^m1.m2 ( )_.v^{() 2}

m

 

1

 

+
m

 

_{1 − 2} . (Eq.

 

2

 

7.2.4 Oblique impact of two bodies

The collision of two bodies is considered to be oblique when the absolute speeds of their centres of gravity are not directed along the straight line that is connecting them. This section of the impact theory is applicable in the cases when the actual striking conditions between the ball and the tennis racket have to be simulated and the speed of the ball being shot is not normal to the tennis racket string surface. Of practical interest in this case is the speed of the ball and the reflection angle β because it is important to know where exactly is the ball going to go after the experiment. To derive the necessary theoretical relations we shall consider two bodies having masses m₁ and m₂, respectively [22]. The speeds of the bodies before the strike are v₁ and v₂ and after the collision, u₁ and u₂, respectively. Applying the theorem for the preservation of the amount of movement of a mechanical system we obtain the following expression:

m₁u₁+ m₂u₂= m₁v₁+ m₂v₂. (Eq. 13) This expression is projected on the tangent and normal lines drawn at the point of contact between the bodies and the following two expressions result form this: m₁u_1n+ m₂u_2n= m₁v_1n+ m₂v_2n, (Eq. 13a)

 

m₁u_1τ+ m₂u_2τ= m₁v_1τ+m₂v_2τ. (Eq. 13b) To be able to determine the unknown values we shall have to add to the above expressions the relations expressing the two Newton impact laws:

 

u
u

 

1
τ

 

−
u

 

2
τ

 

=
1
−
λ
≈
1

 

. (Eq. 14) v_1n − v_2n

 

^{1n −u2n}= −k and _v1τ−v2τ

 

From the second and fourth expression we obtain:

 

u = v_{1τ and} u = v_{2τ, 1τ 2τ}

 

and from the first and the third one: u_1n and u_2n
7.3 Loading the tennis racket
7.3.1 Tennis racket simulation

The tennis racket generally comprises a head, representing a mesh frame (rim), and a body made in the shape of a handle. The tennis racket configuration is very specific and complicated. This results in many difficulties when calculating the load and size of the racket using conventional methods. Therefore, it is more convenient for the purposes of a preliminary study to build a simplified model of the racket.

We can successfully use a surface loaded beam fixed in one of its ends for the purpose. We assume the beam has constant round, rectangular or square shaped cross-sectional area. To establish the extent of adequacy of the suggested simulation model we decided to make the following experimental set-up [23]:

The tennis racket is fixed static at its handle using a vice. Then using a balance, which measures the centre of gravity by means of a spring, the tennis racket is thus loaded at a point, which is located at the longest distance from the point of clamping the handle. This point is located in the tennis racket head area and corresponds to the tip point of its frame. The load is perpendicular to the axis of the tennis racket and is increased with every experiment by an exactly determined value. The displacement of the loaded point from the axis of the tennis racket is measured for every individual load value. A diagram is then drawn up based on this data and the type and nature of this diagram is very close to linear.

Thus, the data derived give grounds to confirm that the simulation model being suggested is acceptable for the needs of the preliminary study.

 

7.3.2 Determining the rigidity of the tennis racket

We know from material resistance reference books that the maximum deflection of the loaded end of a cantilever mounted beam is determined using the following formula:

f
a

 

=
Pl ³ _3EJ (Eq. 15)

In above formulae E is used to express Young’s modulus and J is the basic moment of inertia of the beam cross-section; P is the load force applied, l is the arm of the bending momentum and f_a is the beam loaded point deflection from its axis.

Pl³

 

Hence, EJ =. (Eq. 16)

 

3 f_a

The above formula allows me to calculate the product E.J, which is referred to as beam rigidity for pure bending. This product will be necessary to determine the energy needed for the deformation of the adopted simulation model.

7.3.3 Determining energy distribution during the collision between the ball and the tennis racket

The ball, which hits the tennis racket, is a carrier of mechanical energy, which during the collision with the tennis racket is distributed as follows:
a) Part of it is lost during the collision Δ Ek1 ,
b) A second part is being removed during the tennis racket deformation Δ Ek2
c) A third part is being dissipated by the dissipation forces of the internal friction between the particles of the material the tennis racket is made of Δ Ek3, d) A fourth part is being taken away via the deflected ball Δ Ek4.

The first part of the energy is determined by the formula given in 7.2.1, which uses experimentally defined restitution coefficient. The experiment for determining this coefficient is carried out as follows:

The frame of the tennis racket is fixed statically in horizontal position and the ball is then dropped from a 1-m height and then the height to which the ball bounces is measured. Then to determine k we use the expression from 7.2.1.

The second part of the kinetic energy Δ Ek₂ can be determined by measuring the deflection of the tip point of the tennis racket at the moment when the ball separates from the racket. The dial sensor for measuring the said deflection should therefore be positioned on the side of the reproduction of the impact. If we assume to disregard the third part of the energy Δ Ek₃ for the purposes of the preliminary study (its share is not actually essential), then Δ Ek₂ = Δ Ep, where the right-hand designation indicates the potential energy acquired in the deformed tennis racket. This can be determined in the following way:

ΔEk₂ = ΔEp =¹ cf²_{, (Eq. 17) 2 max}

 

Where f_max is the maximum deflection and the coefficient c can be determined using the expression:

 

^3EJc=_l3. (Eq. 18)

 

Finally, ΔEk₂=³EJ.f³ . It is not precisely correct to read the dimension f_{max 2l3 max}

from the side, which is opposite to the impact, as part of the potential energy being accumulated from this deformation is spent for the back shooting of the ball. If no measurement devices for the respective parameter are available or it is impossible to perform such a measurement the energy Δ Ek₂ or any other part from those mentioned above could be determined using the energy balance [23]. For example, if it is impossible to measure the bouncing speed of the ball this could be calculated in the following way:

ΔEk₄ = Ek₀ −ΔEk₁ − ΔEk₂ − ΔEk₃ , (ΔEk₃≈ 0) (Eq. 19)