## Pole Vault Technique Analysis Essay

The increase in the world record height achieved in pole vaulting can be related to the improved ability of the athletes, in terms of their fitness and technique, and to the change in materials used to construct the pole. For example in 1960 there was a change in vaulting pole construction from bamboo to glass fibre reinforced polymer (GFRP) composites. The lighter GFRP pole enabled the athletes to have a faster run-up, resulting in a greater take-off speed, giving them more kinetic energy to convert into potential energy and hence height. GFRP poles also have a much higher failure stress than bamboo, so the poles were engineered to bend under the load of the athlete, thereby storing elastic strain energy that can be released as the pole straightens, resulting in greater energy efficiency. The bending also allowed athletes to change their vaulting technique from a style that involved the body remaining almost upright during the vault to one where the athlete goes over the bar with their feet upwards. Modern vaulting poles can be made from GFRP and/or carbon fibre reinforced polymer (CFRP) composites. The addition of carbon fibres maintains the mechanical properties of the pole, but allows a reduction in the weight. The number and arrangement of the fibres determines the mechanical properties, in particular the bending stiffness. Vaulting poles are also designed for an individual athlete to take into account each athlete's ability and physical characteristics. The poles are rated by 'weight' to allow athletes to select an appropriate pole for their ability. This paper will review the development of vaulting poles and the requirements to maximize performance. The properties (bending stiffness and pre-bend) and microstructure (fibre volume fraction and lay-up) of typical vaulting poles will be discussed.

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FE simulations of the pole vaulting process were conducted with the commercial FE code abaqus 6.14. The initial boundary value problem of the dynamic equation of motion was solved with an implicit time integration method under consideration of finite deformations. Exemplary FE models of the pole and the vaulter were obtained and generated in abaqus/CAE. To capture the kinematics of the pole vault procedure, a setting was employed that couples pole and vaulter allowing for a relative motion.

### 2.1 Finite element model of the vaulter

A comprehensive overview of the mechanics of pole vaulting can be found in Frère et al. [4] describing the process in four steps: run-up, take-off, pole bending and pole straightening. A crucial part in the jumping process simulation is the description of the vaulter. Muscle work of the vaulter increases the performance [5, 6]. In addition, the moment exerted on the pole by the vaulter influences the vaulting performance [7]—elite vaulters bend the pole such that its effective length^{1} relative to the original length is reduced by ca. 30% [8]. As stated by Ekevad and Lundberg [9], a modeling approach representing the vaulter by a point mass with a fixed position relative to the pole is not sufficient. The complex motion in combination with muscle work needs to be considered.

Properties of segments of vaulter model (data from [11])

^{2}. Specific frames of the video were taken to represent different phases of the vault process. With the aid of CAD software several measurements were taken for each phase to describe the position of the vaulter’s body segments. In accordance with the planar motion of the segments model, a side view was considered in the videos

^{3}. The motion was described relative to the upper pole tip. Thus, it can be used in simulations with poles of different lengths. To do so, three coordinate systems were introduced as shown in Fig. 2. First, the global coordinate system

*x*–

*y*was fixed and located at the lower tip of the pole. Second, a relative coordinate system \(x_{1}\)–\(y_{1}\) was established at the upper tip of the pole. In addition, a relative coordinate system \(x_{2}\)–\(y_{2}\) was utilized with an origin coinciding with the previous one. This coordinate system rotates to maintain the \(x_{2}\)-axis tangential to the pole. The measured quantities are:

Angle \(\varphi\) between ground and a line connecting the tips of the pole,

Height \(h_{\mathrm {p}}\) of upper pole tip,

Angle \(\theta _{0}\) of the rotated coordinate system \(x_{2}\)–\(y_{2}\),

Angle \(\theta _{1}\) between the coordinate system \(x_{2}\)–\(y_{2}\) and segment A, and

Angles \(\theta _{2}\) to \(\theta _{6}\) between the segments A to F.

*m*= 80 kg. Thus, the significant effort to control the individual segments of the vaulter with muscle torques is avoided allowing more computational time on material modeling. The position of the mass center \(\varvec{x}_{m}\) in the coordinate system \(x_{2}\)–\(y_{2}\) was calculated from the measured angles in each phase by

$$\begin{aligned} \varvec{x}_{m}= \begin{bmatrix} x_{m}\\ y_{m}\\ \end{bmatrix} =\frac{\sum _{i=\mathrm {A}}^{\mathrm {F}}\varvec{x}_{i}m_{i}}{\sum _{i=\mathrm {A}}^{\mathrm {F}}m_{i}}, \end{aligned}$$

(1)

*i*th segment of the body and \(m_{i}\) is the mass of the

*i*th segment. The point mass moves relative to the pole in coordinate system \(x_{2}\)–\(y_{2}\) applying abaqus’ connector elements (cf. Sect. 2.3). The different phases of the vault and the position of the mass center (blue square) are illustrated in Fig. 4.

We employed two variables during the pole vault to trigger the relative motion of the point mass: the pole angle \(\varphi\) from take-off to the instance of maximum pole bending, and the relative height \(h_{m,\mathrm {rel}}\) of the point mass from maximum pole bending till pole release^{4}. This is due to motion of the point mass relative to the pole that cannot be implemented as time based. It depends on properties of the pole such as the pole’s length and stiffness—these properties, however, are to be varied during the simulations.

### 2.2 Finite element model of the pole

Modern, elite vaulting poles are manufactured out of lightweight materials consisting mainly of glass fiber-reinforced plastics. The fiberglass pole is rolled on a mandrel and, subsequently, wrapped to stiffen the pole. The perfect pole differs for athletes, depending on their physical properties, abilities and vaulting technique- and plays an essential role in the vaulter’s performance.

$$\begin{aligned} \Psi =\frac{\mu }{2}\left[ {I}_{1}(\hat{\varvec{b}})-3\right] +\frac{\kappa }{2}[J-1]^{2}, \end{aligned}$$

(2)

*J*is the determinant of the deformation gradient \(\varvec{F}\). Neo-Hookean hyperelasticity renders linear elasticity as long as it is valid but also allows for nonlinear behavior which occurs due to the large deflection of the pole. Therefore, it is the more general choice. The material density \(\rho\) and the Poisson’s ratio \(\nu\) were calculated with the rule of mixtures (following [13]) for a fiber volume fraction of 50% as in Davis and Kukureka [14]. The modulus of elasticity was obtained via a flex test simulation

^{5}. This kind of test is used in the pole industry to classify the stiffness of a pole. The pole is pin-supported at both ends and loaded with a mass of 22.7 kg in the center. The deflection in cm gives the flex number (a relative stiffness number) of the pole. The assumed geometry of the equipment, namely the pole, is shown in Fig. 5. Table 2 summarizes the chosen properties. In the study on the pole stiffness (see Sect. 3), the elastic modulus is varied in the neighborhood of the previously calculated value of the flex test.

Summary of the properties of the pole

### 2.3 Coupling of pole and vaulter

The point mass needs to be connected to the pole such that it can move relatively to the upper pole tip according to the vaulter’s motion in different phases of the vault. Moreover, the connection fails once the pole is stretched and the vaulter would release it to clear the bar. An IAAF requirement is that the vaulter must hold the pole in the grip area (no higher than 0.1524 m from the top of any pole or no lower than 0.4572 m from the top of the pole).

abaqus’ connector elements allow constraints involving relative motion of the connected parts via Lagrange multipliers as additional solution variables and also failure of the connection [15]. Through the connector element, a relative position of the point mass was given as a constraint on the upper tip of the pole. Moreover, the connector elements were employed as a sensor to measure the position \([x_{\mathrm {p}},y_{\mathrm {p}}]^{T}\) of the upper pole tip and the height \(h_{m}\) of the mass center. A user subroutine UAMP was used in each increment of the calculation to call the quantities measured by the sensors and enforce relative displacement components [16].

$$\begin{aligned} \varphi =\arcsin \left( \frac{y_{\mathrm {p}}}{\sqrt{x_{\mathrm {p}}^{2}+y_{\mathrm {p}}^{2}}}\right) \quad \text {and}\quad h_{m,\mathrm {rel}}=\frac{h_{m}}{L}. \end{aligned}$$

(3)

$$\begin{bmatrix} x_{m}\\ y_{m}\\ \end{bmatrix} = {\left\{ \begin{array}{ll} f(\varphi )\qquad &\text {if}\qquad \varphi \le {60^\circ }\\ f(h_{m,\mathrm {rel}})\qquad &\text {else}, \end{array}\right. }$$

(4)

*f*is a function that interpolates the relative position of the point mass from the discrete values of the different phases that were obtained in Sect. 2.1. The values calculated with Eq. (4) were further used as an input to feed-forward control the relative motion of the point mass. The UAMP was employed to enforce the relative motion of the point mass until a relative height \(h_{m,\mathrm {rel}}\) was reached, corresponding to the instance when the pole was completely recovered. Then, the connector fails corresponding to the vaulter releasing the pole. The push-off motion of the vaulter is negligible compared to the catapult effect of the pole [17]—regarding the current accuracy of the model. For a fully detailed model, the push-off angle of the vaulter should be taken into account. Eventually, the point mass moves freely under the influence of gravity only.

### 2.4 Initial and boundary conditions of pole vaulting

Regarding the boundary conditions of the simulation, the pole was supported at the lower end fixing all translational degrees of freedom. This represents the contact to the planting box. Furthermore, a gravitational acceleration of \(g=9.81\,\mathrm{m\,s}^{-2}\) was assumed.

Initial conditions of the simulations

Due to the ‘hard contact’ boundary condition at the lower end, such an initial condition results in artificial oscillations of the pole with large amplitudes. All modes of the pole, which is initially at rest, are excited by the impulse. To circumvent these artificial oscillations, we started the simulation at the instance right after the pole was planted and when it started to bend as the vaulter’s mass compressed it. Then, in addition to the initial velocity of the point mass, also a velocity profile \(v_{0,\mathrm {pole}}\) on the pole was necessary that induced the bending. This profile was obtained from a preliminary simulation.

*L*requiring:

- 1.
Zero velocity at the lower end,

- 2.
Velocity at the upper tip equals the preliminary simulation,

- 3.
Maximum velocity node position matches the preliminary simulation, and

- 4.
Maximum velocity value equals the preliminary simulation.

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