Numerical Estimation of the Hand-Arm System Joints Reactions Caused by an Impact Based On Anthropometric Data


This paper is concerned with the numerical analysis of the equations given by the mechanics of the impact of a ball and a hand-arm system for two potential subjects male and female. The influence of the mass of the ball and its velocity on the values of the hand-arm joints reactions is studied.



Introduction
In the dedicated literature, numerous hand-arm studies concerning to the vibrations caused by the hand-held household and professional tools are realized, emphasizing the effects on the joints. In contrast, the weight of the studies regarding the loads and the effects on the wrist, the elbow and the shoulder joints generated by the hand intensive sports (basketball, tennis, volleyball, handball, rugby) is much lower.
Papers are focused on the effects produced by the throwing of the ball and by the ball repeatedly striking the hand in handball [1], the elbow loading and the biomechanical limits of the sportsmen in the case of specific sports as the throwing of some objects of different weights: spear, hammer, metallic balls [2]. King, Kental and Mitchell investigated the hand-arm system load by varying the position of the ball impact on the tennis racket [3].
There are not found any studies regarding the generated effects on the hand-arm joints by receiving the volleyball, more specifically, scientific works about the impact effects.
As a matter of fact, present paper follows up the theoretical study of the impact process of a handarm system bumped by a ball, presented intensively by the paper [4]. The impact system of equations is completed, all these relations are checked up carefully and then they are solved using computer software. On the other hand, the results below obtained may be a good set of values for the medical study in at least two directions: sports medical science and for the study and manufacture of materials (eg plastics) from which different prostheses of parts of the hand-arm system could be made. Regarding to above, there is research studying this issue in particular [5] and in general [6].

Considerations on Mechanical Model of Studied Impact
It is appropriate to do firstly some considerations on the mechanical model widely presented in paper [4], especially for doing the welcome adjustments of this previous study. Before to remind the equations that describe the mechanics of impact, some words about the model are necessary: it is about a classical model of double pendulum representing upper arm, forearm and hand bumped by a ball, in a separate description as it can be seen in Figure 1(a and b). In this representation, the impact reactions in shoulder and elbow are clearly highlight.  In Figure 1(a and b), impact reactions are the following: impact impulse P (very closed by the wrist joint) due to the ball hitting, elbow impact impulse PA, shoulder impact impulse PO, and the special impact moment caused by reaction of ligaments and muscles during impact process ( [4]), respectively MPA, called elbow impact moment and MPO, shoulder impact moment.
It is also important to specify physical and geometric data according to the Figure 1(a and b) and to the equations given by the mechanics of the impact: m1 and l1 are the mass and the length of the upper arm, m2 and l2 are the mass and the length of the forearm, c1 is the distance between the shoulder joint O and the arm mass center C1, c2 is the length AC2 (where C2 is the forearm mass center), b is considered the length between the wrist joint B (practically a rigid joint during impact) and the point of impact C (on the hand), JO is the moment of inertia of the upper arm with respect to the perpendicular axis (Oz), JA is the moment of inertia of the forearm with respect to the perpendicular axis (Az), 2 C J is the moment of inertia of the forearm with respect to the perpendicular axis (C2z), m3 is the mass of the ball and v3 is its velocity, finally P and P being the generalized impact reactions (see [4]). It is important to mention that the subscripts t and n symbolize orthogonal axes, "t" perpendicular to each part of the system (upper arm and forearm respectively) and "n" along to each part ( Figure 1a and b).
According to the reasoning of the impact mechanics ( [4]), the impact equations are the next: With regard to the previous paper [4], the following relations are supplemented, and for this reason they have to be presented one by one. These added relations are equations given by the theorem of the  angular momentum applied in the impact mechanics ( [7], but also [8,9] mostly for a general perspective about the impact mechanics), on the subject "arm" and "forearm" respectively: The last equation is the projection of the theorem of linear momentum applied to the ball bumping the hand along the impact line, using the symbol v3 as resulted velocity of the ball between the velocity before impact ( 0 3 v ) and the velocity after impact (  ). Also, it is determined to consider that the variation of the velocity of the ball (denoted by v3) is given and its mass m3 is known as well.
Before to rewrite the system of the impact equations, it is welcome to specify that all above mentioned relations from (1) to (8) have been checked rigorously, all the small calculation errors from [4] being eliminated.
The unknowns of this impact process are delivered solving the following system of equations: It is specified that all physical and geometric data are provided by paper [4], both for male and female.

Results and discussions
Numerical simulation of impact The system of equations containing the relations from (12) to (20) has been solved keeping into account that the possible angles (corresponding to the real positions of the hand-arm system during the impact with the ball)  and  could have the subsequent values pinned on Table 1. https://doi.org /10.37358/Rev. Chim.1949 Rev. Chim., 71 (7) Configuration of the positions of the hand-arm system during impact As it can seen, Figure 2 offers a suggestive picture of all these positions characterized by the angles  and .
The numerical simulation has put into effect using a computer software based on [10], for six cases (illustrated by Figure 3 to Figure 8). These cases depend on different values of the mass and the velocity of the ball.
The target objective of this paper is to find the impact impulses. The results are caught in the diagrams bellow (Figures 3-8

Conclusions
The analysis of the above results causes the following commentaries. Before, it is necessary to explain that the numerical estimation of the shoulder and elbow impact impulses has a mathematical approach based on more or less realistic physical ball data (meaning the mass and velocity of the ball hitting the hand). For this reason, it is normal to reveal the qualitative aspects of the results and to not insist on the quantitative description of them.
The shoulder impact impulses are greater than the elbow impact impulses both for male and female according to the case A and the case D, for the smallest velocity of ball (v3=0.2 m/s), even if the mass varies from 0.3 kg to 0.6 kg. The elbow impact impulses are greater than the shoulder impact impulses, both for male and female, in case E (m3=0.