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A Numerical Study for the Effect of Ski Vibration on Friction

  • Yunhyoung Nam
  • Do-Nyun KimEmail author
Original Research
  • 38 Downloads

Abstract

In ski jump, minimizing the energy loss when sliding down the track is crucial for improving jump performance. In particular, it is essential to reduce the friction between the ice track and the jumping ski, which is typically achieved by properly waxing the ski base according to weather and ice conditions. Additionally, it might be possible to reduce the friction by controlling the vibration of jumping skis as many experiments and numerical analyses have reported that vibration can reduce the friction in both dry and wet conditions. However, they have been done mostly for small, rigid specimens and therefore the results may not be directly applicable to jumping skis because they have unique vibrational modes due to their slenderness and structural flexibilities. Here, we investigate a potential effect of ski vibration on friction using finite element analysis. Finite element beam models for jumping skis are constructed by measuring the geometry and bending stiffness experimentally. We employ a pressure-dependent friction model between the ski base and the ice track derived from the reported experimental data. Various vibration conditions are tested for four jumping ski models with different bending stiffness profiles. Results demonstrate the possibility of friction reduction by designing the bending stiffness profile of a jumping ski that controls its vibrational modes.

Keywords

Vibration Friction model Friction reduction Ski jump Finite element method 

Introduction

Ski jump is a dynamic winter sports game involving various interactions among athletes, skis, and environment (ice slope and air) during four different phases including in-run, takeoff, flight and landing. Most studies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] related to ski jump have been focused on aerodynamic analysis of an athlete with jumping ski during flight to investigate an optimal posture maximizing the lift-to-drag ratio for flying longer. This is mainly due to the fact that the flying speed is almost 100 km/h and the aerodynamic resistance must be effectively overcome to improve the performance. However, little attention has been paid to the in-run (sliding) state where the friction between the track and the ski base in addition to aerodynamic drag force must be minimized in order to gain the take-off speed as high as possible.

Most common approaches for the friction reduction during in-run are to apply wax layers and groove patterns on the ski base taking the weather and icy track conditions into consideration. It might be possible, in addition to these approaches, to reduce the frictional force by controlling the vibration that jumping skis experience while sliding down the track. Friction reduction due to vibration has been reported in many papers [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. But they have been limited to small and rigid specimens where the externally applied vibration force is transmitted to contact surfaces as it is. However, for slender structures like jumping skis, their inherent vibration modes need to be considered as they vary the location and condition of contact. In this study, we investigate the effect of ski vibration on friction numerically using the finite element method. The geometric and structural properties of commercial jumping skis are measured experimentally and corresponding finite element beam models are constructed. We employ a pressure-dependent friction model on ice derived from the reported experimental data. The degree of friction reduction is calculated by performing transient dynamic analysis for jumping skis under various vibration conditions. Results are expected to provide an important insight into the interaction between the ski and the ice track which might offer an additional way of reducing friction by controlling the dynamic properties of jumping skis.

Method

In this section, we present our finite element model for a jumping ski, the friction model used to describe the interaction between ski plate and icy running slope, and analysis procedure to investigate the effect of ski vibration on friction.

Finite Element Model for Jumping Ski

Jumping skis are composite structures having a unique geometric profile such as long and slender shape with camber and rocker (Fig. 1). As they must meet several structural requirements related to ski jumping performance including lightness, aerodynamically favorable flexibility, low vibration, and high impact resistance, their internal structure and material composition are often highly complicated. Hence, it is not efficient to construct a full three-dimensional finite element model of jumping skis for its use in computationally demanding dynamic analysis where nonlinear frictional contact and vibration are involved. Instead, it is physically reasonable and computationally much more efficient to model a jumping ski as a curved beam structure with axially varying cross-section and stiffness. In particular, jumping skis can be even further approximated on a two-dimensional plane because ski jumping motions are mostly bilaterally symmetric and bending on a plane dominates the structural behavior of jumping skis. Therefore, we use two-dimensional beam finite elements to model jumping skis in this study.
Fig. 1

Shape profile and internal structure of a typical jumping ski

We choose a jumping ski (237 cm long) produced by Elan Sport to construct a finite element model for our study. We first obtain the approximated shape of this jumping ski by measuring the geometric parameters at 80 cross-sectional points along the ski length. Profiles of the top and bottom surfaces, the width, and the thickness are shown in Fig. 2. Then, the approximated distribution of bending stiffness along the ski length is obtained by dividing the ski into 12 parts from the afterbody contact point to the shovel point. The tip deflection of the starting point of each part is measured by applying a vertical force there while clamping the remaining part behind the part’s end point. Mean bending rigidity of each part is estimated using the Euler beam theory (Fig. 3).
Fig. 2

Geometric data of the jumping ski measured at eighty reference points. Blue and red lines in a represent the top and bottom surfaces, respectively

Fig. 3

Bending rigidity profile and finite element model. a Twelve parts of the jumping ski whose bending rigidities are measured. b Schematic representation of the experimental method to measure the bending rigidity of each part. c Measured profile of bending rigidity. d Finite element model for the jumping ski. 43 nodal points are placed on the bottom surface of jumping ski which is in contact with the track during in-run state

Finite element nodes are placed on the bottom surface in order to model the interaction between the ski base and the slope. Each part is discretized using three Hermitian 2-node beam elements resulting in the jumping ski model consisting of 42 beam elements and 43 nodal points (Fig. 3d). We assign, to each element, the experimentally measured bending stiffness of the ski part to which it belongs (Fig. 3c), together with its average width and thickness (Table 1). For the parts outside the afterbody contact and shovel points whose bending stiffness is hardly measured experimentally, the element properties of the nearest part are used, instead.
Table 1

Geometric data of the jumping ski used to generate the finite element model

Ski parts

A

B

C

D

E

F

G

H

I

J

K

L

Length (mm)

230

230

230

230

172.5

172.5

172.5

172.5

140

140

140

140

Average width (mm)

111.2

109.2

107.3

105.4

105.4

107.3

108.9

110.3

111.6

113.0

114.3

114.7

Average thickness (mm)

10.3

15.7

21.0

24.1

23.6

20.8

17.1

13.7

11.2

9.4

7.6

7.0

Friction Model in Icy Condition

Jumping ski in winter games runs on an ice track. The friction between the ski base and the track develops through complicated physical mechanisms during in-run state because the track may exist in multiple phases from solid (ice) to liquid (water) exerting a strong influence on friction. In the beginning of sliding, the track is mostly in solid phase and hence the dry friction due to solid-to-solid asperity contacts dominates the early stage of in-run resulting in a relatively high frictional force. Heat generated by the frictional force melts the top surface of icy track and a thin water layer is, in consequence, formed at the interface. As the water layer becomes thicker, the friction force decreases because water bridges are formed between two solid surfaces and their capillary drag determines the friction force. If the ice melts down even more, the friction increases again with the film thickness because the hydrodynamic friction by viscous shearing of the film becomes the principal friction mechanism. These three friction mechanisms on ice are usually referred to as boundary, mixed, and hydrodynamic friction (or lubrication) regimes, in the order described.

In general, the friction coefficient between a solid material and the ice is modeled as a linear combination of the dry and lubricated frictions as follows [25, 26]
$$\mu = \alpha \mu_{dry} + (1 - \alpha )\frac{{\eta v_{r} }}{{h_{w} p_{r} }}$$
(1)

Here, \(\mu_{dry}\) is the dry friction coefficient without water layer, \(v_{r}\) is the relative velocity of a solid body, \(\eta\) is the viscosity of water layer, \(h_{w}\) is the thickness of water layer, and \(p_{r}\) is the pressure. In this model, the contribution of dry friction with respect to the hydrodynamic friction is controlled by \(\alpha\) ranging from 0 to 1.

Bäurle et al. investigated experimentally the friction between the ski base material and the ice at − 5 °C [27]. They found that the friction coefficient increases with the macroscopic contact area and is independent of the relative velocity between the specimen and the ice. They reported the coefficient of friction for various contact areas at two normal force conditions (84 N and 52 N). It can be easily observed that their results follow the friction model in Eq. (1) very well if we plot the friction coefficients as a function of pressure (normal force divided by macroscopic contact area) for each case (Fig. 4). By fitting the model in Eq. (1) into the converted experimental results, the friction coefficient can be obtained as a function of pressure when given in N/cm2 as follows.
$$\mu = 0.0229 + \frac{96.85}{{p_{r} }}$$
(2)
Fig. 4

Experimentally measured friction coefficients as a function of pressure for two different normal load conditions (blue and red) [27]. Black line shows the fitted data represented by Eq. (2)

In order to use this friction model in our two-dimensional finite element analysis, we rewrite it using the normal contact force (\(F_{n}\)) and the contact length (\(L_{c}\)).
$$\mu = 0.0229 + \frac{96.85}{{F_{n} }}A_{c} = 0.0229 + \frac{48.42}{{F_{n} }}L_{c}$$
(3)

Here, \(A_{c}\) is the contact area and we substitute the contact width, \(w_{c}\), with the one of the specimen used in experiments (0.5 cm). We used this friction model to describe the interaction between the ski base and the in-run track in this study.

While it is almost impossible to achieve a perfect dry condition without any water layer generated when both a solid body and the ice come in contact, Bowden et al. could obtain a generally acceptable value of dry friction coefficient (\(\mu_{dry} = 0.3\)) between the ski base material and the ice with very thin water layer being formed [26, 28]. It can be deduced from this value and Eq. (3) that the dry friction contributes only 7.6% (or \(\alpha = 0.076\)) and therefore, the hydrodynamic friction dominates (92.4%).

Analysis Procedure

The weight of an athlete with ski boots is assumed to be uniformly distributed along the ski binding whose center is located 31 cm behind the balance point (Fig. 5). It is modeled as a rigid block of 32 kg (half the mass of an athlete and boots) under gravity located at the binding site and represented using four bilinear solid elements. In transient dynamic analysis, the gravity load drives and accelerates the jumping ski along the track as well as providing the normal load due to the angle of inclination of the track (\(\gamma\)) that is set to be 36 degree here. Sinusoidal point forces are applied at both ends to induce the ski vibration normal to the track with various amplitudes and frequencies. We do not include any aerodynamic force in our analysis.
Fig. 5

Analysis setup. Here, \(g\), \(\gamma\), \(m\), \(F_{n}\) and \(F_{f}\) represent the acceleration of gravity, the slope angle of the track, the total mass of simulation model, the normal contact force and the frictional force, respectively

To simulate the ski response during in-run state and investigate the effect of vibration on friction, transient dynamic analysis is performed using an implicit time integration method. First, we apply the normal force from the gravity only so that the jumping ski becomes fully contacted with the icy track. Then, sinusoidal forces are imposed at both ends to induce normal vibrations over the ski. Finally, we apply the tangential force from the gravity and allow the ski to slide along the track. Time-variant frictional forces are acting on the contacting nodes according to the friction coefficient in Eq. (3). Since this analysis is highly nonlinear due largely to frictional contact, Bathe method is used for time integration, which is a composite implicit time integration method proven to be more stable and accurate than conventional Newmark method [29, 30, 31]. Here, the time interval (\(\Delta t\)) is set to 0.000125 s considering the frequencies of the applied vibration and the natural frequencies of the jumping ski beam model. Analysis is performed for 6 s, usual elapsed time to slide down a large hill track, using the commercial finite element analysis software, ADINA version 9.0.5 [32].

Numerical Studies

Effect of Vibration Frequency on Friction

We first investigate the effect of vibration frequency on friction by varying the frequency of externally applied loads at ski tips as shown in Fig. 5. In particular, the lowest eight natural frequencies of jumping ski beam model are used in analysis. Normal mode shapes and frequencies obtained by performing normal mode analysis are displayed in Fig. 6. Here, the amplitude of applied loads is fixed to 25 N for all frequencies. Time-averaged frictional force, normal contact force, and contact length (defined as the length sum of beam elements in contact with the track) are calculated and compared with those obtained for the reference case where the external sinusoidal forces are not imposed.
Fig. 6

Normal mode shapes and natural frequencies of the jumping ski beam model

Results clearly demonstrate that the frictional force neither increase nor decrease monotonically with the frequency, but is rather mode-specific (Fig. 7a). In most cases, both frictional force and normal contact force are reduced by vibration and the maximum reduction (about 2.5%) in friction is observed at the fourth natural frequency. It is noteworthy, however, that the friction reduction does not follow the reduction in normal contact force (Fig. 7b). Rather, it is highly correlated with the average contact length of the jumping ski model during analysis (Fig. 7c). This is, in fact, an expected outcome according to the friction model in icy condition employed here. From Eq. (3), the frictional force can be written as
$$F_{f} = 0.0229F_{n} + 48.42L_{c}$$
(4)
and you can see the contact length governs the frictional force due to the dominance of hydrodynamic friction.
Fig. 7

Effect of vibration frequency. Reduction ratios of a the frictional force, b the normal contact force, and c the contact length in comparison to the case where no vibration is applied

Effect of Vibration Amplitude on Friction

In this section, we investigate the effect of vibration amplitude on friction by considering two loading conditions with varying force amplitudes. First, simple harmonic forces are applied whose frequency is fixed at 59 Hz corresponding to the fourth natural frequency at which the friction reduction is the highest in the previous section. The amplitude is varied from 5 N to 25 N. Second, we impose random vibrational forces instead that are reconstructed from a time-varying acceleration profile (Fig. 8) measured experimentally in a real jumping condition [33]. Their magnitudes are scaled such that the externally applied power level is similar to that of each harmonic loading case.
Fig. 8

Acceleration profile measured experimentally by Shionoya et al. in a real jumping condition [33]

Results show that the friction does not monotonically decrease with the force amplitude when single-frequency harmonic forces are applied (Fig. 9a). It is highly correlated with the length of jumping ski in contact with the track exhibiting a nonlinear dependence on the force magnitude (Fig. 9c). Highly nonlinear nature of contact mechanics and ski deformations might be responsible for it. On the other hand, it is observed that the normal contact force decreases monotonically with the amplitude of applied loads, which is probably because the inertia force of jumping ski mostly acts upward due to the track (Fig. 9b). Nevertheless, it has negligible effect on the friction reduction. This non-monotonic dependence of friction to the force amplitude is more evident in case where random vibrational forces are applied (Fig. 10). This suggests that the effect of vibration amplitude on friction also varies with the frequency.
Fig. 9

Effect of vibration amplitude with fixed frequency. Reduction ratios of a the frictional force, b the normal contact force, and c the contact length in comparison to the case where no vibration is applied

Fig. 10

Effect of vibration amplitude with varying frequency. Reduction ratios of a the frictional force, b the normal contact force, and c the contact length in comparison to the case where no vibration is applied

Effect of Bending Stiffness Distribution on Friction

Results in the previous sections demonstrate that the friction reduction has non-monotonic dependencies on both frequency and amplitude of the applied vibratory force probably due to inherent nonlinearity in contact between the ski base and the track as well as in ski deformation. Hence, the distribution of bending stiffness of jumping ski may affect the reduction of friction as it alters the dynamic properties of jumping skis including the natural frequencies and normal mode shapes. To examine this effect, we constructed four jumping ski models with different bending stiffness profiles that are measured from four different commercial jumping skis (Fig. 11). Here, we use the same ski geometry in Fig. 5 for all four models in order to simulate the effect of stiffness distribution only. Random vibrational forces in Fig. 8 are applied with varying amplitudes as in the previous section.
Fig. 11

Bending stiffness profiles of jumping skis

As expected, the distribution of bending stiffness has a significant influence on the reduction of friction (Fig. 12). Notably, we can observe that smoother distributions might be more beneficial to friction reduction. The stiffness distribution that has higher bending stiffness near the binding area and lower stiffness near the shovel shows the least amount of friction reduction irrespective of force amplitude. This observation is probably because the ski vibration similar to the first bending mode in Fig. 6 would be more easily activated for jumping skis with sharper stiffness distribution and this mode is predicted to be disadvantageous for friction reduction as shown in Fig. 7. Hence, we may conclude that jumping skis with a smoothly varying bending stiffness is preferable in the in-run state of ski jump.
Fig. 12

Effect of bending stiffness distribution. Reduction ratios of a the frictional force, b the normal contact force, and c the contact length in comparison to the case where no vibration is applied

Conclusions

In this paper, we perform a set of numerical analysis to investigate the effect of vibration characteristics on friction in ski jump. Finite element models for jumping skis are constructed using beam elements whose spatial coordinates and stiffness values are obtained from experimental measurement of the geometry and bending stiffness distribution of a commercial jumping ski. Friction model in the icy condition is employed whose model parameters are determined from rotational tribometer experiments in literature. We calculate the friction force for jumping skis under various vibratory loading conditions. Results provide some physical insights into the effect of vibration on friction as follows.
  1. (1)

    In icy condition, the contact area between the ski base and the track governs the friction.

     
  2. (2)

    In general, the vibration decreases the friction force. However, it does not monotonically decrease with the frequency and amplitude of applied loads. It is rather mode/amplitude specific probably because of inherent nonlinearity in ski deformation and its contact with the track.

     
  3. (3)

    Smoother bending stiffness distribution seems more beneficial to friction reduction.

     

Notes

Acknowledgements

This research was supported by the Convergent Research Program for Sports Scientification (grant number 2014M3C1B1033983) and the EDucation-research Integration through Simulation On the Net (EDISON) Program (grant number 2014M3C1A6038842) through the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science and ICT.

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Copyright information

© Korean Multi-Scale Mechanics (KMSM) 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringSeoul National UniversitySeoulKorea
  2. 2.Institute of Advanced Machines and DesignSeoul National UniversitySeoulKorea

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