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# The Approximate Solution of Nonlinear Vibration of Tennis Based on Nonlinear Vibration Differential Equation

###### Accepté: 27 Apr 2022
Détails du magazine
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Introduction

The majority of sports enthusiasts favor tennis. Tennis has long since become an important event in the Olympic Games. Therefore, it is of theoretical significance to study the dynamic characteristics of the tennis racket hitting process to guide the design of tennis rackets and improve the level of tennis players [1]. A tennis racket consists of a network cable and a frame. Due to the strong nonlinear vibration characteristics of the racket’s net under the tennis ball’s impact, it is difficult to model the tennis racket mechanically. As a result, many scholars use interlaced beams to replace the tennis racket net while ignoring the nonlinear characteristics of the tennis racket net. Some scholars have studied the serve action of tennis [2]. Some scholars have studied the dynamic characteristics of tennis rackets. Some scholars have discussed the striking center of a tennis racket. The relationship between tennis racket net string tension and hitting speed was studied [3]. None of the above-related studies have considered the nonlinear factors of the net. Based on the above reasons, this paper regards the net of the tennis racket as a cable net, uses the nonlinear vibration theory to study the dynamic characteristics of the tennis racket during the hitting process, and obtains some useful conclusions.

The vibration equation of the net

The tennis racket frame is assumed to be rigid [4]. From the elastic vibration theory, it can be known that the vibration equation of the tennis racket cable net is: $(Hx+ΔHx)∂2w∂x2+(Hy+ΔHy)∂2w∂y2=m∂2w∂t2−q(x,y,t)$

In the formula, Hx, Hy is the initial value of the horizontal component of the upper chord tension in the x, y direction, respectively. ΔHx, ΔHy are the horizontal projections of the chord tension increments in the x, y direction, respectively. m is the mass per unit area of the net. w(x, y, t) is the vibration displacement of the net. q(x, y, t) is the external disturbance force. According to Hooke’s law, the elongation of the net string is: $Δlx=ΔHxlxEAx,Δly=ΔHylyEAy$

lx is the length of the net string in the x direction. ly is the length of the net string in the y direction. Ax, Ay are the cross-sectional areas of the inner chord per unit length in the x, y direction, respectively. The elongation of the chord CD, EF in the x, y direction of the net shown in Figure 1 is: ${ Δlx=∫CD12(∂2w∂x2)2dxΔly=∫EF12(∂2w∂y2)2dy$

From equations (2) and (3), the expression of the net string tension increment ΔHx, ΔHy can be obtained as follows: $ΔHx,EAx2lx∫CD(∂w∂x)2dx,ΔHy,=EAy2ly∫EF(∂w∂y)2dy$

When the net is an ellipse, its boundary area can be expressed by the following equation: $x2a2+y2b2=1$

In the formula, a is the semi-major axis of the ellipse, and b is the semi-minor axis of the ellipse. Suppose the vibration displacement function of the net is as follows: $w(x,y,t)=T(t)(1−x2a2+y2b2)$

Substituting Equation (6) into Equation (4) and integrating within the range of Equation (5), we get: $ΔHx=2EAxT23a2(1−y2b2),ΔHy=2EAyT23b2(1−x2a2)$

Assuming that the part where the tennis racket hits the tennis ball is in the center of the net, we can express q(x, y, t) as follows: $q(x,y,t)=Pδ(x)δ(y)$

In the formula, P is the striking force. δ(x),δ(y) is the Dirac function [5]. We substitute equation (68) into equation (1) and use Galerkin’s principle to get: $d2Tdt2+ω02T+εβT3=N$

Where $ω02=3m(Hxa2+Hyb2),β=5003m(EAxa4+EAyb4),ε=1100,N=3Pπmab$ . Hypothetical tennis racket ball moment when the center of the net produces the maximum displacement a0 under the action of the striking force P is zero [6]. At this point, we can write the initial condition of the nonlinear vibration of the net as: $t=0,T(0)=a0,dT(0)dt=0$

Suppose τ = ωt. We can transform Equation (9) into the following form: $ω2d2Tdτ2+ω02T+εβT3=N$

Suppose: ${ T(τ)=T0+εT1+⋯ω=ω0+εω1+⋯$

We can get the following formula by substituting formula (12) into formula (11): $ω02d2T1dτ2+ω02T1=−2ω0ω1d2T0dτ2−βT03$

Using the first fraction of equation (13) and the initial conditional equation (10), we can obtain: $T0(τ)=Nω02+(a0−Nω02)cos⁡τ$

We substitute equation (14) into the second fraction of equation (13) and use the initial condition equation (10) to obtain: $ω1=β2ω0(34(a0−Nω02)2+3N2ω04)$ $T1(τ)=(βN3ω08+βNω04(a0−Nω02)2−β32ω02(a0−Nω02)3)cos⁡τ−βN3ω08+βNω04(a0−Nω02)2cos⁡2τ+β32ω02(a0−Nω02)3cos⁡τ−3βN2ω04(a0−Nω02)2$

Therefore, we can obtain the function expressions of the natural vibration frequency of the net and the nonlinear vibration displacement of the net as: $ω=ω0+εβ2ω0(34(a0−Nω02)2+3N2ω04)$ $w(x,y,t)=(Nω02+(a0−Nω02)cosωt−εβN3ω08−3εβNω04(a0−Nω02)2+(εβN3ω08+εβNω04(a0−Nω02)2−εβ32ω02(a0−Nω02)3)cosωt+εβN2ω04(a0−Nω02)2cos2ωt+εβ32ω02(a0−Nω02)3cos3ωt}(1−x2a2−y2b2)$

If t0 is the impact duration, t = −t0 / 2, w(0, 0,−t0 / 2) = 0 is the start of the impact on the net. And t = t0 / 2, w(0, 0,t0 / 2) = 0 is the end of the impact on the net. It can be seen that the value condition of time t is ±ωt0/2 = ±π/2. Therefore, the formula for calculating the impact duration is: $t0πω=π/(ω0+εβ2ω0(34(a0−Nω02)+3N2ω04))$

Finally, we use w(0, 0,−t0 / 2) = 0 to determine the value of the maximum displacement a0.

Tennis equation of motion

After ignoring the inertia term, we can see that the relationship between the displacement w0 of the net under the action of the static force P0 and the static force P0 is a nonlinear elastic relationship: $P0=k1w0+k2w03$

In the formula $k1=πab(Hxa2+Hyb2)$ , $k2=59πab(EAxa4+EAyb4)$ Since the net is a nonlinear elastic body, and the tennis ball is a linear elastic body, the impact of the net and the tennis ball is equivalent to the impact of the nonlinear spring and the linear spring [7-8].

We simplify the analysis of tennis. The experiment transforms the nonlinear elasticity of the net into an equivalent linear elasticity [9]. Assume that the equivalent linear elastic coefficient of the net after transformation is k3. When the maximum displacement of the net under the action of the striking force P is a0, the following equation is established by using the law of conservation of energy: $∫0a0(k1w0+k2w03)dw0=∫0a0k3w0dw0$

So: $k3=k1+12k2a02$

Suppose the stiffness of the tennis ball is k4. When the net hits the tennis ball, it is equivalent to 2 springs in series. Its stiffness is: $k=k3k4k3+k4=k4(2k1+k2a02)2(k1+k4)k2a02$

It is assumed that the moment when the center of the net of the tennis racket produces the maximum displacement a0 under the action of the striking force P is zero [10]. We know that the initial conditions for tennis are: $t=0,x(0)a0,dx(0)dt=0$

Suppose the tennis ball has a diameter d0 when it is not deformed [11]. The amount of deformation of the tennis ball after being compressed by the impact force P is l0 = P / k4. So we know that the equation of motion of tennis is: $m0d2xdt2=k((d0−l0)−(x−w(0,0,t)))$

We can transform Equation (25) into the following form: $d2xdt2+λ2x=λ2(d0−l0)+λ2w(0,0,t)$

Where λ2 = k / m0. We substitute Equation (18) into Equation (26), and use the initial condition Equation (24) to obtain the expression of x(t) as: $x(t)=a0cos⁡λt+(d0−l0)(1−cos⁡λt)+(Nω02−εβN3ω08εβNω04(a0−Nω02)2)(1−cos⁡λt)+λ2λ2−ω2(εβN3ω08+εβNω04(a0−Nω02)2−εβ32ω02(a0−Nω02)3)(cos⁡ωt−cos⁡λt)+λ2λ2−ω2(a0−Nω02)(cos⁡ωt−cos⁡λt)+εβNλ22ω04(λ2−4ω2)(a0−Nω02)2(cos⁡2ωt−cos⁡λt)+εβλ232ω04(λ2−9ω2)(a0−Nω02)2(cos⁡3ωt−cos⁡λt)$

When the tennis racket is separated from the tennis ball, the following conditions should be met: $x(t)−w(0,0,t)≥d0$

We can find the separation time of tennis racket and tennis ball by substituting equation (18) and equation (27) into equation (28) and solving equation (28).

Example analysis and discussion

We now take a certain type of tennis racket as an example to analyze the dynamic characteristics of the tennis racket during the hitting process [12-13]. The long axis of the tennis racket net is 2a = 0.33m. The short axis is 2b − 0.25m. The mass per unit area of the net is m = 0.2625kh / m2. The cross-sectional area of a net string is Ax = Ay = 1.5386×10−6m2. The elastic modulus of a net string is E = 1.61011 N / m2. The mass of the tennis ball is m0 5.8×10–2 kg and the stiffness of the tennis ball is k4 = 10857.12N / m. The diameter of a tennis ball is d0 = 0.0651m.

Because the average spacing of the net chords in the x direction of the long axis of the net is 1.65×10–2 m. The average spacing of the net chords in the y direction of the short axis of the net is 1.47×10–2 m. The net string tension in the x direction is 275.6024N, and the net string tension in the y direction is 266.712N. So it can be seen that the average tension of the net in the x direction is Hx 16703.18 N/m. The average tension Hy = 18143.67 N/ m of the net in the y direction [14]. The time the tennis ball stays in the net is generally Δt = 5×10–3 s. So we know the approximate average force P = mv / Δt = 11.6vN of the net on the tennis ball [15-16]. It can be known from Figure 2 that the greater the initial velocity of the tennis ball, the greater the displacement of the net. The greater the initial speed, the greater the initial striking force on the net.

From the analysis of Figure 3, we can see that with the increase of the initial speed of the tennis ball, the change of the tennis motion curve becomes steeper and more complex. As the initial velocity of the tennis ball gradually decreases, the movement curve of the tennis ball tends to be gentle [17]. This means that tennis players who hit the ball faster have a relatively higher success rate in the same situation. From the analysis of Table 1, we can see that as the initial speed of the tennis ball gradually decreases, the separation time of the tennis ball and the tennis racket gradually becomes longer.

Separation time between tennis and tennis rackets

v/(m⋅ s-1) t/ms
66.66 4.323
60 4.382
55 4.424
51.06 4.468
45 4.501
Conclusion

The greater the initial velocity of the tennis ball, the greater the displacement of the net. Tennis players who hit the ball quickly have a relatively high success rate on their serve. As the initial velocity of the tennis ball gradually decreased, the separation time of the tennis ball and the tennis racket gradually became longer.

#### Separation time between tennis and tennis rackets

v/(m⋅ s-1) t/ms
66.66 4.323
60 4.382
55 4.424
51.06 4.468
45 4.501

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