Analyzing the Biomechanical Characteristics of Ski Jumping Take-Off Phase Based on CFD

Abstract

This study aimed to analyze the aerodynamic characteristics of Chinese Nordic combined athletes during the ski jump take-off process, comparing them with elite athletes from the 2009 Nordic World Ski Championships using computational fluid dynamics (CFD) methods. Methods: Using 3D model analysis and continuous relative phase analysis, CFD methods were utilized to assess the mechanical characteristics of athletes during the take-off phase. Results: The analysis revealed that Chinese athletes displayed a lower dominance of the knee joint during the take-off phase, leading to increased air drag. Conclusion: Reduced knee joint dominance and an excessive ankle angle at the initiation of the ski jump take-off contribute to higher air drag. The lean angle of the body and the ankle angle post-take-off significantly affect the resultant lift and drag forces.

1. Introduction

Ski jumping, a critical component of Nordic combined, influences the starting order for the subsequent cross-country skiing event. This sport is divided into four distinct phases: the in-run stage, take-off, flight, and landing [1]. The final score in ski jumping is primarily determined by the athlete’s flight distance. During the take-off phase, athletes must rapidly extend their legs and torso to achieve an optimal body posture within a short timeframe, setting the stage for the flight phase. The take-off phase typically lasts around 300 milliseconds [2], necessitating high coordination and muscular strength in the lower limbs. At take-off, athletes reach horizontal speeds of approximately 25 m/s, and rapid changes in joint angles at such high speeds create significant physical drag. Previous research has demonstrated that kinematic parameters during the take-off phase are crucial for determining the final flight distance [2,3]. Errors during the take-off phase cannot be corrected in the flight phase [4], highlighting the importance of take-off for overall performance. The effectiveness of the take-off phase directly influences the athlete’s final score [5].

To analyze an athlete’s coordination, quantitative methods can be used to measure the relationships among various joints during movement. This approach facilitates direct observation and comparison, offering valuable insights for athletic training [6]. Quantifying limb coordination provides a numerical representation of these relationships [7]. For example, continuous relative phase (CRP) analysis is employed to assess limb coordination. CRP analysis is widely used in sports to evaluate athletes’ performance [8,9,10]. By applying CRP analysis to joint coordination, we gain a clear understanding of how different joints interact throughout a movement. This information is crucial for improving athletic techniques and training methods.

Keizo et al. found that different take-off techniques used by ski jumpers significantly affect the drag experienced during the take-off phase [11]. Although ski jumping is closely linked to aerodynamic characteristics, computational fluid dynamics (CFD) methods are primarily applied to the flight phase [1,12,13,14,15]. Research on the take-off phase is relatively limited.

This study investigates the kinematic and aerodynamic characteristics of Chinese athletes (CAs) in comparison to Nordic World athletes (NWAs) during the take-off phase using various analytical methods. We specifically analyzed the drag of the left and right lower legs (L/RLDs), left and right thighs (L/RTDs), drag of the torso (TOD), total drag (TD), and total lift (TL) at the start moment of the take-off phase (SMT) and the end moment of the take-off phase (EMT). We also calculated the lift-to-drag ratio (L/D) based on the total lift and total drag.

We employed Direct Linear Transformation (DLT) to process images and develop a three-dimensional motion model, which provided the kinematic data of the athletes. Additionally, Continuous Relative Phase (CRP) analysis was used to quantify intra-limb coordination during the take-off phase. Combined with Computational Fluid Dynamics (CFD) analysis, we performed a comprehensive examination of the aerodynamic features of athletes in various postures. By integrating these complementary methods, we created a robust research framework to elucidate the mechanical mechanisms of the take-off phase. This study offers new insights into ski-jumping biomechanics and provides scientific evidence to enhance training and performance.

This study investigates the impact of different joint motion characteristics on aerodynamic properties during the take-off phase using CFD and CRP methods. The findings provide a theoretical foundation for future sports training and research, enhance understanding of the aerodynamic effects of joint movements, and contribute valuable insights to the field of sports biomechanics.

The structure of this manuscript is as follows: First, we present the background and objectives of the study. Next, we detail the research methods and data-processing procedures. We then report the research results and, finally, summarize the main findings and discuss their implications in relation to previous research.

2. Method

This study collects kinematic data from Chinese ski jumpers during competitions and from Nordic elite athletes. Using CRP and CFD analyses, we investigate the biomechanical characteristics of athletes during the take-off phase. The detailed flowchart is shown in Figure 1.

2.1. Test Subjects and Venue

This study was conducted at the National Ski Jumping Training Base in Laiyuan, Baoding, China, to collect data. The subjects were five Chinese Nordic combined athletes, all first-class or higher-level athletes. Video recordings captured the athletes’ take-off actions during ski-jumping training. During testing, each athlete wore the same ski-jumping suit, helmet, and skis. All the athletes performed two full ski jumps at their best, and the data from their optimal performances were selected for analysis.

This study adhered to the ethical principles outlined in the Declaration of Helsinki. All participants provided informed consent prior to testing, ensuring anonymity and confidentiality. Participation was entirely voluntary.

2.2. Data Collection and Processing

This study utilized a three-dimensional (3D) motion-capture system, which included two high-definition cameras, two camera stands, and a 3D calibration framework, operating at a sampling frequency of 50 Hz, to collect data from athletes on-site during competition. The coverage area of the two cameras was approximately 6 m long and 10 m wide.

Using a 3D video analysis system, a 3D coordinate system was established through 3D framework calibration, and the human joints were calibrated with automatic error correction to obtain a 3D human motion model. During the calibration process, the angle between the foot and tibia was defined as the ankle joint angle, the angle between the tibia and femur was defined as the knee joint angle, and the angle between the femur and trunk was defined as the hip joint angle. The kinematic data of the five athletes during the experiment was calculated using a 3D image analysis system (APAS, Ariel Dynamics, Inc., San Diego, CA, USA), including ankle, knee, and hip joint angles, as well as joint angular velocities.

2.3. Continuous Relative Phase (CRP)

After acquiring the kinematic data of the athletes at each moment during the take-off process, we processed the joint angles and joint angular velocities based on CRP analysis. Since the analysis focused on the dominance relationship between different joints (ankle–knee, knee–hip), we standardized the joint angles and angular velocities to account for the varying degrees of freedom among different joints. This standardization resulted in the determination of phase angles for the different joints of the athletes. Using these phase angles, the CRP values between different joints were calculated using the following specific equations [7,16,17]:

$$\overline{\mathsf{\theta }}=2\left[{\displaystyle \frac{\mathsf{\theta }-\mathrm{m}\mathrm{i}\mathrm{n}\left(\mathsf{\theta }\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left(\mathsf{\theta }\right)-\mathrm{m}\mathrm{i}\mathrm{n}\left(\mathsf{\theta }\right)}}\right]-1$$

(1)

$$\begin{array}{ccc}\overline{\mathsf{\omega }}& =& \left[{\displaystyle \frac{\mathsf{\omega }}{\mathrm{m}\mathrm{a}\mathrm{x}(\mid \mathsf{\omega }\mid )}}\right]\end{array}$$

(2)

In Equation (1), θ represents the angle of a specific joint during continuous movement, while $\overline{\mathsf{\theta }}$ represents the standardized value of the joint angle. In Equation (2), ω represents the angular velocity of a specific joint during continuous movement, and $\overline{\omega }$ represents the standardized value of the joint angular velocity during continuous movement.

$$\overline{\mathsf{\phi }}\left(\mathrm{i}\right)={\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}⌈{\displaystyle \frac{\stackrel{\mathrm{‾}}{\mathsf{\omega }}\left(\mathrm{i}\right)}{\stackrel{\mathrm{‾}}{\mathsf{\theta }}\left(\mathrm{i}\right)}}⌉,=\mathrm{1,2},···,\mathrm{n}$$

(3)

Based on the standardized joint angular velocity and angle values obtained from Equations (1) and (2), Equation (3) calculates the phase angle of a joint at each moment in time. In Equation (3), $\overline{\omega }$ represents the phase angle of a specific joint at different moments during the movement.

$$\mathsf{C}\mathsf{R}\mathsf{P}={\phi }_{\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{l}}-{\phi }_{\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{l}}$$

(4)

Equation (4) calculates the value of the CRP by taking the difference between the phase angles of the distal joint and the proximal joint. This allows us to determine the dominant joint at different moments during the movement. When the CRP value shows a decreasing trend, it indicates that the proximal joint is dominant. Conversely, when the CRP value shows an increasing trend, it suggests that the distal joint is dominant. If the CRP value remains relatively stable, it indicates that both joints are moving in synchrony, and neither joint is dominant during that specific period of time [16].

2.4. Fluid Dynamics Calculations

2.4.1. Mesh Density Independence Verification

To meet computational requirements, this study conducted an independence verification of different mesh densities, selecting four types ranging from 3 million to 12 million elements. The verification results, listed in

Table 1. Grid density independence verification results.

Parameters	Grid Density 1	Grid Density 2	Grid Density 3	Grid Density 4	Grid Density 5
Total number of grid	12,000,000	10,000,000	8,000,000	5,000,000	3,000,000
Lift–drag ratio	1.14	1.14	1.14	1.14	1.23

and Figure 2, show that variations in mesh density do not affect the accuracy of the final computational results. Based on the research by Liao Zhangwen et al. [18], which investigated the impact of different mesh generations on result accuracy, this study used a mesh density of approximately 8 million elements to ensure precision in the outcomes.

2.4.2. Establishment of Human Body Model and Fluid Domain

In this paper, we use direct linear transformation (DLT) to analyze the kinematic data during the take-off phase of five Chinese athletes. The “start moment” of the take-off phase is defined as the instant when athletes begin to rapidly extend their lower limbs from the in-run position into take-off. The “end moment” is defined as the stage when athletes have completed take-off and have left the glide path by approximately 5 m. The kinematic data of the athletes at the start moment of the take-off phase (SMT) and the end moment of the take-off phase (EMT) (

Table 2. Kinematic data of athletes at the start and end of the take-off phase.

	Elite Chinese/NW Athletes	Ankle Joint Angle $(\mathbf{Mean} \pm$ SD)	Knee Joint Angle $(\mathbf{Mean} \pm$ SD)	Hip Joint Angle $(\mathbf{Mean} \pm$ SD)
SMT	CA	$61.2 \pm$ 13.8	$90.9 \pm$ 19.3	$61.1 \pm$ 20
	NWA	$40.69 \pm$ 3.67	$95.45 \pm$ 6.91	$60.64 \pm$ 7.36
EMT	CA	$87.9 \pm$ 8.37	$151.19 \pm$ 11.56	$133.09 \pm$ 14.7
	NWA	$56.14 \pm$ 5.56	$180.93 \pm$ 11.86	$149.51 \pm$ 10.95

) are compared with data obtained from Nordic World athletes during the 2009 Nordic World Ski Championships (NWA) by Miroslav Janura et al. [19]. These kinematic data are imported into SolidWorks 2023 SP1.0 modeling software to establish body poses corresponding to the respective kinematic data.

The fluid domain of the human–board system measures 16 m in length, 2 m in width, and 6 m in height, as shown in Figure 3. The side of the fluid domain opposite the human body is designated as an air inlet with a wind speed of 29 m/s, based on the horizontal centroidal velocity during the athlete’s take-off and the fluid velocity used by Huqi [20] to analyze the aerodynamic effects on asymmetrical ski-jumping postures. The wall facing the athlete’s direction is set as an air inlet, the rear wall is designated as an air outlet, and the perimeter walls are treated as smooth boundaries. During grid generation, the mesh around the athlete is refined to enhance the accuracy of the aerodynamic analysis under these conditions.

2.4.3. Governing Equations

This study examines the aerodynamic characteristics of athletes using the k-ε model, a widely adopted approach for fluid turbulence calculations, with its validity supported by previous verification studies [21]. This model is grounded in the Reynolds-Averaged Navier–Stokes (RANS) equations and incorporates two additional equations to close the turbulent stress calculation: one for turbulent kinetic energy (k) and another for the turbulent dissipation rate (ε). The governing equations are as follows:

$${\displaystyle \frac{\partial \left(\mathsf{\rho }\mathrm{k}\right)}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \left(\mathsf{\rho }\mathrm{k}{\mathrm{u}}_{\mathrm{i}}\right)}{\partial {\mathrm{x}}_{\mathrm{i}}}}={\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}\left[(\mathsf{\mu }+{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{t}}}{{\mathsf{\sigma }}_{\mathrm{k}}}})\right.{\displaystyle \frac{\partial \mathrm{k}}{\partial {\mathrm{x}}_{\mathrm{j}}}}]+{\mathrm{P}}_{\mathrm{k}}-\mathsf{\rho }\mathsf{ϵ}$$

(5)

Equation (5) is the k-equation (turbulent kinetic energy equation), which primarily describes the transport and generation of turbulent kinetic energy (k). Here, ρ represents the fluid density; ${\mathrm{u}}_{\mathrm{i}}$ is the velocity component; ${\mathrm{x}}_{\mathrm{i}}$ and ${\mathrm{x}}_{\mathrm{j}}$ are spatial coordinates; μ is the molecular viscosity coefficient; ${\mathsf{\mu }}_{\mathrm{t}}$ is the turbulent viscosity coefficient; $\partial \mathrm{k}$ is the turbulent Prandtl number for the k-equation (typically set to 1.0); ${\mathrm{P}}_{\mathrm{k}}$ represents the production of turbulent kinetic energy; and ϵ denotes the dissipation rate of turbulent kinetic energy.

$${\displaystyle \frac{\partial \left(\mathsf{\rho }\mathsf{ϵ}\right)}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \left(\mathsf{\rho }\mathsf{ϵ}{\mathrm{u}}_{\mathrm{i}}\right)}{\partial {\mathrm{x}}_{\mathrm{i}}}}={\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}\left[\right(\mathsf{\mu }+{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{t}}}{{\mathsf{\sigma }}_{\mathsf{ϵ}}}}\left){\displaystyle \frac{\partial \mathsf{ϵ}}{\partial {\mathrm{x}}_{\mathrm{j}}}}\right]+{\mathrm{C}}_{1\mathsf{ϵ}}{\displaystyle \frac{\mathsf{ϵ}}{\mathrm{k}}}{\mathrm{P}}_{\mathrm{k}}-{\mathrm{C}}_{2\mathsf{ϵ}}\mathsf{\rho }{\displaystyle \frac{{\mathsf{ϵ}}^2}{\mathrm{k}}}$$

(6)

Equation (6) represents the ε-equation (turbulent kinetic energy dissipation rate equation), which describes the transport and generation of the turbulent kinetic energy dissipation rate ε. $\partial \mathsf{ϵ}$ is the turbulent Prandtl number for the ε-equation (typically set to 1.3), and ${\mathrm{C}}_{1\mathsf{ϵ}}$ and ${\mathrm{C}}_{2\mathsf{ϵ}}$ are model constants, commonly taken as 1.44 and 1.92, respectively.

In studying the aerodynamic characteristics of athletes using the k-ε model, the continuity equation ensures mass conservation throughout the analysis. The Navier–Stokes equation describes momentum conservation, while the k-ε model serves to close the turbulent stress term in the Navier–Stokes equation. In this study, these three equations jointly constitute a complete set of fluid dynamics equations, which are used to describe the behavior of fluids during flow.

$${\displaystyle \frac{\partial \mathsf{\rho }}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \left(\mathsf{\rho }{\mathrm{u}}_{\mathrm{i}}\right)}{\partial {\mathrm{x}}_{\mathrm{i}}}}=0$$

(7)

Equation (7) represents the continuity equation, where ρ is the fluid density, ${\mathrm{u}}_{\mathrm{i}}$ is the velocity component, t is time, and ${\mathrm{x}}_{\mathrm{i}}$ is the spatial coordinate.

$${\displaystyle \frac{\partial \left(\mathsf{\rho }{\mathrm{u}}_{\mathrm{i}}\right)}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \left(\mathsf{\rho }{\mathrm{u}}_{\mathrm{i}}{\mathrm{u}}_{\mathrm{j}}\right)}{\partial {\mathrm{x}}_{\mathrm{j}}}}=-{\displaystyle \frac{\partial \mathrm{p}}{\partial {\mathrm{x}}_{\mathrm{i}}}}+{\displaystyle \frac{\partial {\mathsf{\tau }}_{\mathrm{i}\mathrm{j}}}{\partial {\mathrm{x}}_{\mathrm{j}}}}+\mathsf{\rho }{\mathrm{g}}_{\mathrm{i}}$$

(8)

Equation (8) represents the Navier–Stokes equation, where p is the pressure, ${\mathsf{\tau }}_{\mathrm{i}\mathrm{j}}$ is the stress tensor (including viscous stress and turbulent stress), and ${\mathrm{g}}_{\mathrm{i}}$ is the component of gravitational acceleration.

2.4.4. Boundary Conditions and Computational Scenarios

In this study, the inlet velocity was set to 29 m/s, the outlet pressure was set to the normal atmospheric pressure of 101,325 Pa, and other boundaries were set as no-slip walls. The gas was assumed to be incompressible air, and the gravitational acceleration was set to g = 9.81 m/s². This study analyzes the drag and lift forces experienced by athletes based on their different body postures, and the equations are as follows:

$${\mathrm{F}}_{\mathrm{L}}=0.5\times \mathsf{\rho }\times {\mathrm{V}}^2\times \mathrm{S}\times {\mathrm{C}}_{\mathrm{l}}$$

(9)

$$F_D=0.5\times \rho \times A\times C_d\times V^2$$

(10)

Equation (9) is the equation for calculating lift force, where ρ is the density of the fluid, V is the flow velocity, S is the frontal area of the object, and ${\mathrm{C}}_{\mathrm{l}}$ is the lift coefficient. Equation (10) is the equation for calculating drag force, where ρ is the density of the fluid, A is the reference area of the object, ${\mathrm{C}}_{\mathrm{d}}$ is the drag coefficient, and V is the velocity of the object. The ratio of lift-to-drag is defined as the lift-to-drag ratio generated by the athlete.

3. Results

3.1. Analysis of Joint Dominance within Limbs

3.1.1. Analysis of Ankle–Knee Joint Dominance

The results (Figure 4) reveal the dominance characteristics of the ankle and knee joints during the take-off phase of Chinese athletes as follows: Stable CRP values during the 0–20% and 90–100% phases suggest a synchronized movement pattern between the ankle and knee joints, indicating no clear dominance of either joint. During the 20–35%, 50–65%, and 80–90% phases, the CRP values exhibit a continuous increase, signifying that the changes in the ankle joint dominate during these intervals, meaning the ankle joint leads the knee joint in the take-off process. Conversely, during the 35–50% and 65–80% phases, the CRP values display a decreasing trend, indicating that the knee joint predominates during these stages, with the knee joint leading the ankle joint in movement.

3.1.2. Analysis of Knee–Hip Joint Dominance

As shown in Figure 4, the dominance characteristics of the knee and hip joints during the take-off phase of Chinese athletes are evident. During the 0–20% and 90–100% phases, stable CRP values indicate a coordinated movement between the knee and hip joints, with no clear dominance of either joint. During the 20–35%, 50–65%, and 80–90% phases, the CRP values between the knee and hip joints exhibit a decreasing trend, indicating that the hip joint dominates during these intervals, leading the knee joint in the take-off process. Conversely, during the 35–50% and 65–80% phases, the CRP values show an increasing trend, suggesting that the knee joint becomes dominant, leading the hip joint in movement.

3.2. Drag Characteristics at Different Moments

3.2.1. Aerodynamic Characteristics at the Start Moment of the Take-Off Phase (SMT)

As depicted in Figure 5 and Figure 6, the static pressure experienced by athletes at the SMT is predominantly concentrated on the inner thighs and torso. As the fluid flows over the athlete’s body surface, it generates relatively low-pressure zones in the flow area, leading to reduced static pressure on the outer sides of the athletes’ lower limbs during take-off.

The results indicate that the air drag experienced by various limbs of elite athletes varies based on the angle at the SMT. As shown in

Table 3. Drag characteristics at the start moment of the take-off phase (SMT).

	LLD	RLD	LTD	RTD	TOD	TD
	(N)	(N)	(N)	(N)	(N)	(N)
CA	13.26	12.06	0.02	0.01	24.97	108.90
NWA	2.08	2.34	2.88	3.00	3.71	55.40

, during the initial stage of take-off, the drag on the left and right calves of Chinese athletes is significantly greater than that of NW elite athletes, measuring 13.26 N and 12.06 N, respectively. In contrast, the drag on the left and right calves of NW elite athletes is 2.08 N and 2.34 N, respectively. At the start moment of take-off, the drag on the left and right thighs of Chinese athletes is lower than that of NW elite athletes, measuring 0.02 N and 0.01 N, respectively, whereas the drag on the thighs of NW elite athletes is higher, with left and right drags of 2.88 N and 3.00 N, respectively. The discrepancy in drag borne by the trunk is even more pronounced: Chinese athletes experience a drag of 24.97 N on their trunk, compared to approximately 3.71 N for NW elite athletes. The total drag experienced by the athlete–ski system at the start moment of take-off for Chinese and NW elite athletes is 108.90 N and 55.40 N, respectively (Table 3).

3.2.2. Aerodynamic Characteristics at the End Moment of the Take-Off Phase (EMT)

The results reveal significant differences in the air drag experienced by Chinese athletes compared to NW elite athletes during the take-off phase. The discrepancies are most pronounced in the drag experienced by the lower legs, torso, and overall system, while the differences in drag between the left and right thighs of NW elite athletes are relatively minor. Specifically, at the end of take-off, the drag values for the left and right lower legs of Chinese athletes are 74.06 N and 37.02 N, respectively, whereas NW elite athletes experience considerably lower drags of 0.15 N and 0.79 N. Concurrently, the torso drag for Chinese athletes is also significantly higher, at 76.06 N, compared to 10.19 N for NW elite athletes (

Table 4. Drag characteristics at the end moment of the take-off phase (EMT).

	LLD (N)	RLD (N)	LHD (N)	RHD (N)	TOD (N)	TD (N)	TL (N)	L/D
CA	74.06	37.02	8.31	9.54	76.06	170.18	67.10	0.39
NWA	0.15	0.79	6.48	9.47	10.19	84.15	95.18	1.14

). Furthermore, Table 4 shows that the total drag, total lift, and lift-to-drag ratio of the athlete–board system vary between Chinese and NW elite athletes during take-off. For Chinese athletes, the total drag is 170.18 N, the total lift is 67.10 N, and the lift-to-drag ratio at the transition into the early flight phase is 0.39. In contrast, NW elite athletes experience a total drag of 84.15 N, a total lift of 95.18 N, and a lift-to-drag ratio of 1.14 during their transition into the early flight phase.

As shown in Figure 7 and Figure 8, there are notable differences in the static pressure experienced by Chinese and NW elite athletes during the EMT. NW elite athletes, during the completion of take-off, generally lean forward with adequate forward momentum and fully extend their lower limbs. This posture results in relatively lower static pressure across their body segments and the formation of low-pressure zones. In contrast, Chinese athletes do not lean forward as much during the EMT, resulting in comparatively higher static pressure on various body segments.

4. Discussion

4.1. Dominant Characteristics of Limbs during the Take-Off Phase

This study utilizes the CRP method to examine the dominance characteristics of the ankle, knee, and hip joints during the take-off phase in athletes. By comparing these characteristics with those observed in NW elite athletes, the study seeks to offer a theoretical framework for improving athletes’ take-off performance.

Based on the ankle–knee CRP curve, it can be observed that during the take-off phase, the dominant relationship between the ankle and knee joints of Chinese athletes shows that for 40% of the total take-off time, the ankle leads the knee, while for 30% of the time, the knee leads the ankle. This pattern suggests that kinematic changes in the ankle of Chinese elite athletes are relatively more pronounced compared to those in the knee throughout the take-off process, and the knee’s dominance over the ankle is less prominent. As shown in Figure 9 [16], Julien Chardonnens analyzed the dominant characteristics of elite athletes from the Swiss national ski-jumping team during the take-off phase using the CRP method. Chardonnens’ study assessed the coordination of the lower limbs by examining the angles between the limbs and the horizontal plane. Building on this research, we used the DLT method to develop a three-dimensional model and analyzed joint angle changes to explore lower-limb coordination during take-off. Chardonnens’ findings revealed a continuous decrease in CRP values for Swiss elite athletes during the first 75% of take-off, indicating that the thigh dominates the lower leg, with the knee joint leading the ankle joint. This suggests that high-level athletes exhibit significant dominance of the thigh over the lower leg during take-off, attributed to the stability of the lower leg and delayed rotation of the lower leg [16]. Maintaining maximum ankle stability, enhancing lower-limb extension, and increasing knee joint angular velocity during take-off are crucial for optimal intra-limb dominance and performance improvement. Chardonnens also found that, in the latter half of take-off, the lower-leg rotation dominates because the initial rapid extension brings the knee joint angle closer to its optimal position, enhancing forward angular momentum for an optimal flight posture. However, Chinese elite athletes show more frequent changes in ankle and knee joint dominance throughout the take-off phase. In the initial phase (first 20%), their knee and ankle joints move synchronously. During the middle phase, dominance changes frequently, indicating potential issues with ankle angle stability and lower-limb extension. This may prevent the knee joint from reaching its optimal angle, impacting the lift-to-drag ratio in the subsequent flight. Previous studies suggest that elite athletes maintain lower-leg stability and minimal ankle angle changes during take-off, facilitating greater angular momentum and optimal lower-limb extension, resulting in a better flight posture and lift-to-drag ratio after take-off [22,23,24]. Research by Miroslav Janura et al. found that professional ski jumpers have a smaller ankle angle and maintain high stability throughout the take-off phase compared to Nordic combined athletes [19]. Data from Table 2 and Table 3 indicate that an excessive ankle joint angle during take-off may cause athletes to lean back excessively, increasing air drag and affecting acceleration and the horizontal center of mass velocity. Further analysis of the data in Table 4 reveals that a large ankle joint angle increases air drag on the lower leg and torso, negatively impacting the aerodynamic characteristics of the athlete–ski system. An excessive ankle joint angle impedes achieving the optimal lift-to-drag ratio at the end of the take-off phase, which, in turn, affects aerodynamic efficiency during the flight phase that follows. This suboptimal lift-to-drag ratio leads to shorter flight times and reduced jump distances. Previous research indicates that while the knee joint often plays a dominant role during take-off, an excessively large ankle joint angle at the start, combined with significant changes in this angle throughout take-off, increases overall air drag. To enhance the quality of take-off and achieve the best lift-to-drag ratio during the flight phase, it is crucial to minimize the ankle joint angle and maintain stability throughout take-off. This approach will help reduce air drag and ensure optimal aerodynamic performance.

The CRP curve of the knee and hip joints illustrates the dominance characteristics between these two joints during the take-off phase in Chinese elite athletes. This curve shows that the hip joint remains dominant over the knee joint for a longer duration throughout the take-off process (Figure 4). Zhang Dong et al. analyzed air drag on athletes’ bodies in various take-off poses using wind-tunnel training. They concluded that a “knee-driven hip” approach minimizes air drag during the take-off process. Additionally, they suggested that the “synchronous hip–knee variable speed” approach results in the highest drag, likely due to the rapid opening of the hip joint while the lower limbs extend more slowly. This leads to uneven torso drag distribution and an early increase in the torso’s force-bearing area, causing greater drag [25]. The study found that minimal variation in the CRP index values of the knee and hip joints during the initial 20% of the take-off phase indicates that Chinese elite athletes adopt a “synchronous hip–knee variable speed” approach. Throughout take-off, hip joint dominance accounts for 40% of the process, with the knee joint assisting. This movement pattern suggests that Chinese elite athletes might experience an excessively large hip joint angle during take-off, leading to an early and significant increase in torso air drag, thereby affecting their horizontal center-of-mass velocity. This finding highlights the need for optimizing the take-off technique to reduce aerodynamic drag and enhance performance. Janez Vodicar et al. argue that horizontal center-of-mass velocity is the most crucial factor affecting athletes during the take-off phase. Minimizing drag impact on horizontal center-of-mass velocity enables athletes to achieve greater flight distances during the flight phase [3]. Previous studies indicate that hip joint kinematics during take-off significantly impact drag experienced during the jump [26]. Maintaining a smaller hip joint angle during take-off is suggested to improve performance [27]. Although Chinese elite athletes also use a knee-led hip take-off posture, the duration of this phase is relatively short. Based on the data from Table 2 and Table 3, we found that Chinese athletes have smaller knee joint angles during take-off, resulting in lower thigh drag compared to Nordic athletes. This indicates that maintaining the knee joint at approximately 90 degrees during the take-off and sliding phases can more effectively reduce air drag. However, this also suggests that athletes complete the push-off action more quickly within the same timeframe. Data from Table 2 and Table 4 also show that at the end of take-off, Chinese athletes have smaller knee joint angles, correlating with a higher total drag at that moment. Therefore, during the SMT and sliding phases, athletes should reduce the knee joint angle to lower air drag and enhance knee joint dominance during take-off. This approach will help the knee joint achieve the optimal angle by the end of take-off, thereby improving performance. Furthermore, frequent shifts in dominance between the knee and hip joints during this process may generate additional drag, affecting horizontal center-of-mass velocity.

4.2. Aerodynamic Analysis during the Take-Off Phase

4.2.1. Aerodynamic Characteristics during the Start Moment of the Take-Off Phase

The initial phase of the ski jump take-off marks the transition from the approach phase to the take-off phase. Athletes’ varying body postures at the SMT lead to differences in the air drag they encounter. Maximizing or even enhancing the horizontal center-of-mass velocity during the take-off process is crucial for achieving longer flight distances. Alterations in body posture during take-off result in varying degrees of air drag, a pivotal factor influencing the athlete’s horizontal center-of-mass velocity. The approach phase serves as a critical stage for athletes to increase their horizontal center-of-mass velocity. This study aims to analyze the aerodynamic characteristics encountered at the SMT, thereby assessing the magnitude of air drag generated by different postures. As depicted in Table 3, the air drag endured by the athlete–ski system at the SMT for Chinese elite athletes is greater than that of NW elite athletes. Analysis of the kinematic data at the SMT reveals that while the angles of the knee and hip joints during the take-off process in Chinese elite athletes are comparable to those of NW elite athletes, the ankle joint angle exhibits a notable discrepancy. This particular kinematic characteristic results in an increased frontal area of the Chinese athletes’ bodies at the instant of take-off, thereby subjecting them to greater air drag. In comparison to NW elite athletes, Chinese athletes experience relatively lower air drag on their bilateral thighs. This is primarily due to the comparatively smaller knee joint angles exhibited by Chinese athletes, which reduce the drag acting on their thighs during the SMT. Tang Weidi et al. analyzed the impact of different kinematic data on aerodynamic characteristics during the approach phase of ski jumping and found that ankle joint angles significantly affect the drag faced by athletes. They suggested that an ankle joint angle of 43–45° is more conducive to achieving optimal performance [28]. Based on this study, we found that while there are certain similarities in the kinematic data of the knee and hip joints between Chinese and NW elite athletes, the ankle kinematic data differs significantly. Moreover, the increased ankle angle leads to higher air drag experienced by athletes during the take-off process.

4.2.2. Aerodynamic Characteristics at the End Moment of the Take-Off Phase

The body posture of ski jumpers during the EMT significantly influences their ability to swiftly transition into the early flight phase and achieve an optimal lift-to-drag ratio. This, in turn, provides more sufficient flight time for a stable flight. Different body postures during the EMT have varying effects on the amount of drag encountered by the athlete and the lift generated. Xiong Li et al. found that hip angle and body–ski angle play a dominant role in aerodynamic performance, with the ski and torso contributing the most to aerodynamics and torque [13]. Ki-Don Lee et al. suggested that athletes achieve the optimal lift-to-drag ratio of approximately 1.5 when the angle between their body and the ski board is 15° during the flight phase [29]. As shown in Table 4, there are differences in the air drag experienced by various body parts between Chinese and Northwest (NW) elite athletes after take-off. The coordination characteristics and kinematic data during the athletes’ take-off reveal that Chinese athletes have a relatively small proportion of time in which the knee joint dominates during take-off. This results in insufficient push-off and extension of the knee joint, preventing it from reaching the optimal angle. Additionally, the strong dominance of the ankle joint during take-off causes an excessively large ankle angle after take-off, resulting in inadequate forward leaning of the body. Anton and Arndt found that maintaining stability in the ankle joint during take-off allows the athlete’s body to lean forward, gaining forward angular momentum [24]. Compared to NW elite athletes, Chinese athletes tend to have a larger ankle angle and a smaller knee angle, leading to inadequate forward leaning of the body and a larger windward area. Calculations of the air drag experienced by athletes at the EMT (Table 4) reveal that Chinese athletes experience higher air drag in various body segments than NW elite athletes, with significantly higher drag in the bilateral lower legs and torso. This is primarily due to the greater ankle angle during take-off, which results in less forward leaning and higher drag. Our analysis of the drag and lift forces during take-off, with respect to the lift-to-drag ratio generated by different postures at the EMT, shows that NW elite athletes, with smaller ankle angles and fully extended knee angles, achieve better forward leaning, less air drag, and higher lift force compared to Chinese athletes. Consequently, NW elite athletes achieve a higher lift-to-drag ratio (Table 4). Research by Stenseth suggests that disadvantages during the take-off phase are difficult to compensate for in the subsequent flight phase, implying that the body posture after take-off plays a crucial role in the stability and distance of the flight [4]. Mikko Virmavirta’s study on the early flight phase of ski jumpers showed that achieving an optimal lift-to-drag ratio significantly impacts the flight distance [30]. Therefore, NW elite athletes’ body posture at the EMT enables them to enter the stable flight phase faster, providing a longer flight time. In contrast, the kinematic characteristics of Chinese athletes result in higher air drag during take-off and subsequent flight adjustments, potentially leading to an unstable flight and a higher probability of injury.

5. Conclusions

During the take-off phase of ski jumping, adopting a knee-dominated approach while maintaining ankle stability enables athletes to execute the push-off motion more effectively, thereby minimizing air drag. However, an excessively large ankle angle at the SMT can significantly increase drag. Additionally, at the EMT, an excessively large ankle angle combined with insufficient forward leaning not only increases air drag, but also reduces lift generation. This prevents athletes from achieving the optimal lift-to-drag ratio during the flight phase, thereby affecting their performance and ultimately impacting their final score. This study focused solely on the aerodynamic characteristics at the SMT and EMT, without thoroughly investigating the aerodynamic effects of various body postures throughout the entire take-off process. To address these limitations, we employed CRP analysis to explore the dominant characteristics of the ankle, knee, and hip joints throughout the complete take-off process, integrating aerodynamic characteristics from both the SMT and EMT.