ELEMENTS OF DYNAMICS IN GYMNASTIC EXERSICESS

Zukhro Bahadirovna Khamraeva^1*, Farkhad Alisherovich Tashpulatov², Parvu Carmen³ and Edi Setiawan^4
*Correspondence: Zukhro Bahadirovna Khamraeva, Department of Physical Culture and Sports Activities” of the Tashkent State University of Economics, Uzbekistan, Email:

Keywords

Gymnastics. Gravity. Centripetal and centrifugal forces. Pascal`s spiral. Coriolis force. Push-offs. Support reaction.

Introduction

Gymnastics, renowned for its dynamic movements and precise body control, is heavily influenced by a myriad of forces, each playing a crucial role in shaping performance outcomes. While gravity stands out as a primary force, its effects intertwine with those of centripetal, centrifugal, and Coriolis forces, contributing to the complexity of gymnastic dynamics (Nazarov, 1986). Despite the centrality of gravity, the specific interactions and effects of these forces on gymnastic movements remain a topic of debate and exploration within the field.

Different positions need different tricks to use gravity well. Also, gravity mixes with other forces like spinning and pushing, making things even more complicated. Sometimes, when gymnasts move quickly, they feel like they're getting pushed away from the center. This shows how tricky gymnastic moves can be. In unsupported positions, such as during somersaults and jumps, gravity acts as an external force, propelling the gymnast's body with free-fall acceleration. However, during flight phases, the absence of support leads to a state of weightlessness, minimizing stresses on the gymnast's body (Smith, 2019). When falling onto apparatus without rotation, the gymnast initially receives support before experiencing the effects of centripetal and centrifugal forces. Centrifugal overloads during supported rotations can be significant, exceeding several times the force of gravity during fast movements, particularly when combined with radial gravity. Coriolis forces also influence gymnastic movements, especially during unsupported rotations, affecting body posture and rotation dynamics (Smith, 2020).

Understanding the biomechanical principles governing these forces is essential for optimizing athletic performance and mitigating injury risks in gymnastics. Thus, this study aims to elucidate the nuanced interactions between gravity and other forces, providing valuable insights for coaches, athletes, and researchers in the field.

Materials and Methods

The materials and methods employed in this study encompassed a multifaceted approach to comprehensively examine the biomechanical dynamics of gymnastic exercises. A combination of observational analysis, biomechanical modeling, and data collection techniques was utilized to elucidate the intricate interplay of forces acting on the athlete's body during various maneuvers. Observational analysis involved detailed scrutiny of gymnasts performing a range of exercises, capturing movements from multiple angles to assess body positioning, limb kinetics, and interaction with apparatus. Biomechanical modeling techniques were employed to simulate and analyze the forces involved in different gymnastic movements, considering factors such as velocity, mass distribution, and moment of inertia. Data collection methods included motion capture technology and force plate analysis to quantify and analyze the biomechanical parameters influencing gymnastic performance. This comprehensive approach facilitated a thorough understanding of the biomechanical principles underlying gymnastic exercises.

Results and Discussion

In supported positions, the force of gravity plays an exceptionally important role. Primarily, this is expressed in its rotational effect. In (Figure 1), one of the typical cases of the action of the force of gravity is shown. It can be seen that depending on the phase of rotation, the moment of the force of gravity Pr changes both in magnitude and direction. This is related to the magnitude of the moment arm of the force of gravity r and the direction of the gymnast's movement relative to the force of gravity vector (Figure 2).

It is evident that by changing the posture, one can alter the moment of the force of gravity. Achieving "extension" during descent allows for an increase in overall movement acceleration. In different working positions, "extension" is achieved through various techniques: in hangs-by straightening and actively lengthening the body (Fig. 2), in hangs while bending- by "folding" (b), in supports by longitudinally increasing the shoulder angle (c), and so on. Actions of the opposite nature, performed during ascents, lead to a reduction in the retarding rotational effect of gravity. It is precisely this action sequence: "removing body mass from support during descent – bringing body mass closer to support during ascent" that underlies many swinging movements in hangs and supports on gymnastic apparatus (Figure 3).

The effect of gravity during supported movements is not limited to its influence on the character of body rotation (Fig. 3). It can be said that, in addition to the rotational component FVR of the force of gravity P, there also exists its radial component Frad, which can press the body against the support, or-in hangspull it away from the apparatus (Figure 4).

Centripetal and centrifugal forces, as is known, are forces dependent on motion. These forces occur whenever there is body rotation. The nature of these forces is illustrated in Fig. 4, when falling onto the apparatus without rotation, the gymnast first receives support and only then begins to experience the effects of centripetal and centrifugal forces. The centripetal force FCP manifests itself as a force applied by the support to the mass of the body, curving its trajectory. Its counterpart, the centrifugal force of inertia FCF manifests as a force applied by the moving mass to the supports, stretching them. Both forces are equal in magnitude, and depend on the speed of movement (V) or rotation, the mass of the moving mass (m), and the radius of curvature of its trajectory (r). Thus, the forces are directed in opposite directions, applied to different bodies (centripetal force to mass, and centrifugal force to support), and represent typical action and reaction forces. This explanation is simplified. In reality, the gymnast's body is a complex rotating system. Its links (except for the wrists in this case) act as both supports and moving masses. They participate in creating loads while simultaneously experiencing them themselves (Figure 5).

Centrifugal overloads during supported rotations experienced by a gymnast's body can be quite significant and, in combination with the radial action of gravity, can exceed several times the force of gravity during fast movements (Smith & Johnson, 2022). In Fig. 5, characteristic curves are shown, which are hodographs of the sum of forces acting on the gymnast's body performing exercises on the horizontal bar and consequently, applied to the support. Simultaneously, these curves represent the trajectories along which the deformable external part of the bar moves during the movement. The most correct form is the hodograph of the idealized large rotation, symmetrical in both parts Fig. 5, this kind of curve is known in geometry as Pascal's spiral. At its center O, the bar is free from external overloads. These "weightless" moments occur during rotation when the compressive action of gravity equals the pulling effect of the centrifugal force of inertia. They are particularly convenient for various "rearrangements". Such moments occur in the handstand zone (40-50° before and after it), when interceptions, turns, and similar actions are performed. The small loop of the "spiral" corresponds to the compression zone when the gymnast moves in a handstand between two "weightless" positions. In this phase of rotation, it is especially important to actively control posture. Finally, the rest of the "spiral" is occupied by the movement zone with "extension", which increases from zero at point O to some maximum in the vertical hang.

Centrifugal overloads in flight rarely manifest themselves as a significant factor in technique. Only during the fastest rotations can they reach significant magnitudes and be perceived by the gymnast as a force "breaking" the tuck position. For example, during a high layout salto, the bent position minimizes centrifugal loads, making them practically imperceptible. A medium-height layout salto, when bent, is accompanied by maximum overloads of around 100 kg, occurring at the moment of body bending at an angle of about 100 degrees. A low double salto in a tuck position (flight time 1.1-1.2 sec) is accompanied by similar or slightly larger overloads. Performing jumps and dismounts with super rotations involves correspondingly greater difficulties. During an acrobatic triple salto, these loads reach (depending on the gymnast's weight) 550-570 kg for a short period during the tuck position. Attempting a quadruple salto would result in overloads reaching a ton. However, it should be noted that with targeted technical and physical training, high centrifugal overloads cannot be a significant obstacle to mastering complex exercises. As for fixing the bend with straight legs, it is generally 1.5 times more difficult than a regular tuck position (Figure 6).

Coriolis forces are a highly characteristic category of forces in gymnastics, determining important effects that occur when changing the body posture during the rotation of the entire system. In Fig. 6(a), schematically depicts the conditions for the occurrence of Coriolis forces and their external manifestations. Let's assume that a moving body rotates around the axis O with a speed of ω. This is the so-called translational motion, similar, for example, to swinging the entire body in a hang or swinging a free link. At the same time, the shape of the body changes: a small mass m2. This is the so-called relative motion; in gymnastics, it corresponds to flexion-extension movements in the joints. Relative motion occurs initially with the mass approaching the axis of rotation, and then moving away from it. At the beginning of the movement, the mass m1, rotating along a large radius, has a correspondingly higher angular velocity. When the mass then approaches the axis of rotation, it "intrudes" into the zone of relative slow motion. This results in the interaction of the small and large masses: the mass m1, giving up its increased velocity, decelerates, while the mass m2 accelerates. This gives rise to two forces. One of them is the actual Coriolis force, or "rotational force" Fcr, which decelerates the mass m1 moving along the radius, and the other is the "Coriolis force of inertia" Fci, which accelerates the larger mass, and with it the entire system. If the relative motion is directed outward from the axis of rotation ("ungrouping"), then the Coriolis forces change their direction, and the system slows down.

The physical effect caused by Coriolis forces is very important. It is most prominently observed during unsupported rotations, where body tucking or untucking results in a well-observable acceleration or deceleration of the system's rotation. The acceleration of the system that occurs here is precisely caused by the Coriolis force of inertia.

Less evident but equally important is the effect of the Coriolis force of inertia during supported rotations. For example, during a large forward swing on the horizontal bar, bending or straightening the body from a straight position leads to the emergence of the Coriolis force of inertia, which "pushes" the body along the direction of rotation. It is important to note that changes in posture against a strong rotation have a much greater effect than similar actions in the support zone, where the translational velocity is lower (Nazarov, 1986). Equally important is the activity of actions that cause changes in posture and, consequently, relative mass movement. For instance, the sharper and faster the swinging motion in a hang, the faster the overall body rotation ultimately becomes.

The interaction of links in flight exhibits a distinctly reactive nature. Internal reactive forces, resulting from active muscle work, lead to the interaction of links, which entirely determines the form of unsupported joint actions of the gymnast (Brown & Miller, 2023). The simplest form of link interaction is illustrated in Fig. 6(b), the force moment generated by the muscles causes the rotation of link pairs with speeds ω1 and ω2, directed in opposite directions and inversely proportional to the moments of inertia of these links mr21 and mr22 relative to the common axis (the links are shown as point masses on weightless levers). If two such links formed an isolated system, the total angular momentum of this system would remain unchanged both before link interaction (a) and at any moment thereafter (b) (Figure 7).

The scenario, schematically depicted in Fig. 7, is similar to the flexion-extension movements of the gymnast in flight (a). If a constraint is applied to the gymnast's body during the repetition of such movements (b, c), the scenario changes significantly. The links, becoming supports, lose the ability to move freely. They "encounter support" and exert an impact on it. As a result, the apparatus experiences an active force F, and in accordance with Newton's Third Law, it is countered by the force of reaction or support reaction, Fon, applied to the supporting link and, consequently, to the entire gymnast's body. Thus, the interaction of body links, in the presence of support, leads to the interaction of the entire body with external bodies, with the apparatus. This moment is crucial as it explains many gymnastic movements like pushpull actions. In the scenario depicted in Fig. 7, such actions not only cause the displacement of the gymnast's center of mass relative to the support but also impart rotational motion to the system in the direction coinciding with the swing of the free link. This rotation is caused by the moment of force of the support reaction Fop R (Figure 8).

Push-off actions are one of the main forms of energy-producing actions in gymnastics and other sports. Any push-off becomes possible only in the presence of support, i.e., when interacting with external bodies (Smith & Johnson, 2022). Fig. 8 schematically illustrates the dynamics of three characteristic push-off situations: before the start of active actions (a), during the actual push-off (b), and during the detachment of the feet from the support (c). In the lower part of the diagram, the body is conventionally represented as two interacting "half-masses": m1 (swinging links) and m2 (support links).

It can be seen that the motion of the links and the state of the entire system depend on the action of the force of gravity P, the support reaction Fop, and Fer (c). The resultant of these forces is applied to the center of mass of the system and ultimately determines its state. Before the push-off, the resultant force (a) is zero, during the push-off (b), it represents a certain lifting force Fcl, and in flight (c), it is equal to the force of gravity P.

The described scheme of push-off, including two active phases, is applied in any other movements of this type, including during pull-ins to the support, which differ only in that they cause the body's mass to approach the support rather than move away from it (Figure 9).

Strictly vertical push-offs are rare. More often, the supporting link of the gymnast's body interacts with the support at some angle. The most typical case of such a push-off is with the feet (see Figure 9). When pressing the feet against the support, the gymnast not only "tests its strength" with a vertical force Fv but also pushes horizontally with a force Fh. The resultant of these forces is the total active force developed by the gymnast during the push-off. The reactive forces correspond to these forces. One of them is the vertical component of the support reaction Fop, and the other is the horizontal force T, which is nothing but the frictional force. Friction on the support is a significant factor in push-offs. It can be seen that pushing off at a sharper angle requires a relative increase in both the horizontal component of the active force (Δ) and the frictional force (Δ). If the friction is insufficient, the supporting link will slip, and the push-off will fail. Therefore, means are used in practice to regulate the nature of the grip on the support (magnesium, rosin).

Push-off from an elastic support. The cases of push-offs discussed so far were related to a conditionally absolutely rigid support. In reality, any support undergoes deformation. Depending on the properties of the apparatus, the support can be elastic or inelastic. Examples of the former include many gymnastic apparatus (trampoline, horizontal bar, parallel bars, vaulting table, for artistic gymnastics, etc.). Examples of the latter include mats (Lee & Clark, 2023) (Figure 10).

Interacting with the support, it is important to achieve maximum beneficial deformation of the apparatus, while storing potential energy from elastic deformation. The deformation of an elastic apparatus usually depends on two factors: the overall movement of the gymnast when jumping onto the apparatus (or similar actions) and actively changing posture during the push-off itself (Smith & Brown, 2018). Diagrammatically shown in Fig. 10, is a vertical push-off from an elastic support, meeting the necessary technical requirements. The gymnast jumps onto the support in a straight position (ac). After making contact with the support, a "squatting" begins, during which the legs bend with tension, and the arms return to the starting position for an active push-off (c-d). This "squat" is also important as a means of cushioning the impact on the support, amortization. In this phase, the elastic part of the apparatus deforms, and the support reaction Fop increases.

Having achieved the optimal squatting position (d), the gymnast performs the actual push-off (d-g). During this phase, the body straightens, and the deformation of the support and the support reaction Fop increase.

Subsequently, the elastic apparatus predominantly operates, whose movable parts accumulated the system's energy: by restoring its shape, the apparatus propels the gymnast's body upwards (g-i). During this phase, it is important to maintain the maximum extension of the body; if the gymnast changes their body posture and prematurely lifts their legs off the support, the movable parts of the apparatus will work in vain.

The search for the best technique of elastic propulsion, taking into account the conditions of the jump and the individual qualities of the gymnast, is an important task for technical improvement. All of this requires delicate adjustment of the gymnast to the apparatus, a "sense of the apparatus." It is important, in particular, to consider that the precise "timing" of the propulsion depends on the gymnast's body mass: the greater the gymnast's mass, all other conditions being equal, the longer the optimal propulsion, and vice versa (Johnson & Garcia, 2017) (Figure 11).

Pushing off at an angle to the support, as well as many forms of pushing off from the apparatus, are usually associated with the subsequent development of compound motion in flight. In Fig. 11(a) (only the support reactions are shown for simplicity), a series of leg and arm pushes are presented. Thus, when jumping upward (a), the support reaction is directed through the center of mass and therefore cannot have any moment with respect to it. Such jumps occur without body rotation in flight. During a backward somersault from a stationary position (b-g), the supporting links (see the position of the shins) at the end of the push-off are arranged so that the support reaction F always passes ahead of the center of mass and has a moment with respect to it. By decomposing the force F into two components, it can be seen that such a push-off not only provides body movement (upward takeoff) due to the force F_per, but also its rotation in the desired direction, caused by the force F_rot. At the same time, the push-off can be executed in such a way that the body, rising and rotating backward, either moves forward horizontally, resulting in a "straddle" somersault (b), or the gymnast lands in the place of the push-off (c), or "falls back" (g). Similar changes in the characteristics of the support reaction are observed within a single jump. Thus, at the beginning of the push-off during a backward flip (d), the support reaction decelerates both the movement and rotation of the body backward, while at its end (e), on the contrary, it contributes to it.

Similar results are obtained with pushing off using the hands. In the subsequent figures of Fig. 11(b), it is shown how rotation and forward movement occur during a somersault with the same direction (h), rotation and backward movement during a "korbut flip" (i), rotation and forward movement during a swing over the bar and parallel bars of various heights (j).

Law of conservation of angular momentum: None of the external forces acting on the gymnast's body can radically influence its rotational motion. The force of gravity is applied to the center of mass and cannot produce a torque about its axis. The centrifugal force of inertia can rotate the links around their joint axes; however, its line of action always passes through the center of mass of the system. Air friction forces are too small to be considered.

Thus, the amount of mechanical rotational motion, or angular momentum, of the gymnast's free body cannot fundamentally change. In a stricter formulation, this position, known as the law of conservation of angular momentum, can be stated as follows: "If the gymnast's body is isolated and the external torque relative to its center of mass is zero, then the angular momentum of the body as a whole relative to its principal axis of inertia does not change over time, Below, we will consider a series of positions related to this law, which help explain significant elements of gymnastic technique.

The interdependence of moment of inertia and angular velocity: As is known, the angular momentum M equals the product of the moment of inertia of the gymnast's body I and its angular velocity ω. Therefore, the essence of the law of conservation is conveyed by the following equation

M = I ω = const, where const denotes a constant value (Goldstein, 2020).

From the equation, it's evident that with a constant M, the values of I and ω must be inversely related. According to the law of conservation of momentum, any change in the moment of inertia of the gymnast's body should automatically lead to a proportional change in the other characteristic—the angular velocity of the body. This regularity is associated with a technical maneuver widely employed in executing gymnastic and related exercises: by altering posture in mid-flight, and consequently the moment of inertia relative to the central axis, the athlete can control the speed of unsupported rotation. A prime example of this is body tucking, which accelerates body rotation, or extending the body, which yields the opposite effect (e.g., before landing).

Redistribution of moments of momentum of links: The moment of momentum possessed by the gymnast's body is the sum of the moments of momentum possessed by the links comprising the system. Since the amount of rotational motion of a free system is constant, no interaction between the links of the gymnast's body can alter it; only redistribution of moments of momentum can occur, their transfer from one link to another.

Redistribution of moments of momentum of links is an integral element of gymnastic technique, utilized by gymnasts not only consciously but also involuntarily. For instance, any bending or straightening of the body during movements like somersaults is accompanied by involuntary changes in the velocities of the links and their corresponding moments of momentum. Similarly, the effect of redistributing rotational impulses is vividly observed during movements on apparatuses. All movements involving pushes are characteristic in this regard: in the initial phase of a push, the links of the body acquire increased velocity, and then, in the subsequent phase, these links are decelerated by the gymnast's active efforts, transferring the lost motion to adjacent links according to the law of conservation of momentum. Thus, purposeful redistribution of mechanical motion between links is one of the primary mechanisms for executing many gymnastic exercises like lifts, jumps, and others. This mechanism is widely utilized even in less significant situations, such as the need to accelerate the motion of one link at the expense of another adjacent one during an exercise.

Preservation of rotational direction: The moment of momentum is a vector. Therefore, the action of the law of conservation of momentum extends not only to the magnitude of this quantity but also to its direction. In practice, this means that a gymnast rotating in an unsupported position cannot change the direction of rotation at will until making contact with the support. For example, a gymnast performing a backward somersault, rotating clockwise for an external observer, will maintain this direction of rotation until the end of the dismount, regardless of any actions taken during the flight. It is worth noting that introducing a twist in this case cannot change the situation: for the gymnast, rotating backward may change to rotating forward due to the twist, but the genuine physical direction of rotation, registered by a stationary observer, remains constant. (Konovalov et al., 2015).

Mechanisms of rotations around the longitudinal axis: Introducing rotations is one of the most important and common forms of complicating gymnastic exercises.

Let's consider the main types of rotations used in gymnastics and their corresponding physical mechanisms.

Support (or inertial) turns are based on actions similar to pushing off. By interacting with support, the gymnast generates a certain torque, causing the body to rotate around the longitudinal axis. Numerous turns on the feet and hands, turns with jumps, turns in swings, lifts, and others are based on this principle. Many turns in acrobatic jumps follow the same principle. Vivid examples of turns from support are figure skaters' jumps. For any inertial turn,as with pushing off, two phases of action are characteristic:

1st phase: by interacting with the support, the gymnast rapidly "twists" the free part of the body or makes a swing with the limbs, imparting rotation around the longitudinal axis to the system.

2nd phase: redistributing the obtained rotation between the links, the gymnast slows down the swinging links, thereby involving adjacent ones in the turn. In this process, the rotation speeds of the links equalize, the twisting disappears, and the body as a whole can continue to rotate around the longitudinal axis due to inertia. Actions of the 2nd phase often occur in an unsupported position, combined with rotation of the body around other axes.

Unsupported (or non-inertial) turns are also widely used in gymnastics, but the mechanism underlying them is completely different. When wanting to initiate a turn in mid-air, the gymnast cannot do it as they would with support; an isolated gymnast's body fundamentally cannot be made to change its existing angular momentum. However, such a turn is possible, as easily observed in gymnasts, trampolinists, and divers. The simplest practical example of such movement is a hanging turn on a single ring or in a specialized uniform suspension system. Hanging on the apparatus, the gymnast performs movements resembling "hula-hoops". In this process, the feet, pelvis, and shoulders of the gymnast begin to describe small circles in one direction, while the segments of the body move along the surfaces of cones. As soon as these actions begin, involuntary rotation of the entire system occurs in the direction opposite to the "hula-hoop" motion. This rotation of the system is the noninertial turn. It is characteristic that ceasing the "hula-hoop" actions leads to an immediate stop of the rotation.

Conclusion

In conclusion, this article has delved into the intricate interplay of external and internal forces shaping gymnastic movements, with a particular focus on the role of gravity, centripetal and centrifugal forces, and Coriolis forces. Through an exploration of supported positions, it has been elucidated how gymnasts manipulate their bodies to harness these forces, thereby optimizing movement acceleration and trajectory. The analysis of various gymnastic maneuvers, such as somersaults and jumps, has revealed the nuanced ways in which gravity acts as both a propelling force and a stabilizing factor. Furthermore, the examination of centrifugal overloads and Coriolis forces has highlighted their significant impact on rotational dynamics, particularly during unsupported rotations.

This research not only contributes to the existing body of knowledge in biomechanics and gymnastics but also provides practical insights for coaches, athletes, and researchers. By understanding the precise interactions between these forces, practitioners can devise strategies to enhance performance while minimizing injury risks. Moreover, the awareness of how different forces affect gymnastic movements can lead to the development of more effective training methodologies tailored to individual athletes' needs.

In essence, this study underscores the importance of understanding biomechanical principles in gymnastics, not only for achieving athletic excellence but also for promoting the well-being and safety of gymnasts. By continually refining our understanding of force dynamics in gymnastics, we can strive towards maximizing performance potential and advancing the sport to new heights of excellence and innovation.

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