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 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:

In the formula, _{x}_{y}_{x}_{y}

_{x}_{y}_{x}_{y}

From equations (2) and (3), the expression of the net string tension increment Δ_{x}_{y}

When the net is an ellipse, its boundary area can be expressed by the following equation:

In the formula,

Substituting Equation (6) into Equation (4) and integrating within the range of Equation (5), we get:

Assuming that the part where the tennis racket hits the tennis ball is in the center of the net, we can express

In the formula,

Where _{0} under the action of the striking force

Suppose

Suppose:

We can get the following formula by substituting formula (12) into formula (11):

Using the first fraction of equation (13) and the initial conditional equation (10), we can obtain:

We substitute equation (14) into the second fraction of equation (13) and use the initial condition equation (10) to obtain:

Therefore, we can obtain the function expressions of the natural vibration frequency of the net and the nonlinear vibration displacement of the net as:

If _{0} is the impact duration, _{0} / 2, _{0} / 2) = 0 is the start of the impact on the net. And _{0} / 2, _{0} / 2) = 0 is the end of the impact on the net. It can be seen that the value condition of time _{0}/2 = ±

Finally, we use _{0} / 2) = 0 to determine the value of the maximum displacement _{0}.

After ignoring the inertia term, we can see that the relationship between the displacement _{0} of the net under the action of the static force _{0} and the static force _{0} is a nonlinear elastic relationship:

In the formula

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 _{3}. When the maximum displacement of the net under the action of the striking force _{0}, the following equation is established by using the law of conservation of energy:

So:

Suppose the stiffness of the tennis ball is _{4}. When the net hits the tennis ball, it is equivalent to 2 springs in series. Its stiffness is:

It is assumed that the moment when the center of the net of the tennis racket produces the maximum displacement _{0} under the action of the striking force

Suppose the tennis ball has a diameter _{0} when it is not deformed [11]. The amount of deformation of the tennis ball after being compressed by the impact force _{0}_{4}. So we know that the equation of motion of tennis is:

We can transform Equation (25) into the following form:

Where ^{2} = _{0}. We substitute Equation (18) into Equation (26), and use the initial condition Equation (24) to obtain the expression of

When the tennis racket is separated from the tennis ball, the following conditions should be met:

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).

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 2^{2}. The cross-sectional area of a net string is _{x}_{y}^{−6}^{2}. The elastic modulus of a net string is ^{11} ^{2}. The mass of the tennis ball is _{0} 5.8×10^{–2} _{4} = 10857.12_{0} = 0.0651

Because the average spacing of the net chords in the ^{–2} ^{–2} _{x}_{y}^{–3}

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 |

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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