then this offers a method for rapid calculation since the model only requires the measurement of mass, CoM location, and section lengths as inputs. Kreifeldt and Chuang [18] studied human sensitivity to MOI. Ten males and ten females swung a hollow tube with concealed masses, with the tube dimensions set to simulate a tennis racket. Tester experience was not described, so it is unknown if they were sports players. The authors commented that the infrequent need to interact with angular accelerations prevented reliable comparisons of MOI, on the basis that it was an unrecognisable property in how an object felt when swung. However, in ball sports using swung implements the players are interacting frequently with this phenomenon. A study by Brody [19] on MOI sensitivity in tennis rackets demonstrated that non-tennis players required over 25% MOI change to detect a difference. Proficient tennis players required over 2.5% MOI change to detect a difference. The work of Brody therefore provides an initial basis for assessing the efficacy of a beam model for estimating MOI of cricket bats.
Another approach to estimating MOI is to use a first mass moment as a proxy. First mass moment is the product of mass and the distance of the centre of mass from a reference position. In golf, first moment is referred to as swingweight [20]. A reference position of 0.356 m (14 in) from the handle end of the golf club was established in the 1920s by Robert Adams and termed the Lorythmic Scale [21]. The Lorythmic Scale allowed a set of golf club irons to be matched on the property of swingweight. Cross and Nathan [9] showed that a set of irons could also be matched for MOI to within 0.15% if the reference position was changed to 0.47 m from the handle end. The implication is that first moment at a re-defined reference position could be a proxy for MOI. Harper et al. [22] showed that golfers were unable to perceive differences of < ± 3 points in golf club swingweight (first moment) on the Lorythmic scale. Three Lorythmic swingweight points is equal to 0.00432 kg m. With a typical swingweight for a golf iron at 0.155 kg m [9], three swingweight points equates to under 3% of the total swingweight. This indicates that the ability to match MOI on a set of irons to within 0.15% is well below the limit of a player’s perception of swingweight changes. Through the calculation of Weber fractions (a measure of differential sensitivity to a sensory stimulus around a reference value), Kreifeldt and Chuang [18] also noted that values of Weber fraction did not differ widely when comparing test objects with mass-MOI ratios of 1000:1 and 10:1. This finding indicates that sensitivity of golfers to first mass moment changes found by Harper could be similar to cricket bats whose mass–MOI ratio (typically 3.3:1, see Table 1) is of the same magnitude as a golf iron (typically 1.5:1 [9]). This suggests that first mass moment could be used as a suitable proxy for MOI in cricket bats. Therefore, this study developed and tested two methods of estimating MOI for cricket bats using (i) a one-dimensional beam model, and (ii) first moment as a proxy.