Dr. Hab., Professor V.I. Zagrevskiy^{1, 3}
Dr. Hab., Professor O.I. Zagrevskiy^{2, 3
1}Mogilev State A. Kuleshov University, Mogilev, Belarus
²Tyumen State University, Tyumen
³National Research Tomsk State University, Tomsk

Corresponding author: fizkult@teoriya.ru

Abstract

Objective of the study was to offer and substantiate by computation experiments basics of the athletic biomechanical system trajectory modeling using the mass-inertial characteristics and elementary kinematics of the body parts.

Methods of the study. We used for the purposes of the study system-structuring analysis and movement design mathematical/ simulation/ modeling tools to model the biomechanical system kinematics in the computation experiment.

Computation experiment was designed to model the athlete’s musculoskeletal system movements using a biomechanical system movement synthesizing mathematical toolkit. The athlete’s musculoskeletal system movement model may be described as the limited kinematic diagram of the connected bodily elements with cylindrical joints that models a biomechanical system plane rotation process around a contact/ support point

Results and conclusion. The biomechanical system trajectory modeling experiment showed that when the biomechanical system rotates around a contact point, provided the programmed control and startup conditions are the same, then:

• Growths/ falls in masses of the model elements cause no effect on the biomechanical system trajectory;

• Elementary angular velocity angular velocity is directly correlated with the length of element i.e. the higher is the element’s length the higher is the angular velocity and vice versa.

Keywords: biomechanical system trajectory, sport exercise, mass-inertial characteristics, athlete’s body elements.

Background. Modern physical education theory and practice gives room for the belief that individual anthropometric characteristics are correlated with the sport techniques albeit it is seldom if ever substantiated by sound theoretical provisions [1, 2]. We know only in a few study reports making attempts to find and explain correlations between an athletic biomechanical system trajectory and body mass/ length [4, 5]. Presently these and other associating issues are still relevant and of special interest for the sport practice and, hence, there is a need for a theoretical understanding of the phenomena with due biomechanical arguments for one or another athletic technical performance concept [6].

Objective of the study was to offer and substantiate by computation experiments basics of the athletic biomechanical system trajectory modeling using the mass-inertial characteristics and elementary kinematics of the body parts.

Methods of the study. We used for the purposes of the study system-structuring analysis and movement design mathematical/ simulation/ modeling tools to model the biomechanical system kinematics in the computation experiment.

Results and discussion. Computation experiment was designed to model the athlete’s musculoskeletal system movements using a biomechanical system movement synthesizing mathematical toolkit. The athlete’s musculoskeletal system movement model may be described as the limited kinematic diagram of the connected bodily elements with cylindrical joints that models a biomechanical system plane rotation process around a contact/ support point: see Figure 1.

Figure 1. Three-element musculoskeletal system movement kinematics

Let us use the following notations for the model: N - number of elements in the model; i - alphabetic index of every element; (i = 1, 2, ..., N); Li – length of the i-th element; Si – distance from the contact point (rotation axis) of the i-th element versus the mass center; -  i-th  element inclination angle to the Ox axis (generalized coordinates); - generalized speed of the i-th element (i = 1,…, N); and - generalized acceleration of the i-th element (i = 1,…, N).

The biomechanical system movement approximating mathematical model with programmed control at the kinematic level that we developed [3] using the Lagrange formal toolkit, is the following:

              (1)    

Whereas: M1 - frictional moment; Yi - generalized force of the i-th element (i = 1,…, N); and Aij - dynamic coefficients of the model elements (i = 1,…, N; j = 1,…, N).

Dynamic coefficients Aij, Yi of the model elements were computed using the algorithms described in the prior study [3]. Model (1) has no analytical solution, and that is why we used the Runge-Kutta numerical method with the fourth-order accuracy in our computation experiments. We computed the generalized coordinates of every model element and derivatives in time at every integration step using the following algorithm:

             (2)

The computation experiment modeled the second half of a full backward swing on a gymnastics bar. The computation experiment conditions were formulated as follows:

Timing (temporal movement characteristics):

Startup: s; final:  s; integration step s.           (3)

Startup conditions of the movement:

      (radiant/s)                   (4)

Programmed control of the biomechanical system is given in Table 1.

Table 1. Programmed/ controlled kinematics (U₁, U₂) of the model joints

Computation experiment-1 implied unvaried model elements: ) and varied elementary mass-inertial characteristics as follow:

Option I:

Option II:                      (5)

Option III:

(kg/ m2)

Computation experiment-2 Computation experiment -1 implied unvaried elementary mass-inertial characteristics:

( ; ) and varied kinematics:

Option IY: ( );

Option Y – ( );                (6)

Option YI – ( ).

Given hereunder are the modeled synthesized biomechanical system trajectories. Table 2 gives the biomechanical system movement trajectory model produced by Computation experiment-1.

Table 2. Biomechanical system trajectory model produced by computation experiment-1

Figure 2. Biomechanical system movement trajectory model produced by computation experiment-2 with varied elementary mass-inertial characteristics

The computation experiment-2 product shows (Figure 2) that, when the programmed control and the startup movement conditions are the same whilst length of the elements grows, the elementary angular velocity and rotation angle grow as well; and vice versa, when the element gets shorter, the elementary AC and rotation angle fall respectively.

Conclusion. The biomechanical system trajectory modeling experiment showed that when the biomechanical system rotates around a contact point, provided the programmed control and startup conditions are the same, then:

• Growths/ falls in masses of the model elements cause no effect on the biomechanical system trajectory;

• Elementary angular velocity angular velocity is directly correlated with the length of element i.e. the higher is the element’s length the higher is the angular velocity and vice versa.

References

1. Gymnastics. Textbook for technical schools of physical culture. A.T. Brykin, V.M. Smolevskiy [ed.]. Moscow: Fizkultura i sport publ., 1985. 309 p.
2. Evseev S.V. ForExercise machines to build motor actions in gymnastics. Study guide. Leningrad: Lesgaft SCOLIPE publ., 1987. 91 p.
3. Zagrevskiy V.I., Zagrevskiy O.I., Lavshuk D.A. Lagrange and Hamilton's formalism in modeling movements of biomechanical systems. Mogilev: Kuleshov MSU publ., 2018. 296 p.
4. Korenberg V.B. Performance reliability in gymnastics. Moscow: Fizkultura i sport publ., 1970. 192 p.
5. Tadzhiev M.U., Isyanov R.Z. Difficult acrobatic jumping combinations. Tashkent, 1969. 160 p.
6. Sosunovskiy V.S., Zagrevskaya A.I. Kinesiological educational technology in physical education of preschoolers. Teoriya i praktika fiz. kultury. 2020. no.11. pp. 68-70.