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Multiscale Science and Engineering (2019) 1:256–264https://doi.org/10.1007/s42493-018-00005-x

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ORIGINAL RESEARCH

A Numerical Study for the Effect of Ski Vibration on Friction

Yunhyoung Nam1 · Jinsu Gim2 · Taejoon Jeong2 · Byungohk Rhee2 · Do‑Nyun Kim1,3

Received: 19 July 2018 / Accepted: 20 August 2018 / Published online: 11 January 2019 © Korean Multi-Scale Mechanics (KMSM) 2019, corrected publication 2019

AbstractIn 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 vari-ous interactions among athletes, skis, and environment (ice slope and air) during four different phases including in-run, takeoff, flight and landing. Most studies [1–11] related to ski jump have been focused on aerodynamic analysis of an ath-lete 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 atten-tion 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 dur-ing 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 control-ling the vibration that jumping skis experience while slid-ing down the track. Friction reduction due to vibration has been reported in many papers [12–24]. 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 numeri-cally using the finite element method. The geometric and structural properties of commercial jumping skis are meas-ured experimentally and corresponding finite element beam models are constructed. We employ a pressure-dependent friction model on ice derived from the reported experimen-tal data. The degree of friction reduction is calculated by

* Do-Nyun Kim dnkim@snu.ac.kr

1 Department of Mechanical and Aerospace Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea

2 Department of Mechanical Engineering, Ajou University, Suwon, Republic of Korea

3 Institute of Advanced Machines and Design, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea

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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 inter-action between ski plate and icy running slope, and analy-sis procedure to investigate the effect of ski vibration on friction.

Finite Element Model for Jumping Ski

Jumping skis are composite structures having a unique geo-metric 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 vibra-tion, 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 con-tact and vibration are involved. Instead, it is physically rea-sonable 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 symmet-ric 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.

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 meas-ured 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 the-ory (Fig. 3).

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 consist-ing of 42 beam elements and 43 nodal points (Fig. 3d). We assign, to each element, the experimentally measured bend-ing 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.

Friction Model in Icy Condition

Jumping ski in winter games runs on an ice track. The fric-tion 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 fric-tion. In the beginning of sliding, the track is mostly in solid phase and hence the dry friction due to solid-to-solid asper-ity 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 sur-faces and their capillary drag determines the friction force. If the ice melts down even more, the friction increases again

Fig. 1 Shape profile and internal structure of a typical jumping ski

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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 experi-mental 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

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with the film thickness because the hydrodynamic friction by viscous shearing of the film becomes the principal fric-tion 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 mate-rial and the ice is modeled as a linear combination of the dry and lubricated frictions as follows [25, 26]

Here, �dry is the dry friction coefficient without water layer, vr is the relative velocity of a solid body, � is the vis-cosity of water layer, hw is the thickness of water layer, and pr is the pressure. In this model, the contribution of dry fric-tion with respect to the hydrodynamic friction is controlled by � 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

(1)� = ��dry + (1 − �)�vr

hwpr

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.

In order to use this friction model in our two-dimen-sional finite element analysis, we rewrite it using the nor-mal contact force ( Fn ) and the contact length ( Lc).

Here, Ac is the contact area and we substitute the con-tact width, wc , with the one of the specimen used in exper-iments (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 ( �dry = 0.3 ) between the ski base material and

(2)� = 0.0229 +96.85

pr

(3)� = 0.0229 +96.85

Fn

Ac = 0.0229 +48.42

Fn

Lc

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 140Average width (mm) 111.2 109.2 107.3 105.4 105.4 107.3 108.9 110.3 111.6 113.0 114.3 114.7Average 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

Fig. 4 Experimentally measured friction coefficients as a func-tion of pressure for two different normal load conditions (blue and red) [27]. Black line shows the fitted data represented by Eq. (2)

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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 � = 0.076 ) and there-fore, 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 ( � ) 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.

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 inte-gration 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 coef-ficient 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 conven-tional Newmark method [29–31]. Here, the time interval ( Δt ) 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 fric-tion 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 com-pared with those obtained for the reference case where the external sinusoidal forces are not imposed.

Results clearly demonstrate that the frictional force nei-ther 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 fric-tion is observed at the fourth natural frequency. It is note-worthy, 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

and you can see the contact length governs the frictional force due to the dominance of hydrodynamic friction.

Effect of Vibration Amplitude on Friction

In this section, we investigate the effect of vibration ampli-tude 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 reduc-tion is the highest in the previous section. The amplitude

(4)Ff = 0.0229Fn + 48.42Lc

Fig. 5 Analysis setup. Here, g , � , m , Fn and Ff 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

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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 experimen-tally 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.

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 corre-lated 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 neg-ligible effect on the friction reduction. This non-monotonic

dependence of friction to the force amplitude is more evi-dent 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.

Effect of Bending Stiffness Distribution on Friction

Results in the previous sections demonstrate that the fric-tion 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 frequen-cies and normal mode shapes. To examine this effect, we constructed four jumping ski models with different bend-ing stiffness profiles that are measured from four different

Fig. 6 Normal mode shapes and natural frequencies of the jump-ing ski beam model

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

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commercial jumping skis (Fig. 11). Here, we use the same ski geometry in Fig. 5 for all four models in order to simu-late the effect of stiffness distribution only. Random vibra-tional forces in Fig. 8 are applied with varying amplitudes as in the previous section.

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 distri-bution that has higher bending stiffness near the binding area

Fig. 8 Acceleration profile measured experimentally by Shionoya et al. in a real jumping condition [33]

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

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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.

Conclusions

In this paper, we perform a set of numerical analysis to investigate the effect of vibration characteristics on fric-tion in ski jump. Finite element models for jumping skis are constructed using beam elements whose spatial coor-dinates and stiffness values are obtained from experimen-tal 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 pro-vide some physical insights into the effect of vibration on friction as follows.

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

(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) Smoother bending stiffness distribution seems more beneficial to friction reduction.

Acknowledgements This research was supported by the Conver-gent 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.

Fig. 11 Bending stiffness pro-files of jumping skis

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

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