Introduction

Fast spin rate is a critical performance outcome in spin bowling [1]. Spin bowlers impart spin to the cricket ball by applying torque to the ball via an intricate motion of the fingers which are arranged spatially around the circumference of the ball. There are many different types of spin bowling deliveries, but they can generally be divided into two main categories: finger spin and wrist spin. Since the advent of the world’s first smart cricket ball in 2011 [2, 3], the relationships between this torque and spin rate can be quantified for these types of deliveries. From a purely mechanical perspective, the conversion of torque into angular velocity is explained by the following equation:

$$\omega = {\int }_{{t}_{1}}^{{t}_{2}}\alpha \ \mathrm{d}t={I}^{-1}{\int }_{{t}_{1}}^{{t}_{2}}T \ \mathrm{d}t,$$
(1)

where ω denotes the angular velocity (spin rate); α denotes the angular acceleration; t is the time; t1, t2 are the boundaries of the integration window; T is the torque imparted to the ball by the bowler; and I is the moment of inertia of the ball.

A mechanical problem arises when the TR (resultant torque) vector imparted to the ball does not align with the ω vector [4] (Fig. 1). The included angle θ between TR and ω divides TR into two components (Fig. 1), the spin torque TS parallel to ω, and the precession torque TP perpendicular to ω [5]. However, only TS changes the magnitude of ω, whereas TP forces the ω vector to move into the TR vector without affecting the magnitude of ω [5]. Therefore, TP can be considered a wasted or lost torque, incapable of increasing the spin rate of the ball. Because one of the most important performance outcomes in spin bowling is spin rate [1], the mechanical efficiency in generating spin will be a prime consideration for spin bowlers in how they apply technique.

Fig. 1
figure 1

Principle of the relationship of torque and angular velocity vectors; a resultant torque vector and its components and their relationship to the angular velocity vector for a ball to be released with a wrist spin (lefthanded bowler); b, c torques and angular velocity versus time graphs of two bowlers; b wrist spinner, c finger spinner; TR: resultant torque, TS: spin torque, TP: precession torque, ω: angular velocity; the release points are located at the end of each time axis (11.02 s and 13.05 s); inset: TR vector and its components TS and TP, θ denotes the included angle between TR and ω

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Hence, we used the ratio of ωmax to TRmax as a spin efficiency index (SEI), an index that gives a measure of torque conversion into spin rate. As a rule of thumb, 0.3 Nm generates an average of 30 rps, so that the ratio ωmax/TRmax gives a benchmark value of 100 (reciprocal value of ‘bowling potential’ times 100 [6]). The larger ωmax/TRmax, the more efficiently the torque is converted into spin. In addition, we calculated the ratio of TSmax to TPmax as a torque generation index (TEI), rating the bowling deliveries with large TSmax and minimal TPmax as higher performance outcomes.

This study aims to reveal which biomechanical principles apply to the efficient conversion of torque into spin rate for finger-spin and wrist-spin deliveries. Coaches may be able to use these principles to identify and develop spin bowling talent.

Materials and Methods

The Smart Cricket Ball

The second prototype of the smart ball (Fig. 2a) was developed in 2014 [7]. It measures the angular velocity with three single-axis high-speed gyroscopes at a frequency of 815 Hz, transmits the data wirelessly to a laptop or smartphone, and is inductively charged [7]. The electronics are miniaturized and have a maximal diameter of 30 mm (Fig. 2b). As with any smart and instrumented sports equipment, the data are processed and visualized [8], especially in 3D (Fig. 2c). In addition to the measured angular velocity ω, the software calculates the torques TR, TS, and TP; the angular acceleration α, and the power P [5]. Furthermore, it computes four newly discovered skill parameters [5]: the precession p (speed of the movement of the ω-vector, caused by TP; Fig. 2c), the normalized precession pn (angle θ shown in Fig. 1/inset), the efficiency η (ratio of actual to ideal angular kinetic energy, if θ and thus TP were zero), and the ‘frequency’ (ratio of αmax to release ωmax). For a sine wave, the latter skill parameter αmax/ωmax would be 2πf, where f denotes the frequency of the sine wave. The smaller the ratio αmax/ωmax, the longer the period of the torque-generating cycle and the faster is the generated spin rate relative to angular acceleration and torque TR. [5]. Note that α = TS/I and not TR/I; and ω = ∫ α dt. Even if ω is directly calculated from α, the peak values of ω and α, namely αmax and ωmax, do not correlate perfectly, such that the ratio αmax/ωmax is not a constant, and rather depends on the time period during which the ball is accelerated. Consequently, bowling deliveries with maximal ωmax and minimal αmax are rated as higher performance outcomes.

Fig. 2
figure 2

a Smart cricket ball on charging dock; b electronics; c 3D visualization of the vector diagram of the angular velocity (the time is color-coded); d 3D visualization of the center of pressure (where the torque is imparted on the ball)

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Participants

Approvals from corresponding institutional ethics committees were obtained prior to profiling. Between 2012 and 2021, we profiled numerous male spin bowlers (aged 18–48) using the Smart Ball, playing cricket at various levels (from local clubs to first-class cricket). The gender distribution of the participants is explained by the availability and level of performance. Male spin bowlers generally achieve significantly higher standards of performance, including a higher spin rate. All bowlers reported that they were free from injury or dysfunction that would inhibit their bowling performance during the testing. All bowlers were requested to bowl as if under match conditions, whether outside on the oval or inside (indoor gymnasium or 30 m biomechanics lab). The bowlers bowled the ball into a net placed 12 m along the standard pitch length while sighting the target at the wickets.

We used our database of Smart Ball data (412 deliveries) to achieve the above aims. Our database of spin bowlers consists of finger and wrist spinners divided into top, side, and back spinners based on their stock ball. To achieve the above goals, we did not specifically differentiate between finger spinners and wrist spinners as we were looking for the underlying mechanical principles that govern the efficient conversion of torque into spin rate.

Data Processing

Since the physical performance parameters (ω, TR, TS, α, and P) are mathematically related (ω = ∫ α dt; α = TS/I; TS = TR cosθ; P = ω TS), and the torque to spin rate conversion is inherently a different skill factor than suggested by Eq. (1), we correlated the skill performance parameters (TP, p, pn, η, and αmax/ωmax) with the ratios ωmax/TRmax and TSmax/TPmax to identify the greatest influencing factor. We used multiple regressions with five independent variables and two dependent variables, of the general form

$$y = a + b_1 x_1 + b_2 x_2 + b_3 x_3 + b_4 x_4 + b_5 x_5 ,$$
(2)

where x1–5 are the independent variables and y is the dependent.

The degree of influence of an independent variable on a dependent variable was identified from the standardized regression coefficients and from the increase in multiple R2 when adding a fifth independent variable to the previous four [9]. The most influential factor was then further examined in terms of its biomechanical implications for optimal or suboptimal torque to spin rate conversion. For all regression analyses, the R2 and the one-sided p value of the trend (positive or negative gradient) were calculated.

Results

The statistical results of the two multiple regressions are presented in Table 1. In both regressions, the largest contributor to the variance was the ‘frequency’ parameter (αmax/ωmax), followed by the maximum precession torque TPmax. Since TPmax is the denominator of the ratio TSmax/TPmax, which is the TGI, the ideal technique of a spin bowler would be to align the TR vector with the ω-vector and thus to keep TP at zero.

Table 1 Multiple regressions of skill parameters
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Subsequently, the SEI (ωmax/TRmax) and the TGI (TSmax/TPmax) were correlated with the frequency parameter αmax/ωmax with a multiple R2 of 0.7965, and single R2 of 0.4687 and 0.4527, respectively. The combined influence, determined from the sum of the single regression R2 minus the multiple regression R2, was only 0.1249, which is not surprising, since ωmax/TRmax and TSmax/TPmax are indirectly correlated at R2 of only 0.0246 (still significant, p = 0.0007). The individual influences (semi-partial correlations) of ωmax/TRmax and TSmax/TPmax on αmax/ωmax, from the single regression R2 minus the combined influence were 0.3438 and 0.3278, respectively.

The statistical distributions of the ratios ωmax/TRmax, TSmax/TPmax, and αmax/ωmax are shown in Fig. 3a–c. In Fig. 3c, the outliers on the right tail of the histogram are from one single bowler.

Fig. 3
figure 3

Histograms of ωmax/TRmax (a), TSmax/TPmax (b), and αmax/ωmax (c); and correlations of αmax/ωmax against ωmax/TRmax (d), TSmax/TPmax (e); the blue circles identify the data of bowler 5

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The correlations of αmax/ωmax versus ωmax/TRmax and TSmax/TPmax are shown in Fig. 3d, e. In Table 1 and Fig. 3d, αmax/ωmax and ωmax/TRmax are indirectly correlated, which means that high performance values of αmax/ωmax correspond to high performance values of ωmax/TRmax (since low αmax/ωmax values are associated with high performance). Conversely, αmax/ωmax and TSmax/TPmax are directly correlated in Table 1 and Fig. 3e, which means that high performance values of αmax/ωmax correspond to low performance values of TSmax/TPmax. This is a counterintuitive and unexpected result, interpreted as follows:

If the parameter αmax/ωmax decreases (increase in performance by increasing ωmax/TRmax), the parameter TSmax/TPmax decreases (decrease in performance) to values equal to or less than 1, due to increase in TPmax and decrease in TSmax. Figure 4a shows the TS and TP time-series data of four wrist spinners (leg spinners). Table 2 presents the processed data of the four wrist spinners, who have comparable TRmax but slightly different ωmax (faster for bowlers 1 and 2), and significantly different ωmax/TRmax and TSmax/TPmax (better for bowlers 1 and 2). In contrast, bowlers 3 and 4 have excellent TSmax and TPmax data. When calculating the angular impulse ΔL of TS, i.e., ΔLS, then the data of bowlers 1 and 2 are better, even if their TSmax is lower than that of bowlers 3 and 4. The solution to this paradox is obvious: the time-window τ (Table 2), over which the time integral of TS is calculated, is wider for bowlers 1 and 2. This phenomenon can be clearly seen in Fig. 4a, as the increase in TS starts earlier for bowlers 1 and 2 (Fig. 4a, green double arrows), with the decrease of TS occurring simultaneously for all bowlers shortly before releasing the ball. It seems that a wider TS time-window τ cancels out and even outperforms the effect of the smaller TSmax with respect to the angular impulse ΔLS. In addition, the time integrals of TR and TP are also better in bowlers 1 and 2. Hence, a key biomechanical principle for the efficient conversion of torque to spin rate is the angular impulse, which requires the spin bowler to apply torque to the ball well before the release point. However, this practice of increasing the time-window τ of torque application seems to come at the expense of smaller TSmax. The time-window τ correlates with αmax/ωmax at R2 of 0.8009.

Fig. 4
figure 4

Torque versus time plots aligned, so that the ball release points share the same timestamp; a four wrist spinners (bowlers 1–4 in Table 1) and their TS and TP profiles (the green arrows emphasize the timing of the onset of TS), b TS of five bowlers (note that the earlier TS is generated, the smaller the peak TS, but the more torque is converted to spin rate); TS: spin torque, TP: precession torque

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Table 2 Skill parameter values of four wrist spinners
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Considering the outlier data as specific cases, these data were marked with a blue circle in Fig. 3c–e, plotted from torque measurements of Bowler 5 (finger spinner), who generated TRmax values between 0.41 and 0.45 Nm (mean 0.433 Nm) in seven deliveries. These outcomes were only achieved in two out of seven cases (0.41 Nm and 0.42 Nm; mean 0.381 Nm) by Bowler 2 (also finger spinner). However, Bowler 5 showed much worse αmax/ωmax values ranging from 27.3 to 28.1 s–1 (mean, 27.7 s–1), compared to the corresponding values of Bowler 2 ranging from 17.6 to 18.7 s–1 (mean, 19.4 s–1). Accordingly, Bowler’s 5 ωmax/TRmax and TSmax/TPmax average ratios were 71.9 (low) and 2.46 (high), respectively (as expected from the paradox explained above); compared to Bowler’s 2 average ratios of 97.5 (high) and 1.47 (low), respectively. The TS time data are shown in Fig. 4b, characterized by the delayed onset of TS with a short time-window of only 0.130 s. Bowler’s 5 TRmax data ranged between 0.41 Nm and 0.45 Nm, a characteristic of an elite category of wrist spinners, and can be functionally extrapolated to a value of approximately 43 rps; but, at his current level of training, Bowler 5 is unable to convert this level of torque to such a spin rate.

Discussion

Mechanics of Spin Bowling and Performance Outcomes

Spin bowling has been an effective form of bowling since the origins of the sport. The skilled spin bowler imparts spin on the ball to cause it to swerve in the air via the Magnus effect and then deviate off the pitch, factors that act to confound the batter’s ability to score runs or prevent a dismissal. In addition, higher spin rates could potentially increase the amount of swerve and lateral deviation. This is consistent with the observation that experienced spin bowlers usually demonstrate elevated mean spin rates in comparison to amateur and non-professional players. Coaches recognize that spin rate is a crucial factor in achieving success in spin bowling, particularly for those seeking to elevate their performance to a professional level. From a biomechanical perspective, this improvement in performance could be achieved by increasing the efficiency of converting the torque applied to the ball into spin rate. The Smart ball is a unique tool that can measure various parameters of spin bowling efficiency. Therefore, in this study, we used the Smart ball to explore the fundamental biomechanical principles that effectively convert torque into spin rate. With this knowledge, coaches could potentially identify and cultivate spin bowling talent by applying these principles.

Coaching Application: The Time Window

From a practical standpoint, the time frame indicates that angular impulse is a crucial factor in determining the effectiveness of spin production. A spin bowler could increase the angular impulse applied to the ball by increasing the duration of torque application to the ball. One way to improve bowling skills is using specific techniques that focus on increasing the contact time of the fingers on the ball, particularly those using the longest finger [10]. These techniques include utilizing different grips, increasing the range of forearm and wrist motion during the spin torque application, emphasizing spin torque with the middle finger, and making sure that the middle finger is the last to leave the ball's surface [10]. Based on this approach, it is likely that bowlers with longer fingers may have an advantage in applying larger angular impulses due to their longer moment arms and achieving a longer time-window of torque application. Conversely, spin bowlers who apply the spin torque over a relatively shorter time-window would need to increase the magnitude of spin torque as compensation to maintain a similar level of spin rate.

Advantages of Analyzing Bowling Biomechanics with a Smart Ball

These two biomechanical factors of increasing the torque time-window and reducing precession can only be realistically measured with our Smart Ball, as it does not require cameras and reflective markers like a motion analysis system does and, unlike the Kookaburra Smart Ball (Kookaburra, Melbourne, Australia), has three high-speed gyroscopes embedded in the ball.

Working on the premise that elite spin bowlers are the most skillful at imparting spin to the ball, we correlated the skill performance parameters (TP, p, pn, η, αmax/ωmax) with the SEI (ωmax/TRmax) and the TGI (TSmax/TPmax). In both the regressions, the largest contributor to the variance was ‘frequency’ parameter (αmax/ωmax) followed by the maximum precession torque, TPmax. The frequency parameter indicates that the underlying mechanism for converting torque TR into spin rate ω is an early onset of TS that allows a wide time-window τ over which TS can be generated. An intuitive way of understanding the frequency parameter is to consider the units (s−1). Now, if TS were a sine wave, the skill parameter αmax/ωmax would be 2πf, where the variable f denotes the frequency of the sine wave. The reciprocal of f, i.e., 1/f, is identical to the duration of a single sine-wave cycle and is proportional to the time-window τ. Both the time-window and its substitute αmax/ωmax are therefore very useful for performance profiling and talent identification. This distinguishes our Smart Ball from the Kookaburra Smart Ball (Kookaburra, Melbourne, Australia), which only calculates one spin rate data value in the early phase of the ball's flight. The magnetometer data's oscillation frequency appears to be the underlying source of the spin rate information [11], since commercially available low-speed gyroscopes possess the capability to measure spin rates only up to 5.5 rps. Only one specific gyroscope (InvenSense, San Jose, California, USA) can measure up to 11 rps. In addition, a single spin rate data value does not allow calculation of further physical performance parameters (continuous ω, α, TR, TS, and P), let alone skill parameters (p, θ, TP, η, αmax/ωmax). Furthermore, assessing spin bowling performance solely on a single spin rate data value can be misleading, because the spin rate is not only affected by performance, but also by the type of delivery. For instance, topspin deliveries, except for the Googly, produce a higher spin rate than backspin. Similarly, wrist spin deliveries generate more spin rate than finger spin [5]. Choosing young bowlers based on just one spin rate data value would only benefit wrist spin top spinners.

Reassessment of Skill Parameters

In fast bowling, the connection between spin rate and the plane of bowling shoulder circumduction is evident. It has been well established that fast bowlers tend to convert torque to spin rate inefficiently, when the SEI (ωmax/TRmax) is less than 70, and when the frequency parameter (αmax/ωmax) is greater than 25 [12, 13]. When a bowler delivers the ball, the direction of rotation of the bowling arm holding the ball corresponds to topspin. However, the ball is released with a backspin, which causes the angular velocity vector to turn 180° from topspin to backspin. Consequently, there is a decrease in the angular velocity at this transition [12, 13].

Given the usefulness of the skill parameter αmax/ωmax and its connection to converting TR into ω, the list of skill parameters (p, θ, TP, η, αmax/ωmax) needs to be revisited. The precession torque TP seems to have lost its importance, since it cannot be kept to a minimum. On the contrary, it is surpassed and even counteracted by early TS onset, which decreases TSmax and increases TPmax. The normalized precession is still useful for other purposes, namely to assess the angle θ (Fig. 1) in the early stages of torque production. Different spin bowling deliveries show clear differences in the skill parameters pmax, θ, η, and αmax/ωmax [5]. The improvement in skill parameters as an effect of the training intervention showed that θ and η improved significantly, while pmax, TSmax/TPmax, and αmax/ωmax did not change at all.

Comparing the time windows τ of bowlers 1 and 5, 0.233 s and 0.130 s, respectively, then bowler 1 produces the onset of TS 0.103 s earlier than bowler 5. Although the exact mechanism responsible for early TS onset is unknown, this study provides evidence that the conversion of the resultant torque TR (the total torque the bowler produces) into spin rate depends on how early the spin torque TS is generated relative to the release point. The earlier generation of spin torque corresponds with a longer overall time-window, which increases the angular impulse for spin generation, despite paradoxically suffering from a decreased magnitude.

Our study has demonstrated that the smart cricket ball represents a novel and effective tool for analyzing the mechanics of spin bowling. Through its capacity to evaluate multiple skill parameters, spin bowlers can be evaluated on their ability to efficiently convert spin torque into spin rate. Our research findings have revealed that the width of the torque window is associated with spin bowling effectiveness, with the spin angular impulse serving as the underlying principle for this efficiency. These results have significant implications for coaching and players alike. Techniques aimed at optimizing the spin angular impulse for spin bowlers could be explored, particularly given that these players, particularly wrist spinners, are a vital weapon in cricket across all forms of the game, including the lucrative T20 league. The smart ball could be used to provide bowlers with intelligent feedback that can help them improve their skills and excel at the highest levels, thereby increasing the potential for cricket teams to achieve success.