EJp = ω2, (Eq. 29)
F ρ
which makes the unknown quantities totally seven and these are determined by the initial and boundary conditions.
8. Tennis equipment data
From various sources [5], [6], [8], discussions and contacts with specialists and players we were able to identify and select the initial data needed to develop the task. These include the following:
- Ball flying speed throughout the game. Among some of the best players this speed can reach up to 210 – 215 km/h and this is one of the most important parameters for our purpose here.
- Racket length – up to 27 inches;
- Centre of gravity and location with respect to the length. From this point of view rackets are “heavy head” and “light-head” ones;
- Operating area of the string – 500 to 700 cm2;
- String thickness - string diameter 1.25 to 1.75 mm. String positioning is important to the load on the string and the tension forces occurring in it and it could be either uniformly or ununiformly positioned;
- Hardness of the string area of the racket. This is very significant and strongly affects the value of vibrations occurring in the head, rim and handle of the racket;
- Hardness of the body of the racket. This is the degree of yielding (tendency for elastic deformation) of the body of the racket. Important factors for this parameter are the design, materials and manufacturing process. For professional rackets this varies from 50kg to 75 kg.
This quality is proportional to the dynamic inertia, manageability and control and inversely proportional to the length, head size and power.
The dynamic inertia is of significant importance for the high quality tennis rackets. Such rackets are made lighter to ensure better ball control and manageability and at the same time give higher acceleration to the ball when striking it due to their improved dynamic inertia. This involves the use of new materials: titanium, carbon, etc. strings. This parameter usually varies between 290 and 360 [4].
The Power characteristic is important to the loading scheme. It characterises the relation between the energy acquired from the ball and the energy available in the racket during the strike. The aim is to provide maximum ball acceleration with minimum effort of the hand. In professional rackets this characteristic varies within 25 and 80. The power is directly proportional to the length, head size, balance (centre of gravity), weight, body hardness and dynamic inertia. It is inversely proportional to string knitting and tensioning (when knitted strings are pre-tensioned in the range of 25 kg for each single one), the hardness of the string flat, manageability and control [5].
Tennis balls are standard and have a weight of 0.060 kg. Their external diameter is between 65 and 68-mm average and this varies throughout the game as it is reduced as a result of the wear on the external fibre surface. Various designs and makes of materials of balls are available but the standard professional ball is hollow inside, rubber cover and fibre textile surface. Especially important to the design of the test rig is the bouncing feature (damping coefficient) when hitting a solid or elastic surface [6]. This coefficient is not a constant value and varies depending on the contact surface, speed and material of the ball, etc. within a wide range – between 0.35 and 0.75.
The simplest possible preliminary experimental test was carried out to establish and clarify some of the basic parameters of the rackets and ball.
From all values measured and set in Table 1 and Table 2 we drew up the graphs in Figures 1 and 2 underneath showing the relation P/Δ. – Force/deformation. It can be seen from there that this relation is a linear function regardless of the relatively complicated shape (elongated ellipse) of the racket [9].
Table 1 Table 2
No Force P (N) Deformation Δ mm Force P (N) Deformation Δ mm
1. 0.985 0.30 0.985 0.15
2. 1.970 1.97 2.955 0.54
3. 2.955 0.90 4.925 1.00
4. 4.925 1.60 6.895 1.45
5. 5.910 2.20 8.865 4.86
6. 6.895 2.55 10.835 2.25
7. 7.880 2.90 12.805 2.75
8. 9.850 3.70 14.775 3.15
9. 11.820 4.55 16.745 3.62
10. 12.805 4.95 18.715 4.05
We developed this part of the project based on the “collision of two bodies”
Fig. 8 Fig. 9 theory from multi- physics.
Since both the ball and the racket (string) are made of elastic materials we regarded the collision between them as an elastic collision (Figure 8) where the kinetic energy is being partially transformed into potential energy of the elastic deformation. After the strike the ball and the racket move away from each other at a speed the value and direction of which are being determined by two conditions:
- the conservation of the total energy and the conservation of the total impulse of the system [10].
Consideration for this physical process is
necessitated in view of the requirements for
s o eration an quipment
compactness as well as the operation area
calculation and the calculation for the free fall
zone after the strike.
In this section we also determined the
maximum speed of the loading body on the Fig. 10 basis of which all further calculations in the project are to be made. This value is based on our observations of the play of some of the best tennis players in the world.
Σ P1=F=F1 (Eq. 30) The bending momentum on the rim and handle at the moment of the strike is
Mp = F . L (Eq. 31) and is overcome by the momentum of elastic support reactions C and D in the handle (the hand of the player) and
C. L = F (L + l) (Eq. 32)
F1 F Fa
Pi
FbPi PiL
Pi
ΣPi=F=F1 C
Dl
Fig. 11
(Ra)b αRb b Õ
V F (Rb)RaB1 B a
af RaRb
Õ
F
Fig. 12 On the other hand, keeping in mind that at the moment of the collision the racketstring, body and ball system is an elastic system the rim – frame of the racket undergoes certain deformation. These are expressed as a shrinkage of the string area towards the centre, overall bending in the direction of F, elongation of the strings and compression of the ball resulting in a change in its regular spherical shape. Thus, the loading diagram for the arrangement given in Figure 11 will be expressed as given underneath in Figure 12:
The frame – rim around the stri ng area shrinks down to the dimension B1- Fig.12 m. The tension in the strings when the load is applied at the moment of the collision is as shown in Figures 12 n and 12k. As a result of the elongation of the strings and the shrinkage of the rim from dimension B to dimension B1 we can determine the travel distance f needed to stop the ball. In other words, to have the speed of the ball from Vo = 210 km/h to become equal to 0. The number of strings subjected to this load is high and the type of the string area construction is such that it distributed the force F among all available strings. As a result of this the reactive forces Ra, Rb are expressed as the sum of all individual reactive forces, or Ra = Σ Ra.i and Rb = Σ Rb.i.
It is obvious from Figures 12n and 12k that depending on f and the angle Õ, respectively these forces are higher when the angle Õ is larger, or Ra and Rb will increase when the rim-string (string area) system is firmer. The design of the racket and the initial stretching of strings determine this feature. In the general case the rim is made up of two connected curved half beams. As the deformation f of the string area and the ball in the direction of the flight of the ball (“T”) is relatively small then the forces Ra and Rb are significant in value despite of the small ball weight.
The average value of the force F can be determined by the equation for the kinetic energy of the flying ball: ET = mV2 /2, where (U1)=V=Vo=210 km/h. (We assume maximum ball speed of 210 km/h and ball mass = 0.060 kg).
The total work A necessary to stop the ball in the racket is [12]:
A = Fav . f = m . Vo2/2 (Eq. 33) = 0.060 . 58.32/2 = 101.97 Nm.
We assumed f ≤0.08m, hence
Fav = 101.97/0.08 = 1275 N
The deformation f is assumed from measurement using a load force of ≈ 129.9 kg (1275 N). This value varies in value depending on the type of design and kind of racket and the firmness of the string area. In this case we have assumed an average value.
The force F could be calculated approximately from the basic equation in dynamics (and also check the results) [13]:
Fav. = m . a (Eq. 34) V = V0 – at (Eq. 35) f = at2/2
where, m – ball mass m = G/g = 0.06/g kg; V0 – initial speed = 58.3 m/s; V = 0 – final ball speed m/s; f – distance-deformation (m); t - time to stop (sec); a – deceleration (m/s2); 0 = 58.3 – at → t = 58.3/a ; 0.08 = a/2. (58.3/a)2 ;a = 58.32/0.16 = 21243 m/s2 ; t = 58.3/21243 = 0.00274 s
From (5):
F = 21243 . 0.06 = 1275 N
From Figure 12:
F/2Fa(b) = cosÕ Fa(b) = F/cosÕ = 1275/2 cos 513° = 1020.5 N
Where Õmax≈ 51.3° = arc tg 2f/B1 . (B1 ≈ B) then B ≈ 0.200m.
Since the relation Fa(b) = ƒ (F/2cosÕ) is not linear the values for Fa(b) in the initial moment of the strike when the angle Õ is close to 90° are very high thus causing sudden elongation of the loaded strings and compression of the frame. The condition that the movement of the ball in this very short time interval (t = 0.00274 sec) is uniformly retarded is only approximately true. The general view of the diagram indicating the variation of the force F along the travel distance f will be like the curve shown in Figure 13 underneath:
The curve 1 indicates the value of F when the ball stops at the string area and the curve 2 indicates the value of F when the elastically tensioned rim, strings and ball shoot the ball in the backwards direction after the collision. The work A performed to stop the ball is indicated by the area underneath the curve 1down to the x-axis and the work for the backward shooting is indicated by the area underneath curve 2.
Ast = 101.97 = ∫ Ffdf (f = 0.08)
The work A for shooting the ball in the backward direction is less in value due to the energy loss resulting from the friction in the materials of the rim, strings and ball during the deformation process.
Counteracting forces to the force
F are the intrinsic inertia of the racket
and in addition to this the force from F(N)
the hand of the player when he blows 1
to strike the ball [15].
Where, m 1 is the mass of the
racket = 0.320 kg 0.08 f(m)
M is the mass of the hand ≈ 5 kg Fig. 13
The forces causing oscillations in
the racket are the sum of the horizontally acting F, F1 and those occurring in the strings as a result of their action. Dying mechanical oscillations occur in the body of the racket and the following relations can be used to express them:
m . d2x/dt2 = - kx + f (Eq. 36)
and this has the following solution: x = A1eγ1f + A2eγ2f
where, γ2 + r/m . γ + k/m = 0;
A1,2 constants;
f – friction forces for damping oscillations.
Or the expression for dying oscillations at low resistance is:
U1 is ball speed before the strike = 58.3 m/sec
K is the restoration coefficient;
m.x = P + R m.x + c.x +λ.V = 0 if k2 = c/m (Eq. 37)
If we look at the curve 1 in Figure 13 its exact shape or the slope shape will mostly depend on the total elasticity of the racket body (head, rim and handle) and ball system. In other words, this is the total coefficient of elasticity (spring constant) c expressing the force/deformation relation [16].
As the shape, materials and overall design of rackets vary to a large extend then c varies accordingly. In our case, in order to solve the object tasks it is sufficient, due to the only momentary nature of this action to consider not the curve shape in the diagram but only the value of the average force F.
The impulse force isS
=