A Unified Mechanical Framework for Evaluating Stretch–Shortening Cycle Function

The stretch–shortening cycle (SSC) enables explosive movement by allowing muscles and tendons to store and release elastic energy. The reactive strength index (RSI), defined as jump height divided by ground contact time, is widely used to assess SSC performance because it is simple and based on common field measurements. However, RSI mixes variables of incompatible dimensions, behaves inconsistently across tasks, and assigns high values to very short contact times even when little impulse is produced. These limitations indicate the need for a measure that aligns more closely with the mechanical requirements of SSC function. This study introduces the Dynamic Rebound Index (DRI), a dimensionless quantity that characterizes SSC performance using total vertical displacement demand and the time available for acceleration under gravity. DRI was derived from fundamental kinematic relationships and evaluated using modeled inputs spanning representative ranges of contact time, jump height, and drop height. Contour maps, surface plots, standardized differences, cross-sections, and time-varying trajectories were used to compare DRI with RSI across identical conditions. DRI behaved consistently with the mechanics of vertical motion: it increased only when short contact times coincided with large total vertical displacements and scaled predictably with changes in eccentric loading. RSI did not follow these patterns and produced inflated values for stiff, low-displacement rebounds. Cross-sectional and trajectory analyses further showed that DRI distinguished mechanically effective from ineffective short-contact strategies, whereas RSI remained driven primarily by stance duration. These findings support DRI as a more interpretable and generalizable index of SSC performance while preserving the measurement simplicity that contributed to the widespread use of RSI.

Introduction

The ability to evaluate the performance of the stretch–shortening cycle (SSC) is central to understanding explosive human movement. Roughly partitioned into three phases (eccentric, amortization, and concentric, respectively), the SSC enables musculotendon units to store and release elastic energy to allow the body to generate higher forces and velocities with greater mechanical efficiency (Boscoet al., 1982; Komi, 2000). Movements such as sprinting, jumping, hopping, and sudden changes in direction, which depend on rapid force application, are supported by an effective SSC (Cavagnaet al., 1971; Weyandet al., 2000; Weyandet al., 2010). Because of this, identifying a simple and interpretable index of SSC function has long been a priority in both research and applied sport settings (Flanagan & Comyns, 2008).

The reactive strength index (RSI), most commonly defined as the ratio of jump height to ground contact time, has become the most widely used metric for quantifying SSC in research and in practice (Younget al., 1995; Flanagan & Comyns, 2008). Jump height and ground contact time contribute to the appeal of RSI, as they require little effort to measure in the field with simple tools, and on the surface appear to quantify SSC function (Ballet al., 2010; Barr & Nolte, 2011). However, several shortcomings have become apparent. Firstly, RSI conflates variables of incompatible dimensions. Jump height divided by contact time produces values expressed in velocity units (m·s⁻¹) even though, in practice, RSI is interpreted as a dimensionless indicator of SSC performance (Healyet al., 2018). Secondly, the form of RSI is not standardized, with some versions using flight time instead of jump height or simply applying the ratio to countermovement rather than drop jumps (Flanagan & Comyns, 2008; Healyet al., 2018). The result is inconsistent outcomes across studies and testing protocols. Additionally, because contact time appears in the denominator, RSI is especially sensitive to short contact durations which yield disproportionately high RSI values (Healyet al., 2018). Conversely, movements that achieve larger aerial phases through longer push durations are penalized even if impulse is greater (Barr & Nolte, 2011; Walshet al., 2004).

Vertical jump performance is governed by the mechanical demands of opposing gravity and the time available to do so (Bobbert & van Soest, 2001). Thus, any meaningful index of SSC function would have to capture the effectiveness of how the body’s center of mass is accelerated upward against gravity during the stance phase (Fig. 1). For such a metric to be adopted, however, it is necessary to retain the use of accessible, non-specialized, field-based measurements that have contributed to the appeal of RSI.

Fig. 1. Conceptual illustration of the stretch–shortening cycle variables used to compute the Reactive Strength Index and the Dynamic Rebound Index, depicting the drop-landing, amortization, and propulsive phases of a depth jump along with jump height, drop height, and contact time as the quantities used to compute each metric.

This study proposes and evaluates a new dimensionless quantity that satisfies both criteria. The Dynamic Rebound Index (DRI) is designed to provide a unified, mechanics-based measure of SSC performance across tasks that vary in contact time, jump height, and entry velocity. We hypothesize that DRI will exhibit predictable, physically interpretable behavior that distinguishes it from RSI: specifically, that it will (1) remain consistent with the governing kinematics of vertical motion, (2) penalize unrealistically short contacts without sufficient impulse, and (3) generalize across SSC tasks without requiring multiple definitions, all while retaining the field-measurement simplicity that underlies RSI’s practical appeal.

Methods

Theoretical Framework

The framework presented here expresses jump performance in terms of the gravitational and temporal variables that determine vertical displacement. This can be applied across SSC tasks by describing performance as the combined influence of (1) the height achieved during the aerial phase, (2) the time available for force application during ground contact, and (3) the gravitational acceleration that must be overcome. The first step in this framework expresses the jump as a displacement produced under a constant gravitational load and a finite period of acceleration.

When the net vertical force applied to the body’s center of mass (F_net) differs from body weight (W_b = mg), the resulting vertical acceleration (a) can be defined by Newton’s second law as:

(1) $$\begin{array}{r}{F}_{net}=F_{grf}-W_b=ma\end{array}$$

where F_grf is the average ground reaction force during the stance phase and m is body mass and W_b = mg is body weight.

Under the simplifying assumption that the mean net acceleration is constant across the stance phase, the velocity at take-off (v_to) can be expressed as the product of acceleration and the time available for force application (t_c):

(2) $$\begin{array}{r}{v}_{to}=a{t}_c\end{array}$$

The height of the center of mass (h) achieved during flight depends on takeoff velocity and the constant downward acceleration due to gravity (g). From established kinematic relationships, this can be expressed as:

(3) $$\begin{array}{r}h=\frac{v_{to}^2}{2g}=\frac{a^2{t}_c^2}{2g}\end{array}$$

Equation (3) defines the vertical displacement of the center of mass as a function of both the magnitude and duration of net acceleration. This provides a mechanical basis for quantifying performance using quantities measurable in the field: jump height and ground contact time.

Formation of a Dimensionless Group

To evaluate SSC performance independently of measurement units, (3) was rearranged to isolate a dimensionless ratio that links the effective upward displacement demand to the time available for acceleration under gravity:

(4) $$\begin{array}{r}\frac{h}{g{t}_c^2}=\frac{1}{2}{\left(\frac{a}{g}\right)}^2\end{array}$$

Equation (4) expresses a ratio proportional to the effective upward displacement demand relative to the displacement expected under gravity over the same squared stance duration. Because both the numerator and denominator have dimensions of length, this ratio is unitless.

Extension to Drop Jump Conditions

For SSC tasks involving a preceding descent, such as drop jumps, the center of mass possesses a downward velocity at touchdown (v_td). The kinetic energy associated with this velocity must first be dissipated before an upward take-off velocity can be generated. This additional energy requirement can be expressed as an equivalent displacement term using drop height (h_drop), derived from the standard energy–height relationship under gravity and, in typical drop-jump tasks, this equivalent height corresponds to the imposed drop height:

(5) $$\begin{array}{r}{h}_{\text{drop}}=\frac{v_{td}^2}{2g}.\end{array}$$

Because the athlete must both reverse the downward velocity gained from falling and generate the upward velocity required for takeoff within the same stance duration, the combined mechanical demand of the movement is proportional to the sum of the equivalent drop height and the achieved jump height, which appear additively in the governing kinematic relationships. This additive relationship follows directly from the kinematic identity v² = 2gh, which applies independently to both the downward and upward phases of the movement. By incorporating this term into the total vertical displacement demand, the numerator in (4) becomes the sum of the achieved jump height and the equivalent drop height, yielding the complete expression for the Dynamic Rebound Index (DRI):

(6) $$\begin{array}{r}DRI=\frac{h+h_{\text{drop}}}{g{t}_c^2}.\end{array}$$

Equation (6) therefore defines DRI as the ratio of total vertical displacement demand to the gravitational displacement expected over the squared contact duration. The numerator and denominator share identical dimensions, ensuring that DRI remains dimensionless.

Comparison Metric

For comparison, the traditional reactive strength index (RSI) was computed as:

(7) $$\begin{array}{r}RSI=\frac{h}{t_c}.\end{array}$$

Both DRI and RSI rely on the same field-measurable quantities (jump height and ground contact time) but differ fundamentally in their dimensional and mechanical interpretation. Where RSI scales linearly with the inverse of contact time, DRI scales with its square, meaning that unrealistically short contacts cannot inflate performance unless accompanied by proportionally greater vertical displacement.

Theoretical Input Ranges and Analysis

The analysis generated modeled data across representative ranges of ground contact time, jump height, and equivalent drop height to evaluate how DRI behaves relative to RSI under stretch–shortening cycle conditions. These ranges reflect values commonly observed in rebound, countermovement, and drop-jump tasks (Table I):

Table I. Modeled Parameter Ranges used for Generating RSI and DRI Surfaces

Variable	Range	Rationale
Ground contact time (tc)	0.10–0.30 s	Spans fast rebound contacts through longer-duration countermovement contacts
Jump height (h)	0.10–1.00 m	Captures performance from submaximal to elite output ranges
Drop height (hdrop)	0.20, 0.30, 0.40, 0.50 m	Common descent magnitudes in drop-jump testing

The analysis constructed a uniform grid of t_c and h values across the specified ranges and used it to compute RSI and DRI for every combination. The procedure computed DRI separately for each fixed drop height to examine how the metric scales with changes in eccentric loading. The same grid allowed the procedure to compute RSI and establish a baseline for comparison.

The modeled output generated a series of performance landscapes and comparative analyses. Contour plots illustrate how RSI and DRI vary across the t_c–h plane. Three-dimensional surface plots reveal how DRI changes with drop height and confirm that RSI shows no corresponding variation. A standardized difference map identifies regions where the two metrics diverge. Cross-sectional analyses examine both metrics across the full t_c range at a fixed jump height and drop height. A modeled trajectory compares the temporal behavior of RSI and DRI across a sequence of changing values.

All analyses use either Python 3.12 with ‘NumPy’ for numerical computation and ‘Matplotlib’ for visualization or Microsoft Excel, in the case of the more rudimentary plots and analyses. The analysis relies entirely on modeled values derived from the governing equations for RSI and DRI rather than empirical measurements.

Results

All modeled outputs were generated across contact times ranging from 0.10–0.30 s, jump heights ranging from 0.10–1.00 m, and fixed drop heights of 0.20–0.50 m. A uniformly spaced grid of 100 × 100 tc–h combinations was constructed across these ranges, and each combination was treated independently.

Contour Analyses

RSI varied systematically across the modeled range of contact times and jump heights. As shown in Fig. 2, RSI values were highest at the shortest contact times and largest jump heights, forming a steep gradient across the contact-time axis. The contour map showed that maximum RSI values occurred along the lower boundary of contact time and the upper boundary of jump height.

Fig. 2. Contour map of the Reactive Strength Index across the modeled range of contact times and jump heights, showing a systematic increase with larger jump heights and shorter contact times that produces a linear gradient aligned with the ratio structure of the metric.

DRI values changed across contact time, jump height, and drop-height conditions. The contour maps in Fig. 3 illustrated progressive increases in DRI as both jump height and drop height increased. Across all drop-height conditions, DRI increased only when total vertical displacement (h + h_drop) increased, reflecting its dependence on combined eccentric and concentric contributions. Within each drop-height condition, DRI reached its greatest values when short contact times coincided with large vertical displacements.

Fig. 3. Contour maps of the Dynamic Rebound Index across four panels where drop height is held constant at 0.20, 0.30, 0.40, or 0.50 m, illustrating that DRI increases only when short contact times coincide with large jump heights and greater imposed drop heights, revealing its dependence on total vertical displacement demand and stance duration.

RSI surfaces displayed identical shapes across all four drop-height conditions (Fig. 4). For every tc–h coordinate, RSI values were numerically identical regardless of drop height, confirming that drop height did not influence RSI magnitude. Each surface showed the same pattern of highest RSI values at short contact times and large jump heights.

Fig. 4. Three-dimensional surface maps of the Reactive Strength Index across four drop-height conditions, revealing identical surface geometry regardless of drop height and confirming that RSI depends solely on the ratio of jump height to contact time.

DRI surfaces differed across drop-height conditions, as depicted in Fig. 5. Larger drop heights produced broader DRI ranges, higher peak values, and greater curvature within the same tc–h grid. Across all conditions, DRI reached its highest values at the shortest contact times and largest modeled displacements.

Fig. 5. Three-dimensional surface maps of the Dynamic Rebound Index across four drop-height conditions, showing that DRI surface magnitude and curvature increase with drop height, reflecting sensitivity to initial potential energy and total displacement prior to take-off.

A standardized difference map quantified the divergence between the two metrics, shown in Fig. 6 for a representative drop height. The map represented (DRI − RSI) scaled to the pooled standard deviation. Positive values occurred when short contact times combined with large displacements, whereas negative values appeared in regions with short contact times and small displacements.

Fig. 6. Standardized difference map of Dynamic Rebound Index minus Reactive Strength Index for a fixed drop height of 0.30 m across the complete contact-time and jump-height grid, highlighting regions in which DRI exceeds RSI at short contact times paired with large displacements and regions in which RSI exceeds DRI when displacement remains small.

For the cross-sectional comparison, jump height was held constant at 0.4 m while contact time varied from 0.10 to 0.30 s. Both metrics were normalized to their respective maxima to allow direct comparison. Fig. 7 shows these normalized cross-sectional trajectories. Both metrics began at 1.0 at the shortest contact time. As contact time increased, the two curves diverged. RSI declined gradually, retaining more than one-third of its maximum value at 0.30 s. DRI decreased sharply, falling below 0.25 of its maximum by 0.30 s. These patterns indicated that the two metrics responded differently to increasing stance duration despite being derived from identical inputs.

Fig. 7. Normalized cross-sectional trajectories of DRI and RSI across the full contact-time range from 0.10 to 0.30 s with jump height and drop height held constant at representative mid-range values, demonstrating that RSI exhibits a shallow, nearly linear decline while DRI shows a steep nonlinear decrease, revealing the contrasting sensitivity of each metric to stance duration under identical displacement conditions.

Modeled Trajectory Analysis

A modeled sequence of paired changes in contact time and displacement produced distinct temporal profiles for RSI and DRI. As illustrated in Fig. 8, two hypothetical movement strategies were examined. The “stiff” profile consisted of short contact times paired with modest displacements, whereas the “effective” profile combined moderate contact times with progressively larger displacements. RSI increased whenever contact time decreased, regardless of the magnitude of displacement. DRI increased only when short contact times co-occurred with large displacements, and the two strategies converged as contact time increased. These distinctions demonstrated the fundamentally different temporal behaviors generated by the two metrics under the same modeled inputs.

Fig. 8. Modeled trajectories contrasting stiff and effective movement strategies for RSI and DRI, generated by applying identical paired progressions of contact time and vertical displacement across the same sequence of modeled conditions, and showing that RSI favors the stiff, low-displacement profile whereas DRI favors the effective profile that couples short contact times with large total displacement.

Discussion

The purpose of this study was to evaluate whether the Dynamic Rebound Index (DRI) provides a more mechanically coherent representation of stretch–shortening cycle (SSC) performance than the traditionally used Reactive Strength Index (RSI). Three hypotheses guided the analysis, each addressing a distinct property of SSC behavior: alignment with governing mechanics, treatment of short contact times, and generalization across tasks with variable entry velocities. The modeled outputs supported all three hypotheses and provided a consistent basis for interpreting how DRI and RSI differ in their sensitivity to the mechanical determinants of vertical rebound performance.

Hypothesis 1: Alignment With Governing Mechanics

The first hypothesis predicted that DRI would follow the kinematic structure of vertical displacement by scaling with total vertical displacement and the squared time available for force application. This expectation was confirmed by the contour and surface behaviors observed throughout the modeled conditions. The DRI contour maps (Fig. 3) demonstrated that DRI increased only when short contact times were paired with large total vertical displacement demand (h + hdrop). This pattern reflects the requirement for substantial upward acceleration when the body must reverse downward momentum and produce a large aerial displacement within a limited stance duration.

The DRI surfaces (Fig. 5) further reinforced this interpretation. As drop height increased, DRI ranges expanded, and the surfaces exhibited greater curvature, indicating the metric’s sensitivity to eccentric loading. This behavior is consistent with the mechanics of SSC actions, where larger pre-impact velocities impose greater displacement demands on the musculotendon system. The systematic changes observed across drop heights confirm that DRI is grounded in the physical requirements of accelerating the center of mass upward against gravity.

In contrast, RSI did not exhibit these features. The RSI contour maps (Fig. 2) and surfaces (Fig. 4) showed nearly identical geometry across all drop-height conditions, reflecting its insensitivity to the magnitude of the initial potential energy or the total vertical displacement requirement. RSI increased whenever contact time decreased, regardless of whether the movement generated sufficient impulse to produce meaningful upward acceleration. These observations support the first hypothesis: DRI remained consistent with the governing mechanics of vertical motion, whereas RSI varied predominantly with the temporal component of its ratio definition.

Hypothesis 2: Treatment of Short Contact Durations

The second hypothesis predicted that DRI would avoid rewarding unrealistically short contact times unless accompanied by adequate displacement, thereby penalizing stiff, low-impulse rebounds. The results supported this hypothesis across multiple analyses.

The standardized difference map (Fig. 6) showed that RSI produced high values in regions characterized by short contact times and small total displacements, conditions inconsistent with effective SSC function. DRI, in contrast, assigned low values to these same regions. Positive differences (DRI − RSI) occurred only when short contact times co-occurred with large total displacements. This pattern highlights how DRI distinguishes between mechanically effective and ineffective short-contact strategies.

The normalized cross-sectional trajectories (Fig. 7) further clarified these differences. RSI declined gradually as contact time increased, retaining more than one-third of its peak value at 0.30 s. DRI declined sharply over the same interval, falling below 0.25 of its maximum. These trajectories demonstrate that RSI is disproportionately influenced by small decreases in contact time, while DRI maintains sensitivity to the combined influence of displacement and stance duration.

The modeled stiff and effective movement strategies (Fig. 8) provided a final illustration. RSI favored the stiff profile throughout the entire sequence, increasing whenever contact time decreased, regardless of the modest displacement produced. DRI favored the effective profile, increasing only when short contact times were accompanied by large displacements, and the two profiles converged as contact time increased because longer stances allow similar net acceleration. These results confirm the second hypothesis by showing that DRI does not replicate the well-documented artifact in RSI that equates “shorter contact” with “better performance” independent of mechanical effectiveness.

Hypothesis 3: Generalization Across SSC Tasks

The third hypothesis predicted that DRI would generalize across SSC tasks with different entry velocities without requiring multiple metric definitions. This prediction was supported by the clear and systematic influence of drop height on DRI surfaces.

In the DRI surfaces (Fig. 5), increasing drop height expanded the surface curvature and elevated the peak values, indicating a direct and interpretable sensitivity to eccentric loading. Because drop height alters both pre-takeoff velocity and total displacement demand, the resulting changes in DRI align with the expected mechanical consequences of varying entry conditions.

RSI showed none of these characteristics. The RSI surfaces (Fig. 4) were identical across all drop-height conditions, demonstrating that the metric is entirely insensitive to the initial conditions of the movement. This insensitivity prevents RSI from distinguishing between countermovement jumps, drop jumps, and other SSC tasks where entry velocity and eccentric loading differ substantially.

Together, these observations support the third hypothesis: DRI provides a unified representation of SSC behavior across tasks that vary in eccentric demand, whereas RSI requires multiple interpretive frameworks that do not generalize across conditions.

Practical Implementation and Measurement Pathways

The final aim of the study was to ensure that DRI could be implemented using measurement pathways already available in field and laboratory settings. Fig. 9 summarizes the practical computation routes. DRI can be derived using (1) contact time and flight time, (2) contact time and jump height, or (3) force-plate estimates of net impulse and derived takeoff velocity. Because all pathways converge on the same dimensionless formulation, DRI can be incorporated into existing testing workflows without requiring new instrumentation or processing methods. This flexibility enhances the practical utility of the metric and supports its adoption in applied performance environments.

Fig. 9. Illustration of three measurement pathways for computing the Dynamic Rebound Index from common instrumentation, including direct calculation from measured variables, estimation from aerial time, and force-plate-based methods that derive take-off velocity from net impulse.

Limitations

The present analysis relied on modeled data to isolate the mechanical behaviors of RSI and DRI. Although this approach allowed clear comparison of metric properties, empirical validation is needed to evaluate DRI across athlete populations, movement strategies, and SSC task variations. The model assumed constant mean net acceleration during stance and did not incorporate joint-level mechanics or energy losses. Future work should assess the reliability, sensitivity, and minimal detectable change of DRI in real-world testing.

Future Directions

Future research should examine DRI across repeated trials and controlled manipulations of drop height, impulse, and contact time. Establishing normative ranges and test–retest consistency will be important for performance monitoring. Interventions that modify eccentric loading or concentric impulse generation may reveal how changes in DRI correspond to meaningful adaptations. Because DRI can be computed through multiple measurement pathways, the metric may also help unify assessment across technologies that differ in how jump height, contact time, or takeoff velocity are measured.

DRI captured essential features of SSC mechanics that RSI did not. Across all analyses, DRI incorporated both displacement and temporal components, distinguished between effective and ineffective short-contact strategies, and generalized across tasks that varied in eccentric loading and entry velocity. These properties establish DRI as a mechanically interpretable and practically deployable index of SSC performance.

Conflict of Interest

The author declares that they do not have any conflict of interest.