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 length1 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.

Hubbard [10] applied a model of the vaulter consisting of three rigid segments representing different parts of the body. Ekevad and Lundberg [11] extended this approach to six segments in a 2D setting, sequentially connected by joints: forearm, upper arm, head, trunk, thigh and shank (cf. Fig. 2).
The two arms and legs were represented by one equivalent segment. A summary of the properties of the segments is shown in Table  1. In this paper, we extend the approach of Ekevad and Lundberg [11] by additionally accounting for a joint at the neck.
Table 1

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

An exemplary motion of a vaulter was obtained by analyzing videos of a German elite pole vaulter2. 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 videos3. 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 xy 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.

The angles \(\theta _{i}\) between the body segments of the reference pole vault over time are plotted in Fig. 3. A linear interpolation was performed between the discrete values.
At a second stage, the body was reduced to a point mass (center of gravity) with 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)

where \(\varvec{x}_{i}\) is the position of the mass center of the ith segment of the body and \(m_{i}\) is the mass of the ith 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 release4. 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.

As a first step, an isotropic hyperelastic material model of Neo-Hookean type was applied with the strain–energy function

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

(2)

where \(\mu\) is the shear modulus, \({I}_{1}=\mathrm {tr}(\hat{\varvec{b}})\) is the first invariant of \(\hat{\varvec{b}}\), \(\hat{\varvec{b}}=J^{-2/3}\varvec{b}\) is the distortional component of the left Cauchy–Green strain tensor, \(\kappa\) is the bulk modulus and 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 simulation5. 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.
We discretized the pole with continuum elements as well as beam elements and compared the outcome. Continuum elements can represent arbitrary structures and bodies and can resolve complex three-dimensional stress states. Further, they allow for incorporating user defined material laws which might be desired when accounting for microstructure, for example. Structural elements such as beams rely on assumptions like slenderness simplifying the model to one predominant dimension. This leads to a reduced number of degrees of freedom and therefore, higher computational efficiency. abaqus’ Timoshenko beam elements capture large axial strains [15].
Table 2

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].

In the UAMP routine, the pole angle \(\varphi\) and the relative height \(h_{m,\mathrm {rel}}\) of the mass center were calculated via

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

(3)

As the sensor measurements are from the beginning of the time increment, the corresponding values of \(\varphi\) and \(h_{m,\mathrm {rel}}\) at the end of the time increment were extrapolated. Subsequently, the relative position of the point mass in coordinate system \(x_{2}\)–\(y_{2}\) in the next time step was determined as:

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

(4)

depending on the simulation state being in the first or the second part of the vault. Here, 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.

The choice of the initial conditions is not trivial and has to be done carefully. Frequently reported in the literature, is the employment of only the take-off velocity \(v_{0}\) on the vaulter (see e.g., [9, 11, 18]), while no initial velocity was applied to the pole. The initial velocity of the vaulter is split into two components: one in the horizontal direction due to the approach speed and one in the vertical direction due to the jump. The components are calculated with the take-off angle \(\alpha\). Table 3 summarizes exemplary values for initial conditions according to Linthorne [18].
Table 3

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.

In this preliminary simulation, only the take-off velocity \(v_{0}\) of the point mass was applied as an initial condition and the density of the pole was artificially reduced by six orders of magnitude. Therefore, the oscillations still occurred in the beginning of the simulations but the amplitudes were small since the pole had nearly no inertia associated with it. In the following, the oscillations decayed quickly (approximately 0.01 s) and a homogeneous velocity profile developed in the pole (cf. Fig. 6).
The profile has velocity components in the horizontal and vertical directions and is not necessarily perpendicular to the pole. This profile was then extracted and employed as the initial condition for the simulation. The velocity profile’s shape resembles the bending of the pole in its first mode. Thus, an initial velocity profile initiates the real deformation. In the pole vaulting simulation, the profile was approximated with a third order polynomial along the pole length L requiring:
  1. 1.

    Zero velocity at the lower end,

  2. 2.

    Velocity at the upper tip equals the preliminary simulation,

  3. 3.

    Maximum velocity node position matches the preliminary simulation, and

  4. 4.

    Maximum velocity value equals the preliminary simulation.

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